Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
864 conjugacy classes of subgroups; only the 452 normal subgroups are shown.
This group has 864 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. #532 | C2 | direct (N × H) |
| 3. C2×C4⋊Q8 | C2 | semidirect |
| 4. C2×#127 | C2 | semidirect |
| 5. C2×#124 | C2 | semidirect |
| 6. C2×#121 | C2 | semidirect |
| 7. C2×C4×D4 | C2 | semidirect |
| 8. C22×C42 | C2 | semidirect |
| 10. C2×#41 | C22 | semidirect |
| 18. C4⋊Q8 | C22 | semidirect |
| 19. #127 | C22 | semidirect |
| 20. #124 | C22 | semidirect |
| 21. #121 | C22 | semidirect |
| 22. C4×D4 | C22 | semidirect |
| 28. C2×C42 | C22 | semidirect |
| 42. C22×C4 | D4 | semidirect |
| 73. C2×C4 | C2×D4 | semidirect |
| 74. C2×C4 | #41 | semidirect |
| 93. C4 | C2×#41 | semidirect |
| 96. C2 | #532 | 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 |
| #532 | 2 |
| C2×C4⋊Q8 | 2 |
| C2×#127 | 2 |
| C2×#124 | 2 |
| C2×#121 | 2 |
| C2×C4×D4 | 2 |
| C22×C42 | 2 |