Isn't it possible that $D_4$ has also a subgroup with $1$ element?? A consequence of the Lagrange theorem:

Let $G$ a finite group and $H$ a subgroup of G. Then $|H| \mid |G|$.

is that each subgroup $\neq <i_d>$ of $D_4$, which has $8$ elements  , has either $2$ or $4$ elements..
But.... $1 \text{ divides also }8$..Isn't it possible that $D_4$ has also a subgroup with $1$ element??

$D_4$ has a subgroup with one element as does every group: the trivial group $\langle id\rangle$.
