Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
597 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 167 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 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#495 | C2 | semidirect |
| 2. C2×#467 | C2 | semidirect |
| 4. C2×#465 | C2 | semidirect |
| 5. C22×C4×C8 | C2 | semidirect |
| 6. #2850 | C2 | semidirect |
| 7. #1970 | C2 | semidirect |
| 8. #1969 | C2 | semidirect |
| 9. #1968 | C2 | semidirect |
| 10. #1958 | C2 | semidirect |
| 11. #1846 | C2 | semidirect |
| 94. C4×C8 | D4 | semidirect |
| 220. C22 | #495 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#495 | 2 |
| C2×#467 | 2 |
| C2×#466 | 2 |
| C2×#465 | 2 |
| C22×C4×C8 | 2 |
| #2850 | 2 |
| #1970 | 2 |
| #1969 | 2 |
| #1968 | 2 |
| #1958 | 2 |
| #1846 | 2 |