On some imaginary triquadratic number fields $k$ with $\mathrm{Cl}_2(k)\simeq(2, 4)$ or $(2, 2, 2)$

Let $d$ be a square free integer and $L_d:=\mathbb{Q}(\zeta_{8},\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, $\mathrm{Cl}_2(L_d)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.