The Hughes phenomenon of Hyperspectral images (HSIs) with the hundreds of continuous narrow bands makes the computational cost of HSIs processing high. Band selection is an effective way to solve such a problem and a lot of band selection methods have been proposed in recent years. In this paper, a novel hyper-graph regularized subspace clustering with skip connections (HRSC-SC) is proposed for band selection of hyperspectral image, which is a clustering-based band selection method. The networks combine subspace clustering into the convolutional auto-encoder by thinking of it as a self-expressive layer. To make full use of the historical feature maps obtained from the networks and tackle the problem of gradient vanishing caused by multiple nonlinear transformations, the symmetrical skip connections are added to the networks to pass image details from encoder to decoder. Furthermore, the hyper-graph regularization is presented to consider the manifold structure reflecting geometric information within data, which accurately describes the multivariate relationship between data points and makes the results of clustering more accurate so that select the most representative band subset. The proposed HRSC-SC band selection method is compared with the existing robust band selection algorithms on Indian Pines, Salinas-A, and Pavia University HSIs, showing that the results of the proposed method outperform the current state-of-the-art band selection methods. Especially, the overall accuracy of the clustering is the best on three real HSIs compared to other methods when the band selection number is 25, reaching 82.62%, 92.48%, and 96.5% respectively.