Disturbances may create cascading failures in power systems and lead to widespread blackouts. Controlled islanding is an effective approach to mitigate cascading failures by partitioning the power system into a set of disjoint islands. To retain the stability of the power system following disturbances, the islanding strategy should not only be minimally disruptive, but also guarantee post-disturbance stability. In this paper, we study the problem of synthesizing post-disturbance stability-aware controlled islanding strategies. To ensure post-disturbance stability, our computation of islanding strategies takes load-generation balance and transmission line capacity constraints into consideration, leading to a hybrid optimization problem with both discrete and continuous variables. To mitigate the computational challenge incurred when solving the hybrid optimization program, we propose the concepts of hybrid submodularity and hybrid matroid. We show that the islanding problem is equivalent to a hybrid matroid optimization program, whose objective function is hybrid supermodular. Leveraging the supermodularity property, we develop an efficient local search algorithm and show that the proposed algorithm achieves 1/2-optimality guarantee. We compare our approach with a baseline using mixed-integer linear program on IEEE 118-bus, IEEE 300-bus, ActivSg 500-bus, and Polish 2383-bus systems. Our results show that our approach outperforms the baseline in terms of the total cost incurred during islanding across all test cases. Furthermore, our proposed approach can find an islanding strategy for large-scale test cases such as Polish 2383-bus system, whereas the baseline approach becomes intractable.