CRISPR-based gene drives bias inheritance in their favour by inducing double-stranded breaks (DSBs) at wild-type homologous loci and using the drive transgene as a repair template – converting drive heterozygotes into homozygotes. Recent studies have shown that alternate end-joining repair mechanisms produce cut-resistant alleles that rapidly induce drive failure. Multiplexing – simultaneously targeting multiple sites at the wild-type locus – is commonly assumed to overcome this issue since resistance would need to develop at all target sites for the system to fail. This may work for some population suppression drives targeting essential (e.g. viability or fertility) genes if careful design can ensure cut-resistant alleles themselves have low fitness. However, here, models are used to demonstrate that this approach will be ineffective when targeting neutral loci. We then go on to compare the performance of four alternative population level multiplexing approaches with standard individual level multiplexing. Two of these approaches have mechanisms preventing them from becoming linked, thus avoiding multiple simultaneous DSBs and giving a large improvement. Releasing multiple unlinked drives gives a modest improvement while releasing multiple drives that may become linked over time produces a decrease in performance under the conditions tested here. Based on performance and technical feasibility we then take one approach forward for further investigation, demonstrating its robustness to different performance parameters and its potential for controlling very large target populations.