If you are told the Galois group is you know there are three subgroups that have index 2, these correspond to three subfields over degree 2 over by Galois correspondence. Therefore, you just need to find 3 quadratic fields that are distinct. The quantity cannot be a square lest the Galois group fail to be . Thus, is quadradic over . Hence, is quadradic number field. The same argument is used for if you can show that is not a square in . And so on. Afterwards you need to argue that that all these three fields are distinct. Thus, there is some work that needs to be done in this argument but if you can do it, it will show these are the only subfields.
The problem however is this approach is not going to work in general. Let , this is a counterexample.