Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
390 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 124 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 | C25 | C22 |
| 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×C22⋊Q8 | C2 | semidirect |
| 3. #399 | C2 | semidirect |
| 4. #396 | C2 | semidirect |
| 5. #393 | C2 | semidirect |
| 6. #385 | C2 | semidirect |
| 7. #385 | C2 | semidirect |
| 8. #380 | C2 | semidirect |
| 11. C22×Q8 | C22 | semidirect |
| 15. C2×C4⋊C4 | C22 | semidirect |
| 16. C2×C4⋊C4 | C22 | semidirect |
| 17. C2×C4⋊C4 | C22 | semidirect |
| 18. C2×C4⋊C4 | C22 | semidirect |
| 19. C2×C4⋊C4 | C22 | semidirect |
| 29. #95 | C22 | semidirect |
| 30. #95 | C22 | semidirect |
| 31. #95 | C22 | semidirect |
| 32. #95 | C22 | semidirect |
| 37. C24 | Q8 | semidirect |
| 120. C22 | C22⋊Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊Q8 | 2 |
| C22×C22⋊C4 | 2 |
| #399 | 2 |
| #396 | 2 |
| #393 | 2 |
| #385 | 2 |
| #380 | 2 |
| #379 | 2 |