Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
272 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | C22×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×M16 | C2 | semidirect |
| 3. #436 | C2 | semidirect |
| 4. #435 | C2 | semidirect |
| 5. C8×D4 | C2 | semidirect |
| 6. #423 | C2 | semidirect |
| 7. #407 | C2 | semidirect |
| 8. C4×M16 | C2 | semidirect |
| 15. C2×M16 | C22 | semidirect |
| 16. C22×C8 | C22 | semidirect |
| 26. C4⋊C8 | C22 | semidirect |
| 27. C4⋊C8 | C22 | semidirect |
| 28. C22⋊C8 | C22 | semidirect |
| 29. C22⋊C8 | C22 | semidirect |
| 30. #97 | C22 | semidirect |
| 31. C4×C8 | C22 | semidirect |
| 50. M16 | D4 | direct (N × H) |
| 74. D4 | M16 | direct (N × H) |
| 96. C8 | C2×D4 | semidirect |
| 106. C22 | C2×M16 | semidirect |
| 114. C4 | C2×M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16 | 2 |
| C2×C4×D4 | 2 |
| #436 | 2 |
| #435 | 2 |
| C8×D4 | 2 |
| #423 | 2 |
| #407 | 2 |
| C4×M16 | 2 |