TITLE: Why is the generalized quaternion group $Q_n$ not a semi-direct product?
QUESTION [10 upvotes]: Why is the generalized quaternion group $Q_n$ not a semidirect product?

REPLY [6 votes]: One characterization of the generalized quaternion group is:

If $G$ is a non-abelian $p$-group which contains only-one subgroup of order $p$, then $G$ is generalized quaternion group [Hall-Theory of groups; Theorem 12.5.2]. 

So if we try to write the generalized quaternion group $Q_n$ as semi-direct product, then we should have a normal  subgroup $N$, a subgroup $H$, with one necessary condition that $N\cap H=1$; which is not possible because of uniqueness of subgroup of order $p$; it will contained in all subgroups of $Q_n$. Hence the generalized quaternion group is not semi-direct product of smaller $p$-groups.