Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1017 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 1017 conjugacy classes of subgroups — too many to draw. Only its 175 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 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D8 | C2 | semidirect |
| 2. C2×#493 | C2 | semidirect |
| 3. C2×#466 | C2 | semidirect |
| 4. C2×#447 | C2 | semidirect |
| 5. #2433 | C2 | semidirect |
| 6. #2011 | C2 | semidirect |
| 7. #1970 | C2 | semidirect |
| 8. C8×C22⋊C4 | C2 | semidirect |
| 14. C22×D8 | C22 | semidirect |
| 15. C22×D8 | C22 | semidirect |
| 17. C2×#127 | C22 | semidirect |
| 23. C2×#107 | C22 | semidirect |
| 24. C2×#102 | C22 | semidirect |
| 25. C2×#102 | C22 | semidirect |
| 27. C2×C4×C8 | C22 | semidirect |
| 29. #390 | C22 | semidirect |
| 31. #336 | C22 | semidirect |
| 53. C2×D8 | D4 | semidirect |
| 55. C22×C8 | D4 | semidirect |
| 72. C22⋊C8 | D4 | semidirect |
| 126. C22⋊C4 | D8 | semidirect |
| 166. C2×C4 | C2×D8 | semidirect |
| 173. C8 | #120 | semidirect |
| 187. C22 | #493 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D8 | 2 |
| C2×#493 | 2 |
| C2×#466 | 2 |
| C2×#447 | 2 |
| #2433 | 2 |
| #2011 | 2 |
| #1970 | 2 |
| C8×C22⋊C4 | 2 |