Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
5711 conjugacy classes of subgroups; only the 3183 normal subgroups are shown.
This group has 5711 conjugacy classes of subgroups — too many to draw. Only its 3183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator, Frattini, cosocle | C4 | C26 |
| socle | C24 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#41 | C2 | semidirect |
| 2. C22×#135 | C2 | direct (N × H) |
| 3. C23×Q16 | C2 | semidirect |
| 4. C23×SD16 | C2 | semidirect |
| 5. C23×D8 | C2 | semidirect |
| 6. C24×C8 | C2 | semidirect |
| 8. C22×#41 | C22 | semidirect |
| 14. C2×#135 | C22 | direct (N × H) |
| 15. C22×Q16 | C22 | semidirect |
| 16. C22×SD16 | C22 | semidirect |
| 17. C22×D8 | C22 | semidirect |
| 20. C23×C8 | C22 | semidirect |
| 24. C2×#41 | C23 | semidirect |
| 36. #135 | C23 | direct (N × H) |
| 37. C2×Q16 | C23 | semidirect |
| 38. C2×SD16 | C23 | semidirect |
| 39. C2×D8 | C23 | semidirect |
| 42. C22×C8 | C23 | semidirect |
| 49. #41 | C24 | semidirect |
| 62. Q16 | C24 | semidirect |
| 63. SD16 | C24 | semidirect |
| 64. D8 | C24 | semidirect |
| 67. C2×C8 | C24 | semidirect |
| 71. C23 | #135 | direct (N × H) |
| 91. C22 | C2×#135 | direct (N × H) |
| 100. C2 | C22×#135 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C22×#135 | 2 |
| C23×Q16 | 2 |
| C23×SD16 | 2 |
| C23×D8 | 2 |
| C24×C8 | 2 |