Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1955 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1955 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×D8 | C2 | semidirect |
| 2. C22×#127 | C2 | semidirect |
| 3. C22×C4×D4 | C2 | semidirect |
| 4. C2×#459 | C2 | direct (N × H) |
| 5. C22×C4⋊C8 | C2 | semidirect |
| 6. C22×#102 | C2 | semidirect |
| 13. C22×D8 | C22 | semidirect |
| 16. C2×#127 | C22 | semidirect |
| 18. C2×C4×D4 | C22 | semidirect |
| 22. #459 | C22 | direct (N × H) |
| 23. C2×C4⋊C8 | C22 | semidirect |
| 24. C2×#102 | C22 | semidirect |
| 36. C22×D4 | D4 | semidirect |
| 46. C2×D8 | C23 | semidirect |
| 50. #127 | C23 | semidirect |
| 52. C4×D4 | C23 | semidirect |
| 58. C4⋊C8 | C23 | semidirect |
| 59. #102 | C23 | semidirect |
| 80. C2×D4 | C2×D4 | semidirect |
| 88. C22×C4 | D8 | semidirect |
| 124. D4 | C22×D4 | semidirect |
| 130. C2×C4 | C2×D8 | semidirect |
| 146. C22 | #459 | direct (N × H) |
| 154. C4 | C22×D8 | semidirect |
| 159. C2 | C2×#459 | 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×#127 | 2 |
| C22×C4×D4 | 2 |
| C2×#459 | 2 |
| C22×C4⋊C8 | 2 |
| C22×#102 | 2 |