Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
647 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 647 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ 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×#533 | C2 | semidirect |
| 4. #2026 | C2 | semidirect |
| 5. #2025 | C2 | semidirect |
| 6. #2024 | C2 | semidirect |
| 7. #2023 | C2 | semidirect |
| 8. #1735 | C2 | semidirect |
| 9. C2×#336 | C2 | semidirect |
| 36. C2×#41 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#533 | 2 |
| C2×#416 | 2 |
| #2297 | 2 |
| #2026 | 2 |
| #2025 | 2 |
| #2024 | 2 |
| #2023 | 2 |
| #1735 | 2 |
| C2×#336 | 2 |