Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1783 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1783 conjugacy classes of subgroups — too many to draw. Only its 567 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 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D8 | C2 | semidirect |
| 2. C23×M16 | C2 | semidirect |
| 3. C22×#121 | C2 | semidirect |
| 4. C2×#469 | C2 | direct (N × H) |
| 5. C22×#106 | C2 | semidirect |
| 6. C22×#102 | C2 | semidirect |
| 11. C22×D8 | C22 | semidirect |
| 13. C22×M16 | C22 | semidirect |
| 17. C2×#121 | C22 | semidirect |
| 20. #469 | C22 | direct (N × H) |
| 21. C2×#106 | C22 | semidirect |
| 22. C2×#102 | C22 | semidirect |
| 39. C2×D8 | C23 | semidirect |
| 41. C2×M16 | C23 | semidirect |
| 43. C22×C8 | D4 | semidirect |
| 47. #121 | C23 | semidirect |
| 51. #106 | C23 | semidirect |
| 52. #102 | C23 | semidirect |
| 84. C2×C8 | C2×D4 | semidirect |
| 119. C8 | C22×D4 | semidirect |
| 127. C22 | #469 | direct (N × H) |
| 138. C2 | C2×#469 | 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 |
| C23×M16 | 2 |
| C22×#121 | 2 |
| C2×#469 | 2 |
| C22×#106 | 2 |
| C22×#102 | 2 |