A commutator relation

Question

I hope the following is not trivial,

Let $H$ be a subgroup of $G$ s.t. $[H,G]\leq Z(G)$ then can we say that $H$ is normal ?

I think we can not but I could not find counter example. Any counter example is welcome.

Answer 1

No $H$ does not have to be normal. Take $G=D_4=\langle a,b : a^4=1=b^2, ab=ba^{-1} \rangle$. Put $H=\{1,b\}$. Then $[H,G]=Z(G)=G'=\{1,a^2\}$. But $H$ is not normal.