In the field of cancer treatment, drug combination therapy appears to be a promising treatment strategy compared to monotherapy. Recently, plenty of computational models are gradually applied to prioritize synergistic drug combinations. However, the existing prediction models have not fully exploited the multi-way relations between drug combinations and cell lines. Besides, the number of identified drug-drug-cell line triplets is insufficient owning to the high cost of in vitro screening, which affects the ability of models to capture and utilize multi-way relations. To address this challenge, we design the multi-view hypergraph contrastive learning model, termed MHCLSyn, for synergistic drug combination prediction. First, the synergistic drug-drug-cell line triplets are formulated as a drug synergy hypergraph, and three task-specific hypergraphs are designed based on the drug synergy hypergraph. Then, we design a multi-view hypergraph contrastive learning with enhancement schemes, which allows for more expressive and discriminative node representation learning on drug synergy hypergraph. After that, the representations of nodes indicating drug-drug-cell line triplets are inputted to fully connected network for making predictions. Extensive experiments show MHCLSyn achieves better performance than state-of-the-art prediction models on benchmark datasets and is applicable to unseen drug combinations or cell lines. Case study indicates that MHCLSyn is capable of detecting potential synergistic drug combinations.