Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
736 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#953 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#585 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #585 | D3 |
| hypocenter | C3 | #3021 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#585 | C2 | semidirect |
| 2. #5830 | C2 | semidirect |
| 3. C2×#949 | C2 | semidirect |
| 4. #5245 | C2 | semidirect |
| 5. #5241 | C2 | semidirect |
| 6. #5237 | C2 | semidirect |
| 7. #5236 | C2 | semidirect |
| 10. C3×2−1+4 | C22 | semidirect |
| 13. C3×H(5,𝔽2) | C22 | semidirect |
| 18. C22×D12 | C22 | semidirect |
| 20. #949 | C22 | semidirect |
| 21. #947 | C22 | semidirect |
| 23. #940 | C22 | semidirect |
| 27. #926 | C22 | semidirect |
| 28. C2×#208 | C22 | semidirect |
| 29. #837 | C22 | semidirect |
| 30. #836 | C22 | semidirect |
| 31. #833 | C22 | semidirect |
| 32. #585 | D3 | semidirect |
| 50. C3×#41 | D4 | semidirect |
| 54. C2×C3×Q8 | D4 | semidirect |
| 62. C2×C3×D4 | D4 | semidirect |
| 83. 2−1+4 | C2×D3 | semidirect |
| 86. H(5,𝔽2) | C2×D3 | semidirect |
| 153. #41 | #67 | semidirect |
| 157. C2×Q8 | #67 | semidirect |
| 170. C2×D4 | #67 | semidirect |
| 262. C3 | #3021 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#585 | 2 |
| #5830 | 2 |
| C2×#949 | 2 |
| #5245 | 2 |
| #5241 | 2 |
| #5237 | 2 |
| #5236 | 2 |
| #3021 | 3 |