Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
616 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 132 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 |
| 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. C23×D4 | C2 | semidirect |
| 2. C2×#121 | C2 | semidirect |
| 3. C2×#120 | C2 | semidirect |
| 4. C2×C4×D4 | C2 | semidirect |
| 5. #394 | C2 | semidirect |
| 6. #392 | C2 | semidirect |
| 7. #390 | C2 | semidirect |
| 8. #380 | C2 | semidirect |
| 9. #379 | C2 | semidirect |
| 16. C22×D4 | C22 | semidirect |
| 18. C23×C4 | C22 | semidirect |
| 22. C2×C4⋊C4 | C22 | semidirect |
| 23. C2×C4⋊C4 | C22 | semidirect |
| 26. C2×C22⋊C4 | C22 | semidirect |
| 27. C2×C22⋊C4 | C22 | semidirect |
| 28. C2×C22⋊C4 | C22 | semidirect |
| 29. C2×C22⋊C4 | C22 | semidirect |
| 30. C2×C42 | C22 | semidirect |
| 31. #95 | C22 | semidirect |
| 50. C2×D4 | D4 | semidirect |
| 51. C2×D4 | D4 | semidirect |
| 55. C22×C4 | D4 | semidirect |
| 91. C23 | C2×D4 | semidirect |
| 113. C2×C4 | C2×D4 | semidirect |
| 131. C22 | #121 | semidirect |
| 142. C4 | #120 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D4 | 2 |
| C2×#121 | 2 |
| C2×#120 | 2 |
| C2×C4×D4 | 2 |
| #394 | 2 |
| #392 | 2 |
| #390 | 2 |
| #380 | 2 |
| #379 | 2 |