Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
421 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 135 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 | #121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#459 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#486 | C2 | semidirect |
| 2. C2×#461 | C2 | semidirect |
| 4. C2×C8×D4 | C2 | semidirect |
| 5. #1681 | C2 | direct (N × H) |
| 6. C2×#335 | C2 | semidirect |
| 7. C2×#332 | C2 | semidirect |
| 8. C2×#331 | C2 | semidirect |
| 16. #486 | C22 | semidirect |
| 18. #461 | C22 | semidirect |
| 22. C8×D4 | C22 | semidirect |
| 32. #335 | C22 | semidirect |
| 33. #332 | C22 | semidirect |
| 34. #331 | C22 | semidirect |
| 207. C2 | #1681 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#486 | 2 |
| C2×#461 | 2 |
| C2×#459 | 2 |
| C2×C8×D4 | 2 |
| #1681 | 2 |
| C2×#335 | 2 |
| C2×#332 | 2 |
| C2×#331 | 2 |