Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
6560 conjugacy classes of subgroups; only the 3213 normal subgroups are shown.
This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C24×D3 |
| commutator, cosocle | C2×C3 | C26 |
| Frattini | C2 | C25×D3 |
| Fitting | C2×C3×H(5,𝔽2) | C2 |
| socle | C22×C3 | C25 |
| hypercenter | C2×H(5,𝔽2) | D3 |
| hypocenter | C3 | C22×H(5,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×H(5,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×H(5,𝔽2) | C2 | semidirect |
| 2. D3×H(5,𝔽2) | C2 | direct (N × H) |
| 3. C2×#1009 | C2 | semidirect |
| 4. C2×D3×#41 | C2 | semidirect |
| 5. C2×#1004 | C2 | semidirect |
| 6. C22×D3×D4 | C2 | semidirect |
| 8. C24×D3 | C22 | semidirect |
| 9. C3×H(5,𝔽2) | C22 | semidirect |
| 10. C2×C3×#41 | C22 | semidirect |
| 11. C22×C3×D4 | C22 | semidirect |
| 12. C22×#67 | C22 | semidirect |
| 13. #1009 | C22 | semidirect |
| 14. D3×#41 | C22 | semidirect |
| 15. C2×#239 | C22 | semidirect |
| 16. C2×D3×Q8 | C22 | semidirect |
| 17. #1004 | C22 | semidirect |
| 18. C2×#237 | C22 | semidirect |
| 19. C2×D3×D4 | C22 | semidirect |
| 20. C2×D3×D4 | C22 | semidirect |
| 21. C2×D3×D4 | C22 | semidirect |
| 22. C2×#235 | C22 | semidirect |
| 23. C22×D12 | C22 | semidirect |
| 24. C22×C4×D3 | C22 | semidirect |
| 28. C2×H(5,𝔽2) | D3 | direct (N × H) |
| 31. C24×C3 | C23 | semidirect |
| 34. C23×D3 | C23 | semidirect |
| 35. C23×D3 | C23 | semidirect |
| 36. C3×#41 | C23 | semidirect |
| 37. C2×C3×Q8 | C23 | semidirect |
| 38. C2×C3×D4 | C23 | semidirect |
| 39. C2×C3×D4 | C23 | semidirect |
| 40. C22×C3×C4 | C23 | semidirect |
| 41. C2×#67 | C23 | semidirect |
| 42. C2×#67 | C23 | semidirect |
| 43. C22×Dic3 | C23 | semidirect |
| 44. #239 | C23 | semidirect |
| 45. D3×Q8 | C23 | semidirect |
| 46. #237 | C23 | semidirect |
| 47. D3×D4 | C23 | semidirect |
| 48. D3×D4 | C23 | semidirect |
| 49. #235 | C23 | semidirect |
| 50. C2×D12 | C23 | semidirect |
| 51. C2×D12 | C23 | semidirect |
| 53. C2×C4×D3 | C23 | semidirect |
| 54. C2×C4×D3 | C23 | semidirect |
| 55. C2×Dic6 | C23 | semidirect |
| 59. H(5,𝔽2) | C2×D3 | direct (N × H) |
| 64. C2×#41 | C2×D3 | semidirect |
| 71. C22×D4 | C2×D3 | semidirect |
| 75. C23×C3 | C24 | semidirect |
| 82. C22×D3 | C24 | semidirect |
| 83. C3×Q8 | C24 | semidirect |
| 84. C3×D4 | C24 | semidirect |
| 86. C2×C3×C4 | C24 | semidirect |
| 87. #67 | C24 | semidirect |
| 89. C2×Dic3 | C24 | semidirect |
| 90. D12 | C24 | semidirect |
| 92. C4×D3 | C24 | semidirect |
| 93. Dic6 | C24 | semidirect |
| 98. C24 | C22×D3 | semidirect |
| 102. #41 | C22×D3 | semidirect |
| 105. C2×Q8 | C22×D3 | semidirect |
| 112. C2×D4 | C22×D3 | semidirect |
| 113. C2×D4 | C22×D3 | semidirect |
| 118. C22×C4 | C22×D3 | semidirect |
| 126. C2×D3 | H(5,𝔽2) | direct (N × H) |
| 140. C23 | C23×D3 | semidirect |
| 142. Q8 | C23×D3 | semidirect |
| 145. D4 | C23×D3 | semidirect |
| 151. C2×C4 | C23×D3 | semidirect |
| 153. C2×C3 | C2×H(5,𝔽2) | semidirect |
| 156. D3 | C2×H(5,𝔽2) | direct (N × H) |
| 169. C3 | C22×H(5,𝔽2) | semidirect |
| 173. C2 | D3×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×C3×H(5,𝔽2) | 2 |
| D3×H(5,𝔽2) | 2 |
| C2×#1009 | 2 |
| C2×D3×#41 | 2 |
| C2×#1004 | 2 |
| C22×D3×D4 | 2 |
| C22×H(5,𝔽2) | 3 |