Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
381 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ 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. C2×H(5,𝔽2) | C2 | semidirect |
| 2. #575 | C2 | semidirect |
| 3. #535 | C2 | semidirect |
| 4. #455 | C2 | semidirect |
| 5. #453 | C2 | semidirect |
| 6. #421 | C2 | semidirect |
| 7. C2×#100 | C2 | semidirect |
| 13. C2×#41 | C22 | semidirect |
| 18. #137 | C22 | semidirect |
| 20. AGL(1,ℤ/8) | C22 | semidirect |
| 22. C2×M16 | C22 | semidirect |
| 23. #127 | C22 | semidirect |
| 24. #124 | C22 | semidirect |
| 27. #117 | C22 | semidirect |
| 28. #104 | C22 | semidirect |
| 29. #100 | C22 | semidirect |
| 38. #41 | D4 | semidirect |
| 40. C2×Q8 | D4 | semidirect |
| 41. C2×Q8 | C23 | semidirect |
| 53. C2×D4 | D4 | semidirect |
| 55. C2×D4 | C23 | semidirect |
| 65. M16 | C23 | semidirect |
| 70. C42 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2) | 2 |
| #575 | 2 |
| #535 | 2 |
| #455 | 2 |
| #453 | 2 |
| #421 | 2 |
| C2×#100 | 2 |