Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #6295 |
| commutator | C33×C4 | C2×C8 |
| Frattini, hypercenter | C42 | #1113 |
| Fitting, cosocle | C33×C42 | C22 |
| socle, prime radical | C22×C33 | C4⋊C4 |
| hypocenter | C33 | #335 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 21. C3×C3⋊(C3⋊C8) | C8 | semidirect |
| 69. C3⋊(C3⋊C8) | C3⋊C8 | semidirect |
| 128. C33 | #335 | semidirect |
| 206. C32 | #4458 | semidirect |
| 207. C32 | #4479 | semidirect |
| 238. C3 | #10589099 | semidirect |
| 239. C3 | #10589100 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4⋊(C3⋊(C3⋊C8)) | 2 |
| C3×C4×C3⋊(C3⋊C8) | 2 |
| #10589100 | 3 |
| #10589099 | 3 |