I know that there are plenty of infinite groups with no maximal subgroups - classic example is the additive group of rational numbers, $\mathbb{Q}$.
Moreover, I know of the result for finite groups that states that a subgroup $H$ of a (finite) group $G$ is a maximal normal subgroup if and only if $G/H$ is simple.
But does there exist a necessary and sufficient condition for any group, finite or infinite, to have a maximal normal subgroup?
I need to answer the question "is it true that any group has a maximal normal subgroup?", and while I know that $\mathbb{Q}$ has no maximal subgroups, it does have normal ones (namely, $\mathbb{Z}$). So, I would like to ask, in addition, are there any examples of infinite groups that do have maximal subgroups, but none of them are normal?