Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×#67 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C3×#585 | C2 |
| socle | C2×C3 | C23×D4 |
| hypercenter | #585 | D3 |
| hypocenter | C3 | #3585 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#585 | C2 | semidirect |
| 2. C2×#1009 | C2 | semidirect |
| 3. #5837 | C2 | semidirect |
| 4. #5835 | C2 | semidirect |
| 5. #5822 | C2 | semidirect |
| 6. C2×#949 | C2 | semidirect |
| 7. #5819 | C2 | semidirect |
| 10. C3×2−1+4 | C22 | semidirect |
| 12. C3×H(5,𝔽2) | C22 | semidirect |
| 15. C2×C3×#41 | C22 | semidirect |
| 17. #1009 | C22 | semidirect |
| 20. C2×#235 | C22 | semidirect |
| 21. C22×D12 | C22 | semidirect |
| 22. #951 | C22 | semidirect |
| 23. #950 | C22 | semidirect |
| 24. #949 | C22 | semidirect |
| 25. #949 | C22 | semidirect |
| 26. #948 | C22 | semidirect |
| 27. #942 | C22 | semidirect |
| 28. C2×#215 | C22 | semidirect |
| 29. #932 | C22 | semidirect |
| 30. C2×#213 | C22 | semidirect |
| 31. C2×#208 | C22 | semidirect |
| 32. #585 | D3 | semidirect |
| 47. C3×#41 | C23 | semidirect |
| 50. C2×C3×Q8 | C23 | semidirect |
| 54. C2×C3×D4 | C23 | semidirect |
| 68. #216 | C23 | semidirect |
| 69. #215 | C23 | semidirect |
| 70. #214 | C23 | semidirect |
| 71. #213 | C23 | semidirect |
| 73. #208 | C23 | semidirect |
| 74. C2×C3⋊C8 | C23 | semidirect |
| 76. 2−1+4 | C2×D3 | semidirect |
| 80. H(5,𝔽2) | C2×D3 | semidirect |
| 85. C2×#41 | C2×D3 | semidirect |
| 138. #41 | C22×D3 | semidirect |
| 142. C2×Q8 | C22×D3 | semidirect |
| 152. C2×D4 | C22×D3 | semidirect |
| 223. C3 | #3585 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#585 | 2 |
| C2×#1009 | 2 |
| #5837 | 2 |
| #5835 | 2 |
| #5822 | 2 |
| C2×#949 | 2 |
| #5819 | 2 |
| #3585 | 3 |