Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2443 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 2443 conjugacy classes of subgroups — too many to draw. Only its 607 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×#121 | C2 | semidirect |
| 2. #2656 | C2 | direct (N × H) |
| 3. C2×#399 | C2 | semidirect |
| 4. C2×#392 | C2 | semidirect |
| 5. C22×#95 | C2 | semidirect |
| 10. C2×#121 | C22 | semidirect |
| 13. #399 | C22 | semidirect |
| 14. #392 | C22 | semidirect |
| 16. C2×#95 | C22 | semidirect |
| 25. C23×C4 | D4 | semidirect |
| 54. C22×C4 | C2×D4 | semidirect |
| 101. C2 | #2656 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#121 | 2 |
| #2656 | 2 |
| C2×#399 | 2 |
| C2×#392 | 2 |
| C22×#95 | 2 |