Let $G$ be a finite non-abelian $2$-group, $\nu(G)$ denotes the number of conjugacy classes of non-normal subgroups of $G$ and $G^{\prime}$ denotes the derived subgroup of $G$. If $|G^{\prime}|=8$ and $\nu(G)=4$, is it true that $G\cong Q_{32}$?($Q_{32}$ is the generalized quaternion group of order 32).

Remark: By a GAP code I could see that the answer is yes for such groups of small order in the GAP library. To prove it I think that I must show that $G$ has a unique subgroup of order 2. Because by a well-known result, if $G$ is a finite $p$-group and contains a unique subgroup of order $p$, then it is cyclic or generalized quaternion group.

Any comment and help will be greatly appreciated!


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