Is there a standard name for this concept: Let $H \leq G$ be groups. Say $H$ is ?? if there is a finitely-generated group $K \leq G$ such that $H \leq K$. What should one use in place of "??"? I'm also interested in this notion up to isomorphism.
I've used "finitely subgenerated", though I have realized that "finitely supergenerated" probably makes a bit more sense. If you know a reference/situation where such a thing plays a key role, that may also be useful even if no name is given.