Non-abelian group and periods of elements

Prove that a non-abelian group $\displaystyle G$ of order 6 must have at least one element $\displaystyle x$ of period 3.

This is what I have so far.

Let $\displaystyle x$ in $\displaystyle G$ be a non-identity element. $\displaystyle x$ can have period 2, 3, or 6.

Period of $\displaystyle x$ cannot be 6 because that make $\displaystyle G$ cyclic, thus abelian.

If $\displaystyle x$ has period 3, we are done.

If $\displaystyle x$ has period 2, ... (stuck here)

Maybe looking at an element $\displaystyle y$ in $\displaystyle G$ and $\displaystyle y$ is not in $\displaystyle <x>$, but not sure what to do with it.

Thanks in advance for any help.