Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and at least one of its maximal subgroup isn't of prime order. 
Do there exist two distinct non-trivial subgroups $H\_1$ and $H\_2$ of $G$ such that 


* $H\_1$ and $H\_2$ are contained in a common maximal subgroup $M$
* For each maximal subgroup $M'$ of $G$, either $H\_1 \leq M'$ or $H\_2 \leq M'$ (or both)?


If it is hard to answer the question in general, can we answer it for certain classes of finite groups (finite simple groups, symmetric groups,…)?



Finite groups with trivial Frattini subgroup
The alternating group $A\_5$ shows that this cannot hold in general. This group has five maximal subgroups of index $5$, the point stabilizers. If the stated condition holds, then at least three of these would have to contain either $H\_1$ or $H\_2$. But in this group, the intersection of any three point stabilizers is trivial. 


