TITLE: Is a group with only finitely many subgroups of index n (for all n) finitely generated
QUESTION [10 upvotes]: Let $G$ be a group with the following property. For all integers $n$, there are only finitely many subgroups of index $n$.
Question. Is $G$ finitely generated?
The converse is true. That is, if $G$ is finitely generated, then $G$ has only finitely many subgroups of index $n$ (for all $n$). Here's an argument for normal subgroups that you can also make work in the general case.
To give a normal subgroup of index $n$ of $G$ is to give a finite group $H$ of cardinality $n$ and a surjection $G\to H$. There are only finitely many groups of cardinality $n$ and for each finite group $H$ there are only finitely many $G\to H$ (because it suffices to designate the image of each generator of $G$ in $H$). QED

REPLY [4 votes]: Here there is an interesting family:
A profinite group is a compact topological group which is the inverse limite of finite groups. Examples of these are the profinite completion of a group which is defined as follows: For a group $G$ take the inverse limite of the finite quotients $G/N$ and call this group the profinite completion of $G$ which is denoted by $\widehat{G}$. It can be proved that the image of $G$ in $\widehat{G}$ is dense. 
A profinite group is said to be finitely generated if it has a finitely generated subgroup in the abstract sense which is dense. Examples are the profinite completion of finitely generated groups. Infinite profinite groups are not finitely generated. 
You can prove that every finitely generated profinite have for each n just a finite number of open subgroups of index $n$. Then it is a theorem by Nicolov-Segal that every finite index subgrouop of a finitely generated profinite group is open. Then every infinte finitely generated profinite group is a counterexample. Example: The $p$-adic numbers $\mathbb{Z}_p$.