The rapid development of scRNA-seq technology in recent years has enabled us to capture high-throughput gene expression profiles at single-cell resolution, reveal the heterogeneity of complex cell populations, and greatly advance our understanding of the underlying mechanisms in human diseases. Traditional methods for gene co-expression clustering are limited to discovering effective gene groups in scRNA-seq data. In this paper, we propose a novel gene clustering method based on convolutional neural networks called Dual-Stream Subspace Clustering Network (DS-SCNet). DS-SCNet can accurately identify important gene clusters from large scales of single-cell RNA-seq data and provide useful information for downstream analysis. Based on the simulated datasets, DS-SCNet successfully clusters genes into different groups and outperforms mainstream gene clustering methods, such as DBSCAN and DESC, across different evaluation metrics. To explore the biological insights of our proposed method, we applied it to real scRNA-seq data of patients with Alzheimer's disease (AD). DS-SCNet analyzed the single-cell RNA-seq data with 10,850 genes, and accurately identified 8 optimal clusters from 6673 cells. Enrichment analysis of these gene clusters revealed functional signaling pathways including the ILS signaling, the Rho GTPase signaling, and hemostasis pathways. Further analysis of gene regulatory networks identified new hub genes such as ELF4 as important regulators of AD, which indicates that DS-SCNet contributes to the discovery and understanding of the pathogenesis in Alzheimer's disease.