Let $\Omega$ be a finite group, let $G$ be a subgroup of $\Omega$ and let $S$ be a set of subgroups of $\Omega$ such that for $H, H'\in S$ we have $H\cap H' \in S$ and $\langle H, H' \rangle \in S$.
One can show that if $GH = GH'$ then also $GH = G\langle H, H' \rangle$. So given $H \in S$ we can find $H_{\max}\in S$ such that $G H = G H_{\max}$ and $H_{\max}$ is maximal with that property.
We say $H_{\max}$ is $G$-maximal in $S$. One can also show that if $G\cap H = G\cap H'$ then also $G\cap H = G\cap (H\cap H')$.
So given $H\in S$ we can find $H_{\min} \in S$ such that $G\cap H = G\cap H_{\min}$ and $H_{\min}$ is minimal with that property. We say $H_{\min}$ is $G$-minimal in $S$.
In the context of my work I'm interested in groups in $S$ that are $G$-minimal and $G$-maximal at the same time. One can show that they always exist.
So my question is:
What properties do such groups have and is there any literature for this?
An example would be if $\Omega= S_4$, $G = D_8$ and $S$ are all Young-subgroups in $S_4$. Then the groups that are $D_8$-maximal and $D_8$-minimal in $S$ are eaxactly the $S_4$ and all $V_4$'s in $S$.