Give an example(s) of a group $G$ and $H\leq G$ (i.e., $H$ is a subgroup of $G$) where $H$ is not normal in $G$.
What about $S_3$ and $\langle(1\;2)\rangle=\{(1),(1\;2)\}$? [Since $(1\;2\;3)(1\;2)(1\;3\;2)=(2\;3)\not\in\langle(1\;2)\rangle$] Does this work? Any other ideas?
It is not rare to see examples of subgroups of a given group, which are not normal subgroups. I'll give you one from finite groups.
Consider this finite group of order $8$, call it $G$. Then $2\in G$ and $H=\{0,1\}$ is a subgroup of $G$. But $H$ is not normal, since $$2H=\{2,4\}\ne\{2,3\}=H2.$$