Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
233 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 21 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 | #11757410 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊((C3⋊S4)⋊C3) | — | 3 | 1 | no | maximal |
| 3 | C3⋊(((C22⋊C9)⋊C2)⋊C3) | — | 3 | 2 | no | maximal |
| 4 | #10619630 | — | 3 | 1 | no | maximal |
| 5 | #92199 | — | 4 | 1 | no | maximal |
| 6 | C3×C22⋊H(3,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 7 | C3×#1094 | D3 | 1 | 2 | yes | complemented |
| 8 | C22×C32⋊C9 | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×#4169 | — | 3 | 1 | no | — |
| 10 | #66316 | — | 3 | 1 | no | — |
| 11 | #6857 | — | 4 | 1 | no | Sylow |
| 12 | C3⋊(C3⋊S4) | — | 3 | 1 | no | — |
| 13 | C3×#638 | — | 9 | 1 | no | — |
| 14 | C3⋊((C22⋊C9)⋊C2) | — | 3 | 2 | no | — |
| 15 | (C3⋊S4)⋊C3 | — | 9 | 3 | no | — |
| 16 | ((C22⋊C9)⋊C2)⋊C3 | — | 9 | 6 | no | — |
| 17 | #6233 | — | 3 | 1 | no | — |
| 18 | C3⋊AGL(1,ℤ/9) | — | 12 | 2 | no | — |
| 19 | C3⋊((C3⋊D3)⋊C3) | — | 12 | 1 | no | — |
| 20 | C2×C32⋊C9 | — | 3 | 1 | no | — |
| 21 | #4169 | — | 6 | 1 | no | — |
| 22 | C22×C33 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 23 | C2×C3×C3⋊D3 | — | 9 | 1 | no | — |
| 24 | C32×A4 | — | 3 | 1 | no | — |
| 25 | C3×C3⋊Dic3 | — | 9 | 1 | no | — |
| 26 | C22×C3×C9 | — | 3 | 1 | no | — |
| 27 | C2×C3⋊D9 | — | 3 | 1 | no | — |
| 28 | C22⋊H(3,𝔽3) | — | 3 | 3 | no | — |
| 29 | #1094 | — | 3 | 6 | no | — |
| 30 | C3×C22⋊C9 | — | 3 | 2 | no | — |
| 31 | C3⋊Dic9 | — | 3 | 1 | no | — |
| 32 | C3×3−1+2 | — | 4 | 2 | no | — |
| 33 | C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 34 | C32⋊C9 | S4 | 1 | 1 | yes | characteristic, complemented |
| 35 | C3⋊S4 | — | 9 | 3 | no | — |
| 36 | C3⋊S4 | — | 27 | 2 | no | — |
| 37 | C3⋊S4 | — | 27 | 1 | no | — |
| 38 | #638 | — | 9 | 1 | no | — |
| 39 | C3×#67 | — | 27 | 2 | no | — |
| 40 | C3×#67 | — | 27 | 1 | no | — |
| 41 | C3×#67 | — | 27 | 1 | no | — |
| 42 | (C22⋊C9)⋊C2 | — | 9 | 6 | no | — |
| 43 | #611 | — | 27 | 1 | no | — |
| 44 | C2×C33 | — | 3 | 1 | no | — |
| 45 | C3⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 46 | C3×C3⋊D3 | — | 18 | 1 | no | — |
| 47 | C2×C3×C9 | — | 3 | 1 | no | — |
| 48 | C2×C3×C9 | — | 6 | 1 | no | — |
| 49 | C3⋊D9 | — | 6 | 1 | no | — |
| 50 | C3⋊D9 | — | 12 | 2 | no | — |
| 51 | AGL(1,ℤ/9) | — | 36 | 6 | no | — |
| 52 | (C3⋊D3)⋊C3 | — | 36 | 3 | no | — |
| 53 | C22×C32 | — | 3 | 2 | no | — |
| 54 | C22×C32 | — | 3 | 1 | no | — |
| 55 | C22×C32 | — | 3 | 1 | no | — |
| 56 | C22×C32 | #272 | 1 | 1 | yes | characteristic |
| 57 | C2×C3⋊D3 | — | 9 | 1 | no | — |
| 58 | C2×C3×D3 | — | 27 | 1 | no | — |
| 59 | C2×C3×D3 | — | 27 | 1 | no | — |
| 60 | C2×C3×D3 | — | 27 | 2 | no | — |
| 61 | C3×A4 | — | 3 | 3 | no | — |
| 62 | C3×A4 | — | 9 | 2 | no | — |
| 63 | C3×A4 | — | 9 | 1 | no | — |
| 64 | C3⋊Dic3 | — | 9 | 1 | no | — |
| 65 | C3×Dic3 | — | 27 | 1 | no | — |
| 66 | C3×Dic3 | — | 27 | 1 | no | — |
| 67 | C3×Dic3 | — | 27 | 2 | no | — |
| 68 | C22×C9 | — | 9 | 1 | no | — |
| 69 | C2×D9 | — | 27 | 1 | no | — |
| 70 | C22⋊C9 | — | 3 | 6 | no | — |
| 71 | Dic9 | — | 27 | 1 | no | — |
| 72 | C33 | — | 12 | 1 | no | — |
| 73 | C33 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 74 | 3−1+2 | — | 12 | 6 | no | — |
| 75 | H(3,𝔽3) | — | 12 | 3 | no | — |
| 76 | C3×C9 | — | 3 | 1 | no | — |
| 77 | C3×C9 | — | 12 | 2 | no | — |
| 78 | S4 | — | 81 | 3 | no | — |
| 79 | C3×D4 | — | 81 | 1 | no | — |
| 80 | #67 | — | 27 | 1 | no | — |
| 81 | #67 | — | 27 | 2 | no | — |
| 82 | #67 | — | 27 | 1 | no | — |
| 83 | C2×C32 | — | 3 | 1 | no | — |
| 84 | C2×C32 | — | 3 | 1 | no | — |
| 85 | C2×C32 | — | 6 | 1 | no | — |
| 86 | C2×C32 | — | 9 | 1 | no | — |
| 87 | C2×C32 | — | 9 | 2 | no | — |
| 88 | C3⋊D3 | — | 18 | 1 | no | — |
| 89 | C3⋊D3 | — | 36 | 3 | no | — |
| 90 | C3⋊D3 | — | 108 | 1 | no | — |
| 91 | C3⋊D3 | — | 108 | 2 | no | — |
| 92 | C3×D3 | — | 54 | 1 | no | — |
| 93 | C3×D3 | — | 54 | 2 | no | — |
| 94 | C3×D3 | — | 54 | 1 | no | — |
| 95 | C2×C9 | — | 9 | 1 | no | — |
| 96 | C2×C9 | — | 18 | 1 | no | — |
| 97 | D9 | — | 36 | 6 | no | — |
| 98 | D9 | — | 54 | 1 | no | — |
| 99 | C22×C3 | — | 9 | 1 | no | — |
| 100 | C22×C3 | — | 81 | 1 | no | — |
| 101 | C22×C3 | #4183 | 1 | 2 | yes | — |
| 102 | C22×C3 | #4183 | 1 | 1 | yes | characteristic |
| 103 | C22×C3 | #4184 | 1 | 1 | yes | characteristic |
| 104 | C2×D3 | — | 27 | 1 | no | — |
| 105 | C2×D3 | — | 27 | 1 | no | — |
| 106 | C2×D3 | — | 27 | 2 | no | — |
| 107 | A4 | — | 9 | 3 | no | — |
| 108 | C3×C4 | — | 81 | 1 | no | — |
| 109 | Dic3 | — | 27 | 1 | no | — |
| 110 | Dic3 | — | 27 | 1 | no | — |
| 111 | Dic3 | — | 27 | 2 | no | — |
| 112 | C32 | — | 3 | 1 | no | — |
| 113 | C32 | — | 3 | 1 | no | — |
| 114 | C32 | — | 3 | 2 | no | — |
| 115 | C32 | — | 12 | 3 | no | — |
| 116 | C32 | — | 36 | 1 | no | — |
| 117 | C32 | — | 36 | 2 | no | — |
| 118 | C32 | #6258 | 1 | 1 | yes | characteristic |
| 119 | C9 | — | 9 | 1 | no | — |
| 120 | C9 | — | 12 | 6 | no | — |
| 121 | D4 | — | 81 | 1 | no | Sylow |
| 122 | C2×C3 | — | 3 | 1 | no | — |
| 123 | C2×C3 | — | 3 | 1 | no | — |
| 124 | C2×C3 | — | 3 | 2 | no | — |
| 125 | C2×C3 | — | 9 | 1 | no | — |
| 126 | C2×C3 | — | 18 | 1 | no | — |
| 127 | C2×C3 | — | 162 | 1 | no | — |
| 128 | D3 | — | 54 | 1 | no | — |
| 129 | D3 | — | 54 | 1 | no | — |
| 130 | D3 | — | 54 | 2 | no | — |
| 131 | D3 | — | 324 | 3 | no | — |
| 132 | C22 | — | 81 | 1 | no | — |
| 133 | C22 | #92199 | 1 | 1 | yes | characteristic, complemented |
| 134 | C4 | — | 81 | 1 | no | — |
| 135 | C3 | — | 9 | 1 | no | — |
| 136 | C3 | — | 36 | 3 | no | — |
| 137 | C3 | #10619751 | 1 | 1 | yes | characteristic |
| 138 | C3 | #10619757 | 1 | 1 | yes | characteristic |
| 139 | C3 | #66366⋊C2 | 1 | 2 | yes | — |
| 140 | C2 | — | 3 | 1 | no | — |
| 141 | C2 | — | 162 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #11757410 | C2 |
| Frattini | C33 | C3⋊S4 |
| Fitting | C22×C32⋊C9 | D3 |
| socle | C22×C32 | #272 |
| LCS | G ▷ #11757410 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11757410 | C2 | semidirect |
| 7. C3×#1094 | D3 | semidirect |
| 8. C22×C32⋊C9 | D3 | semidirect |
| 34. C32⋊C9 | S4 | semidirect |
| 133. C22 | #92199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11757410 | 2 |
| C3⋊((C3⋊S4)⋊C3) | 3 |
| C3⋊(((C22⋊C9)⋊C2)⋊C3) | 3 |
| #10619630 | 3 |
| #92199 | 4 |