Detecting offensive content in internet memes is challenging as it needs additional contextual knowledge. While previous works have only focused on detecting offensive memes, classifying them further into implicit and explicit categories depending on their severity is still a challenging and underexplored area. In this work, we present an end-to-end multitask model for addressing this challenge by empirically investigating two correlated tasks simultaneously: (i) offensive meme detection and (ii) explicit-implicit offensive meme detection by leveraging the two self-supervised pre-trained models. The first pre-trained model, referred to as the “knowledge encoder,” incorporates contextual knowledge of the meme. On the other hand, the second model, referred to as the “fine-grained information encoder”, is trained to understand the obscure psycho-linguistic information of the meme. Our proposed model utilizes contrastive learning to integrate these two pre-trained models, resulting in a more comprehensive understanding of the meme and its potential for offensiveness. To support our approach, we create a large-scale dataset, CM-Off-Meme, as there is no publicly available such dataset for the code-mixed Hindi-English (Hinglish) domain. Empirical evaluation, including both qualitative and quantitative analysis, on the CM-Off-Meme dataset demonstrates the effectiveness of the proposed model in terms of cross-domain generalization.