As soon as it looks obvious one way, it looks obvious the other way: if $G$ is a group, $M$ is maximal subgroup and $N$ is subgroup of G, is $M\cap N$ maximal subgroup of $N$? What if $N\lhd G$?
In last case I thought: $M/(M\cap N)\cong MN/N\le G/N\;$ ...but I can't continue.
Other condition: same question if we assume $N\lhd G$ is of finite index...? Then in the above we have $G/N\;$ finite group, so $\;M/N\le G/N\;$ also finite...but still stuck.
Any help/direction will be thanked.