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Revealing relationships between genes and disease phenotypes is a critical problem in biomedical studies. This problem has been challenged by the heterogeneity of diseases. Patients of a perceived same disease may form multiple subgroups, and different subgroups have distinct sets of important genes. It is hence imperative to discover the latent subgroups and reveal the subgroup-specific important genes. Some heterogeneity analysis methods have been proposed in recent literature. Despite considerable successes, most of the existing studies are still limited as they cannot accommodate data contamination and ignore the interconnections among genes. Aiming at these shortages, we develop a robust structured heterogeneity analysis approach to identify subgroups, select important genes as well as estimate their effects on the phenotype of interest. Possible data contamination is accommodated by employing the Huber loss function. A sparse overlapping group lasso penalty is imposed to conduct regularization estimation and gene identification, while taking into account the possibly overlapping cluster structure of genes. This approach takes an iterative strategy in the similar spirit of K-means clustering. Simulations demonstrate that the proposed approach outperforms alternatives in revealing the heterogeneity and selecting important genes for each subgroup. The analysis of Cancer Cell Line Encyclopedia data leads to biologically meaningful findings with improved prediction and grouping stability.