Normal not implies central factor

Statement
A normal subgroup of a group need not be a central factor.

Related facts

 * Normal not implies direct factor
 * Central factor not implies direct factor
 * Abelian normal not implies central

Proof
Suppose $$G$$ is the dihedral group of order eight, $$H$$ is the cyclic subgroup of order four in $$G$$, and $$K_1$$ and $$K_2$$ are the two Klein four-subgroups. Then, the three subgroups $$H$$, $$K_1$$, and $$K_2$$ are all normal in $$G$$. However, none of them are central factors. In fact, they are all self-centralizing subgroups of $$G$$.