Most finite groups empirically are 2-groups (in the sense of being a [p-group][1] with $p=2$ not [in the other sense of the word][2]). There are a lot of them. Conjecturally almost all finite groups are 2-groups. That is it is conjectured 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 specific 2-groups like $(Z/(2Z))^k$, when dealing with "largish" finite groups all these weird 2-groups don't seem to often show up. 


  [1]: https://en.wikipedia.org/wiki/P-group
  [2]: https://en.wikipedia.org/wiki/2-group