Give an example of a group $G$ and subgroups $H$ and $K$ such that $HK=\{hk\mid h\in H,k\in K\}$ is not a subgroup of $G.$

Question

Pretty much the question in the title. I had an answer and wanted to know if I did it correctly-

I picked $G$ as the $U(5) - \{1\} = \{2,3,4\}$ under the operation multiplication modulo 5.

$H=\{2\}$ and $K=\{3\}$. Then $HK = \{1\}$ which is not $G$. Would this be correct? I looked up a few solutions online, and they all seemed to be more complicated than this. Am I missing something here?

Thank you.