Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
804 conjugacy classes of subgroups; only the 474 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 474 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C24×C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C32×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×H(5,𝔽2) | C2 | semidirect |
| 2. C32×AGL(1,ℤ/8) | C2 | semidirect |
| 3. C32×#135 | C2 | semidirect |
| 4. C2×C32×D8 | C2 | semidirect |
| 5. C32×#131 | C2 | semidirect |
| 6. C3×#576 | C3 | direct (N × H) |
| 9. C32×#41 | C22 | semidirect |
| 11. C2×C32×D4 | C22 | semidirect |
| 12. C32×Q16 | C22 | semidirect |
| 13. C32×SD16 | C22 | semidirect |
| 14. C32×D8 | C22 | semidirect |
| 15. C32×D8 | C22 | semidirect |
| 16. C32×M16 | C22 | semidirect |
| 17. C2×C32×C8 | C22 | semidirect |
| 18. C3×H(5,𝔽2) | C2×C3 | semidirect |
| 19. C3×AGL(1,ℤ/8) | C2×C3 | semidirect |
| 20. C3×#135 | C2×C3 | semidirect |
| 21. C2×C3×D8 | C2×C3 | semidirect |
| 22. C3×#131 | C2×C3 | semidirect |
| 34. #576 | C32 | direct (N × H) |
| 37. C3×#41 | C22×C3 | semidirect |
| 39. C2×C3×D4 | C22×C3 | semidirect |
| 40. C3×Q16 | C22×C3 | semidirect |
| 41. C3×SD16 | C22×C3 | semidirect |
| 42. C3×D8 | C22×C3 | semidirect |
| 43. C3×D8 | C22×C3 | semidirect |
| 44. C3×M16 | C22×C3 | semidirect |
| 45. C2×C3×C8 | C22×C3 | semidirect |
| 52. H(5,𝔽2) | C2×C32 | semidirect |
| 53. AGL(1,ℤ/8) | C2×C32 | semidirect |
| 54. #135 | C2×C32 | semidirect |
| 55. C2×D8 | C2×C32 | semidirect |
| 56. #131 | C2×C32 | semidirect |
| 73. #41 | C22×C32 | semidirect |
| 75. C2×D4 | C22×C32 | semidirect |
| 76. Q16 | C22×C32 | semidirect |
| 77. SD16 | C22×C32 | semidirect |
| 78. D8 | C22×C32 | semidirect |
| 79. D8 | C22×C32 | semidirect |
| 80. M16 | C22×C32 | semidirect |
| 81. C2×C8 | C22×C32 | semidirect |
| 88. C32 | #576 | direct (N × H) |
| 109. C3 | C3×#576 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×H(5,𝔽2) | 2 |
| C32×AGL(1,ℤ/8) | 2 |
| C32×#135 | 2 |
| C2×C32×D8 | 2 |
| C32×#131 | 2 |
| C3×#576 | 3 |