Consider an Ising ferromagnet defined on a triangular lattice. This is similar to square lattice, except every site now has six nearest neighbors instead of four. Triangular lattice ferromagnet has a phase transition with the same critical exponents as the square lattice ferromagnet.
This is not the case for the triangular lattice antiferromagnet. Behavior is very different, since not all neighboring sites can simultaneously have the lowest energy. Links are unsatisfied. The model is said to be frustrated.
Dealing with frustration is a difficult problem!