Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
444 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#532 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | #3213 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #532 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#532 | C2 | semidirect |
| 2. C2×#875 | C2 | semidirect |
| 3. #5089 | C2 | semidirect |
| 4. #5074 | C2 | semidirect |
| 5. #5040 | C2 | semidirect |
| 6. #5036 | C2 | semidirect |
| 7. #5016 | C2 | semidirect |
| 8. #4926 | C2 | semidirect |
| 13. C3×C4⋊Q8 | C22 | semidirect |
| 14. C3×#127 | C22 | semidirect |
| 19. C3×C4×D4 | C22 | semidirect |
| 21. C2×#214 | C22 | semidirect |
| 30. #810 | C22 | semidirect |
| 32. C4⋊(C3⋊C8) | C22 | semidirect |
| 33. #803 | C22 | semidirect |
| 34. #532 | D3 | semidirect |
| 80. C4⋊Q8 | C2×D3 | semidirect |
| 81. #127 | C2×D3 | semidirect |
| 90. C4×D4 | C2×D3 | semidirect |
| 234. C3 | #3213 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#532 | 2 |
| C2×#875 | 2 |
| #5089 | 2 |
| #5074 | 2 |
| #5040 | 2 |
| #5036 | 2 |
| #5016 | 2 |
| #4926 | 2 |
| #3213 | 3 |