Precision medicine fundamentally aims to establish causality between dysregulated biochemical mechanisms and cancer subtypes. Omics-based cancer subtyping has emerged as a revolutionary approach, as different level of omics records the biochemical products of multistep processes in cancers. This paper focuses on fully exploiting the potential of multi-omics data to improve cancer subtyping outcomes, and hence developed MoCLIM, a representation learning framework. MoCLIM independently extracts the informative features from distinct omics modalities. Using a unified representation informed by contrastive learning of different omics modalities, we can well-cluster the subtypes, given cancer, into a lower latent space. This contrast can be interpreted as a projection of inter-omics inference observed in biological networks. Experimental results on six cancer datasets demonstrate that our approach significantly improves data fit and subtyping performance in fewer high-dimensional cancer instances. Moreover, our framework incorporates various medical evaluations as the final component, providing high interpretability in medical analysis.