Sample groups Klein V, $Z_4$, $S_3$, $D_4$. These four groups: Klein V, $Z_4$, $S_3$, $D_4$ were probably the most interesting examples used to solve for counterexamples so far.  They're so useful that I can most likely guess that one of the four would be the answer on any homework problem.  However I cannot but wonder, what are the unique properties for every one of these that you all can list?  I would like to have a list in the back of my head just in case.  
FYI: $Z_4$ is Z mod 4.  $S_3$ describes the permutations of an equilateral triangle and $D_4$ describes the permutations of a square.  
 A: Each of these groups can be characterized as being smallest with respect to some interesting property, where smallest in this answer means by order:
$V_4$: smallest non-cyclic group, smallest split extension of nontrivial groups (namely $\mathbb{Z}_2$ by $\mathbb{Z}_2$), smallest group with exponent strictly smaller than order, smallest group with $|\text{Aut}(G)| > |G|$
$\mathbb{Z}_4$: smallest cyclic group of non-prime order, smallest cyclic non-simple group, smallest non-split extension (also of $\mathbb{Z}_2$ by $\mathbb{Z}_2$)
$S_3$: smallest non-abelian group, smallest non-nilpotent group, smallest group with a non-normal subgroup (namely a Sylow $2$-subgroup $\langle (12) \rangle \le S_3$), smallest (nontrivial) semidirect product
$D_4$: smallest $p$-group that is a semidirect product, smallest nonabelian nilpotent group (along with $Q_8$)
$Q_8$: smallest nonabelian group with all subgroups normal, smallest group that is not a direct or semidirect product of abelian groups
A: More particularly, $D_4$ describes the symmetries of a square. Klein-4 is a good example of a non-cyclic abelian group isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$. $S_3$ is isomorphic $D_3$ which is the group of symmetries of an equilateral triangle. $S_3$ is the set of permutations on { 1, 2, 3 }
