Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1631 conjugacy classes of subgroups; only the 863 normal subgroups are shown.
This group has 1631 conjugacy classes of subgroups — too many to draw. Only its 863 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 | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C2×D4 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ 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×Q8 | C2 | semidirect |
| 4. C2×C4×SD16 | C2 | direct (N × H) |
| 5. C22×#106 | C2 | semidirect |
| 6. C22×#103 | C2 | semidirect |
| 8. C22×C4×C8 | C2 | semidirect |
| 12. C22×SD16 | C4 | direct (N × H) |
| 17. C2×C4×Q8 | C22 | semidirect |
| 26. C4×SD16 | C22 | direct (N × H) |
| 27. C2×#106 | C22 | semidirect |
| 28. C2×#103 | C22 | semidirect |
| 30. C2×C4×C8 | C22 | semidirect |
| 34. C22×Q8 | C2×C4 | semidirect |
| 48. C2×SD16 | C2×C4 | direct (N × H) |
| 51. C22×C8 | C2×C4 | semidirect |
| 55. C4×Q8 | C23 | semidirect |
| 67. #106 | C23 | semidirect |
| 68. #103 | C23 | semidirect |
| 70. C4×C8 | C23 | semidirect |
| 79. C2×Q8 | C22×C4 | semidirect |
| 95. C22×C4 | SD16 | direct (N × H) |
| 102. SD16 | C22×C4 | direct (N × H) |
| 104. C2×C8 | C22×C4 | semidirect |
| 127. Q8 | C23×C4 | semidirect |
| 140. C2×C4 | C2×SD16 | direct (N × H) |
| 148. C8 | C23×C4 | semidirect |
| 153. C22 | C4×SD16 | direct (N × H) |
| 163. C4 | C22×SD16 | direct (N × H) |
| 167. C2 | C2×C4×SD16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×SD16 | 2 |
| C22×C4×Q8 | 2 |
| C22×C4×D4 | 2 |
| C2×C4×SD16 | 2 |
| C22×#106 | 2 |
| C22×#103 | 2 |
| C22×#102 | 2 |
| C22×C4×C8 | 2 |