Subgroups of $\{\sigma^k,\tau \sigma^k\mid 0\le k \le 7\}$ I am reading in Dummit and Foote, page $579$. We have the following group:
$$G = \langle \tau,\sigma\mid \sigma^8,\tau^2,\sigma \tau = \tau \sigma^3\rangle = \{\sigma^k,\tau\sigma^k\mid 0\le k\le 7\}$$
which is just $Gal\left( \mathbb Q(i,\sqrt[8]{2})/\mathbb Q\right)$.
It is stated that: "determining the subgroups of this group is a straightforward problem". 
How so? Is there an easy way to determine the subgroups? Here are the subgroups:


The Galois group of $K/\mathbb{Q}$ with $K:=\mathbb{Q}(\sqrt[8]{2},\zeta)$ has order $16$ and is an extension of $(\mathbb Z_8)^\times$ by $\mathbb Z_4$. For all groups of order $16$, in particular for this Galois group, the subgroups have been computed, see here, or here and similar references. One can use the classification of groups of order $1,2,4,8$ and other facts about $2$-groups. One also can use GAP. Doing the computations by hand may take some time. 
For the Galois group see also this MSE-question:
Finding the Galois group over $\Bbb{Q}$.
