Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1011 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 479 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C25 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#578 | C2 | semidirect |
| 2. #3417 | C2 | direct (N × H) |
| 3. C2×#510 | C2 | semidirect |
| 4. C2×#508 | C2 | semidirect |
| 5. C22×Q32 | C2 | semidirect |
| 6. C2×#504 | C2 | semidirect |
| 9. #578 | C22 | semidirect |
| 14. #510 | C22 | semidirect |
| 15. #508 | C22 | semidirect |
| 18. C2×Q32 | C22 | semidirect |
| 21. #504 | C22 | semidirect |
| 113. C2 | #3417 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#578 | 2 |
| #3417 | 2 |
| C2×#510 | 2 |
| C2×#508 | 2 |
| C22×Q32 | 2 |
| C2×#504 | 2 |