Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
1316 conjugacy classes of subgroups; only the 375 normal subgroups are shown.
This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 375 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 | C22×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C23 |
| Fitting | C22×C33×C4 | C2 |
| hypercenter | C22×C4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #121 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C33×C4 | C2 | semidirect |
| 2. C2×#6312 | C2 | semidirect |
| 3. C2×C4⋊(C3⋊(C3⋊D3)) | C2 | semidirect |
| 4. #88859 | C2 | semidirect |
| 5. C4⋊(C3⋊(C3⋊Dic3)) | C2 | semidirect |
| 15. C22×C32×C4 | D3 | semidirect |
| 22. C22×C33 | D4 | semidirect |
| 28. C33×C4 | D4 | semidirect |
| 44. C22×C3×C4 | C3⋊D3 | semidirect |
| 51. C22×C32 | D12 | semidirect |
| 57. C32×C4 | #67 | semidirect |
| 60. C33 | #121 | semidirect |
| 76. C22×C4 | C3⋊(C3⋊D3) | semidirect |
| 81. C22×C3 | C4⋊(C3⋊D3) | semidirect |
| 87. C3×C4 | #638 | semidirect |
| 89. C32 | #930 | semidirect |
| 108. C22 | C4⋊(C3⋊(C3⋊D3)) | semidirect |
| 112. C4 | #6312 | semidirect |
| 113. C3 | #64139 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C33×C4 | 2 |
| C2×#6312 | 2 |
| C2×C4⋊(C3⋊(C3⋊D3)) | 2 |
| #88859 | 2 |
| C4⋊(C3⋊(C3⋊Dic3)) | 2 |
| #64139 | 3 |