Graph-based Subspace Clustering （SC） has become a popular technique for processing high-dimensional data efficiently. However， existing methods suffer from the following problems： the constructed graph neglects to establish associations with clustering and fails to capture intrinsic correlated structure of the data. To address these issues， a new SC method was proposed， called Graph regularized Elastic Net Subspace Clustering （GENSC）. GENSC employed L₂ norm regularization to enhance the connectivity among samples with the correlated structure， and utilized L₁ norm regularization to discard the connectivity among samples from different subspaces. Simultaneously， a nearest neighbor graph of the representation was constructed to capture the intrinsic local structure among samples， and a rank constraint was incorporated to encourage the learned graph to have clear clustering structure. GENSC integrated L₂ norm， L₁ norm， and rank constraint into a general framework which was solved by an iterative optimization algorithm. Experimental results on nine real-world datasets demonstrate that on ChinaCXRSet， the accuracy and Normalized Mutual Information （NMI） values of GENSC exceeded the second-best method by 9.03 and 7.61 percentage points， respectively， and the clustering Purity reached the best； on UMIST， the accuracy， NMI， and Purity values of GENSC exceeded the second-best method by 4.15， 3.17 and 5.21 percentage points， respectively， validating the effectiveness of GENSC.