Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
220 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×C16 | C2 | 1 | 14 | yes | maximal, direct product factor |
| 3 | C23×C4 | C4 | 1 | 1 | yes | characteristic |
| 4 | C22×C8 | C4 | 1 | 7 | yes | — |
| 5 | C22×C8 | C22 | 1 | 7 | yes | — |
| 6 | C2×C16 | C22 | 1 | 28 | yes | direct product factor |
| 7 | C24 | C8 | 1 | 1 | yes | characteristic |
| 8 | C22×C4 | C8 | 1 | 7 | yes | — |
| 9 | C22×C4 | C2×C4 | 1 | 7 | yes | — |
| 10 | C2×C8 | C2×C4 | 1 | 21 | yes | — |
| 11 | C2×C8 | C23 | 1 | 7 | yes | — |
| 12 | C16 | C23 | 1 | 8 | yes | direct product factor |
| 13 | C23 | C16 | 1 | 8 | yes | direct product factor |
| 14 | C23 | C2×C8 | 1 | 7 | yes | — |
| 15 | C2×C4 | C2×C8 | 1 | 21 | yes | — |
| 16 | C2×C4 | C22×C4 | 1 | 7 | yes | — |
| 17 | C8 | C22×C4 | 1 | 7 | yes | — |
| 18 | C8 | C24 | 1 | 1 | yes | characteristic |
| 19 | C22 | C2×C16 | 1 | 28 | yes | direct product factor |
| 20 | C22 | C22×C8 | 1 | 7 | yes | — |
| 21 | C4 | C22×C8 | 1 | 7 | yes | — |
| 22 | C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 23 | C2 | C22×C16 | 1 | 14 | yes | direct product factor |
| 24 | C2 | C23×C8 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C8 | C24 |
| socle, prime radical | C24 | C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C16 | C2 | direct (N × H) |
| 6. C2×C16 | C22 | direct (N × H) |
| 12. C16 | C23 | direct (N × H) |
| 13. C23 | C16 | direct (N × H) |
| 19. C22 | C2×C16 | direct (N × H) |
| 23. C2 | C22×C16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C8 | 2 |
| C22×C16 | 2 |