What are some compact (Hausdorff) groups?

Question

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?

Answer 1

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.

Answer 2

A solenoid is an interesting example that arises as a closed subgroup of a product of circles.