There are five different groups of order 8. Each has at least one normal subgroup of order 2, and at least one normal subgroup of order 4. Wherever we have a group with a normal subgroup, the group can be regarded as an extension of the normal subgroup by its quotient group. This table lists all 14 ways a group of order 8 has a proper normal subgroup; and for each, it links to a page showing how to build it as an extension of that normal subgroup.
| type of extension resulting group |
× | ⋊ | ↑ | ⋊↑ | total |
|---|---|---|---|---|---|
| C8 | C2↑C4 C4↑C2 |
2 | |||
| C4×C2 | C2×C4 C4×C2 |
C2↑C22 C22↑C2 C4↑C2 |
5 | ||
| C23 | C2×C22 C22×C2 |
2 | |||
| D8 (dihedral) |
C4⋊C2 C22⋊C2 |
C2↑C22 |
3 | ||
| Q8 (quaternion) |
C2↑C22 |
C4⋊↑C2 |
2 | total | 4 | 2 | 7 | 1 | 14 |
This page provides examples for Extensions of Groups
See also my main index page for groups.
Copyright N.S.Wedd 2008