\begin{tikzcd}
                                                                &  & \mathbb{Q} \arrow[rrrrdd] \arrow[rrdd] \arrow[lldd, dashed] \arrow[dd] &                                           &                                                              &  &                                             \\
                                                                &  &                                                                        &                                           &                                                              &  &                                             \\
{\mathbb{Q}(\sqrt[3] 2)} \arrow[rrrrdd] \arrow[rrdd] \arrow[dd] &  & \mathbb Q(\sqrt 3) \arrow[rrrrdd] \arrow[lldd, dashed]                 &                                           & \mathbb Q(\zeta_3 \sqrt 3) \arrow[rrdd] \arrow[lldd, dashed] &  & \mathbb Q(\zeta_3) \arrow[dd] \arrow[lldd]  \\
                                                                &  &                                                                        &                                           &                                                              &  &                                             \\
{\mathbb Q(\sqrt[3] 2, \sqrt 3)} \arrow[rrrdd]                  &  & {\mathbb Q(\zeta_3\sqrt 3 \sqrt[3] 2)} \arrow[rdd]                     &                                           & {\mathbb Q(\zeta_3, \sqrt[3] 2)} \arrow[ldd]                 &  & {\mathbb Q(\zeta_3, \sqrt 3)} \arrow[llldd] \\
                                                                &  &                                                                        &                                           &                                                              &  &                                             \\
                                                                &  &                                                                        & {\mathbb Q(\zeta_3, \sqrt[3] 2, \sqrt 3)} &                                                              &  &                                            
\end{tikzcd}