The availability of recognized types of magic squares is related to the order of the squares. Orders are generally divided as odd and even, with even orders being further split into singly even (4k+2) and doubly even, (4k), k≥1. One type, Franklin, requires a further split of doubly even to doubly doubly even, (8k).
Order 3: See
Order 4: See type summary. The pandiagonal are all most-perfect, (compact + complete).
Order 5: See type summary. There are 16 ultramagic, (associative + pandiagonal).