Union of two subgroups ($H\cup K\le G⟺H\subseteq K\text{ or }K\subseteq H$)

The union of two subgroups is a subgroup exactly when one contains the other. Otherwise choose $h\in H\setminus K$ and $k\in K\setminus H$. Their product cannot lie in $H$, since $h^{-1}(hk)=k$ would then lie in $H$, and cannot lie in $K$, since $(hk)k^{-1}=h$ would then lie in $K$. The union therefore fails closure. Unlike the union, the subgroup generated by both subgroups always exists.