It was the “new group” menu.
On the other hand, I managed to work around it. I deleted 1 group and then created 2 new ones with one person inside, and the dropdown list went beyond 12 groups.
I think it’s related to something very specific. I had one person in the 12th group and I removed them. After that, when I wanted to add a person to a group, it didn’t work anymore.
Ideally, I’d need to be able to reproduce it, but I’m not really keen on creating a problem. If it happens again, I’ll try to take note of everything as I go.