Determine the Galois group over Q of the splitting field of the question. List all of the subgroups of the Galois group. List all of the subfield of the splitting field of the equation.
Find the subgroups of it and correspond them to the intermediate fields.
For instance, if you choose a generator of the above group as , then one of the subgroups of it is .
The reason why I choose the above generator (among them) is because 3 has order 6 in ( Meanwhile, 2 has order 3 in ).
Since , the corresponding intermediate field is .
I'll leave it to you to find the remaining subgroups of and their corresponding intermediate fields.