Let $K$ be a number field. If $d$  be the smallest even integer such that $\Bbb Q (\zeta_d) \subset K,$  then I wanted to prove that if $d'>d$ then $\Bbb Q (\zeta_{d'}) \not\subset H(K),$ where $H(K)$ is Hilbert class field $K.$

I understand that it is not true in general. Can I see this working with some assumptions?

Ps. thanks to the comment of  Franz Lemmermeyer.