Question

Prove that order of each subgroup of a finite group is a divisor of the order of the group.

Answer 1

The smallest example is the alternating group G = A4, which has 12 elements but no subgroup of order 6. A "Converse of Lagrange's Theorem" (CLT) group is a finite group with the property that for every divisor of the order of the group, there is a subgroup of that order.
Hope it helps you!