Finite Group DatabasebetaFinite Group DatabasebetaOrder 12564.
36 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×D349 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊#13063703 | — | 3 | 4 | no | maximal |
| 3 | C32×C349 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×D349 | D3 | 1 | 4 | yes | — |
| 5 | #13063703 | — | 9 | 1 | no | Hall |
| 6 | C3×C349 | Dic3 | 1 | 4 | yes | complemented |
| 7 | D349 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 8 | C349 | C3⋊Dic3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C3⋊Dic3 | — | 349 | 1 | no | maximal, Hall |
| 10 | C2×C32 | — | 349 | 1 | no | — |
| 11 | Dic3 | — | 1047 | 4 | no | — |
| 12 | C32 | #13063703 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C2×C3 | — | 349 | 4 | no | — |
| 14 | C4 | — | 3141 | 1 | no | Sylow |
| 15 | C3 | C3⋊#13063703 | 1 | 4 | yes | complemented |
| 16 | C2 | — | 349 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C349 | C4 |
| prime radical, cosocle | C32×D349 | C2 |
| LCS | G ▷ C32×C349 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C32×C349 | C4 | semidirect |
| 6. C3×C349 | Dic3 | semidirect |
| 8. C349 | C3⋊Dic3 | semidirect |
| 12. C32 | #13063703 | semidirect |
| 15. C3 | C3⋊#13063703 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×D349 | 2 |
| C3⋊#13063703 | 3 |
| C3⋊Dic3 | 349 |