Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1088 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 181 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, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C22⋊Q8 | C2 |
| prime radical | C23×D12 | C2 |
| hypercenter | C2×C22⋊Q8 | D3 |
| hypocenter | C3 | #2435 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×C22⋊Q8 | C2 | semidirect |
| 3. C2×#947 | C2 | semidirect |
| 4. C4×#217 | C2 | semidirect |
| 5. #4991 | C2 | semidirect |
| 6. #4988 | C2 | semidirect |
| 7. #4955 | C2 | semidirect |
| 10. C22×C3×Q8 | C22 | semidirect |
| 17. C2×C3×C4⋊C4 | C22 | semidirect |
| 18. C2×C3×C4⋊C4 | C22 | semidirect |
| 22. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 23. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 24. C2×C4×Dic3 | C22 | semidirect |
| 25. C3⋊#95 | C22 | semidirect |
| 28. C2×C22⋊Q8 | D3 | semidirect |
| 62. C22×Q8 | C2×D3 | semidirect |
| 70. C2×C4⋊C4 | C2×D3 | semidirect |
| 71. C2×C4⋊C4 | C2×D3 | semidirect |
| 138. C22×C3 | #124 | semidirect |
| 198. C22 | #947 | semidirect |
| 208. C3 | #2435 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D12 | 2 |
| C2×C3×C22⋊Q8 | 2 |
| C2×#947 | 2 |
| C4×#217 | 2 |
| #4991 | 2 |
| #4988 | 2 |
| #4955 | 2 |
| #2435 | 3 |