# Direct factor is not upper join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor)notsatisfying a subgroup metaproperty (i.e., upper join-closed subgroup property).

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## Statement

We can have a subgroup of a group , and intermediate subgroups and such that is a direct factor of as well as a direct factor of , but is not a direct factor of the join of subgroups .

## Proof

### Example of the dihedral group

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

Consider the dihedral group of order eight:

.

Let:

.

Then:

- is a direct factor of , which is a Klein four-group and is the internal direct product of and .
- is a direct factor of , which is a Klein four-group and is the internal direct product of and .
- is not a direct factor of . In fact, since is the center of , it intersects every nontrivial normal subgroup nontrivially (nilpotent implies center is normality-large).