A series of highly connected metal-organic frameworks (MOFs), [Co8 (O)(OH)4 (H2 O)4 (ina)8 ](NO3 )2 ⋅2 C2 H5 OH⋅4 H2 O (1), [Co8 (O)(OH)4 (H2 O)4 (pba)8 ](NO3 )2 ⋅8 C2 H5 OH⋅28 H2 O (2), and [Co8 (O)(OH)4 (H2 O)4 (pbba)8 ](NO3 )2 ⋅guest (3), in which ina=isonicotinate, pba=4-pyridylbenzoate, and pbba=4-(pyridine-4-yl)phenylbenzoate, is reported. These MOFs contain a new secondary building unit (SBU), with a square Co4 (μ4 -O) central unit having the rare μ4 -O(2-) motif, which is decorated by the other four peripheral cobalt atoms through μ3 -OH in a windmill-like shape. This SBU holds 16 divergent connecting organic ligands, pyridyl-carboxylates, to form three different frameworks. The high porosity of desolvated 2 is shown by the efficient gas absorption of N2 , CO2 , CH4 , and H2 . In addition, 1 and 2 exhibit unusual canted antiferromagnetic behavior with spin-glass-like relaxation, with blocking temperatures that are fairly high, 20 K (1) and 10 K (2), for cobalt materials. The relationship between the metal clusters and linkers has been studied, in which the size and rotational degrees of freedom of the ligands are found to control the topology, gas sorption, and magnetic properties.