I am asking whether the converse of the following theorem is true:
( $H$ is normal) or ( $T$ is normal) $\implies$ $HT$ is subgroup of $G$.
My motivation is that I am interested in normal subgroups. I recently had a funny coincidence while trying to solve a question that asked me to give an example of subgroups $H$ and $T$ such that $HT$ is not a subgroup. I had to try non-normal subgroups in non-abelian groups, but the first non-abelian group I tried, the Quaternions $Q8$ , consists only of normal subgroups. But I am done with the question of finding example of subgroups $H$ and $T$ such that $HT$ is not a subgroup , when I tried $D_{4}$.