Are two finite groups of the same order always isomorphic? Are two finite groups of the same order always isomorphic? Some simple example would be great!

Just an added note: 
All finite groups of prime order $p$ are isomorphic, and they are all isomorphic to $\langle \mathbb Z_p, +_p\rangle$, where $+_p$ is addition modulo p.
This makes a nice exercise to prove it. I'm sure you can search math.stackexchange to check your proof and compare with others.
