Many groups have the property of having only finitely many conjugacy classes of finite subgroups. Examples include hyperbolic groups, CAT(0) groups, mapping class groups, and automorphism groups of free groups. Do subgroups of these groups inherit this property? The answer turns out to be no. I will describe how combinatorial Morse theory can be used to produce couterexamples. This is joint work with Noel Brady and Matt Clay.