Flag manifolds have the form $G/C(S)$ where $C(S)$ is the centralizer (in $G$) of its center $S$ (a torus).

Clearly, the Wolf spaces $\mathrm{SO}(p+4)/\mathrm{SO}(p)\mathrm{SO}(4)$ ($p\geqslant3$) don't have this form since $\mathrm{SO}(p)\mathrm{SO}(4)$ has discrete center.

Flag manifolds have the form $G/C(S)$ where $C(S)$ is the centralizer (in $G$) of its center $S$ (a torus).

Most $G/H$ in the list don't have this form, for $H$ has discrete center in each case except the complex Grassmannians $\mathrm{SU}(p+2)/\mathrm S(\mathrm{U}(p)\times\mathrm{U}(2))$ and $\mathrm{SO}(6)/\mathrm{SO}(2)\mathrm{SO}(4)$.