Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
214 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11722067 |
| commutator, Fitting, cosocle | C33×C16 | C22 |
| Frattini, hypercenter | C16 | #1113 |
| socle | C2×C33 | D16 |
| hypocenter | C33 | D32 |
| LCS | G ▷ C33×C16 ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#63626 | C2 | semidirect |
| 2. C3×C3⋊(C3⋊C32) | C2 | semidirect |
| 7. #63626 | D3 | semidirect |
| 11. C3⋊(C3⋊C32) | D3 | semidirect |
| 68. C33 | D32 | semidirect |
| 101. C32 | D96 | semidirect |
| 102. C32 | #4519 | semidirect |
| 119. C3 | #10589070 | semidirect |
| 120. C3 | #10589071 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63626 | 2 |
| C3×C3⋊(C3⋊C32) | 2 |
| #10589071 | 3 |
| #10589070 | 3 |