Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
532 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 141 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 | #3205 |
| 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×#874 | C2 | semidirect |
| 3. #5088 | C2 | semidirect |
| 4. #5073 | C2 | semidirect |
| 5. #5036 | C2 | semidirect |
| 6. #5015 | C2 | semidirect |
| 7. #4925 | C2 | semidirect |
| 13. C3×C4⋊Q8 | C22 | semidirect |
| 14. C3×#127 | C22 | semidirect |
| 19. C3×C4×D4 | C22 | semidirect |
| 21. C2×#213 | C22 | semidirect |
| 27. #809 | C22 | semidirect |
| 28. C4⋊(C3⋊C8) | C22 | semidirect |
| 29. C4×C3⋊C8 | C22 | semidirect |
| 30. #532 | D3 | semidirect |
| 72. C4⋊Q8 | C2×D3 | semidirect |
| 77. #127 | C2×D3 | semidirect |
| 82. C4×D4 | C2×D3 | semidirect |
| 103. C2×C3×C4 | D8 | semidirect |
| 186. C2×C4 | #213 | semidirect |
| 218. C3 | #3205 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#532 | 2 |
| C2×#874 | 2 |
| #5088 | 2 |
| #5073 | 2 |
| #5036 | 2 |
| #5015 | 2 |
| #4925 | 2 |
| #3205 | 3 |