Graph classification is a critical research problem in many applications from different domains. In order to learn a graph classification model, the most widely used supervision component is an output layer together with classification loss (e.g.,cross-entropy loss together with softmax or margin loss). In fact, the discriminative information among instances are more fine-grained, which can benefit graph classification tasks. In this paper, we propose the novel Label Contrastive Coding based Graph Neural Network (LCGNN) to utilize label information more effectively and comprehensively. LCGNN still uses the classification loss to ensure the discriminability of classes. Meanwhile, LCGNN leverages the proposed Label Contrastive Loss derived from self-supervised learning to encourage instance-level intra-class compactness and inter-class separability. To power the contrastive learning, LCGNN introduces a dynamic label memory bank and a momentum updated encoder. Our extensive evaluations with eight benchmark graph datasets demonstrate that LCGNN can outperform state-of-the-art graph classification models. Experimental results also verify that LCGNN can achieve competitive performance with less training data because LCGNN exploits label information comprehensively.