As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on the diagonal by $U(1) \times U(1)$, embedded in the diagonal as $(e^{i \theta_1}, e^{i \theta_2}, e^{-i(\theta_1 + \theta_2})$ then we get the full flag manifold of $SU(3)$. However, we can also embed $U(1)$ into $SU(3)$ on the diagonal as $(e^{i \theta}, e^{-i \theta}, 1)$. What is the corresponding quotient? Is it somehow pathological?