Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
780 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 140 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 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C22×C33 | C23 |
| Fitting | C2×C33×C4 | C22 |
| hypercenter | C2×C4 | D3×C3⋊D3 |
| hypocenter | C33 | #121 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4⋊(C3⋊(C3⋊D3)) | C2 | semidirect |
| 2. C2×C32×C4×D3 | C2 | semidirect |
| 3. C2×#6291 | C2 | semidirect |
| 4. C3×C4⋊(C3⋊Dic3) | C2 | semidirect |
| 5. #88804 | C2 | semidirect |
| 21. C2×C3×C4×D3 | D3 | semidirect |
| 24. C4⋊(C3⋊Dic3) | D3 | semidirect |
| 33. C2×C32×D3 | D4 | semidirect |
| 34. C33×C4 | D4 | semidirect |
| 69. C2×C4×D3 | C3⋊D3 | semidirect |
| 91. C2×C3×D3 | D12 | semidirect |
| 94. C32×C4 | D12 | semidirect |
| 95. C32×C4 | #67 | semidirect |
| 100. C33 | #121 | semidirect |
| 156. C2×D3 | C4⋊(C3⋊D3) | semidirect |
| 160. C3×C4 | #626 | semidirect |
| 161. C3×C4 | #638 | semidirect |
| 165. C32 | #895 | semidirect |
| 166. C32 | #930 | semidirect |
| 198. C4 | #6291 | semidirect |
| 200. C3 | #63909 | semidirect |
| 201. C3 | #64139 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊(C3⋊(C3⋊D3)) | 2 |
| C2×C32×C4×D3 | 2 |
| C2×#6291 | 2 |
| C3×C4⋊(C3⋊Dic3) | 2 |
| #88804 | 2 |
| #64139 | 3 |
| #63909 | 3 |