With LLM usage rapidly increasing, their vulnerability to jailbreaks that create harmful outputs are a major security risk. As new jailbreaking strategies emerge and models are changed by fine-tuning, continuous testing for security vulnerabilities is necessary. Existing Red Teaming methods fall short in cost efficiency, attack success rate, attack diversity, or extensibility as new attack types emerge. We address these challenges with Modular And Diverse Malicious Attack MiXtures (MAD-MAX) for Automated LLM Red Teaming. MAD-MAX uses automatic assignment of attack strategies into relevant attack clusters, chooses the most relevant clusters for a malicious goal, and then combines strategies from the selected clusters to achieve diverse novel attacks with high attack success rates. MAD-MAX further merges promising attacks together at each iteration of Red Teaming to boost performance and introduces a similarity filter to prune out similar attacks for increased cost efficiency. The MAD-MAX approach is designed to be easily extensible with newly discovered attack strategies and outperforms the prominent Red Teaming method Tree of Attacks with Pruning (TAP) significantly in terms of Attack Success Rate (ASR) and queries needed to achieve jailbreaks. MAD-MAX jailbreaks 97% of malicious goals in our benchmarks on GPT-4o and Gemini-Pro compared to TAP with 66%. MAD-MAX does so with only 10.9 average queries to the target LLM compared to TAP with 23.3. WARNING: This paper contains contents which are offensive in nature.