Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
356 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 111 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 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3358 | C2 | semidirect |
| 2. #3287 | C2 | semidirect |
| 3. D4×Q16 | C2 | semidirect |
| 4. #1709 | C2 | semidirect |
| 5. #1682 | C2 | semidirect |
| 6. #1669 | C2 | semidirect |
| 7. #1667 | C2 | semidirect |
| 8. #1641 | C2 | semidirect |
| 9. #1627 | C2 | semidirect |
| 10. C8×Q16 | C2 | semidirect |
| 11. C8×D8 | C2 | semidirect |
| 117. Q16 | D8 | semidirect |
| 164. D4 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3358 | 2 |
| #3287 | 2 |
| D4×Q16 | 2 |
| #1709 | 2 |
| #1682 | 2 |
| #1669 | 2 |
| #1667 | 2 |
| #1641 | 2 |
| #1627 | 2 |
| C8×Q16 | 2 |
| C8×D8 | 2 |