Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4060 conjugacy classes of subgroups; only the 893 normal subgroups are shown.
This group has 4060 conjugacy classes of subgroups — too many to draw. Only its 893 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 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×C4×Q8 | C2 |
| hypercenter | C4×Q8 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #548 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#89080 | C2 | semidirect |
| 2. C2×Q8×C3⋊(C3⋊D3) | C2 | semidirect |
| 3. C33×C4×Q8 | C2 | semidirect |
| 4. #11723938 | C2 | semidirect |
| 5. #11723936 | C2 | semidirect |
| 6. #11723916 | C2 | semidirect |
| 7. C4×C4⋊(C3⋊(C3⋊D3)) | C2 | semidirect |
| 25. C32×C4×Q8 | D3 | semidirect |
| 32. C33×Q8 | D4 | semidirect |
| 64. C2×C3⋊(C3⋊D3) | #41 | semidirect |
| 72. C3×C4×Q8 | C3⋊D3 | semidirect |
| 79. C32×Q8 | D12 | semidirect |
| 127. C4×Q8 | C3⋊(C3⋊D3) | semidirect |
| 129. C33 | #548 | semidirect |
| 132. C3×Q8 | C4⋊(C3⋊D3) | semidirect |
| 174. C32 | #5576 | semidirect |
| 178. Q8 | C4⋊(C3⋊(C3⋊D3)) | semidirect |
| 205. C3 | #10596047 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#89080 | 2 |
| C2×Q8×C3⋊(C3⋊D3) | 2 |
| C33×C4×Q8 | 2 |
| #11723938 | 2 |
| #11723936 | 2 |
| #11723916 | 2 |
| C4×C4⋊(C3⋊(C3⋊D3)) | 2 |
| #10596047 | 3 |