Example of $1-1$ Correspondence with Subgroups of Factor Group

Question

I am working out an example to deomstrate the one-one correspondence between $\{\text{subgroups of}\ D_4/N\}$ and $\{\text{subgroups of $D_4$ that contain $N$}\}$ but I am short one in $D_4$.

$G=D_4$

$N=\{e,r^2\} \trianglelefteq D_4$

$D_4/N=\{N,rN,sN,(rs)N\}$

Subgroups of $D_4/N$: $\ H_1=\{N\},\ H_2=D_4/N,\ H_3=\{N,(rs)N\},\ H_4=\{N,sN\}$

Subgroups of $D_4$ which contain $N$: $\ D_4,\ N, \ \langle r \rangle$

I am definitely overlooking something but I don't know what. I should have four subgroups of $D_4$ that contain $N$ since there are four subgroups of the quotient group $D_4/N$.

Edit: I was missing $K=\{e,s,r^2,r^2s\}$

Answer 1

You're missing $\langle r^2, s\rangle$ and $\langle r^2, rs\rangle$.

Answer 2

You should have five subgroups of $D_4/N$ and five subgroups of $D_4$ which contain $N$.

Of the subgroups of $D_4/N$, you're missing $\{N,rN\}$.

Of the subgroups of $D_4$ which contain $N$, you're missing two: $\langle r^2,s\rangle$ and $\langle r^2,rs\rangle$. (As Nishant said.)

Contrary to what you claim in the comments, $\langle r^2,rs\rangle$ is not the whole $D_4$, it is only $\{e,r^2,rs,r^3s\}$.