Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2171 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 2171 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 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#127 | C2 | semidirect |
| 2. C22×#121 | C2 | semidirect |
| 3. C22×#117 | C2 | semidirect |
| 4. #2619 | C2 | direct (N × H) |
| 5. C2×#394 | C2 | semidirect |
| 6. C2×#392 | C2 | semidirect |
| 7. C2×#390 | C2 | semidirect |
| 8. C2×#382 | C2 | semidirect |
| 14. C2×#127 | C22 | semidirect |
| 16. C2×#121 | C22 | semidirect |
| 18. C2×#117 | C22 | semidirect |
| 26. #394 | C22 | semidirect |
| 27. #392 | C22 | semidirect |
| 28. #390 | C22 | semidirect |
| 29. #382 | C22 | semidirect |
| 45. C23×C4 | D4 | semidirect |
| 66. C2×C42 | D4 | semidirect |
| 106. C22×C4 | C2×D4 | semidirect |
| 119. C42 | C2×D4 | semidirect |
| 190. C2 | #2619 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#127 | 2 |
| C22×#121 | 2 |
| C22×#117 | 2 |
| #2619 | 2 |
| C2×#394 | 2 |
| C2×#392 | 2 |
| C2×#390 | 2 |
| C2×#382 | 2 |