Originally Posted by

**aliceinwonderland** Well, I might be able to help u too :)

As u said, the possible proper subgroups of D4 are all abelian (groups of order 2, 4). If D4 are the internal direct product of its nontrivial proper subgroups, then D4 becomes abelian. Contradiction.

As Drexel28 said,

$\displaystyle D_4 \cong C_4 \ltimes_\phi C_2$, where $\displaystyle \phi:C_2 \rightarrow \text{Aut}(C_4) $ and the assoicated action is $\displaystyle x \cdot h = h^{-1}$ for all h in $\displaystyle C_4$ and x in $\displaystyle C_2$ such that $\displaystyle xhx^{-1} = h^{-1}$.