Finite subgroup of infinite group

Question

Let $G$ be an infinite order group. Let $H$ be its finite order subgroup. Then what we say about $H$? For example, $H$ is always cyclic or we have a group $G$ having finite subgroup $H$ which is not cyclic.

Answer 1

Any finite group is be a subgroup of some infinite group. Simply consider $G=H\times\mathbb{Z}$. In fact there's an infinite group having all finite groups (up to isomorphism) as subgroups. Simply take $G=\bigoplus H_i$ over all (up to isomorphism) finite groups $\{H_i\}$, which is a countable set.

And so not much can be said in such an extremely general case.