Given $G$, $|G|=p^n$, $H,K\le G$ s.t. $|H|=|K|=p^k$ for some $k<n$.
Can we say that $H\simeq K$? I think it's true and I tried to prove it building by hand the isomorphism $\psi:H\longrightarrow K$ and working by induction on $k$: in fact every $p$-group (such $K$ and $H$ are) contains an element of order $p$; call them $h\in H$ and $k\in K$: I can make correspond these two elements in the isomorphism $\psi$.
Then I can consider the quotient groups $H/\langle h\rangle$ and $K/\langle k\rangle$ that are isomorphic by inductive hypotesis... and then? Can I conclude that $H$ and $K$ are isomorphic too?