Trying to brush up on my group theory before next year...

One thing I've always had trouble with is proving two groups are (or in this case aren't) isomorphic.

So here's my Q.

Show that $\displaystyle C_8$ and $\displaystyle C_4 \times C_2$ are not isomorphic.

My thoughts... I'm thinking something along the lines of, an element of order 6 in the group $\displaystyle C_4 \times C_2$ is just the identity element since if you have $\displaystyle g \in C_4 \times C_2$ then $\displaystyle g^6 = g^{4+2} = g^4 \cdot g^2 = e \cdot e = e$? Whereas an element of order 6 in the group $\displaystyle C_8$ is not.

Think this is wrong so any help appreciated.