Example of two subgroups $H$ and $K$ of a non-abelian group $G$ such that $HK$ is not a subgroup of $G$? I need to disprove that the set $HK$ is not a subgroup of $G$ if $G$ is non-abelian. Does anyone have a trivial or easy counterexample? I cannot think of one...

Take $G=S_3,H=\left<(1,2)\right>$ and $K=\left<(2,3)\right>$. Then $H$ and $K$ are subgroups of $G($each containing two elements$)$ and $$HK=\{1,(12),(23),(132)\}$$a set of size $4$. Therefore, $HK$ is not a subgroup of $G$$($by Lagrange's Theorem$)$ since $4$ does not divide $6$.
