If $HK$ is a subgroup but not equal to $G$ (where $H,K<G$), then are $H$ and $K$ normal in $HK$?

Question

I have asked a very similar question already and many have answered it also.

If $HK$ is a subgroup of $G$ (where $H$ and $K$ are subgroups of $G$), then are $H$ and $K$ normal in $HK$?

But all the answers include trivial examples where HK becomes the whole group and one answers gives and example of semidirect product. But I am a newbie on this topic and only know a handful of things like external direct product, internal direct product. So, it would be helpful if one can provide example using these concepts only and a non trivial one where $HK$ is a proper subgroup of $G$.