Let $G$ be a finite group different from a cyclic $p-$group and $\Phi(G)=M_i\cap M_j$, where $M_i$ and $M_j$ are two arbitrary distinct maximal subgroups of $G$ and $i, j \geq 1$. Is it possible to characterize such $G$? ($\Phi(G)$ denotes the Frattini subgroup of $G$, which is the intersection of all maximal subgroups of $G$).
Examples: (a) Cyclic groups of order $p^nq^m$ are examples of such groups, where $p, q$ are distinct primes. Also quaternion group of order 8 is another example.
(b) Alternating groups $A_n; n\geq 5$, have no such property because as well as we know, there are different non-trivial intersections of any two point stabilizers in $A_n$ and on the other hand $\Phi(A_n)=1$.
Remark: According to an exercise we know that if $p$ is a prime divisor of $|G|$, ${\rm Syl}_p(G)$ is the set of all Sylow $p-$subgroups of $G$ and $P\in {\rm Syl}_p(G)$ then $P\nleq \Phi(G)$. Combining this fact with the condition on $G$ implies that each Sylow subgroup of $G$ contained in exactly one maximal subgroup of $G$. May be this point can help.
Thank's a lot!