Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2847 conjugacy classes of subgroups; only the 1007 normal subgroups are shown.
This group has 2847 conjugacy classes of subgroups — too many to draw. Only its 1007 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| socle | C24 | C24 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4×D4 | C2 | semidirect |
| 3. C4×#127 | C2 | direct (N × H) |
| 4. C2×#390 | C2 | semidirect |
| 5. C2×#384 | C2 | semidirect |
| 6. C2×C43 | C2 | semidirect |
| 9. C2×#127 | C4 | direct (N × H) |
| 12. C2×C4×D4 | C22 | semidirect |
| 18. #390 | C22 | semidirect |
| 19. #384 | C22 | semidirect |
| 20. C43 | C22 | semidirect |
| 24. C22×D4 | C2×C4 | semidirect |
| 32. #127 | C2×C4 | direct (N × H) |
| 39. C2×C42 | C2×C4 | semidirect |
| 40. C2×C42 | D4 | semidirect |
| 52. C2×D4 | C22×C4 | semidirect |
| 69. C42 | C22×C4 | semidirect |
| 70. C42 | C2×D4 | semidirect |
| 87. C2×C4 | C4×D4 | semidirect |
| 88. C2×C4 | #127 | direct (N × H) |
| 106. C4 | C2×C4×D4 | semidirect |
| 107. C4 | C2×#127 | direct (N × H) |
| 109. C2 | C4×#127 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#127 | 2 |
| C22×C4×D4 | 2 |
| C4×#127 | 2 |
| C2×#390 | 2 |
| C2×#384 | 2 |
| C2×C43 | 2 |