Galois group of $x^8-2$ and intermediate fields

Question

Let $p(x)=x^8-2$ be a polynomial over $\mathbb{Q}$. I am to find its Galois group $G$ and the correspondence between subfields of its splitting field $\mathbb{K}_p$ and subgroups of $G$. It is easy to prove that $\mathbb{K}_p=\mathbb{Q}[w,2^{\frac{1}{8}}]$, where $w=e^{\frac{2\pi i}{8}}$ and $2^{\frac{1}{8}}$ is a positive real root of $p$. Each automorphism $f\in G$ maps $w$ to some power $w^a$, where $a\in \mathbb{Z}_8^*$ (an invertible one) and maps $2^{\frac{1}{8}}$ to some root $2^{\frac{1}{8}}e^{2\pi i b/8}$ of $p$ (where $b\in \mathbb{Z}_8$). Let us denote each such automorphism by $f_{a,b}$. I have proved that $G$ is $\{f_{a,b}:(a,b)\in\{(1,0),(1,2),(1,4),(1,6),(3,1),(3,3),(3,5),(3,7),(5,1),(5,3),(5,5),(5,7),(7,0),(7,2),(7,4),(7,6)\}\}$. I know how to multiply them: $$f_{a,b}\cdot f_{c,d}=f_{ac,ad+b}.$$ Could you help me to find all intermediate fields and (or) all subgroups $H\subset G$ to build the correspondence? Is it true that it can be done without some special technique?