Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
745 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 745 conjugacy classes of subgroups — too many to draw. Only its 175 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 | C22×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×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×#417 | C2 | semidirect |
| 4. C22×#102 | C2 | semidirect |
| 5. C22×#100 | C2 | semidirect |
| 6. C2×#408 | C2 | semidirect |
| 7. #2051 | C2 | direct (N × H) |
| 8. C2×#343 | C2 | semidirect |
| 23. C2×#108 | C22 | semidirect |
| 25. #417 | C22 | semidirect |
| 30. C2×#102 | C22 | semidirect |
| 32. C2×#100 | C22 | semidirect |
| 34. #408 | C22 | semidirect |
| 36. #343 | C22 | semidirect |
| 302. C2 | #2051 | 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 |
| C22×#108 | 2 |
| C2×#417 | 2 |
| C22×#102 | 2 |
| C22×#100 | 2 |
| C2×#408 | 2 |
| #2051 | 2 |
| C2×#343 | 2 |