Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2019 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 2019 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×SD16 | C2 | semidirect |
| 2. C23×D8 | C2 | semidirect |
| 3. C22×#127 | C2 | semidirect |
| 4. C22×#124 | C2 | semidirect |
| 5. C2×#496 | C2 | direct (N × H) |
| 6. C22×#97 | C2 | semidirect |
| 12. C22×SD16 | C22 | semidirect |
| 14. C22×D8 | C22 | semidirect |
| 17. C2×#127 | C22 | semidirect |
| 19. C2×#124 | C22 | semidirect |
| 22. #496 | C22 | direct (N × H) |
| 23. C2×#97 | C22 | semidirect |
| 42. C2×SD16 | C23 | semidirect |
| 44. C2×D8 | C23 | semidirect |
| 45. C22×C8 | D4 | semidirect |
| 48. #127 | C23 | semidirect |
| 50. #124 | C23 | semidirect |
| 54. #97 | C23 | semidirect |
| 86. C2×C8 | C2×D4 | semidirect |
| 118. C8 | C22×D4 | semidirect |
| 125. C22 | #496 | direct (N × H) |
| 135. C2 | C2×#496 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×SD16 | 2 |
| C23×D8 | 2 |
| C22×#127 | 2 |
| C22×#124 | 2 |
| C2×#496 | 2 |
| C22×#97 | 2 |