Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2162 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 2162 conjugacy classes of subgroups — too many to draw. Only its 607 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | #1113 |
| commutator | C2×C33 | C2×C42 |
| Frattini | C23 | C2×#1113 |
| Fitting, socle, prime radical | C24×C33 | C22 |
| cosocle | C23×C33 | C23 |
| hypocenter | C33 | C2×#95 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11722082 | C2 | direct (N × H) |
| 6. C22×C3×C3⋊Dic3 | C4 | semidirect |
| 18. C2×C3×C3⋊Dic3 | C2×C4 | semidirect |
| 24. C22×C3⋊Dic3 | Dic3 | semidirect |
| 62. C2×C3⋊Dic3 | C2×Dic3 | semidirect |
| 71. C2×C33 | #95 | semidirect |
| 119. C33 | C2×#95 | semidirect |
| 156. C2×C32 | C3⋊#95 | semidirect |
| 197. C32 | C2×C3⋊#95 | semidirect |
| 215. C2×C3 | #63579 | semidirect |
| 230. C3 | C2×#63579 | semidirect |
| 231. C2 | #11722082 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C3⋊Dic3 | 2 |
| #11722082 | 2 |
| C2×#63579 | 3 |