Below is a list of all possible groups per order up to isomorphism.

Groups of prime order:

• All groups of prime order are isomorphic to a cyclic group of that order.

Groups of prime square order:

• All groups of order $p^2$ , where $p$ is a prime, are isomorphic to one of the following:
  • $C_{p^2}$ (Abelian): cyclic group of order $p^2$ .
  • $C_p\times C_p$ (Abelian): elementary abelian group of order $p^2$ .

Groups of order 1:

• trivial group (i.e. $\{ e\}$ ).

Groups of order 6:

• $C_6$ (Abelian): cyclic group of order 6.
• $S_3$ (non-Abelian): symmetric group where $n=3$ .

Groups of order 8:

• $C_8$ (Abelian): cyclic group of order 8.
• $C_4\times C_2$ (Abelian): direct product of two groups of a cyclic group of order 4 and a cyclic group of order 2.
• $C_2\times C_2\times C_2$ (Abelian): direct product of three groups of a cyclic group of order 2.
• $D_4$ (non-Abelian): octic group; dihedral group of degree 4.
• $Q_8$ (non-Abelian): quaternion group.

Groups of order 10:

• $C_{10}$ (Abelian): cyclic group of order 10.
• $D_5$ (non-Abelian): dihedral group of degree 5.

Groups of order 12:

• $C_{12}$ (Abelian): cyclic group of order 12.
• $C_2\times C_6$ (Abelian).
• $A_4$ (non-Abelian): alternating group of degree 4.
• $D_6$ (non-Abelian): dihedral group of degree 6.
• $\Dic(C_6)$ (non-Abelian): dicyclic group of order 12. This is a generalized quaternion group $Q_{12}$ .

Groups of order 14:

• $C_{14}$ (Abelian): cyclic group of order 14.
• $D_7$ (non-Abelian): dihedral group of degree 7.

Groups of order 15:

• $C_{15}$ (Abelian): cyclic group of order 15.

References

PJ

Pedersen, John: Groups of small order. http://www.math.usf.edu/~eclark/algctlg/small_groups.html