Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
85 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3⋊(#41⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #985 | — | 3 | 1 | no | maximal |
| 5 | #919 | — | 3 | 1 | no | maximal |
| 6 | C3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 7 | #624 | — | 4 | 1 | no | maximal |
| 8 | #239 | — | 3 | 1 | no | — |
| 9 | #239 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #41⋊C3 | — | 3 | 1 | no | — |
| 11 | GL(2,𝔽3) | — | 3 | 1 | no | — |
| 12 | GL(2,𝔽3) | — | 6 | 1 | no | — |
| 13 | #226 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C3×Q16 | — | 3 | 1 | no | — |
| 15 | #215 | — | 3 | 1 | no | — |
| 16 | #215 | — | 3 | 1 | no | — |
| 17 | D24 | — | 3 | 1 | no | — |
| 18 | D3×C8 | — | 3 | 1 | no | — |
| 19 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 20 | C3×Dic3 | — | 4 | 1 | no | — |
| 21 | C3×Dic3 | — | 4 | 1 | no | — |
| 22 | #135 | — | 9 | 1 | no | Sylow |
| 23 | C3×Q8 | — | 3 | 1 | no | — |
| 24 | C3×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 25 | D12 | — | 3 | 1 | no | — |
| 26 | D12 | — | 3 | 1 | no | — |
| 27 | D12 | — | 6 | 1 | no | — |
| 28 | C4×D3 | — | 3 | 1 | no | — |
| 29 | C4×D3 | — | 6 | 1 | no | — |
| 30 | C4×D3 | — | 12 | 1 | no | — |
| 31 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 32 | SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 33 | C3×C8 | — | 3 | 1 | no | — |
| 34 | C3⋊C8 | — | 3 | 1 | no | — |
| 35 | C2×C32 | — | 4 | 1 | no | — |
| 36 | C3⋊D3 | — | 4 | 2 | no | — |
| 37 | #41 | — | 3 | 1 | no | — |
| 38 | #41 | — | 9 | 1 | no | — |
| 39 | Q16 | — | 3 | 1 | no | — |
| 40 | SD16 | — | 9 | 1 | no | — |
| 41 | SD16 | — | 9 | 1 | no | — |
| 42 | D8 | — | 9 | 1 | no | — |
| 43 | C2×C8 | — | 9 | 1 | no | — |
| 44 | C2×D3 | — | 3 | 1 | no | — |
| 45 | C2×D3 | — | 6 | 1 | no | — |
| 46 | C2×D3 | — | 12 | 1 | no | — |
| 47 | C2×D3 | — | 24 | 1 | no | — |
| 48 | C3×C4 | — | 3 | 1 | no | — |
| 49 | C3×C4 | — | 6 | 1 | no | — |
| 50 | C3×C4 | — | 12 | 1 | no | — |
| 51 | Dic3 | — | 4 | 1 | no | — |
| 52 | Dic3 | S4 | 1 | 1 | yes | characteristic |
| 53 | C32 | — | 4 | 1 | no | Sylow |
| 54 | Q8 | — | 3 | 1 | no | — |
| 55 | Q8 | D32 | 1 | 1 | yes | characteristic |
| 56 | D4 | — | 9 | 1 | no | — |
| 57 | D4 | — | 9 | 1 | no | — |
| 58 | D4 | — | 18 | 1 | no | — |
| 59 | C2×C4 | — | 9 | 1 | no | — |
| 60 | C2×C4 | — | 18 | 1 | no | — |
| 61 | C8 | — | 3 | 1 | no | — |
| 62 | C8 | — | 9 | 1 | no | — |
| 63 | C2×C3 | — | 4 | 1 | no | — |
| 64 | C2×C3 | — | 8 | 1 | no | — |
| 65 | C2×C3 | C2×S4 | 1 | 1 | yes | characteristic |
| 66 | D3 | — | 6 | 1 | no | — |
| 67 | D3 | — | 12 | 2 | no | — |
| 68 | D3 | — | 12 | 1 | no | — |
| 69 | D3 | — | 24 | 2 | no | — |
| 70 | C22 | — | 9 | 1 | no | — |
| 71 | C22 | — | 18 | 1 | no | — |
| 72 | C4 | — | 3 | 1 | no | — |
| 73 | C4 | — | 3 | 1 | no | — |
| 74 | C4 | — | 6 | 1 | no | — |
| 75 | C3 | — | 4 | 1 | no | — |
| 76 | C3 | — | 8 | 1 | no | — |
| 77 | C3 | #985 | 1 | 1 | yes | characteristic, complemented |
| 78 | C2 | — | 18 | 1 | no | — |
| 79 | C2 | — | 36 | 1 | no | — |
| 80 | C2 | D3×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C22 |
| Fitting | C3×Q8 | C2×D3 |
| socle | C2×C3 | C2×S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊(#41⋊C3) | C2 | semidirect |
| 2. C3⋊GL(2,𝔽3) | C2 | semidirect |
| 3. C3×#226 | C2 | semidirect |
| 9. #239 | D3 | semidirect |
| 13. #226 | D3 | semidirect |
| 77. C3 | #985 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊(#41⋊C3) | 2 |
| C3⋊GL(2,𝔽3) | 2 |
| C3×#226 | 2 |
| #985 | 3 |
| #919 | 3 |
| #624 | 4 |