Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1351 conjugacy classes of subgroups; only the 527 normal subgroups are shown.
This group has 1351 conjugacy classes of subgroups — too many to draw. Only its 527 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 | C22×#121 | C2 |
| 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 |
|---|---|---|
| 2. C22×#117 | C2 | semidirect |
| 3. C2×#482 | C2 | direct (N × H) |
| 4. C22×#107 | C2 | semidirect |
| 5. C22×#106 | C2 | semidirect |
| 6. C22×#102 | C2 | semidirect |
| 7. C22×#102 | C2 | semidirect |
| 8. C22×C22⋊C8 | C2 | semidirect |
| 17. C2×#117 | C22 | semidirect |
| 24. #482 | C22 | direct (N × H) |
| 25. C2×#107 | C22 | semidirect |
| 26. C2×#106 | C22 | semidirect |
| 27. C2×#102 | C22 | semidirect |
| 28. C2×#102 | C22 | semidirect |
| 29. C2×C22⋊C8 | C22 | semidirect |
| 54. #117 | C23 | semidirect |
| 64. #107 | C23 | semidirect |
| 65. #106 | C23 | semidirect |
| 66. #102 | C23 | semidirect |
| 67. #102 | C23 | semidirect |
| 68. C22⋊C8 | C23 | semidirect |
| 159. C22 | #482 | direct (N × H) |
| 173. C2 | C2×#482 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#121 | 2 |
| C22×#117 | 2 |
| C2×#482 | 2 |
| C22×#107 | 2 |
| C22×#106 | 2 |
| C22×#102 | 2 |
| C22×C22⋊C8 | 2 |