TITLE: What are some compact (Hausdorff) groups?
QUESTION [7 upvotes]: I just realized today that I don't know any compact groups that aren't profinite groups or Lie groups. 
Generalizing from these, a product of compact groups is again a compact group, a closed subgroup of such a product is then again a compact group, and a quotient of such a group by a closed normal subgroup is again a compact group. 
So now these are all the examples of compact groups I know.
What are some interesting and/or counterintuitive examples arising in this fashion? What other compact groups are out there?

REPLY [5 votes]: You''ve generated all examples. A corollary of the Peter-Weyl theorem is that every compact (Hausdorff) group is a closed subgroup of a product of $U(n)$s.