Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1842 conjugacy classes of subgroups; only the 1026 normal subgroups are shown.
This group has 1842 conjugacy classes of subgroups — too many to draw. Only its 1026 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C24 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×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 |
|---|---|---|
| 11. C4×#107 | C4 | direct (N × H) |
| 13. C42×C8 | C4 | semidirect |
| 57. #107 | C42 | direct (N × H) |
| 59. C4×C8 | C42 | semidirect |
| 86. C42 | #107 | direct (N × H) |
| 112. C8 | C43 | semidirect |
| 122. C4 | C4×#107 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#107 | 2 |
| C2×C42×C8 | 2 |
| C42×C4⋊C4 | 2 |
| C4×#340 | 2 |