A question about finitely generated $p$-groups

Question

My question is about finitely generated $p$-groups. In general, a subgroup of a finitely generated group is not necessarily finitely generated. But, my question is about finite and finitely generated $p$-groups. More specifically: if $G$ is a finitely generated $p$-group, say, $m$-generated and $U$ is a finitely generated subgroup of $G$, then is $U$ at most $m$-generated? If not, can $U$ be generated by a number of elements that depends only on $m$?

Answer 1

A wreath product of a cyclic group of order $p$ with a cyclic group of order $p^k$ is 2-generated, but the base group of the wreath product requires $p^k$ generators. So the answer to both questions is no.