TITLE: What groups have a second maximal subgroup below exactly four maximal subgroups?
QUESTION [7 upvotes]: 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.)

REPLY [4 votes]: I think that with some work you can characterize the examples where $G$ is solvable as follows. (I did not check carefully, so you can take this with a grain of salt.)
If $H$ has trivial core in $G$ and $[H,G]$ is $M_4$, then one of the following holds:
1) $G$ is the semidirect product $H(V+V)$, where $V$ is an irreducible $F_3[H]$-module such that the only elements of $GL(V)$ commuting with $H$ are the scalar transformations $1$ and $-1$.
(This case occurs when every maximal subgroup containing H has nontrivial core in $G$. Your example $C_2:C_3xC_3$ is of this type.  This condition should be sufficient and you can construct tons of examples of this type without too much trouble.) 
2) $G$ is the semidirect product $MV$, where $V$ is an irreducible $F_3[M]$-module.  There is a $1$-dimensional subspace $W$ of $V$ such that $H=N_M(W)=C_M(W)$, and $C_V(H)=W$.
  In particular, no element of $M$ has $W$ as an eigenspace with eigenvalue $-1$.
(This case occurs when some maximal $M$ containing H has trivial core in $G$.  Your example $S_3$ is of this type.  I don't know if there are lots of examples with $H$ maximal in $M$)
I can recommend that if you are interested in understanding how to find intervals of type M_n, you look at the work of Andrea Lucchini, including his joint paper with Robert Baddeley in Journal of Algebra.  There is also recent work of Michael Aschbacher on the interval representation problem in general that is most certainly worth examining.