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).