Modular multilevel converter (MMC) is one of the most attractive topologies in medium-voltage motor drives. Unfortunately, the open-circuit (OC) faults of the submodule (SM) switching devices affect seriously the MMC's reliable operation. Under the condition of a low modulation index, OC faults have a more severe impact, and the fault characteristics change fundamentally, bringing great challenges to the fault diagnosis strategy. Therefore, this article proposes a multipoint SM OC fault diagnosis strategy under a low modulation index, which consists of fault detection (FD) and fault location (FL). In more detail, FD improves the existing sliding-mode observer and realizes a rapid detection of the faulty arm and fault type. As for FL, it achieves the recovery of arm current and the differentiation of the SM capacitor-voltage data. Moreover, the OC fault diagnosis of SM can be achieved quickly and accurately through the strategy proposed in this article while using a low modulation index. In addition, this proposed strategy has fault tolerance and can reach the stability degree of MMC in a short time. Finally, the multipoint and the multitype fault diagnosis are carried out through simulation and experiment to verify the effectiveness of the proposed strategy.