The Sylow theorems do more than prove non-simplicity; they also help organise the early classification of finite groups. Small orders give a training ground for recognizing cyclic groups, direct products, dihedral groups, and quaternion behavior. In this lesson, we focus in the groups of small order. The aim is not to memorize a table blindly, but to understand why order, element orders, commutativity, and Sylow subgroups distinguish the main examples.

Let $n$ be a positive integer. To \textbf{classify groups of order} $n$ means to list one representative from each isomorphism class of groups having order $n$.