Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
414 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#3713 |
| commutator, Fitting, cosocle | C33×C8 | C23 |
| Frattini, hypercenter | C8 | D33 |
| socle | C2×C33 | C2×D8 |
| prime radical | C3×#63808 | C2 |
| hypocenter | C33 | #510 |
| LCS | G ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×D3×Dic12 | C2 | semidirect |
| 3. #11722068 | C2 | semidirect |
| 4. #11722013 | C2 | semidirect |
| 5. #11722008 | C2 | semidirect |
| 6. #11722001 | C2 | semidirect |
| 7. C3×#63550 | C2 | semidirect |
| 27. D3×Dic12 | D3 | semidirect |
| 36. #63550 | D3 | semidirect |
| 262. C33 | #510 | semidirect |
| 367. C32 | #4918 | semidirect |
| 368. C32 | #5179 | semidirect |
| 369. C32 | #5195 | semidirect |
| 407. C3 | #10590636 | semidirect |
| 408. C3 | #10591042 | semidirect |
| 409. C3 | #10591047 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63808 | 2 |
| C3×D3×Dic12 | 2 |
| #11722068 | 2 |
| #11722013 | 2 |
| #11722008 | 2 |
| #11722001 | 2 |
| C3×#63550 | 2 |
| #10591047 | 3 |
| #10591042 | 3 |
| #10590636 | 3 |