Prove that a non-abelian group of order 6 must have at least one element of period 3.

This is what I have so far.

Let in be a non-identity element. can have period 2, 3, or 6.

Period of cannot be 6 because that make cyclic, thus abelian.

If has period 3, we are done.

If has period 2, ... (stuck here)

Maybe looking at an element in and is not in , but not sure what to do with it.

Thanks in advance for any help.