Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
214 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #442 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1762 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#347 | C2 | semidirect |
| 5. #1773 | C2 | semidirect |
| 6. #1389 | C2 | semidirect |
| 7. #1384 | C2 | semidirect |
| 12. #442 | C4 | semidirect |
| 13. #441 | C4 | semidirect |
| 21. #360 | C4 | semidirect |
| 22. #359 | C4 | semidirect |
| 47. M32 | C2×C4 | semidirect |
| 49. C2×C16 | C2×C4 | semidirect |
| 51. #108 | C2×C4 | semidirect |
| 55. #106 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2943 | 2 |
| #2145 | 2 |
| #2143 | 2 |
| C2×#347 | 2 |
| #1773 | 2 |
| #1389 | 2 |
| #1384 | 2 |