Galois extension, lattice of subfields

[mathsss2]

K=\mathbb{Q}(\sqrt{2+\sqrt{2}}) is a Galois extension of \mathbb{Q} [I showed this]. Determine \text{Gal}(K/\mathbb{Q}) and describe the lattice of subfields \mathbb{Q} \subset F \subset K.

I found that \text{Gal}(K/\mathbb{Q})=\mathbb{Z}_4. I do not know how to draw the lattice of subfields, so far I have this: [see attachment], but I do not think it is right. I need help with the lattice of subfields.

 

[Office_Shredder]

Recall that the lattice of subfields is directly correlated to the lattice of subgroups of the galois group. This should make it easier