Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2292 conjugacy classes of subgroups; only the 316 normal subgroups are shown.
This group has 2292 conjugacy classes of subgroups — too many to draw. Only its 316 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×D3×C3⋊D3 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×C42 | C22 |
| hypercenter | C42 | D3×C3⋊D3 |
| hypocenter | C33 | #532 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#89015 | C2 | semidirect |
| 2. C42×C3⋊(C3⋊D3) | C2 | semidirect |
| 3. C3×#64078 | C2 | semidirect |
| 4. C32×#874 | C2 | semidirect |
| 5. #11723531 | C2 | semidirect |
| 6. #11723499 | C2 | semidirect |
| 7. #11723434 | C2 | semidirect |
| 32. #64078 | D3 | semidirect |
| 33. C3×#874 | D3 | semidirect |
| 45. C4×C3⋊(C3⋊D3) | D4 | semidirect |
| 91. C33×C4 | #41 | semidirect |
| 105. #874 | C3⋊D3 | semidirect |
| 198. C32×C4 | #237 | semidirect |
| 199. C32×C4 | #239 | semidirect |
| 215. C33 | #532 | semidirect |
| 291. C3×C4 | #3796 | semidirect |
| 294. C3×C4 | #3830 | semidirect |
| 303. C32 | #5716 | semidirect |
| 304. C32 | #5725 | semidirect |
| 338. C4 | #89015 | semidirect |
| 341. C3 | #10593907 | semidirect |
| 342. C3 | #10596187 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#89015 | 2 |
| C42×C3⋊(C3⋊D3) | 2 |
| C3×#64078 | 2 |
| C32×#874 | 2 |
| #11723531 | 2 |
| #11723499 | 2 |
| #11723434 | 2 |
| #10596187 | 3 |
| #10593907 | 3 |