Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1827 conjugacy classes of subgroups; only the 823 normal subgroups are shown.
This group has 1827 conjugacy classes of subgroups — too many to draw. Only its 823 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 | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C23×D8 | C2 |
| LCS | G ▷ 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 |
|---|---|---|
| 2. C22×C4×D4 | C2 | semidirect |
| 3. C2×#442 | C2 | direct (N × H) |
| 4. C22×#106 | C2 | semidirect |
| 5. C22×#102 | C2 | semidirect |
| 6. C22×#97 | C2 | semidirect |
| 9. C22×D8 | C4 | semidirect |
| 14. C2×C4×D4 | C22 | semidirect |
| 18. #442 | C22 | direct (N × H) |
| 19. C2×#106 | C22 | semidirect |
| 20. C2×#102 | C22 | semidirect |
| 21. C2×#97 | C22 | semidirect |
| 25. C22×D4 | C2×C4 | semidirect |
| 32. C2×D8 | C2×C4 | semidirect |
| 35. C22×C8 | C2×C4 | semidirect |
| 39. C4×D4 | C23 | semidirect |
| 45. #106 | C23 | semidirect |
| 46. #102 | C23 | semidirect |
| 47. #97 | C23 | semidirect |
| 56. C2×D4 | C22×C4 | semidirect |
| 68. D8 | C22×C4 | semidirect |
| 70. C2×C8 | C22×C4 | semidirect |
| 88. D4 | C23×C4 | semidirect |
| 100. C8 | C23×C4 | semidirect |
| 105. C22 | #442 | direct (N × H) |
| 115. C2 | C2×#442 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D8 | 2 |
| C22×C4×D4 | 2 |
| C2×#442 | 2 |
| C22×#106 | 2 |
| C22×#102 | 2 |
| C22×#97 | 2 |