Give an example of three different groups with eight elements

Question

Give an example of three different groups with eight elements. Why are the groups different?

One particular answer that I found was the groups $\mathbb{Z}_8$, $\mathbb{Z}_4 \times\mathbb{Z}_2$, and $\mathbb{Z}_2\times \mathbb{Z}_2\times\mathbb{Z}_2$. Would someone be able to explain this answer easily so I can understand? Feel free to give an a different answer if you think you have a better one or you believe this one is wrong.

Thanks!

Answer 1

Hint: Yes, those three groups will work. To see why they're different, recall that isomorphisms will preserve the number of elements of a particular order. Try figuring out the number of elements of each order in each group.

Answer 2

There are two more, both of them non-abelian. (That is, multiplication isn't commutative.) The easy one is the group of symmetries of a square.