Three metal-organic frameworks (MOFs) {[Co4(OH)2(adc)6(H2O)5][Co2(OH)(btrb)]2·8H2O}n (1), {[Co4(OH)2(btrb)(nip)3(H2O)3]·4H2O}n (2·4H2O) and {[Co4(OH)2(btrb)(btc)2(H2O)2]·9H2O}n (3·9H2O) (btrb = 1,4-bis(1,2,4-triazol-4-ylmethyl)benzene, H2adc = 1,3-adamantanedicarboxylic acid, H2nip = 5-nitroisophthalic acid and H3btc = 1,3,5-benzenetricarboxylic acid) were synthesized by a hydrothermal method. 1 shows an unusual (3,4)-connected 2D network with a point symbol of (4·62)2(42·62·82) based on [Co4(μ3-OH)2] and [Co2(μ3-OH)] units. 2 is a 6-connected pcu topology based on the [Co4(μ3-OH)2] unit. 3 is a rare (3,8)-connected 3D framework with a rare point symbol of (4·5·6)2(42·56·616·72·82) based on the [Co4(μ3-OH)2] unit. The magnetic measurements reveal that 1-3 exhibit antiferromagnetic interactions between the adjacent cobalt(ii) centers. 1-3 are good photocatalysts for the degradation of methylene blue (MB). The 1,2-position nitrogen atoms of the 4-substituted 1,2,4-triazole ligand act as strong σ-donators to form a quite stable coordination quadrangle with two metal ions. The btrb ligand containing two 4-substituted 1,2,4-triazole rings is favorable to form tetranuclear metal units. By the regulation of the multicarboxylate ligands, the connection number of [Co4(μ3-OH)2] clusters increases from 4 in 1 to 6 in 2 and then 8 in 3. This work presents a novel but effective method to design and regulate the connection number (4, 6 and 8 connected in this work) of MOFs with 4-substituted 1,2,4-triazole and rigid multicarboxylate ligands.