Subgroup structure of groups of order 8
This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 8.
View subgroup structure of group families | View subgroup structure of groups of a particular order |View other specific information about groups of order 8
The list
| Group | Second part of GAP ID | Subgroup structure page | Lattice of subgroups picture |
|---|---|---|---|
| Cyclic group:Z8 | 1 | subgroup structure of cyclic group:Z8 | |
| Direct product of Z4 and Z2 | 2 | subgroup structure of direct product of Z4 and Z2 | |
| Dihedral group:D8 | 3 | subgroup structure of dihedral group:D8 | 
|
| Quaternion group | 4 | subgroup structure of quaternion group | 
|
| Elementary abelian group:E8 | 5 | subgroup structure of elementary abelian group:E8 |
Subgroup/quotient relationships
Subgroup relationships
Quotient relationships
Numerical information on counts of subgroups
Number of subgroups per isomorphism type
The number in each column is the number of subgroups in the given group of that isomorphism type:
| Group | Second part of GAP ID | Hall-Senior number | cyclic group:Z2 | cyclic group:Z4 | Klein four-group | Total (row sum + 2, for trivial group and whole group) |
|---|---|---|---|---|---|---|
| cyclic group:Z8 | 1 | 3 | 1 | 1 | 0 | 4 |
| direct product of Z4 and Z2 | 2 | 2 | 3 | 2 | 1 | 8 |
| dihedral group:D8 | 3 | 4 | 5 | 1 | 2 | 10 |
| quaternion group | 4 | 5 | 1 | 3 | 0 | 6 |
| elementary abelian group:E8 | 5 | 1 | 7 | 0 | 7 | 16 |
Number of normal subgroups per isomorphism type
| Group | Second part of GAP ID | Hall-Senior number | cyclic group:Z2 | cyclic group:Z4 | Klein four-group | Total (row sum + 2, for trivial group and whole group) |
|---|---|---|---|---|---|---|
| cyclic group:Z8 | 1 | 3 | 1 | 1 | 0 | 4 |
| direct product of Z4 and Z2 | 2 | 2 | 3 | 2 | 1 | 8 |
| dihedral group:D8 | 3 | 4 | 1 | 1 | 2 | 6 |
| quaternion group | 4 | 5 | 1 | 3 | 0 | 6 |
| elementary abelian group:E8 | 5 | 1 | 7 | 0 | 7 | 16 |
Number of subgroups of various kinds per order
| Group | Second part of GAP ID | Hall-Senior number | Subgroups of order 2 | Normal subgroups of order 2 | Subgroups of order 4 | Normal subgroups of order 4 |
|---|---|---|---|---|---|---|
| cyclic group:Z8 | 1 | 3 | 1 | 1 | 1 | 1 |
| direct product of Z4 and Z2 | 2 | 2 | 3 | 3 | 3 | 3 |
| dihedral group:D8 | 3 | 4 | 5 | 1 | 3 | 3 |
| quaternion group | 4 | 5 | 1 | 1 | 3 | 3 |
| elementary abelian group:E8 | 5 | 1 | 7 | 7 | 7 | 7 |
Possibilities for maximal subgroups
| Collection of isomorphism classes of maximal subgroups | Groups |
|---|---|
| cyclic group:Z4 only | cyclic group:Z8, quaternion group |
| Klein four-group only | elementary abelian group:E8 |
| cyclic group:Z4 and Klein four-group | direct product of Z4 and Z2, dihedral group:D8 |