Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1443 conjugacy classes of subgroups; only the 527 normal subgroups are shown.
This group has 1443 conjugacy classes of subgroups — too many to draw. Only its 527 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, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C23×D8 | C2 | semidirect |
| 3. C22×#131 | C2 | semidirect |
| 4. C2×#472 | C2 | direct (N × H) |
| 5. C22×#108 | C2 | semidirect |
| 6. C22×#100 | C2 | semidirect |
| 10. C2×AGL(1,ℤ/8) | C22 | semidirect |
| 16. C22×D8 | C22 | semidirect |
| 18. C2×#131 | C22 | semidirect |
| 24. #472 | C22 | direct (N × H) |
| 25. C2×#108 | C22 | semidirect |
| 26. C2×#100 | C22 | semidirect |
| 44. AGL(1,ℤ/8) | C23 | semidirect |
| 50. C2×D8 | C23 | semidirect |
| 52. #131 | C23 | semidirect |
| 65. #108 | C23 | semidirect |
| 66. #100 | C23 | semidirect |
| 164. C22 | #472 | direct (N × H) |
| 177. C2 | C2×#472 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×AGL(1,ℤ/8) | 2 |
| C23×D8 | 2 |
| C22×#131 | 2 |
| C2×#472 | 2 |
| C22×#108 | 2 |
| C22×#100 | 2 |