What are the groups containing dihedral group $D_4$ of order $8$?

Question

I'm a little embarrassed to ask this but I couldn't answer it myself.

I am looking the groups that contains $D_4$ and larger than $D_4$. Here is what I think:

We cannot say $D_4 \subseteq D_n, n\ge 5$ because their group operation are different. But what else? $D_4$ doesn't lie in $S_4$.

Any help is appreciated.

Answer 1

They can't be abelian.

What about products? For any group $G,$ $$G×D_4,G*D_4.$$

Direct and free, respectively. Don't think you can always do a semi-direct or a wreath; but sometimes.

Also, many dihedral groups contain embedded $D_4$'s (some contain more than one copy of $D_4$).

$S_n,n\ge4$ contains $D_4.$