Finite Group DatabasebetaFinite Group DatabasebetaOrder 1800.
64 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×#1039⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊#6034 | — | 3 | 1 | no | maximal |
| 3 | #10597074 | — | 3 | 3 | no | maximal |
| 4 | C2×C3×C52⋊C3 | C4 | 1 | 1 | yes | characteristic |
| 5 | C3×#1039 | D3 | 1 | 1 | yes | characteristic |
| 6 | #1039⋊C3 | D3 | 1 | 3 | yes | — |
| 7 | C3×C52⋊C3 | C8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | #6034 | — | 9 | 1 | no | Hall |
| 9 | C2×C3×C52 | Dic3 | 1 | 1 | yes | characteristic |
| 10 | C2×C52⋊C3 | Dic3 | 1 | 3 | yes | — |
| 11 | C3⋊#170 | — | 15 | 2 | no | — |
| 12 | #1039 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 13 | C3×C52 | C3⋊C8 | 1 | 1 | yes | characteristic, complemented |
| 14 | C52⋊C3 | C3⋊C8 | 1 | 3 | yes | complemented |
| 15 | C3⋊(C3⋊C8) | — | 25 | 1 | no | maximal, Hall |
| 16 | C3×Dic5 | — | 15 | 2 | no | — |
| 17 | C2×C52 | C3⋊Dic3 | 1 | 1 | yes | characteristic |
| 18 | #170 | — | 45 | 2 | no | — |
| 19 | C32×C4 | — | 25 | 1 | no | — |
| 20 | C2×C3×C5 | — | 3 | 2 | no | — |
| 21 | C52 | C3⋊(C3⋊C8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 22 | C3⋊C8 | — | 75 | 1 | no | — |
| 23 | C3⋊C8 | — | 75 | 3 | no | — |
| 24 | Dic5 | — | 15 | 2 | no | — |
| 25 | C2×C32 | — | 25 | 1 | no | — |
| 26 | C3×C5 | — | 3 | 2 | no | — |
| 27 | C3×C4 | — | 25 | 3 | no | — |
| 28 | C3×C4 | — | 25 | 1 | no | — |
| 29 | C2×C5 | — | 3 | 2 | no | — |
| 30 | C32 | — | 25 | 1 | no | Sylow |
| 31 | C8 | — | 225 | 1 | no | Sylow |
| 32 | C2×C3 | — | 25 | 3 | no | — |
| 33 | C2×C3 | #64480 | 1 | 1 | yes | characteristic |
| 34 | C5 | — | 3 | 2 | no | — |
| 35 | C4 | — | 25 | 1 | no | — |
| 36 | C3 | — | 25 | 3 | no | — |
| 37 | C3 | #10597074 | 1 | 1 | yes | characteristic, complemented |
| 38 | C2 | C3⋊#64480 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#64480 |
| commutator, hypocenter | C3×C52⋊C3 | C8 |
| Fitting, socle | C2×C3×C52 | Dic3 |
| prime radical | C2×C3×C52⋊C3 | C4 |
| cosocle | C3×#1039⋊C3 | C2 |
| LCS | G ▷ C3×C52⋊C3 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C3×C52⋊C3 | C8 | semidirect |
| 13. C3×C52 | C3⋊C8 | semidirect |
| 14. C52⋊C3 | C3⋊C8 | semidirect |
| 21. C52 | C3⋊(C3⋊C8) | semidirect |
| 37. C3 | #10597074 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#1039⋊C3 | 2 |
| C3⋊#6034 | 3 |
| #10597074 | 3 |
| C3⋊(C3⋊C8) | 25 |