Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2382 conjugacy classes of subgroups; only the 870 normal subgroups are shown.
This group has 2382 conjugacy classes of subgroups — too many to draw. Only its 870 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 | C22⋊C4 |
| commutator | C22 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C42×M16 | C2 | semidirect |
| 4. C4×#343 | C2 | semidirect |
| 14. C4×#100 | C4 | direct (N × H) |
| 36. C4×M16 | C2×C4 | semidirect |
| 46. #343 | C2×C4 | semidirect |
| 87. #100 | C42 | direct (N × H) |
| 127. M16 | C2×C42 | semidirect |
| 136. C42 | #100 | direct (N × H) |
| 194. C4 | C4×#100 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C42×D4 | 2 |
| C2×C4×#100 | 2 |
| C42×M16 | 2 |
| C4×#343 | 2 |