Direct factor is not upper join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor) not satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property).
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Statement

We can have a subgroup $H$ of a group $G$, and intermediate subgroups $K_{1}$ and $K_{2}$ such that $H$ is a direct factor of $K_{1}$ as well as a direct factor of $K_{2}$, but $H$ is not a direct factor of the join of subgroups $\langle K_{1},K_{2}$.

Proof

Example of the dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Consider the dihedral group of order eight:

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

Let:

$H=\langle a^{2}\rangle ,K_{1}=\langle a^{2},x\rangle ,K_{2}=\langle a^{2},ax\rangle$.

Then:

• $H$ is a direct factor of $K_{1}$, which is a Klein four-group and is the internal direct product of $H$ and $\langle x\rangle$.
• $H$ is a direct factor of $K_{2}$, which is a Klein four-group and is the internal direct product of $H$ and $\langle ax\rangle$.
• $H$ is not a direct factor of $\langle K_{1},K_{2}\rangle =G$. In fact, since $H$ is the center of $G$, it intersects every nontrivial normal subgroup nontrivially (nilpotent implies center is normality-large).