Example: Most finite groups empirically are 2-groups. There are a lot of them. Conjecturally almost all finite groups are 2-groups (in the sense that if you count all groups up to isomorphism with at most $n$ elements, then the fraction of those which are 2-groups goes to 1 as n goes to infinity). In practice, while we often encounter small 2-groups and a few specifici 2 groups like $(Z/(2Z))^k$, when dealing with "largish" finite groups all these weird 2-groups don't seem to often show up.