The task of detecting common and unique characteristics among different cancer subtypes is an important focus of research that aims to improve personalized therapies. Unlike current approaches mainly based on predictive techniques, our study aims to improve the knowledge about the molecular mechanisms that descriptively led to cancer, thus not requiring previous knowledge to be validated. Here, we propose an approach based on contrast set mining to capture high-order relationships in cancer transcriptomic data. In this way, we were able to extract valuable insights from several cancer subtypes in the form of highly specific genetic relationships related to functional pathways affected by the disease. To this end, we have divided several cancer gene expression databases by the subtype associated with each sample to detect which gene groups are related to each cancer subtype. To demonstrate the potential and usefulness of the proposed approach we have extensively analysed RNA-Seq gene expression data from breast, kidney, and colon cancer subtypes. The possible role of the obtained genetic relationships was further evaluated through extensive literature research, while its prognosis was assessed via survival analysis, finding gene expression patterns related to survival in various cancer subtypes. Some gene associations were described in the literature as potential cancer biomarkers while other results have been not described yet and could be a starting point for future research.