Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1847 conjugacy classes of subgroups; only the 863 normal subgroups are shown.
This group has 1847 conjugacy classes of subgroups — too many to draw. Only its 863 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 | D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C2×D4 |
| prime radical | C23×D8 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4×D4 | C2 | semidirect |
| 3. C2×C4×D8 | C2 | direct (N × H) |
| 4. C22×#107 | C2 | semidirect |
| 5. C22×#102 | C2 | semidirect |
| 6. C22×C4×C8 | C2 | semidirect |
| 9. C22×D8 | C4 | direct (N × H) |
| 14. C2×C4×D4 | C22 | semidirect |
| 18. C4×D8 | C22 | direct (N × H) |
| 19. C2×#107 | C22 | semidirect |
| 20. C2×#102 | C22 | semidirect |
| 21. C2×C4×C8 | C22 | semidirect |
| 25. C22×D4 | C2×C4 | semidirect |
| 33. C2×D8 | C2×C4 | direct (N × H) |
| 36. C22×C8 | C2×C4 | semidirect |
| 40. C4×D4 | C23 | semidirect |
| 46. #107 | C23 | semidirect |
| 47. #102 | C23 | semidirect |
| 48. C4×C8 | C23 | semidirect |
| 57. C2×D4 | C22×C4 | semidirect |
| 64. C22×C4 | D8 | direct (N × H) |
| 71. D8 | C22×C4 | direct (N × H) |
| 73. C2×C8 | C22×C4 | semidirect |
| 92. D4 | C23×C4 | semidirect |
| 98. C2×C4 | C2×D8 | direct (N × H) |
| 106. C8 | C23×C4 | semidirect |
| 111. C22 | C4×D8 | direct (N × H) |
| 119. C4 | C22×D8 | direct (N × H) |
| 123. C2 | C2×C4×D8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D8 | 2 |
| C22×C4×D4 | 2 |
| C2×C4×D8 | 2 |
| C22×#107 | 2 |
| C22×#102 | 2 |
| C22×C4×C8 | 2 |