We know that, $\vec \nabla \cdot \vec B=0$ but $\vec \nabla \cdot \vec H\neq 0$, if $\vec \nabla \cdot \vec M \neq 0$. Does it mean that, in those cases $\vec H$-field has poles although $\vec B$-field does not?

It is always true that $\boldsymbol{\nabla}\cdot \textbf{B}=0$ (implying that there are no magnetic monopoles). However, $\boldsymbol{\nabla}\cdot \textbf{H}\neq 0$ when $\boldsymbol{\nabla}\cdot \textbf{M} \neq 0$. Does it mean that, in those cases $\textbf{H}$-field has poles although $\textbf{B}$-field does not?