Let $G$ be a finite group and $M$ be a nontrivial proper subgroup of $G$ with the following conditions:

a) If $H$ is a subgroup of $G$ such that $M\lneqq H\lneqq G$, then $H$ contains at least one minimal subgroup of $G$ say $L$, such that $M\cap L=1$.

b) If $K$ is a subgroup of $G$ such that $M\cap K=1$ then $K\cong Z_p$ for some prime $p$.

Can we say that $M$ is a maximal subgroup of $G$?

Let $G$ be a finite group and $M$ be a nontrivial proper subgroup of $G$ with the following conditions:

a) If $H$ is a subgroup of $G$ such that $M\lneqq H\lneqq G$, then $H$ contains at least one minimal subgroup of $G$ say $L$, such that $M\cap L=1$.

b) If $K$ is a subgroup of $G$ such that $M\cap K=1$ then $K\cong Z_p$ for some prime $p$.

Can we say that $M$ is a maximal subgroup of $G$?

Please read my comments after Derek Holt's answer.