Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
253 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 253 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 | #11757458 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊((C3⋊S4)⋊C3) | — | 3 | 1 | no | maximal |
| 3 | C3⋊(((C22⋊C9)⋊C2)⋊C3) | — | 3 | 1 | no | maximal |
| 4 | #10619980 | — | 3 | 1 | no | maximal |
| 5 | #10619746 | — | 3 | 1 | no | maximal |
| 6 | #92199 | — | 4 | 1 | no | maximal |
| 7 | C22×C3×3−1+2 | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×C3⋊AGL(1,ℤ/9) | — | 3 | 1 | no | — |
| 9 | C3×C22⋊H(3,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 10 | C3×#1094 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C3⋊(3−1+2⋊C4) | — | 3 | 1 | no | — |
| 12 | C32⋊(C22⋊C9) | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | #6857 | — | 4 | 1 | no | Sylow |
| 14 | C3⋊(C3⋊S4) | — | 3 | 1 | no | — |
| 15 | C3×#638 | — | 9 | 1 | no | — |
| 16 | C3⋊((C22⋊C9)⋊C2) | — | 3 | 1 | no | — |
| 17 | C3⋊((C22⋊C9)⋊C2) | — | 3 | 1 | no | — |
| 18 | (C3⋊S4)⋊C3 | — | 9 | 3 | no | — |
| 19 | ((C22⋊C9)⋊C2)⋊C3 | — | 9 | 3 | no | — |
| 20 | #6233 | — | 3 | 1 | no | — |
| 21 | #611⋊C3 | — | 9 | 3 | no | — |
| 22 | C2×C3×3−1+2 | — | 3 | 1 | no | — |
| 23 | C3⋊AGL(1,ℤ/9) | — | 6 | 1 | no | — |
| 24 | C3⋊AGL(1,ℤ/9) | — | 12 | 1 | no | — |
| 25 | C3⋊((C3⋊D3)⋊C3) | — | 12 | 1 | no | — |
| 26 | #4169 | — | 12 | 1 | no | — |
| 27 | C22×C33 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 28 | C2×C3×C3⋊D3 | — | 9 | 1 | no | — |
| 29 | C32×A4 | — | 3 | 1 | no | — |
| 30 | C3×C3⋊Dic3 | — | 9 | 1 | no | — |
| 31 | C22×3−1+2 | — | 3 | 3 | no | — |
| 32 | C22×C3×C9 | — | 3 | 1 | no | — |
| 33 | C2×C3⋊D9 | — | 3 | 1 | no | — |
| 34 | C2×AGL(1,ℤ/9) | — | 9 | 3 | no | — |
| 35 | C22⋊H(3,𝔽3) | — | 3 | 3 | no | — |
| 36 | #1094 | — | 3 | 3 | no | — |
| 37 | C3×C22⋊C9 | — | 3 | 1 | no | — |
| 38 | C3×C22⋊C9 | — | 3 | 1 | no | — |
| 39 | C3⋊Dic9 | — | 3 | 1 | no | — |
| 40 | 3−1+2⋊C4 | — | 9 | 3 | no | — |
| 41 | C3×3−1+2 | — | 4 | 1 | no | — |
| 42 | C3×3−1+2 | S4 | 1 | 1 | yes | characteristic, complemented |
| 43 | C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 44 | C32⋊C9 | — | 4 | 1 | no | — |
| 45 | C3⋊S4 | — | 9 | 3 | no | — |
| 46 | C3⋊S4 | — | 27 | 1 | no | — |
| 47 | C3⋊S4 | — | 27 | 1 | no | — |
| 48 | C3⋊S4 | — | 27 | 1 | no | — |
| 49 | #638 | — | 9 | 1 | no | — |
| 50 | C3×#67 | — | 27 | 1 | no | — |
| 51 | C3×#67 | — | 27 | 1 | no | — |
| 52 | C3×#67 | — | 27 | 1 | no | — |
| 53 | C3×#67 | — | 27 | 1 | no | — |
| 54 | (C22⋊C9)⋊C2 | — | 9 | 3 | no | — |
| 55 | (C22⋊C9)⋊C2 | — | 27 | 1 | no | — |
| 56 | #611 | — | 9 | 3 | no | — |
| 57 | C2×C33 | — | 3 | 1 | no | — |
| 58 | C3⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 59 | C3×C3⋊D3 | — | 18 | 1 | no | — |
| 60 | C2×3−1+2 | — | 3 | 3 | no | — |
| 61 | C2×3−1+2 | — | 6 | 3 | no | — |
| 62 | C2×C3×C9 | — | 3 | 1 | no | — |
| 63 | C2×C3×C9 | — | 6 | 1 | no | — |
| 64 | C3⋊D9 | — | 6 | 1 | no | — |
| 65 | C3⋊D9 | — | 12 | 1 | no | — |
| 66 | C3⋊D9 | — | 12 | 1 | no | — |
| 67 | AGL(1,ℤ/9) | — | 18 | 3 | no | — |
| 68 | AGL(1,ℤ/9) | — | 36 | 3 | no | — |
| 69 | (C3⋊D3)⋊C3 | — | 36 | 3 | no | — |
| 70 | C22×C32 | — | 3 | 1 | no | — |
| 71 | C22×C32 | — | 3 | 1 | no | — |
| 72 | C22×C32 | — | 3 | 1 | no | — |
| 73 | C22×C32 | — | 3 | 1 | no | — |
| 74 | C22×C32 | #272 | 1 | 1 | yes | characteristic |
| 75 | C2×C3⋊D3 | — | 9 | 1 | no | — |
| 76 | C2×C3×D3 | — | 27 | 1 | no | — |
| 77 | C2×C3×D3 | — | 27 | 1 | no | — |
| 78 | C2×C3×D3 | — | 27 | 1 | no | — |
| 79 | C2×C3×D3 | — | 27 | 1 | no | — |
| 80 | C3×A4 | — | 3 | 3 | no | — |
| 81 | C3×A4 | — | 9 | 1 | no | — |
| 82 | C3×A4 | — | 9 | 1 | no | — |
| 83 | C3×A4 | — | 9 | 1 | no | — |
| 84 | C3⋊Dic3 | — | 9 | 1 | no | — |
| 85 | C3×Dic3 | — | 27 | 1 | no | — |
| 86 | C3×Dic3 | — | 27 | 1 | no | — |
| 87 | C3×Dic3 | — | 27 | 1 | no | — |
| 88 | C3×Dic3 | — | 27 | 1 | no | — |
| 89 | C22×C9 | — | 3 | 3 | no | — |
| 90 | C2×D9 | — | 9 | 3 | no | — |
| 91 | C22⋊C9 | — | 3 | 3 | no | — |
| 92 | C22⋊C9 | — | 9 | 1 | no | — |
| 93 | Dic9 | — | 9 | 3 | no | — |
| 94 | C33 | — | 12 | 1 | no | — |
| 95 | C33 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 96 | 3−1+2 | — | 3 | 3 | no | — |
| 97 | 3−1+2 | — | 12 | 3 | no | — |
| 98 | H(3,𝔽3) | — | 12 | 3 | no | — |
| 99 | C3×C9 | — | 3 | 1 | no | — |
| 100 | C3×C9 | — | 12 | 1 | no | — |
| 101 | C3×C9 | — | 12 | 1 | no | — |
| 102 | S4 | — | 81 | 3 | no | — |
| 103 | C3×D4 | — | 81 | 1 | no | — |
| 104 | #67 | — | 27 | 1 | no | — |
| 105 | #67 | — | 27 | 1 | no | — |
| 106 | #67 | — | 27 | 1 | no | — |
| 107 | #67 | — | 27 | 1 | no | — |
| 108 | C2×C32 | — | 3 | 1 | no | — |
| 109 | C2×C32 | — | 3 | 1 | no | — |
| 110 | C2×C32 | — | 6 | 1 | no | — |
| 111 | C2×C32 | — | 9 | 1 | no | — |
| 112 | C2×C32 | — | 9 | 1 | no | — |
| 113 | C2×C32 | — | 9 | 1 | no | — |
| 114 | C3⋊D3 | — | 18 | 1 | no | — |
| 115 | C3⋊D3 | — | 36 | 3 | no | — |
| 116 | C3⋊D3 | — | 108 | 1 | no | — |
| 117 | C3⋊D3 | — | 108 | 1 | no | — |
| 118 | C3⋊D3 | — | 108 | 1 | no | — |
| 119 | C3×D3 | — | 54 | 1 | no | — |
| 120 | C3×D3 | — | 54 | 1 | no | — |
| 121 | C3×D3 | — | 54 | 1 | no | — |
| 122 | C3×D3 | — | 54 | 1 | no | — |
| 123 | C2×C9 | — | 3 | 3 | no | — |
| 124 | C2×C9 | — | 6 | 3 | no | — |
| 125 | D9 | — | 18 | 3 | no | — |
| 126 | D9 | — | 36 | 3 | no | — |
| 127 | D9 | — | 108 | 1 | no | — |
| 128 | C22×C3 | — | 9 | 1 | no | — |
| 129 | C22×C3 | — | 81 | 1 | no | — |
| 130 | C22×C3 | #4183 | 1 | 1 | yes | characteristic |
| 131 | C22×C3 | #4183 | 1 | 1 | yes | characteristic |
| 132 | C22×C3 | #4183 | 1 | 1 | yes | characteristic |
| 133 | C22×C3 | #4184 | 1 | 1 | yes | characteristic |
| 134 | C2×D3 | — | 27 | 1 | no | — |
| 135 | C2×D3 | — | 27 | 1 | no | — |
| 136 | C2×D3 | — | 27 | 1 | no | — |
| 137 | C2×D3 | — | 27 | 1 | no | — |
| 138 | A4 | — | 9 | 3 | no | — |
| 139 | C3×C4 | — | 81 | 1 | no | — |
| 140 | Dic3 | — | 27 | 1 | no | — |
| 141 | Dic3 | — | 27 | 1 | no | — |
| 142 | Dic3 | — | 27 | 1 | no | — |
| 143 | Dic3 | — | 27 | 1 | no | — |
| 144 | C32 | — | 3 | 1 | no | — |
| 145 | C32 | — | 3 | 1 | no | — |
| 146 | C32 | — | 3 | 1 | no | — |
| 147 | C32 | — | 3 | 1 | no | — |
| 148 | C32 | — | 12 | 3 | no | — |
| 149 | C32 | — | 36 | 1 | no | — |
| 150 | C32 | — | 36 | 1 | no | — |
| 151 | C32 | — | 36 | 1 | no | — |
| 152 | C32 | #6258 | 1 | 1 | yes | characteristic |
| 153 | C9 | — | 3 | 3 | no | — |
| 154 | C9 | — | 12 | 3 | no | — |
| 155 | C9 | — | 36 | 1 | no | — |
| 156 | D4 | — | 81 | 1 | no | Sylow |
| 157 | C2×C3 | — | 3 | 1 | no | — |
| 158 | C2×C3 | — | 3 | 1 | no | — |
| 159 | C2×C3 | — | 3 | 1 | no | — |
| 160 | C2×C3 | — | 3 | 1 | no | — |
| 161 | C2×C3 | — | 9 | 1 | no | — |
| 162 | C2×C3 | — | 18 | 1 | no | — |
| 163 | C2×C3 | — | 162 | 1 | no | — |
| 164 | D3 | — | 54 | 1 | no | — |
| 165 | D3 | — | 54 | 1 | no | — |
| 166 | D3 | — | 54 | 1 | no | — |
| 167 | D3 | — | 54 | 1 | no | — |
| 168 | D3 | — | 324 | 3 | no | — |
| 169 | C22 | — | 81 | 1 | no | — |
| 170 | C22 | #92199 | 1 | 1 | yes | characteristic, complemented |
| 171 | C4 | — | 81 | 1 | no | — |
| 172 | C3 | — | 9 | 1 | no | — |
| 173 | C3 | — | 36 | 3 | no | — |
| 174 | C3 | #66366⋊C2 | 1 | 1 | yes | characteristic |
| 175 | C3 | #66366⋊C2 | 1 | 1 | yes | characteristic |
| 176 | C3 | #66365⋊C2 | 1 | 1 | yes | characteristic |
| 177 | C3 | #10619762 | 1 | 1 | yes | characteristic |
| 178 | C2 | — | 3 | 1 | no | — |
| 179 | C2 | — | 162 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #11757458 | C2 |
| Frattini | C33 | C3⋊S4 |
| Fitting | C22×C3×3−1+2 | D3 |
| socle | C22×C32 | #272 |
| LCS | G ▷ #11757458 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11757458 | C2 | semidirect |
| 7. C22×C3×3−1+2 | D3 | semidirect |
| 10. C3×#1094 | D3 | semidirect |
| 12. C32⋊(C22⋊C9) | D3 | semidirect |
| 42. C3×3−1+2 | S4 | semidirect |
| 170. C22 | #92199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11757458 | 2 |
| C3⋊((C3⋊S4)⋊C3) | 3 |
| C3⋊(((C22⋊C9)⋊C2)⋊C3) | 3 |
| #10619980 | 3 |
| #10619746 | 3 |
| #92199 | 4 |