Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
274 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×AGL(1,ℤ/8) | C2 | semidirect |
| 2. #532 | C2 | semidirect |
| 3. C2×C4×D4 | C2 | semidirect |
| 4. #463 | C2 | semidirect |
| 5. #461 | C2 | semidirect |
| 6. #459 | C2 | semidirect |
| 7. #423 | C2 | semidirect |
| 8. #417 | C2 | semidirect |
| 16. C4⋊Q8 | C22 | semidirect |
| 17. #127 | C22 | semidirect |
| 18. #124 | C22 | semidirect |
| 30. C4⋊C8 | C22 | semidirect |
| 31. C4⋊C8 | C22 | semidirect |
| 32. #103 | C22 | semidirect |
| 33. #102 | C22 | semidirect |
| 43. C2×D4 | D4 | semidirect |
| 120. C4 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×AGL(1,ℤ/8) | 2 |
| #532 | 2 |
| C2×C4×D4 | 2 |
| #463 | 2 |
| #461 | 2 |
| #459 | 2 |
| #423 | 2 |
| #417 | 2 |