Why didn't Snape work out Voldemort had Horcruxes?

Question

Snape is repeatedly referred to as a highly intelligent wizard, most skilled in the Dark Arts. We don't know for sure, but it seems probable he would know about Horcruxes, at least in theory.

He basically possesses all the same facts about the Harry-Voldemort connection as Dumbledore (that Harry can speak Parseltongue, that his mind is connected to Voldemort's etc), and about Riddle's diary: Dumbledore is not secretive about its origin at the end of Harry Potter and the Chamber of Secrets even with the Weasleys. While for Harry preserving a piece of someone's soul in a diary looks like a random act of magic, Snape would definitely recognize it as very peculiar case, same as Dumbledore did. Snape surely didn't know about Slughorn's memory, but he knew about Voldemort's ambitions in general as Voldemort stated himself at the end of Harry Potter and the Goblet of Fire:

"And then I ask myself, but how could they have believed I would not rise again? They, who knew the steps I took, long ago, to guard myself against mortal death?"

(Which to me sounds like Death Eaters could know about Horcruxes, but as Dumbledore keeps telling us they didn't, let's just believe him). Still it implies Snape knew the Dark Lord's final aim - immortality.

It seems out of character for Snape not to use his brains, put two and two together and get the same conclusions as Dumbledore came to.
Why, I'm even surprised Dumbledore himself didn't expect Snape to realize the truth.

UPD: Adding quotes that prove Snape didn't know about the Horcruxes (all from Snape's memories in the Deathly Hallows)

“He does not need protection. The Dark Lord has gone – ” “The Dark Lord will return, and Harry Potter will be in terrible danger when he does.”

Here we see Snape thinks Voldemort's gone for good, so he does not know about Horcruxes.

“Did you think that breaking the ring would break the curse?” “Something like that…I was delirious, no doubt…” said Dumbledore.

Here we see Dumbledore conceals the real reason he broke the stone. He doesn't want Snape to know it's a Horcruxe.

“What are you doing with Potter, all these evenings you are closeted together?” Snape asked abruptly.

Dumbledore looked weary...

"...I spend time with Harry because I have things to discuss with him, information I must give him before it is too late.”

“Information,” repeated Snape. “You trust him… you do not trust me.”

“It is not a question of trust. I have, as we both know, limited time. It is essential that I give the boy enough information for him to do what he needs to do.”

“And why may I not have the same information?”

“I prefer not to put all of my secrets in one basket, particularly not a basket that spends so much time dangling on the arm of Lord Voldemort.”

This does not require a comment I believe. Same as the next one:

!“But what must he do?”

“That is between Harry and me. Now listen closely, Severus... There will come a time when Lord Voldemort will seem to fear for the life of his snake.”

“For Nagini?” Snape looked astonished.

“Precisely. If there comes a time when Lord Voldemort stops sending that snake forth to do his bidding, but keeps it safe beside him under magical protection, then, I think, it will be safe to tell Harry.”

“Tell him what?”

Dumbledore took a deep breath and closed his eyes.

“Tell him that on the night Lord Voldemort tried to kill him, when Lily cast her own life between them as a shield, the Killing Curse rebounded upon Lord Voldemort, and a fragment of Voldemort’s soul was blasted apart from the whole, and latched itself onto the only living soul left in that collapsed building. Part of Lord Voldemort lives inside Harry, and it is that which gives him the power of speech with snakes, and a connection with Lord Voldemort’s mind that he has never understood. And while that fragment of soul, unmissed by Voldemort, remains attached to and protected by Harry, Lord Voldemort cannot die.”

“So the boy…the boy must die?”

Nothing about Harry being a Horcrux, or Nagini being a Horcrux, or Voldemort having Horcruxes at all. Snape did not know and Dumbledore tried to phrase everything the way it would not specify the kind of magic.

UPD: here I found related topics that, though do not answer my question, may be interesting, so I leave the links here:

Why didn't any Death Eaters create their own horcruxes?

How did R.A.B get to know about Voldemort's Horcruxes?

Answer 1

Horcruxes were a fairly obscure topic, even among Dark wizards.

Though Snape had an interest in the Dark Arts, Horcruxes aren’t a common topic in Dark Arts. When Hermione searches the books in the restricted section, none of them give details on Horcruxes, and only one of them even mentioned Horcruxes.

“I haven’t found one single explanation of what Horcruxes do!’ she told him. ‘Not a single one! I’ve been right through the restricted section and even in the most horrible books, where they tell you how to brew the most gruesome potions – nothing! All I could find was this, in the introduction to Magick Moste Evile – listen – “of the Horcrux, wickedest of magical inventions, we shall not speak nor give direction” … I mean, why mention it, then?’ she said impatiently, slamming the old book shut; it let out a ghostly wail. ‘Oh, shut up,’ she snapped, stuffing it back into her bag.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 18 (Birthday Surprises)}}

In addition, Dumbledore didn’t expect the Death Eaters, at least some of whom would be as knowledgeable in the Dark Arts as Snape, if not more since Snape defected early on, to know about Horcruxes or figure out that the Dark Lord had made them from what he’d said.

“Then you told me, two years later, that on the night that Voldemort returned to his body, he made a most illuminating and alarming statement to his Death Eaters. “I, who have gone further than anybody along the path that leads to immortality.” That was what you told me he said. “Further than anybody.” And I thought I knew what that meant, though the Death Eaters did not. He was referring to his Horcruxes, Horcruxes in the plural, Harry, which I do not believe any other wizard has ever had.”
_{^{Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)}}

Even people involved with the Dark Arts didn’t typically seem to be able to figure out the Dark Lord had Horcruxes. Lucius Malfoy, who also would’ve had some interest in the Dark Arts since he had Dark objects around his manor, didn’t figure out the diary he was given to keep and knew the purpose of was a Horcrux - he believed it was just ‘cleverly enchanted’, as Dumbledore said.

“Of course, Lucius did not know what the diary really was. I understand that Voldemort had told him the diary would cause the Chamber of Secrets to reopen, because it was cleverly enchanted. Had Lucius known he held a portion of his master’s soul in his hands he would undoubtedly have treated it with more reverence – but instead he went ahead and carried out the old plan for his own ends: by planting the diary upon Arthur Weasley’s daughter, he hoped to discredit Arthur, have me thrown out of Hogwarts and get rid of a highly incriminating object in one stroke.”
_{^{Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)}}

Just being knowledgeable about the Dark Arts doesn’t mean someone will necessarily know about Horcruxes. Tom Riddle most likely found out about them from books that were at one point kept in the Hogwarts library, but were later removed and hidden away by Dumbledore.

“Well,’ said Hermione, ‘I’ve been researching that.’

‘How?’ asked Harry. ‘I didn’t think there were any books on Horcruxes in the library?’

‘There weren’t,’ said Hermione, who had turned pink. ‘Dumbledore removed them all, but he – he didn’t destroy them.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)}}

Hermione suspects that Tom Riddle learned everything he needed to know about Horcruxes from the books she thinks may have been in the library until Dumbledore became Headmaster.

“This is the one that gives explicit instructions on how to make a Horcrux. Secrets of the Darkest Art – it’s a horrible book, really awful, full of evil magic. I wonder when Dumbledore removed it from the library … if he didn’t do it until he was Headmaster, I bet Voldemort got all the instruction he needed from here.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)}}

Slughorn could have also learned about Horcruxes from those books, as he was a teacher at Hogwarts before Dumbledore became Headmaster. However, we know Dumbledore became Headmaster before Snape began attending Hogwarts since he was the one who arranged for Remus Lupin (who was in Snape’s year) to attend Hogwarts despite being a werewolf. Since Dumbledore was already Headmaster by then, the books on Horcruxes at the Hogwarts library would’ve almost certainly been removed before Snape could read them.

Snape likely stopped studying Dark Arts when he switched sides.

Though Snape was fascinated by the Dark Arts in his teenage years, he likely didn’t continue to study them in his later life. He defected to Dumbledore when the Dark Lord chose to go after Lily, and had remained loyal to him since then. Soon after that, Dumbledore gave him a teaching post at Hogwarts. It’s unlikely that Dumbledore would approve of him continuing to study the Dark Arts, so he probably wouldn’t have been studying them (at least not as much) after leaving the Death Eaters. Lily died at 21 and he was around her age, so he’d have switched sides at about 20, meaning it’s likely he gave up studying the Dark Arts at around that age as well.

“Hide them all, then,’ he croaked. ‘Keep her – them – safe. Please.’

‘And what will you give me in return, Severus?’

‘In – in return?’ Snape gaped at Dumbledore, and Harry expected him to protest, but after a long moment he said, ‘Anything.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

That means he most likely stopped studying the Dark Arts just a few years after graduating from Hogwarts. Therefore, it seems even less likely that he’d know about Horcruxes. Even the Death Eaters who would have continued involvement in the Dark Arts didn’t know about Horcruxes. Snape, who changed sides to work with Dumbledore, a man who disapproved of Dark magic and probably wouldn’t want him to keep studying it, would have even less opportunity to possibly learn about Horcruxes. The only way he’d likely be able to find out about Horcruxes after switching sides was if Dumbledore told him about them, which he didn’t.