Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
864 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 152 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 | #127 |
| 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×C3×C4⋊(C3⋊D3) | C2 | semidirect |
| 3. C2×#6292 | C2 | semidirect |
| 4. C32×C4×Dic3 | C2 | semidirect |
| 18. C2×C4⋊(C3⋊D3) | D3 | semidirect |
| 22. C3×C4×Dic3 | D3 | semidirect |
| 28. C33×C4 | D4 | semidirect |
| 29. C32×Dic3 | D4 | semidirect |
| 61. C4×Dic3 | C3⋊D3 | semidirect |
| 77. C32×C4 | D12 | semidirect |
| 78. C32×C4 | #67 | semidirect |
| 79. C3×Dic3 | D12 | semidirect |
| 81. C33 | #127 | semidirect |
| 129. C3×C4 | #626 | semidirect |
| 130. C3×C4 | C4⋊(C3⋊D3) | semidirect |
| 131. Dic3 | C4⋊(C3⋊D3) | semidirect |
| 133. C32 | #874 | semidirect |
| 134. C32 | #940 | semidirect |
| 159. C4 | #6292 | semidirect |
| 161. C3 | #63911 | semidirect |
| 162. C3 | #64083 | 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×C3×C4⋊(C3⋊D3) | 2 |
| C2×#6292 | 2 |
| C32×C4×Dic3 | 2 |
| #64083 | 3 |
| #63911 | 3 |