Connection between subgroups

Question

Let M be any maximal subgroup of a finite group $G$ and $N$ is a normal subgroup of $G$ and $N$ is not a subgroup of $M$. So, $NM \leq G$. Does that necessarily mean $|NM|=|G|$? why yes or no?

My guessing is no for example if $G=Z_{12}$, $N=\langle2\rangle$,$M=\langle3\rangle$. Am I right?

Thanks

Edit: Im interested to this problem because I want to see if there is any case that $|NM|$ is not equal $|G|$ under stated conditions.

Answer 1

Hint: since $N$ is normal, $MN$ is a subgroup with $M \subseteq MN \subseteq G$. Now use that $M$ is maximal and $N \not\subseteq M$ to infer that $G=MN$.