Condition which implies that finitely generated subgroups of a finitely generated group have fewer generators

Question

I'm still working on quandles, and have made some progress. I now want to understand finitely generated things. To this end, I want to, again, consider finitely generated groups. Let $G$ be a finitely generated group, and let $H\leq G$. I know that it's possible for $H$ to have more generators than $G$ does, for instance, in the case of the finitely generated free group. Are there any natural conditions one might put on $G$ to guarantee that a $H\leq G$ has fewer generators than $G$ does? This would naturally delete the class of free groups from consideration.