Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
406 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#41 | C2 | semidirect |
| 2. C22×SD16 | C2 | semidirect |
| 3. C22×D8 | C2 | semidirect |
| 4. C2×#121 | C2 | semidirect |
| 5. #449 | C2 | direct (N × H) |
| 6. C2×#103 | C2 | semidirect |
| 7. C2×#102 | C2 | semidirect |
| 8. C2×C22⋊C8 | C2 | semidirect |
| 12. C2×#41 | C22 | semidirect |
| 21. C2×SD16 | C22 | semidirect |
| 23. C2×D8 | C22 | semidirect |
| 27. #121 | C22 | semidirect |
| 30. #103 | C22 | semidirect |
| 31. #102 | C22 | semidirect |
| 32. C22⋊C8 | C22 | semidirect |
| 41. C2×Q8 | D4 | semidirect |
| 52. C2×D4 | D4 | semidirect |
| 89. Q8 | C2×D4 | semidirect |
| 97. D4 | C2×D4 | semidirect |
| 139. C2 | #449 | 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 |
| C22×SD16 | 2 |
| C22×D8 | 2 |
| C2×#121 | 2 |
| #449 | 2 |
| C2×#103 | 2 |
| C2×#102 | 2 |
| C2×C22⋊C8 | 2 |