If a group $G$ has a nontrivial centre, must every subgroup of index $3$ be normal?
$S_3$ yields an example of a group with a non-normal subgroup of index $3$, although it has a trivial centre. Moreover, for finite $G$, it's well-known that if $p$ is the smallest prime dividing $|G|$, then any subgroup of index $p$ is normal. Hence the answer to this question is "yes" if $G$ is a finite group of odd order divisible by $3$.
I'm considering dihedral groups as possible counterexamples, but haven't come up with anything.