Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2347 conjugacy classes of subgroups; only the 1551 normal subgroups are shown.
This group has 2347 conjugacy classes of subgroups — too many to draw. Only its 1551 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 | C2 | C23×C42 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2275 | C2 | direct (N × H) |
| 3. C2×C4×C4⋊C4 | C2 | semidirect |
| 4. C2×C4×C22⋊C4 | C2 | semidirect |
| 5. C2×#376 | C2 | semidirect |
| 6. C22×#95 | C2 | semidirect |
| 9. C2×C4×D4 | C4 | semidirect |
| 15. C4×C4⋊C4 | C22 | semidirect |
| 16. C4×C22⋊C4 | C22 | semidirect |
| 17. #376 | C22 | semidirect |
| 21. C2×#95 | C22 | semidirect |
| 26. C23×C4 | C2×C4 | semidirect |
| 32. C4×D4 | C2×C4 | semidirect |
| 33. C2×C4⋊C4 | C2×C4 | semidirect |
| 35. C2×C22⋊C4 | C2×C4 | semidirect |
| 38. C2×C42 | C2×C4 | semidirect |
| 49. C2×D4 | C42 | semidirect |
| 61. C22×C4 | C22×C4 | semidirect |
| 66. C4⋊C4 | C22×C4 | semidirect |
| 67. C22⋊C4 | C22×C4 | semidirect |
| 69. C42 | C22×C4 | semidirect |
| 80. D4 | C2×C42 | semidirect |
| 109. C2 | #2275 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×D4 | 2 |
| #2275 | 2 |
| C2×C4×C4⋊C4 | 2 |
| C2×C4×C22⋊C4 | 2 |
| C2×#376 | 2 |
| C22×#95 | 2 |