When thinking about groups, I often find that it helps me to think in terms of some "natural examples" which arise outside of group theory. E.g. to my mind general abelian groups are like vector spaces, infinite nonabelian groups are like matrix groups over $\mathbb R$ or $\mathbb C$, and finite abelian groups are products of cyclic groups. The first two of these are familiar to me from linear algebra and the third from elementary number theory.
With finite nonabelian groups I can't think of any such examples, and large swathes of group theory (e.g. the Sylow Theorems) seem only concerned with this case. This is not for lack of groups: I can read about symmetric groups, alternating groups, matrix groups over finite fields, everything discussed at this question, but none of these strike me as "natural" since I don't know of any purpose for them other than as examples and counterexamples in group theory itself. I imagine this has more to do with my own ignorance than with the groups themselves, hence this question.
So what I'm looking for are examples of classes of finite nonabelian groups appearing prominently in some subject outside of group theory, with a description of what role they play in that subject. I would prefer classes of groups to individual groups, and I would prefer relatively accessible subjects over those that are somewhat niche or require a lot of group theory to understand. In particular I'm not interested in the sort of technical, applications-focused answers such as the ones found here.