Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
668 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 668 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×C3×C4⋊(C3⋊D3) | C2 | semidirect |
| 2. C2×C32×C4×D3 | C2 | semidirect |
| 3. C2×#6290 | C2 | semidirect |
| 4. C4⋊(C3⋊(C3⋊Dic3)) | C2 | semidirect |
| 5. #88803 | C2 | semidirect |
| 18. C2×C4⋊(C3⋊D3) | D3 | semidirect |
| 20. C2×C3×C4×D3 | D3 | semidirect |
| 33. C2×C32×D3 | D4 | semidirect |
| 34. C33×C4 | D4 | semidirect |
| 67. C2×C4×D3 | C3⋊D3 | semidirect |
| 87. C2×C3×D3 | D12 | semidirect |
| 90. C32×C4 | #67 | semidirect |
| 91. C32×C4 | #67 | semidirect |
| 97. C33 | #121 | semidirect |
| 148. C2×D3 | C4⋊(C3⋊D3) | semidirect |
| 151. C3×C4 | #625 | semidirect |
| 152. C3×C4 | #638 | semidirect |
| 158. C32 | #930 | semidirect |
| 159. C32 | #938 | semidirect |
| 190. C4 | #6290 | semidirect |
| 192. C3 | #63908 | semidirect |
| 193. C3 | #64139 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4⋊(C3⋊D3) | 2 |
| C2×C32×C4×D3 | 2 |
| C2×#6290 | 2 |
| C4⋊(C3⋊(C3⋊Dic3)) | 2 |
| #88803 | 2 |
| #64139 | 3 |
| #63908 | 3 |