Let $G$ be a group. We will call $H$ an essential subgroup of $G$, if it is a subgroup of $G$, and $H \cap K \neq \{0\}$ for any $K \neq \{0\}$ subgroup of $G$.
In other words, a non trivial subgroup is essential, if it's interesection with any non trivial subgroup is a non trivial subgroup.
Does there exist an essential subgroup $H$ in $G$, such that it is not normal in $G$? If not, could we perhaps prove that a group $H$ being essential in $G$ implies that it is also normal in $G$?