Three-stage Stackelberg game models and one centralized decision model was established to considered the cap-and-trade regulation and government green subsidy policies in a three-level low-carbon supply chain, which 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 associated with different game 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 social welfare are highest under the centralized decision model. The optimal green investment subsidy rate of the government decreases with consumer green preference and increases with carbon emission reduction. When the parameter values satisfy the 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.