Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
393 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C2×#120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#124 | C2 | semidirect |
| 2. C2×C22⋊Q8 | C2 | semidirect |
| 4. C22×C22⋊C4 | C2 | semidirect |
| 5. #397 | C2 | semidirect |
| 6. #394 | C2 | semidirect |
| 7. #394 | C2 | semidirect |
| 8. #393 | C2 | semidirect |
| 9. #390 | C2 | semidirect |
| 10. #388 | C2 | semidirect |
| 11. #388 | C2 | semidirect |
| 12. #386 | C2 | semidirect |
| 13. #386 | C2 | semidirect |
| 14. #379 | C2 | semidirect |
| 15. C4×C22⋊C4 | C2 | semidirect |
| 17. C22×Q8 | C22 | semidirect |
| 26. C2×C4⋊C4 | C22 | semidirect |
| 27. C2×C4⋊C4 | C22 | semidirect |
| 28. C2×C4⋊C4 | C22 | semidirect |
| 43. C2×C22⋊C4 | C22 | semidirect |
| 44. C2×C22⋊C4 | C22 | semidirect |
| 45. C2×C22⋊C4 | C22 | semidirect |
| 46. C2×C22⋊C4 | C22 | semidirect |
| 47. C2×C42 | C22 | semidirect |
| 48. C2×C42 | C22 | semidirect |
| 49. #95 | C22 | semidirect |
| 50. #95 | C22 | semidirect |
| 51. #95 | C22 | semidirect |
| 52. #95 | C22 | semidirect |
| 53. #95 | C22 | semidirect |
| 54. #95 | C22 | semidirect |
| 114. C22⋊C4 | D4 | semidirect |
| 220. C22 | #124 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#124 | 2 |
| C2×C22⋊Q8 | 2 |
| C2×#120 | 2 |
| C22×C22⋊C4 | 2 |
| #397 | 2 |
| #394 | 2 |
| #393 | 2 |
| #390 | 2 |
| #388 | 2 |
| #386 | 2 |
| #379 | 2 |
| C4×C22⋊C4 | 2 |