Deep neural networks (DNNs) based approaches have achieved superior performance in single image super-resolution (SR). To obtain better visual quality, DNNs for SR are generally designed with massive computation overhead. To accelerate network inference under resource constraints, we propose a classification-based dynamic network for efficient super-resolution (CDNSR), which combines the classification and SR networks in a unified framework. Specifically, CDNSR decomposes a large image into a number of image-patches, and uses a classification network to categorize them into different classes based on the restoration difficulty. Each class of image-patches will be handled by the SR network that corresponds to the difficulty of this class. In particular, we design a new loss to trade off between the computational overhead and the reconstruction quality. Besides, we apply contrastive learning based knowledge distillation to guarantee the performance of SR networks and the quality of reconstructed images. Extensive experiments show that CDNSR significantly outperforms the other SR networks and backbones on image quality and computational overhead.