Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
547 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 547 conjugacy classes of subgroups — too many to draw. Only its 139 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. C2×C4×D8 | C2 | semidirect |
| 2. C2×C8×D4 | C2 | semidirect |
| 3. #2537 | C2 | semidirect |
| 4. #2075 | C2 | semidirect |
| 5. #1999 | C2 | semidirect |
| 6. #1893 | C2 | semidirect |
| 7. #1834 | C2 | semidirect |
| 8. #1802 | C2 | semidirect |
| 64. C4×C8 | D4 | semidirect |
| 78. C2×D4 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D8 | 2 |
| C2×C8×D4 | 2 |
| #2537 | 2 |
| #2075 | 2 |
| #1999 | 2 |
| #1893 | 2 |
| #1834 | 2 |
| #1802 | 2 |