Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces :  
$$
G/H = SO(n)\big/S\big({(O_{t_1})}_\Delta ^{d_1} \times \cdots \times {(O_{t_s})}_\Delta ^{d_s}  \big),
 $$  
$$
G/H = SU(n)\big/S\big( (U_{t_1})_\Delta ^{d_1}\times \cdots \times (U_{t_s})_\Delta ^{d_s} \big).
$$
Note that in both the cases $H$ may NOT be connected. At least some reference is also helpful.

Notations: For a group $H$, we denote by $H_\Delta^n$ the diagonally embedded copy of $H$ in the $n$-fold direct product $H^n$.