List of small groups

Example

Up to isomorphism, the groups of order $n \le 15$ are the following. Here $C_n$ is the cyclic group of order $n$ and $\times$ is the direct product. The group $D_{2m}$ is the dihedral group of order $2m$, and $Q_8$ is the quaternion group. The group $A_4$ is the alternating group on $4$ letters, and $\mathrm{Dic}_{12} = C_3 \rtimes C_4$ is the dicyclic group of order $12$.

1. Orders $1$, $2$, $3$, $5$, $7$, $11$, $13$: only $C_n$.
2. Order $4$: $C_4$ and the Klein four-group $C_2 \times C_2$.
3. Order $6$: $C_6$ and $D_6 \cong S_3$, the symmetric group.
4. Order $8$: $C_8$, $C_4 \times C_2$, $C_2 \times C_2 \times C_2$, $D_8$ and $Q_8$.
5. Order $9$: $C_9$ and $C_3 \times C_3$.
6. Order $10$: $C_{10}$ and $D_{10}$.
7. Order $12$: $C_{12}$, $C_6 \times C_2$, $A_4$, $D_{12}$ and $\mathrm{Dic}_{12}$.
8. Order $14$: $C_{14}$ and $D_{14}$.
9. Order $15$: only $C_{15}$.

So the numbers of groups of orders $1, 2, \dots, 15$ are $1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1$. For order $16$ there are already $14$ groups.