Central factor is not finite-intersection-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) not satisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Intersection-closed subgroup property (?), .
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Statement

An intersection of central factors need not be a central factor.

Proof

We construct a counterexample as follows. Let $G=D\times C$ where $D$ is the dihedral group of order 8, and $C$ is the cyclic group on 2 elements. Let $a$ be an element of order 4 in $D$, and $x$ a reflection in $D$. Further, let $y$ be the generator of $C$.

Look at the subgroups $H=<x,a>$ and $K=<xy,a>$. Both these subgroups are actually automorphs of each other, and moreover, they both have the same centralizer: the group $<a^2,y>$. However, the group $H\cap K$, which is just $a$, is not a central factor.

Note that both $H$ and $K$ are direct factors, so the proof shows that an intersection of direct factors need not be a central factor.