This paper establishes three-stage Stackelberg game models and one centralized decision model considering the cap-and-trade regulation and government green subsidy policies in a three-level low-carbon supply chain composed of the government, one manufacturer and one retailer. The influences of the consumer’s green preference, carbon trading price, carbon reduction cost coefficient on the optimal decisions and social welfare levels of different channel power models are analyzed. In addition, the proposed game models are coordinated by using two-part tariff contract and low-price promotion strategy respectively. Numerical examples are conducted to verify the derived conclusions. The results indicate that: Strong consumer green preference is beneficial for improving product greenness, channel members’ profits, and the whole supply chain’s social welfare. The greenness of the product and the level of social welfare in the supply chain are highest under the centralized decision model and lower in the decentralized decision models. The optimal green investment subsidy rate of the government decreases when consumer green preference is strong and increases when carbon emission reduction is more difficult. When the parameter values satisfy certain conditions, the perfect coordination of the low-carbon supply chain can be achieved be means of two-part tariff contract and low-price promotion strategy.