Automatic modulation classification (AMC) plays an important role in intelligent wireless communications. With the rapid development of deep learning in recent years, neural network-based automatic modulation classification methods have become increasingly mature. However, the high complexity and large number of parameters of neural networks make them difficult to deploy in scenarios and receiver devices with strict requirements for low latency and storage. Therefore, this paper proposes a lightweight neural network-based AMC framework. To improve classification performance, the framework combines complex convolution with residual networks. To achieve a lightweight design, depthwise separable convolution is used. To compensate for any performance loss resulting from a lightweight design, a hybrid data augmentation scheme is proposed. The simulation results demonstrate that the lightweight AMC framework reduces the number of parameters by approximately 83.34% and the FLOPs by approximately 83.77%, without a degradation in performance.