Is there an example of fields $F_1$, $F_2$, and $F_3$ such that $\mathbb{Q}\subset F_1\subset F_2\subset F_3$ such that $[F_3:\mathbb{Q}]=8$ and each field is Galois over all its subfields but $F_2$ is not Galois over $\mathbb{Q}$?
I know of $F_3$ is Galois over $\mathbb{Q}$, then it will automatically be Galois over all other subfields, but that's about it. What's a good field to investigate here? Thanks.