# Subgroup structure of direct product of D8 and Z2

From Groupprops

This article gives specific information, namely, subgroup structure, about a particular group, namely: direct product of D8 and Z2.

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This article discusses the subgroup structure of the direct product of D8 and Z2.

A presentation for the group that we use is:

.

The group has the following subgroups:

- The trivial group. (1)
- The cyclic group of order two. This equals the commutator subgroup, is central, and is also the set of squares. Isomorphic to cyclic group:Z2. (1)
- The subgroups and . These are both central subgroups of order two, but are related by an outer automorphism. Isomorphic to cyclic group:Z2. (2)
- The subgroups , , , , , , , and . These are all related by automorphisms, and are all 2-subnormal subgroups. They come in four conjugacy classes, namely the class , the class , the class , and the class . Isomorphic to cyclic group:Z2. (8)
- The subgroup . This is the center, hence is a characteristic subgroup. Isomorphic to Klein four-group. (1)
- The subgroups , , , and . These are all normal subgroups, but are related by outer automorphisms. Isomorphic to Klein four-group. (4)
- The subgroups , , , , , , , . These subgroups are all 2-subnormal subgroups and are related by outer automorphisms, and they come in four conjugacy classes of size two. Isomorphic to Klein four-group. (8)
- The subgroups and . They are both normal and are related via an outer automorphism. Isomorphic to cyclic group:Z4. (2)
- The subgroups and . These are normal and are related by outer automorphisms. Isomorphic to elementary abelian group of order eight. (2)
- The subgroups , , and . These are all normal and are related by an outer automorphism. Isomorphic to dihedral group:D8. (4)
- The subgroup . This is a characteristic subgroup. Isomorphic to direct product of Z4 and Z2. (1)
- The whole group. (1)

## Tables for quick information

### 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 | 11 | 7 | 3 | 1 | |

Cyclic group:Z4 | 2 | 2 | 2 | 0 | |

Klein four-group | 13 | 9 | 5 | 1 | |

Direct product of Z4 and Z2 | 1 | 1 | 1 | 1 | |

Dihedral group:D8 | 4 | 4 | 4 | 0 | |

Elementary abelian group of order eight | 2 | 2 | 2 | 0 | |

Direct product of D8 and Z2 | 1 | 1 | 1 | 1 | |

Total | -- | 35 | 27 | 19 | 5 |

### Table listing number of subgroups by order

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

1 | 1 | 1 | 1 | |

11 | 7 | 3 | 1 | |

15 | 11 | 7 | 1 | |

7 | 7 | 7 | 1 | |

1 | 1 | 1 | 1 | |

Total | 35 | 27 | 19 | 5 |

## The commutator subgroup (type (2))

This is the two-element subgroup generated by .

### Subgroup-defining functions yielding this subgroup

### Subgroup properties satisfied by this subgroup

On account of being a commutator subgroup as well as an agemo subgroup, this subgroup is a verbal subgroup. Thus, it satisfies the following subgroup properties:

- Fully invariant subgroup
- Image-closed fully invariant subgroup
- Characteristic subgroup
- Image-closed characteristic subgroup

It also satisfies the following properties:

### Subgroup properties not satisfied by this subgroup

- Intermediately characteristic subgroup
- Normal-isomorph-free subgroup: There are other normal subgroups isomorphic to it.
- Complemented normal subgroup, lattice-complemented subgroup, permutably complemented subgroup: This follows from the general phenomenon that nilpotent implies center is normality-large.

## The center (type (5))

This is a Klein four-subgroup comprising the identity, , and .

### Subgroup-defining functions yielding this subgroup

### Subgroup properties satisfied by this subgroup

- Characteristic subgroup
- Central subgroup, central factor, transitively normal subgroup
- Characteristic central factor
- Normality-large subgroup: This is more general, since nilpotent implies center is normality-large.

### Subgroup properties not satisfied by this subgroup

- Verbal subgroup
- Fully invariant subgroup
- Normal-isomorph-free subgroup
- Lattice-complemented subgroup, permutably complemented subgroup: This follows from the fact that nilpotent implies center is normality-large.

## The unique characteristic subgroup of order eight (type (10))

This is a subgroup generated by and , and is the direct product of a cyclic group of order four generated by and a cyclic group of order two generated by .

### Subgroup-defining functions yielding this subgroup

- Maximal among abelian characteristic subgroups: It is the unique maximal among abelian characteristic subgroups. In general, there need not be a unique subgroup maximal among abelian characteristic subgroups.
`Further information: Maximal among abelian characteristic subgroups may be multiple and isomorphic` - The unique constructibly critical subgroup.

### Subgroup properties satisfied by this subgroup

- Isomorph-free subgroup
- Intermediately characteristic subgroup
- Maximal characteristic subgroup
- Characteristic subgroup
- Self-centralizing subgroup
- Maximal among abelian characteristic subgroups
- Maximal among abelian normal subgroups
- Abelian critical subgroup, constructibly critical subgroup, critical subgroup