Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
847 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 847 conjugacy classes of subgroups — too many to draw. Only its 247 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 | #121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#108 | C2 | semidirect |
| 3. C2×#418 | C2 | semidirect |
| 4. C22×#100 | C2 | semidirect |
| 5. C2×#405 | C2 | semidirect |
| 6. #1978 | C2 | direct (N × H) |
| 12. C22×D8 | C4 | semidirect |
| 19. C2×#108 | C22 | semidirect |
| 21. #418 | C22 | semidirect |
| 24. C2×#100 | C22 | semidirect |
| 26. #405 | C22 | semidirect |
| 47. C2×D8 | C2×C4 | semidirect |
| 210. C2 | #1978 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D8 | 2 |
| C22×#108 | 2 |
| C2×#418 | 2 |
| C22×#100 | 2 |
| C2×#405 | 2 |
| #1978 | 2 |