If $G$ is a finite $p$-group, it is trivial that every subgroup has index $p^r$ for some integer $r$. If $G$ is infinite, this is not true as the index can be infinite.
If $G$ is an infinite $p$-group, what is possible to say about $[G:H]$ if :
- $H$ is of finite index ?
- $H$ is maximal and of finite index ?