It is well known that if $G$ is a finitely presented subgroup then all its finite index subgroups are finitely presented also. I was wondering if it is possible to extend this,
What classes $\mathcal{C}$ of finitely generated/presented groups are there such if $G \in \mathcal{C}$ then every subgroup of $G$ is finitely generated/presented?
This is, in some ways, the opposite to what happens in finitely generated free groups, where a normal subgroup is of finite index if and only if it is finitely generated.
Examples of such classes of groups are the Tarski monster groups, as well as finitely generated abelian groups. I was therefore wondering if there were more interesting examples (one could argue that you cannot get more interesting that Tarski monster groups...but anyway), such as amenable, or hyperbolic, with added conditions. This is (conceivably) not too much to ask - f.g. abelian groups are amenable, while the Tarski monster groups were constructed using a sort of layered small cancellation theory (so if they aren't hyperbolic, some of them at least "look like" they are.)