Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
280 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C8 | #157 |
| commutator, hypocenter | C32 | C3×C82 |
| Frattini | C2×C4 | C2×C3×#157 |
| Fitting | C2×C33×C8 | C4 |
| socle, prime radical | C22×C33 | C42 |
| cosocle | C4×C3⋊Dic3 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C3×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C8×#622 | C3 | direct (N × H) |
| 13. C3×#622 | C8 | direct (N × H) |
| 14. C33×C8 | C8 | semidirect |
| 15. C3×C3⋊(C3⋊C8) | C8 | semidirect |
| 37. #622 | C3×C8 | direct (N × H) |
| 40. C32×C8 | C3×C8 | semidirect |
| 41. C3⋊(C3⋊C8) | C3×C8 | semidirect |
| 67. C33 | C82 | semidirect |
| 77. C3×C8 | #622 | direct (N × H) |
| 102. C32 | C3×C82 | semidirect |
| 107. C8 | C3×#622 | direct (N × H) |
| 120. C3 | C8×#622 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×#622 | 2 |
| C3×C8×C3⋊Dic3 | 2 |
| C8×#622 | 3 |
| C3×C82 | 9 |