If $HK$ is a subgroup of $G$ (where $H$ and $K$ are subgroups of $G$), then are $H$ and $K$ normal in $HK$?
Consider the (nondirect) internal semidirect product $G=H\rtimes K$. We have $G=HK$, $H\unlhd G$, $H\cap K=\{e\}$, and $K\le G$, but $K\not\unlhd HK=G$.