Give an example of a finite non-abelian group $G$ which contains a subgroup $H_0 \neq (e)$ such that $H_0 \subset H$ for all subgroups $H \neq (e)$ of $G$.
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Consider the Quaternion group $Q_8$ and the subgroup $H_0=\{1,-1\}$
iam_agf
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2$Q_8={\pm 1, \pm i, \pm j, \pm k}$ and $H_1={\pm1, \pm i}=$, $H_2={\pm1, \pm j}=
$ and $H_3={\pm1, \pm k}= – Jul 25 '17 at 03:34$ .Finally $H_o \subset H_s$, for all $s=1,2,3$ -
You're welcome ;-) – iam_agf Jul 25 '17 at 03:35