Multi-label classification (MLC) is a prediction task where each sample can have more than one label. We propose a novel contrastive learning boosted multi-label prediction model based on a Gaussian mixture variational autoencoder (C-GMVAE), which learns a multimodal prior space and employs a contrastive loss. Many existing methods introduce extra complex neural modules to capture the label correlations, in addition to the prediction modules. We find that by using contrastive learning in the supervised setting, we can exploit label information effectively, and learn meaningful feature and label embeddings capturing both the label correlations and predictive power, without extra neural modules. Our method also adopts the idea of learning and aligning latent spaces for both features and labels. More specifically, C-GMVAE imposes a Gaussian mixture structure on the latent space, to alleviate posterior collapse and over-regularization issues, in contrast to previous works based on a unimodal prior. C-GMVAE outperforms existing methods on multiple public datasets and can often match other models' full performance with only 50\% of the training data. Furthermore, we show that the learnt embeddings provide insights into the interpretation of label-label interactions.