Every Finite Simple Group (which is not Cyclic) has Even Order, and the Order of every Finite Simple noncommutative group is Doubly Even, i.e., divisible by 4 (Feit and Thompson 1963).
See also Burnside Problem, Finite Group, Order (Group), Simple Group
Feit, W. and Thompson, J. G. ``Solvability of Groups of Odd Order.'' Pacific J. Math. 13, 775-1029, 1963.