Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
450 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 120 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. C2×#123 | C2 | semidirect |
| 2. C2×#121 | C2 | semidirect |
| 3. C2×#120 | C2 | semidirect |
| 4. C22×C22⋊C4 | C2 | semidirect |
| 5. #399 | C2 | semidirect |
| 6. #392 | C2 | semidirect |
| 7. #390 | C2 | semidirect |
| 8. C4×C22⋊C4 | C2 | semidirect |
| 12. C22×D4 | C22 | semidirect |
| 18. C2×C4⋊C4 | C22 | semidirect |
| 19. C2×C4⋊C4 | C22 | semidirect |
| 25. C2×C22⋊C4 | C22 | semidirect |
| 26. C2×C22⋊C4 | C22 | semidirect |
| 27. C2×C42 | C22 | semidirect |
| 28. #95 | C22 | semidirect |
| 47. C22×C4 | D4 | semidirect |
| 60. C22⋊C4 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#123 | 2 |
| C2×#121 | 2 |
| C2×#120 | 2 |
| C22×C22⋊C4 | 2 |
| #399 | 2 |
| #392 | 2 |
| #390 | 2 |
| C4×C22⋊C4 | 2 |