Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
717 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 717 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D8 | C2 | semidirect |
| 2. C22×D16 | C2 | semidirect |
| 3. C2×#466 | C2 | semidirect |
| 4. #2184 | C2 | direct (N × H) |
| 5. C2×#357 | C2 | semidirect |
| 6. C2×C22⋊C16 | C2 | semidirect |
| 12. C22×D8 | C22 | semidirect |
| 13. C22×D8 | C22 | semidirect |
| 17. C2×D16 | C22 | semidirect |
| 18. C22×C16 | C22 | semidirect |
| 20. #466 | C22 | semidirect |
| 21. C2×#107 | C22 | semidirect |
| 23. #357 | C22 | semidirect |
| 24. C22⋊C16 | C22 | semidirect |
| 43. C2×D8 | D4 | semidirect |
| 45. C2×D8 | C23 | semidirect |
| 57. C2×C16 | C23 | semidirect |
| 59. #107 | C23 | semidirect |
| 95. D8 | C2×D4 | semidirect |
| 121. C23 | D16 | semidirect |
| 160. C22 | C2×D16 | semidirect |
| 171. C2 | #2184 | 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×D16 | 2 |
| C2×#466 | 2 |
| #2184 | 2 |
| C2×#357 | 2 |
| C2×C22⋊C16 | 2 |