Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
739 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 739 conjugacy classes of subgroups — too many to draw. Only its 147 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 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#135 | C2 | semidirect |
| 2. C2×#465 | C2 | semidirect |
| 3. C2×C8×D4 | C2 | semidirect |
| 4. #2407 | C2 | semidirect |
| 5. #2056 | C2 | semidirect |
| 6. #2054 | C2 | semidirect |
| 7. #2001 | C2 | semidirect |
| 8. #1999 | C2 | semidirect |
| 9. #1968 | C2 | semidirect |
| 10. #1807 | C2 | semidirect |
| 11. #1788 | C2 | semidirect |
| 62. C2×SD16 | D4 | semidirect |
| 68. C22×C8 | D4 | semidirect |
| 194. C23 | #135 | semidirect |
| 231. C2×C4 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#135 | 2 |
| C2×#465 | 2 |
| C2×C8×D4 | 2 |
| #2407 | 2 |
| #2056 | 2 |
| #2054 | 2 |
| #2001 | 2 |
| #1999 | 2 |
| #1968 | 2 |
| #1807 | 2 |
| #1788 | 2 |