Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1611 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1611 conjugacy classes of subgroups — too many to draw. Only its 567 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 |
|---|---|---|
| 1. C23×SD16 | C2 | semidirect |
| 2. C23×M16 | C2 | semidirect |
| 3. C22×C22⋊Q8 | C2 | semidirect |
| 5. C2×#468 | C2 | direct (N × H) |
| 6. C22×#107 | C2 | semidirect |
| 7. C22×#103 | C2 | semidirect |
| 8. C22×#102 | C2 | semidirect |
| 14. C22×SD16 | C22 | semidirect |
| 16. C22×M16 | C22 | semidirect |
| 20. C2×C22⋊Q8 | C22 | semidirect |
| 28. #468 | C22 | direct (N × H) |
| 29. C2×#107 | C22 | semidirect |
| 30. C2×#103 | C22 | semidirect |
| 31. C2×#102 | C22 | semidirect |
| 54. C2×SD16 | C23 | semidirect |
| 56. C2×M16 | C23 | semidirect |
| 58. C22×C8 | D4 | semidirect |
| 62. C22⋊Q8 | C23 | semidirect |
| 72. #107 | C23 | semidirect |
| 73. #103 | C23 | semidirect |
| 74. #102 | C23 | semidirect |
| 116. C2×C8 | C2×D4 | semidirect |
| 164. C8 | C22×D4 | semidirect |
| 173. C22 | #468 | direct (N × H) |
| 187. C2 | C2×#468 | 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 |
| C23×M16 | 2 |
| C22×C22⋊Q8 | 2 |
| C22×#121 | 2 |
| C2×#468 | 2 |
| C22×#107 | 2 |
| C22×#103 | 2 |
| C22×#102 | 2 |