It is well known that finite $p$-groups have (normal) subgroups of all possible orders. Now, what can we say about subgroups containing a given non-normal subgroup? i.e.
Let $G$ be a group of order $p^n$ and let $H$ be a non-normal subgroup of $G$ of order $p^m$. Does there exist a (normal) subgroup of $G$ containing $H$ of order $p^i$, for $i=m,\ldots,n$? If not, can you show a counterexample?
Remark: I ask non-normality for $H$ because if it was normal, I could quotient out by it.
Thank you very much in advance!