<p>This paper is dedicated to the finite-time synchronization (FTS) of fractional-order inertial memristive neural networks (FOIMNNs) with distributed delays and impulse effects. Firstly, the finite time convergence principle of fractional order pulse system is established, and the obtained settling time is not only related to the system coefficients, but also affected by the pulse intensity and pulse interval. Secondly, two hybrid control schemes are designed for FOIMNNs with different orders: one is fractional-order quantized state feedback controller and the other is fractional-order quantized adaptive controller. Thirdly, the algebraic conditions for the FTS of several new FOIMNNs are derived without transforming the second-order terms into first-order terms, using Lyapunov functionals and fractional-order differential inequalities. Finally, the validity of the theoretical results is verified by simulation examples.</p>