Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
226 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| socle | C2 | C2×#121 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #575 | C2 | semidirect |
| 2. C2×#135 | C2 | semidirect |
| 3. C2×#131 | C2 | semidirect |
| 4. #473 | C2 | semidirect |
| 5. #472 | C2 | semidirect |
| 6. #471 | C2 | semidirect |
| 7. C2×#108 | C2 | semidirect |
| 8. #413 | C2 | semidirect |
| 20. C2×Q16 | C22 | semidirect |
| 21. C2×SD16 | C22 | semidirect |
| 22. C2×D8 | C22 | semidirect |
| 32. #108 | C22 | semidirect |
| 33. #108 | C22 | semidirect |
| 34. #101 | C22 | semidirect |
| 35. #100 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #575 | 2 |
| C2×#135 | 2 |
| C2×#131 | 2 |
| #473 | 2 |
| #472 | 2 |
| #471 | 2 |
| C2×#108 | 2 |
| #413 | 2 |