Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
19680 conjugacy classes of subgroups; only the 17016 normal subgroups are shown.
This group has 19680 conjugacy classes of subgroups — too many to draw. Only its 17016 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C24 |
| commutator, Frattini, cosocle | C2 | C26×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C32×H(5,𝔽2) | C2 | direct (N × H) |
| 2. C22×C32×#41 | C2 | semidirect |
| 3. C23×C32×D4 | C2 | semidirect |
| 4. C22×C3×H(5,𝔽2) | C3 | direct (N × H) |
| 5. C25×C32 | C22 | semidirect |
| 6. C32×H(5,𝔽2) | C22 | direct (N × H) |
| 7. C2×C32×#41 | C22 | semidirect |
| 8. C22×C32×Q8 | C22 | semidirect |
| 9. C22×C32×D4 | C22 | semidirect |
| 10. C22×C32×D4 | C22 | semidirect |
| 11. C23×C32×C4 | C22 | semidirect |
| 12. C2×C3×H(5,𝔽2) | C2×C3 | direct (N × H) |
| 13. C22×C3×#41 | C2×C3 | semidirect |
| 14. C23×C3×D4 | C2×C3 | semidirect |
| 17. C24×C32 | C23 | semidirect |
| 18. C32×#41 | C23 | semidirect |
| 19. C2×C32×Q8 | C23 | semidirect |
| 20. C2×C32×D4 | C23 | semidirect |
| 21. C2×C32×D4 | C23 | semidirect |
| 23. C22×C32×C4 | C23 | semidirect |
| 24. C22×H(5,𝔽2) | C32 | direct (N × H) |
| 25. C25×C3 | C22×C3 | semidirect |
| 26. C3×H(5,𝔽2) | C22×C3 | direct (N × H) |
| 27. C2×C3×#41 | C22×C3 | semidirect |
| 28. C22×C3×Q8 | C22×C3 | semidirect |
| 29. C22×C3×D4 | C22×C3 | semidirect |
| 30. C22×C3×D4 | C22×C3 | semidirect |
| 31. C23×C3×C4 | C22×C3 | semidirect |
| 36. C23×C32 | C24 | semidirect |
| 37. C32×Q8 | C24 | semidirect |
| 38. C32×D4 | C24 | semidirect |
| 40. C2×C32×C4 | C24 | semidirect |
| 41. C2×H(5,𝔽2) | C2×C32 | direct (N × H) |
| 42. C22×#41 | C2×C32 | semidirect |
| 43. C23×D4 | C2×C32 | semidirect |
| 46. C24×C3 | C23×C3 | semidirect |
| 47. C3×#41 | C23×C3 | semidirect |
| 48. C2×C3×Q8 | C23×C3 | semidirect |
| 49. C2×C3×D4 | C23×C3 | semidirect |
| 50. C2×C3×D4 | C23×C3 | semidirect |
| 52. C22×C3×C4 | C23×C3 | semidirect |
| 55. C22×C32 | H(5,𝔽2) | direct (N × H) |
| 59. C25 | C22×C32 | semidirect |
| 60. H(5,𝔽2) | C22×C32 | direct (N × H) |
| 61. C2×#41 | C22×C32 | semidirect |
| 62. C22×Q8 | C22×C32 | semidirect |
| 63. C22×D4 | C22×C32 | semidirect |
| 64. C22×D4 | C22×C32 | semidirect |
| 65. C23×C4 | C22×C32 | semidirect |
| 70. C23×C3 | C24×C3 | semidirect |
| 71. C3×Q8 | C24×C3 | semidirect |
| 72. C3×D4 | C24×C3 | semidirect |
| 74. C2×C3×C4 | C24×C3 | semidirect |
| 76. C2×C32 | C2×H(5,𝔽2) | direct (N × H) |
| 80. C24 | C23×C32 | semidirect |
| 81. #41 | C23×C32 | semidirect |
| 82. C2×Q8 | C23×C32 | semidirect |
| 83. C2×D4 | C23×C32 | semidirect |
| 84. C2×D4 | C23×C32 | semidirect |
| 86. C22×C4 | C23×C32 | semidirect |
| 89. C22×C3 | C3×H(5,𝔽2) | direct (N × H) |
| 93. C32 | C22×H(5,𝔽2) | direct (N × H) |
| 98. C23 | C24×C32 | semidirect |
| 99. Q8 | C24×C32 | semidirect |
| 100. D4 | C24×C32 | semidirect |
| 102. C2×C4 | C24×C32 | semidirect |
| 104. C2×C3 | C2×C3×H(5,𝔽2) | direct (N × H) |
| 108. C22 | C32×H(5,𝔽2) | direct (N × H) |
| 112. C3 | C22×C3×H(5,𝔽2) | direct (N × H) |
| 114. C2 | C2×C32×H(5,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×H(5,𝔽2) | 2 |
| C22×C32×#41 | 2 |
| C23×C32×D4 | 2 |
| C22×C3×H(5,𝔽2) | 3 |