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We presentIRAM 30 m and APEX telescope observations of CO(1-0) and CO(2-1) lines in 36 group-dominant early-type galaxies, completing our molecular gas survey of dominant galaxies in the Complete Local-volume Groups Sample. We detect CO emission in 12 of the galaxies at >4σ significance, with molecular gas masses in the range ∼0.01-6x10⁸M_☉, as well as CO in absorption in the non-dominant group member galaxy NGC 5354.In total 21 of the 53 CLoGS dominant galaxies are detected in CO and we confirm our previous findings that they have low star formation rates (0.01-1M_☉/yr) but short depletion times (>1Gyr) implying rapid replenishment of their gas reservoirs. Comparing molecular gas mass with radio luminosity, we find that a much higher fraction of our group-dominant galaxies (60±16%) are AGN-dominated than is the case for the general population of ellipticals, but that there is no clear connection between radio luminosity and the molecular gas mass. Using data from the literature, we find that at least 27 of the 53 CLoGS dominant galaxies contain HI, comparable to the fraction of nearby non-cluster early type galaxies detected in HI and significantly higher that the fraction in the Virgo cluster. We see no correlation between the presence of an X-ray detected intra-group medium and molecular gas in the dominant galaxy, but find that the HI-richest galaxies are located in X-ray faint groups. Morphological data from the literature suggests the cold gas component most commonly takes the form of a disk, but many systems show evidence of galaxy-galaxy interactions, indicating that they may have acquired their gas through stripping or mergers. We provide improved molecular gas mass estimates for two galaxies previously identified as being in the centres of cooling flows, NGC 4636 and NGC 5846, and find that they are relatively molecular gas poor compared to our other detected systems.