Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1736 conjugacy classes of subgroups; only the 573 normal subgroups are shown.
This group has 1736 conjugacy classes of subgroups — too many to draw. Only its 573 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C22×C3×#41 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C22×#41 | D3 |
| hypocenter | C3 | C2×#550 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×#41 | C2 | semidirect |
| 2. C22×#239 | C2 | semidirect |
| 3. #5834 | C2 | direct (N × H) |
| 4. C2×Q8×Dic3 | C2 | semidirect |
| 5. C2×#940 | C2 | semidirect |
| 6. C2×#938 | C2 | semidirect |
| 7. C2×#930 | C2 | semidirect |
| 8. C2×C4×#67 | C2 | semidirect |
| 11. C2×C3×#41 | C22 | semidirect |
| 18. C2×#239 | C22 | semidirect |
| 22. Q8×Dic3 | C22 | semidirect |
| 23. #940 | C22 | semidirect |
| 24. #938 | C22 | semidirect |
| 25. #930 | C22 | semidirect |
| 26. C4×#67 | C22 | semidirect |
| 32. C22×#41 | D3 | semidirect |
| 42. C2×C3×Q8 | D4 | semidirect |
| 70. C2×#41 | C2×D3 | semidirect |
| 97. C3×Q8 | C2×D4 | semidirect |
| 108. C2×Dic3 | #41 | semidirect |
| 118. C2×Q8 | #67 | semidirect |
| 158. Dic3 | C2×#41 | semidirect |
| 168. Q8 | C2×#67 | semidirect |
| 186. C2×C3 | #550 | semidirect |
| 208. C3 | C2×#550 | semidirect |
| 211. C2 | #5834 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#41 | 2 |
| C22×#239 | 2 |
| #5834 | 2 |
| C2×Q8×Dic3 | 2 |
| C2×#940 | 2 |
| C2×#938 | 2 |
| C2×#930 | 2 |
| C2×C4×#67 | 2 |
| C2×#550 | 3 |