Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1287 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1287 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 |
| prime radical | C23×D4 | C22 |
| 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. C22×#125 | C2 | semidirect |
| 3. C2×#478 | C2 | direct (N × H) |
| 4. C22×#107 | C2 | semidirect |
| 5. C22×#106 | C2 | semidirect |
| 6. C22×C4⋊C8 | C2 | semidirect |
| 14. C2×#125 | C22 | semidirect |
| 24. #478 | C22 | direct (N × H) |
| 25. C2×#107 | C22 | semidirect |
| 26. C2×#106 | C22 | semidirect |
| 27. C2×C4⋊C8 | C22 | semidirect |
| 48. #125 | C23 | semidirect |
| 62. #107 | C23 | semidirect |
| 63. #106 | C23 | semidirect |
| 64. C4⋊C8 | C23 | semidirect |
| 142. C22 | #478 | direct (N × H) |
| 156. C2 | C2×#478 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#125 | 2 |
| C22×C4×D4 | 2 |
| C2×#478 | 2 |
| C22×#107 | 2 |
| C22×#106 | 2 |
| C22×C4⋊C8 | 2 |
| C22×#102 | 2 |