Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
274 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C22×D4 |
| prime radical | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×#41 | C2 | semidirect |
| 3. C2×C4×D4 | C2 | semidirect |
| 4. #442 | C2 | semidirect |
| 5. #440 | C2 | semidirect |
| 6. C4×SD16 | C2 | semidirect |
| 7. C4×D8 | C2 | semidirect |
| 8. #428 | C2 | semidirect |
| 9. #418 | C2 | semidirect |
| 10. #417 | C2 | semidirect |
| 11. C4×M16 | C2 | semidirect |
| 15. AGL(1,ℤ/8) | C4 | direct (N × H) |
| 20. C4×Q8 | C22 | semidirect |
| 21. C4×Q8 | C22 | semidirect |
| 28. C4×D4 | C22 | semidirect |
| 29. C4×D4 | C22 | semidirect |
| 36. #107 | C22 | semidirect |
| 37. #106 | C22 | semidirect |
| 38. #103 | C22 | semidirect |
| 41. #102 | C22 | semidirect |
| 42. #97 | C22 | semidirect |
| 43. C4×C8 | C22 | semidirect |
| 46. #41 | C2×C4 | semidirect |
| 51. C2×D4 | C2×C4 | semidirect |
| 64. SD16 | C2×C4 | semidirect |
| 65. D8 | C2×C4 | semidirect |
| 67. M16 | C2×C4 | semidirect |
| 95. Q8 | C22×C4 | semidirect |
| 101. D4 | C22×C4 | semidirect |
| 123. C8 | C22×C4 | semidirect |
| 143. C4 | AGL(1,ℤ/8) | direct (N × H) |
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 |
| C4×#41 | 2 |
| C2×C4×D4 | 2 |
| #442 | 2 |
| #440 | 2 |
| C4×SD16 | 2 |
| C4×D8 | 2 |
| #428 | 2 |
| #418 | 2 |
| #417 | 2 |
| C4×M16 | 2 |