Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1728 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1728 conjugacy classes of subgroups — too many to draw. Only its 557 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 | C2×C3×C4⋊Q8 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×C4⋊Q8 | D3 |
| hypocenter | C3 | C2×#532 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#239 | C2 | semidirect |
| 2. C2×C3×C4⋊Q8 | C2 | semidirect |
| 3. C2×#947 | C2 | semidirect |
| 4. #5725 | C2 | direct (N × H) |
| 5. C2×#895 | C2 | semidirect |
| 6. C2×#893 | C2 | semidirect |
| 7. C2×#874 | C2 | semidirect |
| 8. C2×C42×D3 | C2 | semidirect |
| 12. C2×#239 | C22 | semidirect |
| 19. C3×C4⋊Q8 | C22 | semidirect |
| 22. #947 | C22 | semidirect |
| 26. #895 | C22 | semidirect |
| 27. #893 | C22 | semidirect |
| 28. #874 | C22 | semidirect |
| 29. C42×D3 | C22 | semidirect |
| 32. C2×C4⋊Q8 | D3 | semidirect |
| 56. C2×C4×D3 | D4 | semidirect |
| 74. C4⋊Q8 | C2×D3 | semidirect |
| 95. C2×C3×C4 | #41 | semidirect |
| 108. C4×D3 | C2×D4 | semidirect |
| 147. C3×C4 | C2×#41 | semidirect |
| 170. C2×C4 | #239 | semidirect |
| 174. C2×C3 | #532 | semidirect |
| 192. C4 | C2×#239 | semidirect |
| 193. C3 | C2×#532 | semidirect |
| 196. C2 | #5725 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#239 | 2 |
| C2×C3×C4⋊Q8 | 2 |
| C2×#947 | 2 |
| #5725 | 2 |
| C2×#895 | 2 |
| C2×#893 | 2 |
| C2×#874 | 2 |
| C2×C42×D3 | 2 |
| C2×#532 | 3 |