The \(\varvec{2}\)-class group of certain fields of the form \(\pmb {\mathbb {Q}(\sqrt{2},\sqrt{-p}, \sqrt{d})}\)

Let \(p\ge 5\) $p\ge 5$ be a prime number such that \(p\equiv 3\pmod 8\) $p\equiv 3(mod8)$ and d an odd positive squarefree integer. Put \(\mathbb {K}=\mathbb {Q}(\sqrt{2},\sqrt{-p}, \sqrt{d})\) $\mathbb{K}=\mathbb{Q}(\sqrt{2},\sqrt{-p},\sqrt{d})$ . In this paper, we compute the 2-rank of the class group of the fields \(\mathbb {K}=\mathbb {Q}(\sqrt{2},\sqrt{-p}, \sqrt{d})\) $\mathbb{K}=\mathbb{Q}(\sqrt{2},\sqrt{-p},\sqrt{d})$ . Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic and of type (2, 2) or (2, 4).