Existing hyperspectral image (HSI) super-resolution (SR) methods struggle to effectively capture the complex spectral-spatial relationships and low-level details, while diffusion models represent a promising generative model known for their exceptional performance in modeling complex relations and learning high and low-level visual features. The direct application of diffusion models to HSI SR is hampered by challenges such as difficulties in model convergence and protracted inference time. In this work, we introduce a novel Group-Autoencoder (GAE) framework that synergistically combines with the diffusion model to construct a highly effective HSI SR model (DMGASR). Our proposed GAE framework encodes high-dimensional HSI data into low-dimensional latent space where the diffusion model works, thereby alleviating the difficulty of training the diffusion model while maintaining band correlation and considerably reducing inference time. Experimental results on both natural and remote sensing hyperspectral datasets demonstrate that the proposed method is superior to other state-of-the-art methods both visually and metrically.