Abelian normal not implies central

Statement
It is possible to have a group $$G$$ and an abelian normal subgroup $$H$$ of $$G$$ (i.e., $$H$$ is an abelian group and is a normal subgroup of $$G$$) that is not a central subgroup of $$G$$ (i.e., $$H$$ is not contained in the center of $$G$$).

Similar facts

 * Maximal among abelian normal implies self-centralizing in nilpotent: This shows that in a nilpotent group and in particular in a group of prime power order, any subgroup that is maximal among abelian normal subgroups is a self-centralizing subgroup. In particular, if the whole group is not abelian, it cannot be a central subgroup. Examples include dihedral group:D8, quaternion group, and many others.
 * Maximal among abelian normal implies self-centralizing in supersolvable: The result also holds for supersolvable groups, such as the symmetric group of degree three.
 * Normal not implies central factor
 * Abelian-quotient not implies cocentral
 * Nilpotent and every abelian characteristic subgroup is central implies class at most two

Opposite facts

 * Normal subgroup of least prime order is central
 * Abelian normal implies commutator-in-center

Example of the dihedral group
If we take $$G$$ to be the dihedral group of order eight, and $$H$$ to be any of the three maximal subgroups of $$G$$, then $$H$$ is abelian and normal in $$G$$ but is not central in $$G$$.