# Subgroup structure of generalized quaternion group:Q16

From Groupprops

This article gives specific information, namely, subgroup structure, about a particular group, namely: generalized quaternion group:Q16.

View subgroup structure of particular groups | View other specific information about generalized quaternion group:Q16

The generalized quaternion group:Q16, denoted is a group of order , is a generalized quaternion group. It can be described by the following presentation:

.

Note that from these relations, and . This in turn forces that , forcing to have order two. We shall denote this element of order two, which is clearly central, as .

Here is a list of subgroups:

- The trivial subgroup. Isomorphic to trivial group. (1)
- The center, which is a subgroup of order two, generated by . Isomorphic to cyclic group:Z2. (1)
- The cyclic subgroup of order four generated by . Isomorphic to cyclic group:Z4. (1)
- The four cyclic subgroups of order four, namely: , , and . These come in two conjugacy classes of 2-subnormal subgroups, one conjugacy class comprising and and the other comprising and . Isomorphic to cyclic group:Z4. (4)
- The cyclic subgroup of order eight, generated by . This is characteristic; in fact, it equals the centralizer of commutator subgroup. Isomorphic to cyclic group:Z8. (1)
- Two quaternion groups of order eight, namely and . Isomorphic to quaternion group. (2)
- The whole group. (1)

### Table classifying isomorphism types of subgroups

Group name | GAP ID | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|

Trivial group | 1 | 1 | 1 | 1 | |

Cyclic group:Z2 | 1 | 1 | 1 | 1 | |

Cyclic group:Z4 | 5 | 3 | 1 | 1 | |

Cyclic group:Z8 | 1 | 1 | 1 | 1 | |

Quaternion group | 2 | 2 | 2 | 0 | |

Generalized quaternion group:Q16 | 1 | 1 | 1 | 1 | |

Total | -- | 11 | 9 | 7 | 5 |

### Table listing number of subgroups by order

Group order | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|

1 | 1 | 1 | 1 | |

1 | 1 | 1 | 1 | |

5 | 3 | 1 | 1 | |

3 | 3 | 3 | 1 | |

1 | 1 | 1 | 1 | |

Total | 11 | 9 | 7 | 5 |