Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
6459 conjugacy classes of subgroups; only the 3223 normal subgroups are shown.
This group has 6459 conjugacy classes of subgroups — too many to draw. Only its 3223 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#41 | C2 | semidirect |
| 2. C2×#550 | C2 | direct (N × H) |
| 3. C22×#127 | C2 | semidirect |
| 4. C22×#121 | C2 | semidirect |
| 5. C22×C4×Q8 | C2 | semidirect |
| 6. C22×C4×D4 | C2 | semidirect |
| 8. C22×#41 | C22 | semidirect |
| 14. #550 | C22 | direct (N × H) |
| 15. C2×#127 | C22 | semidirect |
| 16. C2×#121 | C22 | semidirect |
| 17. C2×C4×Q8 | C22 | semidirect |
| 18. C2×C4×D4 | C22 | semidirect |
| 25. C2×#41 | C23 | semidirect |
| 26. C22×Q8 | D4 | semidirect |
| 40. #127 | C23 | semidirect |
| 41. #121 | C23 | semidirect |
| 42. C4×Q8 | C23 | semidirect |
| 43. C4×D4 | C23 | semidirect |
| 54. C2×Q8 | C2×D4 | semidirect |
| 66. C22×C4 | #41 | semidirect |
| 85. Q8 | C22×D4 | semidirect |
| 93. C2×C4 | C2×#41 | semidirect |
| 102. C22 | #550 | direct (N × H) |
| 110. C4 | C22×#41 | semidirect |
| 112. C2 | C2×#550 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C2×#550 | 2 |
| C22×#127 | 2 |
| C22×#121 | 2 |
| C22×C4×Q8 | 2 |
| C22×C4×D4 | 2 |