We present IRAM 30m 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 > 4 \sigma significance , with molecular gas masses in the range \sim 0.01-6 \times 10 ^ { 8 } M _ { \odot } , 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 ( 0.01-1 M _ { \odot } ~ { } yr ^ { -1 } ) but short depletion times ( < 1 Gyr ) 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 \pm 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 H i , comparable to the fraction of nearby non-cluster early type galaxies detected in H i 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 H i -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 .