In this article, a dynamically tunable multiple plasmon-induced transparency (PIT) effect in monolayer graphene structure system with rectangular defect cavities is investigated both theoretically and numerically. Because the graphene of our structure exists in a continuous form, the Fermi level of the graphene can be dynamically tuned by simply applying a bias voltage. The expressions of the theoretical transmittance are correctly deduced, and the fitting theoretical results are very consistent with the numerical simulation data. When the Fermi level of the graphene is increased from 0.8 eV to 1.2 eV, the group index of the dual-PIT system is controlled between 383 and 766. Alternatively, the group index of the triple-PIT system is maintained between 445 and 812. Moreover, the maximum group index can reach 812 at 1.2 eV, which shows that it can be designed as an excellent slow light device. Therefore, the proposed structures and results may provide strong guidance towards multichannel optical filters, dynamically tunable and excellent slow light and light storage devices.