Composite magic squares are only possible if the order of the magic square has two divisors, for which magic squares exist. This implies n=9 as minimal order. Since the maximum order of a magic square has been set arbitrarily to n=20 on these pages, only magic squares of orders 9, 12, 15, 16,18 and 20 are possible.
All algorithms are described in detail in my PDF book. (chapter: Composite Magic Squares)