Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1492 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1492 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, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C23×C3×Q8 | C2 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | #3486 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×Q8 | C2 | semidirect |
| 2. C2×#1007 | C2 | semidirect |
| 3. #5817 | C2 | semidirect |
| 4. #5816 | C2 | semidirect |
| 5. Q8×#67 | C2 | semidirect |
| 6. #5812 | C2 | semidirect |
| 7. #5791 | C2 | semidirect |
| 15. C2×#239 | C22 | semidirect |
| 16. C2×D3×Q8 | C22 | semidirect |
| 19. #947 | C22 | semidirect |
| 20. #946 | C22 | semidirect |
| 21. Q8×Dic3 | C22 | semidirect |
| 22. #944 | C22 | semidirect |
| 29. C23×Q8 | D3 | semidirect |
| 38. C2×C3×Q8 | D4 | semidirect |
| 104. C2×Q8 | #67 | semidirect |
| 126. C22×C3 | 2−1+4 | semidirect |
| 164. C22 | #1007 | semidirect |
| 171. C3 | #3486 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×Q8 | 2 |
| C2×#1007 | 2 |
| #5817 | 2 |
| #5816 | 2 |
| Q8×#67 | 2 |
| #5812 | 2 |
| #5791 | 2 |
| #3486 | 3 |