Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
576 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C2×C3×#121 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C2×#121 | D3 |
| hypocenter | C3 | #2377 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #5221 | C2 | semidirect |
| 3. #5220 | C2 | semidirect |
| 4. #5220 | C2 | semidirect |
| 5. #5212 | C2 | semidirect |
| 6. #5211 | C2 | semidirect |
| 7. #4980 | C2 | semidirect |
| 8. #4951 | C2 | semidirect |
| 9. #4950 | C2 | semidirect |
| 15. C3×#121 | C4 | semidirect |
| 48. C2×C3×D4 | C2×C4 | semidirect |
| 50. C22×C3×C4 | C2×C4 | semidirect |
| 61. C3×C4⋊C4 | C2×C4 | semidirect |
| 62. C3×C22⋊C4 | C2×C4 | semidirect |
| 76. #121 | Dic3 | semidirect |
| 136. C2×D4 | C2×Dic3 | semidirect |
| 145. C22×C4 | C2×Dic3 | semidirect |
| 151. C4⋊C4 | C2×Dic3 | semidirect |
| 158. C22⋊C4 | C2×Dic3 | semidirect |
| 251. C3 | #2377 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#121 | 2 |
| #5221 | 2 |
| #5220 | 2 |
| #5212 | 2 |
| #5211 | 2 |
| #4980 | 2 |
| #4951 | 2 |
| #4950 | 2 |
| #2377 | 3 |