Subfields of splitting field of $x^{15}-1$ over $\mathbb Q$

Question

I'd like to find the subfields of splitting field of $x^{15}-1$ over $\mathbb Q$

I solved the followings step.

1. The splitting field L of $x^{15}-1$ is same as the splitting field of $(x^3-1)(x^5-1)$. And $\mathbb Q(w_{15})=\mathbb Q(w_3,w_5)$

2. So, i find the galois group of $Gal(L/\mathbb Q)$ is $\langle f \rangle \times \langle g\rangle$ isomorphic to $\mathbb Z_2 \times \mathbb Z_4$ where $$f:\mathbb Q(w_3) \rightarrow \mathbb Q(w_3),$$ $f(w_3)=w_3^2,$ $f(a)=a$, $\forall a \in \mathbb Q$

$$g:\mathbb Q(w_5) \rightarrow \mathbb Q(w_5),$$ $g(w_5)=w_5^2$, $g(a)=a$, $\forall a \in \mathbb Q$

3. So i'd like to find the subfields of $L$ by using galois theory and result of (2).

But, i stuck here.

For example, one of the subgroup of $\langle f \rangle \times \langle g \rangle$, for $\langle (id,g^2) \rangle$ , what is the corresponding subfield of $L$?

How can i find it? If there is another method, please help.