I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal subgroups. (In other words, $[H, G] \cong M_4$.)

I have used GAP to search for such groups and, to my surprise, I could find only three: $S_3$, $C_3 \times C_3$, and $(C_3 \times C_3) : C_2$. So far, all other examples reduce to these after modding out by a normal subgroup (so they are not examples if we require $H$ be core-free).

I've searched through most (but not all) of the SmallGroups library. Though, I can't promise my code is free of bugs that may be causing me to miss something.

Question: Does anyone know of other finite groups, besides $S_3$, $C_3 \times C_3$, and $(C_3 \times C_3) : C_2$, with an upper interval isomorphic to $M_4$? (If not, I would welcome any ideas that could help explain why this should be a rare phenomenon.)

I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal subgroups. (In other words, $[H, G] \cong M_4$.)

I have used GAP to search for such groups and, to my surprise, I could find only three: $S_3$, $C_3 \times C_3$, and $(C_3 \times C_3) : C_2$. So far, all other examples reduce to these after modding out by a normal subgroup (so they are not examples if we require $H$ be core-free).

I've searched through most of the groups of order less than 960. Though, I can't promise my GAP code is free of bugs that may be causing me to miss something.

Question: Does anyone know of other finite groups, besides $S_3$, $C_3 \times C_3$, and $(C_3 \times C_3) : C_2$, with an upper interval isomorphic to $M_4$? (If not, I would welcome any ideas that could help explain why this should be a rare phenomenon.)