With the development of satellite photography, meteorologists are inclined to rely on methods for the automatic and efficient classification of weather images. However, many popular networks require numerous parameters and a lengthy inference time, making them unsuitable for real-time classification tasks. To solve these problems, a lightweight convolutional network termed the Channel-Dilation-Concatenation network (CDC-net) is constructed for meteorological satellite image classification. When extracting features, CDC-net utilizes depth-wise convolution rather than standard convolution. Additionally, a FeatureCopy operation was employed instead of a half-convolution operation. CDC-net extracts high-dimensional features and contains a local importance-based pooling layer, reducing the network's depth, the number of network parameters and inference time. Based on these techniques, the CDC-net achieves an accuracy of 93.56% on the Large-Scale Satellite Cloud Image Database for Meteorological Research, with a GPU inference time of 3.261 ms and 1.12 million parameters. Because many weather images reveal multiple weather patterns, multiple labels are necessary. Therefore, we propose a prediction method and conduct experiments on multi-label data. Experiments on single-label and multi-label meteorological satellite image datasets demonstrate the superiority of the CDC-net over other structures. Thus, the proposed CDC-net can provide a faster and lighter solution in meteorological satellite image classification.