Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
940 conjugacy classes of subgroups; only the 452 normal subgroups are shown.
This group has 940 conjugacy classes of subgroups — too many to draw. Only its 452 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator, Frattini, cosocle | C22 | C25 |
| socle | C23 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#41 | C2 | semidirect |
| 2. C24×C4 | C2 | semidirect |
| 3. #525 | C2 | direct (N × H) |
| 4. C2×#123 | C2 | semidirect |
| 5. C2×C22⋊Q8 | C2 | semidirect |
| 6. C2×#121 | C2 | semidirect |
| 7. C2×#120 | C2 | semidirect |
| 8. C2×C4×D4 | C2 | semidirect |
| 9. C2×#117 | C2 | semidirect |
| 12. C2×#41 | C22 | semidirect |
| 13. C22×Q8 | C22 | semidirect |
| 16. C22×D4 | C22 | semidirect |
| 22. C23×C4 | C22 | semidirect |
| 23. C23×C4 | C22 | semidirect |
| 24. #123 | C22 | semidirect |
| 25. C22⋊Q8 | C22 | semidirect |
| 26. #121 | C22 | semidirect |
| 27. #120 | C22 | semidirect |
| 28. C4×D4 | C22 | semidirect |
| 29. #117 | C22 | semidirect |
| 30. C2×C4⋊C4 | C22 | semidirect |
| 31. C2×C4⋊C4 | C22 | semidirect |
| 34. C2×C22⋊C4 | C22 | semidirect |
| 35. C2×C42 | C22 | semidirect |
| 44. C2×Q8 | C23 | semidirect |
| 50. C2×D4 | C23 | semidirect |
| 56. C22×C4 | D4 | semidirect |
| 66. C22×C4 | C23 | semidirect |
| 67. C4⋊C4 | C23 | semidirect |
| 68. C4⋊C4 | C23 | semidirect |
| 71. C22⋊C4 | C23 | semidirect |
| 72. C42 | C23 | semidirect |
| 82. C23 | #41 | semidirect |
| 100. C2×C4 | C2×D4 | semidirect |
| 118. C22 | C2×#41 | semidirect |
| 128. C2 | #525 | 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 |
| C24×C4 | 2 |
| #525 | 2 |
| C2×#123 | 2 |
| C2×C22⋊Q8 | 2 |
| C2×#121 | 2 |
| C2×#120 | 2 |
| C2×C4×D4 | 2 |
| C2×#117 | 2 |