Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1155 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 1155 conjugacy classes of subgroups — too many to draw. Only its 287 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×M16 | C2 | semidirect |
| 3. C22×#102 | C2 | semidirect |
| 4. C2×#380 | C2 | semidirect |
| 5. #1895 | C2 | direct (N × H) |
| 6. C2×#340 | C2 | semidirect |
| 12. C22×M16 | C22 | semidirect |
| 16. C2×#121 | C4 | semidirect |
| 23. C2×#102 | C22 | semidirect |
| 25. #380 | C22 | semidirect |
| 27. #340 | C22 | semidirect |
| 54. C22×C8 | D4 | semidirect |
| 58. #121 | C2×C4 | semidirect |
| 127. C2×C8 | C2×D4 | semidirect |
| 222. C2 | #1895 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C22×#121 | 2 |
| C22×#102 | 2 |
| C2×#380 | 2 |
| #1895 | 2 |
| C2×#340 | 2 |