Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
893 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 893 conjugacy classes of subgroups — too many to draw. Only its 263 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C2×#120 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C8 | C2 | semidirect |
| 2. C2×C8×D4 | C2 | semidirect |
| 4. #1869 | C2 | semidirect |
| 5. #1850 | C2 | semidirect |
| 6. #1779 | C2 | semidirect |
| 7. C8×C22⋊C4 | C2 | semidirect |
| 12. C23×C8 | C22 | semidirect |
| 16. C2×C4⋊C8 | C22 | semidirect |
| 19. C2×C22⋊C8 | C22 | semidirect |
| 20. C2×C4×C8 | C22 | semidirect |
| 25. #336 | C22 | semidirect |
| 38. C22×C8 | D4 | semidirect |
| 43. #120 | C8 | direct (N × H) |
| 60. C24 | C2×C8 | semidirect |
| 65. C2×D4 | C2×C8 | semidirect |
| 94. C22⋊C4 | C2×C8 | semidirect |
| 106. C23 | C22×C8 | semidirect |
| 123. C2×C4 | C22×C8 | semidirect |
| 129. C8 | #120 | direct (N × H) |
| 138. C22 | C8×D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C8 | 2 |
| C2×C8×D4 | 2 |
| C4×#120 | 2 |
| #1869 | 2 |
| #1850 | 2 |
| #1779 | 2 |
| C8×C22⋊C4 | 2 |