Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
5914 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C26×C32 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22⋊(C22⋊(C22⋊C9)) | C3 | 1 | 3 | yes | maximal, complemented |
| 3 | #88907 | — | 4 | 21 | no | maximal |
| 4 | C25×C32 | — | 3 | 21 | no | — |
| 5 | C26×C3 | — | 3 | 1 | no | — |
| 6 | C26×C3 | C32 | 1 | 1 | yes | characteristic |
| 7 | C24×C32 | — | 3 | 210 | no | — |
| 8 | C24×C32 | A4 | 1 | 21 | yes | — |
| 9 | C22⋊(C22⋊C9) | — | 4 | 63 | no | — |
| 10 | #1094 | — | 16 | 21 | no | — |
| 11 | C25×C3 | — | 3 | 63 | no | — |
| 12 | C25×C3 | — | 3 | 21 | no | — |
| 13 | C23×C32 | — | 3 | 360 | no | — |
| 14 | C23×C32 | — | 3 | 105 | no | — |
| 15 | C26 | 3−1+2 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | C24×C3 | — | 3 | 630 | no | — |
| 17 | C24×C3 | — | 3 | 210 | no | — |
| 18 | C24×C3 | — | 3 | 21 | no | — |
| 19 | C24×C3 | C3×A4 | 1 | 21 | yes | — |
| 20 | C22×C32 | — | 3 | 210 | no | — |
| 21 | C22×C32 | C22⋊A4 | 1 | 21 | yes | — |
| 22 | C22⋊C9 | — | 16 | 63 | no | — |
| 23 | C25 | — | 3 | 21 | no | — |
| 24 | 3−1+2 | — | 64 | 1 | no | Sylow |
| 25 | C23×C3 | — | 3 | 1080 | no | — |
| 26 | C23×C3 | — | 3 | 360 | no | — |
| 27 | C23×C3 | — | 3 | 315 | no | — |
| 28 | C23×C3 | — | 3 | 105 | no | — |
| 29 | C2×C32 | — | 3 | 21 | no | — |
| 30 | C24 | — | 3 | 210 | no | — |
| 31 | C24 | #1094 | 1 | 21 | yes | complemented |
| 32 | C22×C3 | — | 3 | 630 | no | — |
| 33 | C22×C3 | — | 3 | 210 | no | — |
| 34 | C22×C3 | — | 3 | 21 | no | — |
| 35 | C22×C3 | C3×C22⋊A4 | 1 | 21 | yes | — |
| 36 | C32 | C22⋊(C22⋊A4) | 1 | 1 | yes | characteristic |
| 37 | C9 | — | 64 | 3 | no | — |
| 38 | C23 | — | 3 | 360 | no | — |
| 39 | C23 | — | 3 | 105 | no | — |
| 40 | C2×C3 | — | 3 | 63 | no | — |
| 41 | C2×C3 | — | 3 | 21 | no | — |
| 42 | C22 | — | 3 | 210 | no | — |
| 43 | C22 | #88907 | 1 | 21 | yes | complemented |
| 44 | C3 | — | 3 | 1 | no | — |
| 45 | C3 | C3×C22⋊(C22⋊A4) | 1 | 1 | yes | characteristic |
| 46 | C2 | — | 3 | 21 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C22⋊(C22⋊A4) |
| commutator, socle, cosocle | C26×C3 | C32 |
| Fitting, prime radical | C26×C32 | C3 |
| hypercenter | C32 | C22⋊(C22⋊A4) |
| hypocenter | C26 | 3−1+2 |
| LCS | G ▷ C26×C3 ▷ C26 | |
| UCS | C1 ◁ C3 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22⋊(C22⋊(C22⋊C9)) | C3 | semidirect |
| 15. C26 | 3−1+2 | semidirect |
| 31. C24 | #1094 | semidirect |
| 43. C22 | #88907 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C26×C32 | 3 |
| C22⋊(C22⋊(C22⋊C9)) | 3 |
| #88907 | 4 |