Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
3260 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×#41 | C2 | 1 | 56 | yes | maximal, direct product factor |
| 2 | C23×Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C23×D4 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C24×C4 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C25 | C22 | 1 | 3 | yes | — |
| 6 | C2×#41 | C22 | 1 | 448 | yes | direct product factor |
| 7 | C22×Q8 | C22 | 1 | 28 | yes | complemented |
| 8 | C22×D4 | C22 | 1 | 84 | yes | complemented |
| 9 | C23×C4 | C22 | 1 | 3 | yes | — |
| 10 | C23×C4 | C22 | 1 | 1 | yes | characteristic |
| 11 | C23×C4 | C22 | 1 | 84 | yes | complemented |
| 12 | C24 | — | 2 | 24 | no | — |
| 13 | C24 | C23 | 1 | 42 | yes | — |
| 14 | C24 | C23 | 1 | 1 | yes | characteristic |
| 15 | #41 | C23 | 1 | 512 | yes | direct product factor |
| 16 | C2×Q8 | C23 | 1 | 112 | yes | complemented |
| 17 | C2×D4 | C23 | 1 | 336 | yes | complemented |
| 18 | C22×C4 | C23 | 1 | 42 | yes | — |
| 19 | C22×C4 | C23 | 1 | 14 | yes | — |
| 20 | C22×C4 | C23 | 1 | 336 | yes | complemented |
| 21 | C23 | — | 2 | 168 | no | — |
| 22 | C23 | #41 | 1 | 8 | yes | direct product factor |
| 23 | C23 | C24 | 1 | 84 | yes | — |
| 24 | C23 | C24 | 1 | 7 | yes | — |
| 25 | Q8 | C24 | 1 | 64 | yes | complemented |
| 26 | D4 | C24 | 1 | 192 | yes | complemented |
| 27 | C2×C4 | C24 | 1 | 28 | yes | — |
| 28 | C2×C4 | C24 | 1 | 84 | yes | — |
| 29 | C2×C4 | C24 | 1 | 192 | yes | complemented |
| 30 | C22 | — | 2 | 168 | no | — |
| 31 | C22 | C2×#41 | 1 | 28 | yes | direct product factor |
| 32 | C22 | C25 | 1 | 24 | yes | — |
| 33 | C22 | C25 | 1 | 7 | yes | — |
| 34 | C4 | C25 | 1 | 8 | yes | — |
| 35 | C4 | C25 | 1 | 24 | yes | — |
| 36 | C2 | — | 2 | 24 | no | — |
| 37 | C2 | C22×#41 | 1 | 14 | yes | direct product factor |
| 38 | C2 | C26 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22 |
| commutator, Frattini, cosocle | C2 | C26 |
| socle | C24 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#41 | C2 | direct (N × H) |
| 2. C23×Q8 | C2 | semidirect |
| 3. C23×D4 | C2 | semidirect |
| 4. C24×C4 | C2 | semidirect |
| 6. C2×#41 | C22 | direct (N × H) |
| 7. C22×Q8 | C22 | semidirect |
| 8. C22×D4 | C22 | semidirect |
| 11. C23×C4 | C22 | semidirect |
| 15. #41 | C23 | direct (N × H) |
| 16. C2×Q8 | C23 | semidirect |
| 17. C2×D4 | C23 | semidirect |
| 20. C22×C4 | C23 | semidirect |
| 22. C23 | #41 | direct (N × H) |
| 25. Q8 | C24 | semidirect |
| 26. D4 | C24 | semidirect |
| 29. C2×C4 | C24 | semidirect |
| 31. C22 | C2×#41 | direct (N × H) |
| 37. C2 | C22×#41 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41 | 2 |
| C23×Q8 | 2 |
| C23×D4 | 2 |
| C24×C4 | 2 |