Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
260 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #11757763 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊(D9⋊A4) | — | 3 | 2 | no | maximal |
| 3 | C22×C9⋊D9 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10619770 | — | 3 | 2 | no | maximal |
| 5 | #92309 | — | 4 | 1 | no | maximal |
| 6 | C3×C9⋊A4 | D3 | 1 | 2 | yes | — |
| 7 | C22×C92 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×C9⋊D9 | — | 3 | 1 | no | — |
| 9 | C2×C9⋊D9 | — | 3 | 1 | no | — |
| 10 | (C3×A4)⋊C9 | D3 | 1 | 2 | yes | — |
| 11 | C9⋊(C22⋊C9) | — | 2 | 2 | no | — |
| 12 | (C22⋊C9)⋊C9 | — | 2 | 2 | no | — |
| 13 | #6896 | — | 4 | 1 | no | Sylow |
| 14 | A4×C3⋊D3 | — | 9 | 1 | no | — |
| 15 | C22×C3⋊D9 | — | 3 | 2 | no | — |
| 16 | C22×C3⋊D9 | — | 3 | 2 | no | — |
| 17 | D9⋊A4 | — | 9 | 6 | no | — |
| 18 | C3⋊AGL(1,ℤ/9) | — | 12 | 2 | no | — |
| 19 | C2×C92 | — | 3 | 1 | no | — |
| 20 | C9⋊D9 | — | 3 | 1 | no | — |
| 21 | C9⋊D9 | A4 | 1 | 1 | yes | characteristic, complemented |
| 22 | #4169 | — | 12 | 2 | no | — |
| 23 | C32×A4 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 24 | C22×C3×C9 | C3×D3 | 1 | 2 | yes | — |
| 25 | C22×C3×C9 | C3×D3 | 1 | 2 | yes | — |
| 26 | C2×C3⋊D9 | — | 9 | 2 | no | — |
| 27 | C2×C3⋊D9 | — | 9 | 2 | no | — |
| 28 | C2×C3⋊D9 | — | 9 | 2 | no | — |
| 29 | C2×C3⋊D9 | — | 9 | 2 | no | — |
| 30 | C3×C22⋊C9 | — | 2 | 2 | no | — |
| 31 | C3×C22⋊C9 | — | 2 | 2 | no | — |
| 32 | C9⋊A4 | — | 3 | 6 | no | — |
| 33 | C3×3−1+2 | — | 4 | 2 | no | — |
| 34 | C9⋊C9 | — | 8 | 2 | no | — |
| 35 | C9⋊C9 | — | 8 | 2 | no | — |
| 36 | C32⋊C9 | — | 4 | 2 | no | — |
| 37 | C92 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 38 | C22×C3⋊D3 | — | 9 | 1 | no | — |
| 39 | D3×A4 | — | 27 | 2 | no | — |
| 40 | D3×A4 | — | 27 | 2 | no | — |
| 41 | C22×D9 | — | 9 | 6 | no | — |
| 42 | C22×D9 | — | 27 | 2 | no | — |
| 43 | C3×C3⋊D3 | — | 36 | 1 | no | — |
| 44 | C2×C3×C9 | — | 3 | 2 | no | — |
| 45 | C2×C3×C9 | — | 3 | 2 | no | — |
| 46 | C3⋊D9 | — | 3 | 2 | no | — |
| 47 | C3⋊D9 | — | 3 | 2 | no | — |
| 48 | C3⋊D9 | — | 9 | 2 | no | — |
| 49 | C3⋊D9 | — | 9 | 2 | no | — |
| 50 | AGL(1,ℤ/9) | — | 36 | 6 | no | — |
| 51 | C22×C32 | C3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 52 | C2×C3⋊D3 | — | 27 | 1 | no | — |
| 53 | C2×C3⋊D3 | — | 27 | 1 | no | — |
| 54 | C3×A4 | — | 3 | 2 | no | — |
| 55 | C3×A4 | — | 3 | 2 | no | — |
| 56 | C22×C9 | — | 3 | 2 | no | — |
| 57 | C22×C9 | AGL(1,ℤ/9) | 1 | 6 | yes | complemented |
| 58 | C2×D9 | — | 27 | 6 | no | — |
| 59 | C2×D9 | — | 27 | 6 | no | — |
| 60 | C2×D9 | — | 27 | 6 | no | — |
| 61 | C2×D9 | — | 27 | 6 | no | — |
| 62 | C22⋊C9 | — | 6 | 2 | no | — |
| 63 | C22⋊C9 | — | 6 | 2 | no | — |
| 64 | C33 | — | 4 | 1 | no | — |
| 65 | 3−1+2 | — | 12 | 6 | no | — |
| 66 | C3×C9 | — | 8 | 2 | no | — |
| 67 | C3×C9 | — | 8 | 2 | no | — |
| 68 | C3×C9 | D3×A4 | 1 | 2 | yes | — |
| 69 | C3×C9 | D3×A4 | 1 | 2 | yes | — |
| 70 | C22×D3 | — | 27 | 2 | no | — |
| 71 | C22×D3 | — | 27 | 2 | no | — |
| 72 | C2×A4 | — | 81 | 1 | no | — |
| 73 | C2×C32 | — | 3 | 1 | no | — |
| 74 | C3⋊D3 | — | 9 | 1 | no | — |
| 75 | C3⋊D3 | — | 27 | 1 | no | — |
| 76 | C3×D3 | — | 108 | 2 | no | — |
| 77 | C3×D3 | — | 108 | 2 | no | — |
| 78 | C2×C9 | — | 3 | 6 | no | — |
| 79 | C2×C9 | — | 3 | 6 | no | — |
| 80 | D9 | — | 9 | 6 | no | — |
| 81 | D9 | — | 27 | 2 | no | — |
| 82 | D9 | — | 27 | 6 | no | — |
| 83 | D9 | — | 27 | 6 | no | — |
| 84 | C22×C3 | C3⋊AGL(1,ℤ/9) | 1 | 2 | yes | — |
| 85 | C22×C3 | #4207 | 1 | 2 | yes | — |
| 86 | C2×D3 | — | 81 | 2 | no | — |
| 87 | C2×D3 | — | 81 | 2 | no | — |
| 88 | C2×D3 | — | 81 | 2 | no | — |
| 89 | C2×D3 | — | 81 | 2 | no | — |
| 90 | A4 | — | 9 | 1 | no | — |
| 91 | C32 | — | 12 | 2 | no | — |
| 92 | C32 | — | 12 | 2 | no | — |
| 93 | C32 | A4×C3⋊D3 | 1 | 1 | yes | characteristic |
| 94 | C9 | — | 3 | 2 | no | — |
| 95 | C9 | — | 24 | 2 | no | — |
| 96 | C9 | — | 24 | 2 | no | — |
| 97 | C9 | D9⋊A4 | 1 | 6 | yes | complemented |
| 98 | C23 | — | 81 | 1 | no | Sylow |
| 99 | C2×C3 | — | 3 | 2 | no | — |
| 100 | C2×C3 | — | 3 | 2 | no | — |
| 101 | C2×C3 | — | 324 | 1 | no | — |
| 102 | D3 | — | 27 | 2 | no | — |
| 103 | D3 | — | 27 | 2 | no | — |
| 104 | D3 | — | 81 | 2 | no | — |
| 105 | D3 | — | 81 | 2 | no | — |
| 106 | C22 | — | 243 | 1 | no | — |
| 107 | C22 | — | 243 | 1 | no | — |
| 108 | C22 | #92309 | 1 | 1 | yes | characteristic, complemented |
| 109 | C3 | — | 36 | 1 | no | — |
| 110 | C3 | C3⋊(D9⋊A4) | 1 | 2 | yes | — |
| 111 | C3 | #10620070 | 1 | 2 | yes | — |
| 112 | C2 | — | 3 | 1 | no | — |
| 113 | C2 | — | 81 | 1 | no | — |
| 114 | C2 | — | 243 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C22×C92 | C2×C3 |
| Frattini | C32 | A4×C3⋊D3 |
| socle | C22×C32 | C3×C3⋊D3 |
| LCS | G ▷ C22×C92 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11757763 | C2 | semidirect |
| 3. C22×C9⋊D9 | C3 | semidirect |
| 7. C22×C92 | C2×C3 | semidirect |
| 21. C9⋊D9 | A4 | semidirect |
| 37. C92 | C2×A4 | semidirect |
| 57. C22×C9 | AGL(1,ℤ/9) | semidirect |
| 97. C9 | D9⋊A4 | semidirect |
| 108. C22 | #92309 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11757763 | 2 |
| C3⋊(D9⋊A4) | 3 |
| C22×C9⋊D9 | 3 |
| #10619770 | 3 |
| #92309 | 4 |