Summary: A regression model with skew-normal errors provides a useful extension for traditional normal regression models when the data involve asymmetric outcomes. Moreover, data that arise from a heterogeneous population can be efficiently analysed by a finite mixture of regression models. These observations motivate us to propose a novel finite mixture of median regression model based on a mixture of the skew-normal distributions to explore asymmetrical data from several subpopulations. With the appropriate choice of the tuning parameters, we establish the theoretical properties of the proposed procedure, including consistency for variable selection method and the oracle property in estimation. A productive nonparametric clustering method is applied to select the number of components, and an efficient EM algorithm for numerical computations is developed. Simulation studies and a real data set are used to illustrate the performance of the proposed methodologies.