Clustering methods based on deep neural networks have proven promising for clustering real-world data because of their high representational power. In this paper, we propose a systematic taxonomy of clustering methods that utilize deep neural networks. We base our taxonomy on a comprehensive review of recent work and validate the taxonomy in a case study. In this case study, we show that the taxonomy enables researchers and practitioners to systematically create new clustering methods by selectively recombining and replacing distinct aspects of previous methods with the goal of overcoming their individual limitations. The experimental evaluation confirms this and shows that the method created for the case study achieves state-of-the-art clustering quality and surpasses it in some cases.