Please improve the title if it is not suitable.


Does there exist a finite group $G$ with the following properties?


$G$ has a minimal subgroup say $L$ isomorphic to $\mathbb{Z}\_{p}$ and a maximal subgroup say $M$ isomorphic to $\mathbb{Z}\_{pq}$ containing properly $L$ ($p$ and $q$ are some distinct primes), such that $L\ntrianglelefteq G$, $M\ntrianglelefteq G$ and $M$ is the only proper subgroup of $G$ containing properly $L$. 



Finite nonabelian groups with abelian maximal subgroups
How about $G = {\rm SL}(2,3) = Q\_8.C\_3$ with $|L|=3$ and $|M|=6$?


There are similar examples of the form $N.C\_p$, with $N$ an extraspecial $q$-group exponent $q$, where the $C\_p$ subgroup acts irreducibly on $N/Z(N)$.


