Cauchy’s theorem for the prime 6

Before you think I have gone totally crackers:

Cauchy’s theorem says that a finite group whose order is divisible by a prime number p contains a subgroup which is cyclic of order p.

My co-authors and I have proved some similar results, of which the one referred to in the title is the following:

A finite grup whose order is divisible by 6 contains a subgroup which is either cyclic of order 6, dihedral of order 6, or isomorphic to the alternating group of degree 4 (with order 12).

When the more general theorem is proved and the paper written, I hope to elaborate on this. But my question for now is: have you seen this before?