Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1112 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C3×Q8⋊C9 | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C22×C32×Q8 | D3 |
| socle | C23×C32 | S4 |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×Q8⋊C9 | C2 | semidirect |
| 2. C2×C3⋊#3689 | C2 | direct (N × H) |
| 6. C2×C3×Q8⋊C9 | C22 | semidirect |
| 7. C3⋊#3689 | C22 | direct (N × H) |
| 13. C22×Q8⋊C9 | D3 | semidirect |
| 18. C3×Q8⋊C9 | C23 | semidirect |
| 31. C2×Q8⋊C9 | C2×D3 | semidirect |
| 37. C22×C3×Q8 | D9 | semidirect |
| 61. Q8⋊C9 | C22×D3 | semidirect |
| 70. C2×C3×Q8 | C2×D9 | semidirect |
| 94. C22×Q8 | C3⋊D9 | semidirect |
| 110. C3×Q8 | C22×D9 | semidirect |
| 130. C2×Q8 | C2×C3⋊D9 | semidirect |
| 137. C22×C3 | #3689 | semidirect |
| 156. Q8 | C22×C3⋊D9 | semidirect |
| 162. C2×C3 | C2×#3689 | semidirect |
| 170. C22 | C3⋊#3689 | direct (N × H) |
| 174. C3 | C22×#3689 | semidirect |
| 177. C2 | C2×C3⋊#3689 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×Q8⋊C9 | 2 |
| C2×C3⋊#3689 | 2 |
| C22×#3755 | 3 |
| C22×#3689 | 3 |
| C23×C3⋊D9 | 4 |