The rank of the 2-class group of some fields with large degree

Let $n\ge 3$ be an integer and $d$ an odd square-free integer. We shall compute the rank of the $2$-class group of $L_{n,d}:=\mathbb{Q}({\zeta }_{2^n},\sqrt{d})$, when all the prime divisors of $d$ are congruent to $\pm 3(\mathrm{m}\mathrm{o}\mathrm{d}8)$ or $9(\mathrm{m}\mathrm{o}\mathrm{d}16)$.