Attributed graph clustering, which learns node representation from node attribute and topological graph for clustering, is a fundamental but challenging task for graph analysis. Recently, methods based on graph contrastive learning (GCL) have obtained impressive clustering performance on this task. Yet, we observe that existing GCL-based methods 1) fail to benefit from imprecise clustering labels; 2) require a post-processing operation to get clustering labels; 3) cannot solve out-of-sample (OOS) problem. To address these issues, we propose a novel attributed graph clustering network, namely Self-supervised Contrastive Attributed Graph Clustering (SCAGC). In SCAGC, by leveraging inaccurate clustering labels, a self-supervised contrastive loss, which aims to maximize the similarities of intra-cluster nodes while minimizing the similarities of inter-cluster nodes, are designed for node representation learning. Meanwhile, a clustering module is built to directly output clustering labels by contrasting the representation of different clusters. Thus, for the OOS nodes, SCAGC can directly calculate their clustering labels. Extensive experimental results on four benchmark datasets have shown that SCAGC consistently outperforms 11 competitive clustering methods.