Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3973 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 3973 conjugacy classes of subgroups — too many to draw. Only its 607 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 | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×D4 | C2 | semidirect |
| 2. C22×#123 | C2 | semidirect |
| 3. C22×#120 | C2 | semidirect |
| 4. #2613 | C2 | direct (N × H) |
| 5. C2×#394 | C2 | semidirect |
| 6. C2×#392 | C2 | semidirect |
| 7. C2×#381 | C2 | semidirect |
| 8. C2×#379 | C2 | semidirect |
| 14. C23×D4 | C22 | semidirect |
| 15. C23×D4 | C22 | semidirect |
| 18. C2×#123 | C22 | semidirect |
| 20. C2×#120 | C22 | semidirect |
| 21. C22×C4⋊C4 | C22 | semidirect |
| 24. C22×C22⋊C4 | C22 | semidirect |
| 25. C22×C22⋊C4 | C22 | semidirect |
| 26. C22×C22⋊C4 | C22 | semidirect |
| 27. #394 | C22 | semidirect |
| 28. #392 | C22 | semidirect |
| 29. #381 | C22 | semidirect |
| 30. #379 | C22 | semidirect |
| 31. C2×#95 | C22 | semidirect |
| 35. C25 | D4 | semidirect |
| 51. C22×D4 | C23 | semidirect |
| 53. C23×C4 | D4 | semidirect |
| 62. C2×C4⋊C4 | C23 | semidirect |
| 70. C2×C22⋊C4 | C23 | semidirect |
| 71. C2×C22⋊C4 | C23 | semidirect |
| 72. C2×C22⋊C4 | C23 | semidirect |
| 73. #95 | C23 | semidirect |
| 92. C24 | C2×D4 | semidirect |
| 117. C22×C4 | C2×D4 | semidirect |
| 203. C2 | #2613 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D4 | 2 |
| C22×#123 | 2 |
| C22×#120 | 2 |
| #2613 | 2 |
| C2×#394 | 2 |
| C2×#392 | 2 |
| C2×#381 | 2 |
| C2×#379 | 2 |