Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1251 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1251 conjugacy classes of subgroups — too many to draw. Only its 487 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 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#126 | C2 | semidirect |
| 2. #2823 | C2 | direct (N × H) |
| 3. C2×#398 | C2 | semidirect |
| 4. C2×#396 | C2 | semidirect |
| 6. C2×#384 | C2 | semidirect |
| 9. C2×#126 | C22 | semidirect |
| 17. #398 | C22 | semidirect |
| 18. #396 | C22 | semidirect |
| 20. #384 | C22 | semidirect |
| 43. C2×C42 | D4 | semidirect |
| 73. C42 | C2×D4 | semidirect |
| 113. C2 | #2823 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#126 | 2 |
| #2823 | 2 |
| C2×#398 | 2 |
| C2×#396 | 2 |
| C2×#394 | 2 |
| C2×#384 | 2 |