Complemented normal not implies direct factor

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., complemented normal subgroup) need not satisfy the second subgroup property (i.e., direct factor)
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Statement

It is possible to have a group $G$ and a complemented normal subgroup $H$ such that $H$ is not a direct factor of $G$. In other words, $H$ has a permutable complement in $G$, but no normal complement in $G$.

Examples

Here are some examples where the ambient group is quite important and easy to understand:

Here are some examples where the ambient group is somewhat more complicated:

Here are some examples where the ambient group is even more complicated:

Proof

Further information: dihedral group:D8, cyclic maximal subgroup of dihedral group:D8

We can take $G$ to be the dihedral group $D_{8}$ given as:

$\langle a,x\mid a^{4}=x^{2}=e,xax=a^{-1}\rangle$

and $H$ to be its unique cyclic maximal subgroup $\langle a\rangle$. Then $\langle x\rangle$ is a permutable complement to $H$ in $G$, but $H$ has no normal complement, so is not a direct factor.