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