For groups of finite order there are only a finite number of groups, up to isomorphism. For example, ifprime, thenthe cyclic group of orderIfis not prime then more possibilities exist, but there is still a finite number of possible groups up to isomorphism.

For groups up to order 8 these are:

Order | Groups |

1 | |

2 | |

3 | |

4 | |

5 | |

6 | |

7 | |

8 | |

9 | |

10 | |

11 | |

12 | |

13 | |

14 | |

15 |

Various methods exist for finding these groups – using Sylow's theorems, analysis of the conjugacy class equation, Lagrange's equation.