Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
888 conjugacy classes of subgroups; only the 366 normal subgroups are shown.
This group has 888 conjugacy classes of subgroups — too many to draw. Only its 366 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C22×D8 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C42×D8 | C2 | semidirect |
| 3. C4×#366 | C2 | semidirect |
| 4. C4×#357 | C2 | semidirect |
| 5. C42×C16 | C2 | semidirect |
| 9. C4×D16 | C4 | direct (N × H) |
| 25. C4×D8 | C2×C4 | semidirect |
| 33. #366 | C2×C4 | semidirect |
| 34. #357 | C2×C4 | semidirect |
| 36. C4×C16 | C2×C4 | semidirect |
| 53. D16 | C42 | direct (N × H) |
| 75. D8 | C2×C42 | semidirect |
| 85. C42 | D16 | direct (N × H) |
| 89. C16 | C2×C42 | semidirect |
| 118. C4 | C4×D16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D16 | 2 |
| C42×D8 | 2 |
| C4×#366 | 2 |
| C4×#357 | 2 |
| C42×C16 | 2 |