Why didn't Snape see Dumbledore's Army in Occlumency lessons with Harry?

Question

In Harry Potter and the Order of the Phoenix, why didn't Snape see Dumbledore's Army in Harry's mind during Occlumency lessons?

Answer 1

Because the memories Snape saw were pretty miscellaneous.

The memories which emerge during Legilimency appear to be fairly random, unless the Legilimens is actively searching for a particular memory or thought within their target's mind. When Dumbledore was questioning Morfin and Kreacher he was looking for specific information. When Snape was teaching Harry Occlumency he wasn't searching for anything in particular. He was just testing Harry's defences. As such, the memories which Snape sees are pretty sporadic and random.

He was finding it increasingly difficult to disentangle separate memories from the rush of images and sound that Snape kept calling forth.
_{(Order of the Phoenix, Chapter 26, Seen and Unforseen).}

It's worth pointing out that Snape did see a DA lesson, even if he only saw the make-out session at the end rather than the lesson itself.

A hundred Dementors were closing in on him beside the dark lake...Cho Chang was drawing nearer to him under the mistletoe...
No, said a voice in Harry’s head, as the memory of Cho drew nearer, you’re not watching that, you’re not watching it, it’s private -
_{(Order of the Phoenix, Chapter 24, Occlumency).}

We don't have a complete record of all the things Snape saw in the Occlumency lessons. Some of the lessons aren't described at all in the books. So he may have seen a DA lesson in progress. If he had done it wouldn't have been a big deal. We know that Snape was loyal to Dumbledore, not Umbridge. Snape wouldn't have betrayed Harry's secret to her. He may have passed comment on the DA, asking what it was and whether Harry was getting up to anything he shouldn't. It would've been fairly easy for Harry to lie at this point and pretend that the meetings had happened in the pre-Umbridge era.

We don't know if Snape told Dumbledore about the DA, even supposing that he saw any of the memories. Dumbledore probably knew about the meetings anyway, however. Anything that Snape told Dumbledore would've been in addition to the information he'd received from Aberforth and Mundungus. It wouldn't have told him anything new. Nevertheless, the most logical explanation is that, amongst the random stream of memories that Snape saw, the DA meetings never came up.

However, there is another explanation for why Snape may not have seen the meetings. Although the Legimilency process is fairly random it isn't totally random. Snape was more likely to see the memories that Harry was most eager to hide.

“Get up!” said Snape sharply. “Get up! You are not trying, you are making no effort, you are allowing me access to memories you fear, handing me weapons!”
_{(Order of the Phoenix, Chapter 24, Occlumency).}

Most of the memories Snape saw were from Harry's childhood. It was less likely that Snape would see the DA lessons when so many of Harry's memories were of childhood embarrassments.

He had just been forced, yet again, to relive a stream of very early memories he had not even realized he still had, most of them concerning humiliations Dudley and his gang had inflicted upon him in primary school.
_{(Order of the Phoenix, Chapter 26, Seen and Unforseen).}

There's no particular reason why Harry would fear Snape seeing the DA meetings (no more so than any other teacher discovering them) and they certainly weren't embarrassing or humiliating. These were the sorts of memories that Snape ended up seeing so it's probably unlikely that the DA meetings would've surfaced during the Occlumency lessons.

Answer 2

It's also likely that he already knew about it as Sirius and Molly Weasley already knew about it so the Order would have been notified:

“She [Molly] says on no account whatsoever are you to take part in an illegal secret Defense Against the Dark Arts group. She says you’ll be expelled for sure and your future will be ruined. She says there will be plenty of time to learn how to defend yourself later and that you are too young to be worrying about that right now. She also” (Sirius’s eyes turned to the other two) “advises Harry and Hermione not to proceed with the group, though she accepts that she has no authority over either of them and simply begs them to remember that she has their best interests at heart. She would have written all this to you, but if the owl had been intercepted you’d all have been in real trouble, and she can’t say it for herself because she’s on duty tonight.”

^{Harry Potter and the Order of the Phoenix}

Therefore, his loyalty remained... true-ish and he didn't do anything about it...

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Note that this is an addition to @TheDarkLord's answer, but was far too long for a comment...