Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1614 conjugacy classes of subgroups; only the 362 normal subgroups are shown.
This group has 1614 conjugacy classes of subgroups — too many to draw. Only its 362 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C22⋊C16 | C2 | semidirect |
| 3. C22×#340 | C2 | semidirect |
| 4. C2×#1347 | C2 | direct (N × H) |
| 5. C23×D8 | C4 | semidirect |
| 12. C2×#466 | C4 | semidirect |
| 14. C22×#107 | C4 | semidirect |
| 17. C2×C22⋊C16 | C22 | semidirect |
| 19. C2×#340 | C22 | semidirect |
| 20. #1347 | C22 | direct (N × H) |
| 27. C22×D8 | C2×C4 | semidirect |
| 50. #466 | C2×C4 | semidirect |
| 53. C2×#107 | C2×C4 | semidirect |
| 57. C22⋊C16 | C23 | semidirect |
| 59. #340 | C23 | semidirect |
| 86. C2×D8 | C22×C4 | semidirect |
| 119. #107 | C22×C4 | semidirect |
| 254. C22 | #1347 | direct (N × H) |
| 273. C2 | C2×#1347 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#466 | 2 |
| C22×C22⋊C16 | 2 |
| C22×#340 | 2 |
| C2×#1347 | 2 |