Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
496 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 30 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 | C3×#66368 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C32×#638 | — | 3 | 1 | no | maximal |
| 3 | C3×(C3⋊S4)⋊C3 | — | 3 | 1 | no | maximal |
| 4 | C3×((C22⋊C9)⋊C2)⋊C3 | — | 3 | 2 | no | maximal |
| 5 | #10619762 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 6 | C3×#4183 | — | 4 | 1 | no | maximal |
| 7 | C22×C34 | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×C32×C3⋊D3 | — | 3 | 1 | no | — |
| 9 | C32×C3⋊Dic3 | — | 3 | 1 | no | — |
| 10 | C3×C22⋊H(3,𝔽3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C3×#1094 | D3 | 1 | 2 | yes | complemented |
| 12 | #66368 | — | 2 | 1 | no | — |
| 13 | #66368 | — | 2 | 1 | no | — |
| 14 | #66368 | — | 2 | 2 | no | — |
| 15 | #66368 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C3×AΣL(1,𝔽27) | — | 4 | 1 | no | Sylow |
| 17 | C3×C3⋊S4 | — | 3 | 1 | no | — |
| 18 | C3×#638 | — | 3 | 2 | no | — |
| 19 | C3×#638 | — | 3 | 1 | no | — |
| 20 | C3×#638 | — | 3 | 1 | no | — |
| 21 | C32×#67 | — | 9 | 3 | no | — |
| 22 | C32×#67 | — | 9 | 1 | no | — |
| 23 | (C3⋊S4)⋊C3 | — | 3 | 2 | no | — |
| 24 | (C3⋊S4)⋊C3 | — | 3 | 1 | no | — |
| 25 | C3×(C22⋊C9)⋊C2 | — | 3 | 2 | no | — |
| 26 | ((C22⋊C9)⋊C2)⋊C3 | — | 3 | 4 | no | — |
| 27 | ((C22⋊C9)⋊C2)⋊C3 | — | 3 | 2 | no | — |
| 28 | C2×C34 | — | 3 | 1 | no | — |
| 29 | C32×C3⋊D3 | — | 6 | 1 | no | — |
| 30 | C3×AGL(1,ℤ/9) | — | 12 | 2 | no | — |
| 31 | C3×(C3⋊D3)⋊C3 | — | 12 | 1 | no | — |
| 32 | #4183 | — | 4 | 1 | no | — |
| 33 | C22×C33 | — | 2 | 1 | no | — |
| 34 | C22×C33 | — | 3 | 3 | no | — |
| 35 | C22×C33 | — | 3 | 2 | no | — |
| 36 | C22×C33 | — | 3 | 1 | no | — |
| 37 | C22×C33 | — | 6 | 3 | no | — |
| 38 | C22×C33 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 39 | C22×C33 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 40 | C2×C3×C3⋊D3 | — | 3 | 1 | no | — |
| 41 | C2×C3×C3⋊D3 | — | 3 | 2 | no | — |
| 42 | C2×C3×C3⋊D3 | — | 3 | 1 | no | — |
| 43 | C2×C32×D3 | — | 9 | 3 | no | — |
| 44 | C2×C32×D3 | — | 9 | 1 | no | — |
| 45 | C32×A4 | — | 3 | 1 | no | — |
| 46 | C3×C3⋊Dic3 | — | 3 | 2 | no | — |
| 47 | C3×C3⋊Dic3 | — | 3 | 1 | no | — |
| 48 | C3×C3⋊Dic3 | — | 3 | 1 | no | — |
| 49 | C32×Dic3 | — | 9 | 3 | no | — |
| 50 | C32×Dic3 | — | 9 | 1 | no | — |
| 51 | C22⋊H(3,𝔽3) | — | 2 | 1 | no | — |
| 52 | C22⋊H(3,𝔽3) | — | 3 | 2 | no | — |
| 53 | C22⋊H(3,𝔽3) | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 54 | #1094 | — | 2 | 2 | no | — |
| 55 | #1094 | — | 3 | 4 | no | — |
| 56 | #1094 | C3×D3 | 1 | 2 | yes | complemented |
| 57 | C3×C22⋊C9 | — | 3 | 2 | no | — |
| 58 | C34 | S4 | 1 | 1 | yes | characteristic, complemented |
| 59 | C3×3−1+2 | — | 4 | 2 | no | — |
| 60 | C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 61 | AΣL(1,𝔽27) | — | 4 | 1 | no | — |
| 62 | AΣL(1,𝔽27) | — | 8 | 1 | no | — |
| 63 | AΣL(1,𝔽27) | — | 8 | 2 | no | — |
| 64 | AΣL(1,𝔽27) | — | 8 | 1 | no | — |
| 65 | C3⋊S4 | — | 3 | 1 | no | — |
| 66 | C3×S4 | — | 27 | 1 | no | — |
| 67 | C32×D4 | — | 27 | 1 | no | — |
| 68 | #638 | — | 3 | 1 | no | — |
| 69 | C3×#67 | — | 9 | 3 | no | — |
| 70 | C3×#67 | — | 9 | 6 | no | — |
| 71 | C3×#67 | — | 9 | 3 | no | — |
| 72 | C3×#67 | — | 9 | 2 | no | — |
| 73 | C3×#67 | — | 9 | 1 | no | — |
| 74 | C3×#67 | — | 9 | 1 | no | — |
| 75 | (C22⋊C9)⋊C2 | — | 3 | 2 | no | — |
| 76 | C2×C33 | — | 3 | 3 | no | — |
| 77 | C2×C33 | — | 3 | 2 | no | — |
| 78 | C2×C33 | — | 3 | 1 | no | — |
| 79 | C2×C33 | — | 3 | 1 | no | — |
| 80 | C2×C33 | — | 3 | 1 | no | — |
| 81 | C2×C33 | — | 6 | 1 | no | — |
| 82 | C2×C33 | — | 6 | 3 | no | — |
| 83 | C2×C33 | — | 6 | 1 | no | — |
| 84 | C2×C33 | — | 6 | 2 | no | — |
| 85 | C2×C33 | — | 6 | 6 | no | — |
| 86 | C2×C33 | — | 6 | 3 | no | — |
| 87 | C3×C3⋊D3 | — | 6 | 1 | no | — |
| 88 | C3×C3⋊D3 | — | 6 | 2 | no | — |
| 89 | C3×C3⋊D3 | — | 6 | 1 | no | — |
| 90 | C3×C3⋊D3 | — | 12 | 1 | no | — |
| 91 | C32×D3 | — | 18 | 3 | no | — |
| 92 | C32×D3 | — | 18 | 1 | no | — |
| 93 | AGL(1,ℤ/9) | — | 12 | 2 | no | — |
| 94 | AGL(1,ℤ/9) | — | 12 | 4 | no | — |
| 95 | (C3⋊D3)⋊C3 | — | 12 | 1 | no | — |
| 96 | (C3⋊D3)⋊C3 | — | 12 | 2 | no | — |
| 97 | C3×D9 | — | 12 | 2 | no | — |
| 98 | C22×C32 | — | 3 | 1 | no | — |
| 99 | C22×C32 | — | 3 | 1 | no | — |
| 100 | C22×C32 | — | 3 | 3 | no | — |
| 101 | C22×C32 | — | 3 | 6 | no | — |
| 102 | C22×C32 | — | 3 | 3 | no | — |
| 103 | C22×C32 | — | 6 | 1 | no | — |
| 104 | C22×C32 | — | 6 | 3 | no | — |
| 105 | C22×C32 | — | 6 | 6 | no | — |
| 106 | C22×C32 | — | 6 | 3 | no | — |
| 107 | C22×C32 | — | 6 | 1 | no | — |
| 108 | C22×C32 | — | 27 | 1 | no | — |
| 109 | C22×C32 | #272 | 1 | 1 | yes | characteristic |
| 110 | C22×C32 | #272 | 1 | 2 | yes | — |
| 111 | C22×C32 | C3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 112 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 113 | C2×C3×D3 | — | 9 | 3 | no | — |
| 114 | C2×C3×D3 | — | 9 | 6 | no | — |
| 115 | C2×C3×D3 | — | 9 | 3 | no | — |
| 116 | C2×C3×D3 | — | 9 | 1 | no | — |
| 117 | C2×C3×D3 | — | 9 | 2 | no | — |
| 118 | C2×C3×D3 | — | 9 | 1 | no | — |
| 119 | C3×A4 | — | 3 | 1 | no | — |
| 120 | C3×A4 | — | 6 | 1 | no | — |
| 121 | C3×A4 | — | 9 | 1 | no | — |
| 122 | C3×A4 | — | 18 | 1 | no | — |
| 123 | C32×C4 | — | 27 | 1 | no | — |
| 124 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 125 | C3×Dic3 | — | 9 | 3 | no | — |
| 126 | C3×Dic3 | — | 9 | 6 | no | — |
| 127 | C3×Dic3 | — | 9 | 3 | no | — |
| 128 | C3×Dic3 | — | 9 | 2 | no | — |
| 129 | C3×Dic3 | — | 9 | 1 | no | — |
| 130 | C3×Dic3 | — | 9 | 1 | no | — |
| 131 | C22⋊C9 | — | 3 | 2 | no | — |
| 132 | C22⋊C9 | — | 6 | 2 | no | — |
| 133 | C33 | — | 2 | 1 | no | — |
| 134 | C33 | — | 3 | 3 | no | — |
| 135 | C33 | — | 3 | 2 | no | — |
| 136 | C33 | — | 3 | 1 | no | — |
| 137 | C33 | — | 6 | 3 | no | — |
| 138 | C33 | — | 12 | 1 | no | — |
| 139 | C33 | C3×S4 | 1 | 1 | yes | characteristic, complemented |
| 140 | C33 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 141 | 3−1+2 | — | 4 | 2 | no | — |
| 142 | 3−1+2 | — | 8 | 2 | no | — |
| 143 | 3−1+2 | — | 12 | 4 | no | — |
| 144 | H(3,𝔽3) | — | 4 | 1 | no | — |
| 145 | H(3,𝔽3) | — | 8 | 1 | no | — |
| 146 | H(3,𝔽3) | — | 12 | 2 | no | — |
| 147 | C3×C9 | — | 12 | 2 | no | — |
| 148 | S4 | — | 27 | 1 | no | — |
| 149 | C3×D4 | — | 27 | 2 | no | — |
| 150 | C3×D4 | — | 27 | 1 | no | — |
| 151 | C3×D4 | — | 27 | 1 | no | — |
| 152 | #67 | — | 9 | 3 | no | — |
| 153 | #67 | — | 9 | 1 | no | — |
| 154 | C2×C32 | — | 3 | 1 | no | — |
| 155 | C2×C32 | — | 3 | 1 | no | — |
| 156 | C2×C32 | — | 3 | 3 | no | — |
| 157 | C2×C32 | — | 3 | 6 | no | — |
| 158 | C2×C32 | — | 3 | 3 | no | — |
| 159 | C2×C32 | — | 3 | 2 | no | — |
| 160 | C2×C32 | — | 3 | 1 | no | — |
| 161 | C2×C32 | — | 3 | 1 | no | — |
| 162 | C2×C32 | — | 6 | 3 | no | — |
| 163 | C2×C32 | — | 6 | 1 | no | — |
| 164 | C2×C32 | — | 6 | 3 | no | — |
| 165 | C2×C32 | — | 6 | 1 | no | — |
| 166 | C2×C32 | — | 6 | 3 | no | — |
| 167 | C2×C32 | — | 6 | 6 | no | — |
| 168 | C2×C32 | — | 6 | 2 | no | — |
| 169 | C2×C32 | — | 6 | 1 | no | — |
| 170 | C2×C32 | — | 6 | 6 | no | — |
| 171 | C2×C32 | — | 6 | 3 | no | — |
| 172 | C2×C32 | — | 6 | 2 | no | — |
| 173 | C2×C32 | — | 6 | 1 | no | — |
| 174 | C2×C32 | — | 6 | 6 | no | — |
| 175 | C2×C32 | — | 6 | 6 | no | — |
| 176 | C2×C32 | — | 6 | 6 | no | — |
| 177 | C2×C32 | — | 6 | 6 | no | — |
| 178 | C2×C32 | — | 54 | 1 | no | — |
| 179 | C3⋊D3 | — | 6 | 1 | no | — |
| 180 | C3⋊D3 | — | 12 | 1 | no | — |
| 181 | C3×D3 | — | 18 | 3 | no | — |
| 182 | C3×D3 | — | 18 | 6 | no | — |
| 183 | C3×D3 | — | 18 | 3 | no | — |
| 184 | C3×D3 | — | 18 | 1 | no | — |
| 185 | C3×D3 | — | 18 | 2 | no | — |
| 186 | C3×D3 | — | 18 | 1 | no | — |
| 187 | C3×D3 | — | 108 | 1 | no | — |
| 188 | D9 | — | 12 | 2 | no | — |
| 189 | C22×C3 | — | 2 | 1 | no | — |
| 190 | C22×C3 | — | 3 | 3 | no | — |
| 191 | C22×C3 | — | 3 | 2 | no | — |
| 192 | C22×C3 | — | 3 | 1 | no | — |
| 193 | C22×C3 | — | 6 | 3 | no | — |
| 194 | C22×C3 | — | 27 | 2 | no | — |
| 195 | C22×C3 | — | 27 | 1 | no | — |
| 196 | C22×C3 | — | 27 | 1 | no | — |
| 197 | C22×C3 | #4183 | 1 | 1 | yes | characteristic, complemented |
| 198 | C22×C3 | C3×#272 | 1 | 1 | yes | characteristic |
| 199 | C2×D3 | — | 9 | 3 | no | — |
| 200 | C2×D3 | — | 9 | 1 | no | — |
| 201 | A4 | — | 9 | 1 | no | — |
| 202 | A4 | — | 18 | 1 | no | — |
| 203 | C3×C4 | — | 27 | 2 | no | — |
| 204 | C3×C4 | — | 27 | 1 | no | — |
| 205 | C3×C4 | — | 27 | 1 | no | — |
| 206 | Dic3 | — | 9 | 3 | no | — |
| 207 | Dic3 | — | 9 | 1 | no | — |
| 208 | C32 | — | 3 | 3 | no | — |
| 209 | C32 | — | 3 | 1 | no | — |
| 210 | C32 | — | 3 | 1 | no | — |
| 211 | C32 | — | 3 | 6 | no | — |
| 212 | C32 | — | 3 | 3 | no | — |
| 213 | C32 | — | 6 | 1 | no | — |
| 214 | C32 | — | 6 | 3 | no | — |
| 215 | C32 | — | 6 | 6 | no | — |
| 216 | C32 | — | 6 | 3 | no | — |
| 217 | C32 | — | 6 | 1 | no | — |
| 218 | C32 | — | 12 | 1 | no | — |
| 219 | C32 | — | 24 | 1 | no | — |
| 220 | C32 | — | 36 | 1 | no | — |
| 221 | C32 | — | 72 | 1 | no | — |
| 222 | C32 | #6258 | 1 | 1 | yes | characteristic |
| 223 | C32 | #6258 | 1 | 2 | yes | — |
| 224 | C32 | C3×C3⋊S4 | 1 | 1 | yes | characteristic |
| 225 | C9 | — | 12 | 2 | no | — |
| 226 | C9 | — | 24 | 2 | no | — |
| 227 | D4 | — | 27 | 1 | no | Sylow |
| 228 | C2×C3 | — | 3 | 3 | no | — |
| 229 | C2×C3 | — | 3 | 2 | no | — |
| 230 | C2×C3 | — | 3 | 1 | no | — |
| 231 | C2×C3 | — | 3 | 1 | no | — |
| 232 | C2×C3 | — | 3 | 1 | no | — |
| 233 | C2×C3 | — | 6 | 1 | no | — |
| 234 | C2×C3 | — | 6 | 3 | no | — |
| 235 | C2×C3 | — | 6 | 2 | no | — |
| 236 | C2×C3 | — | 6 | 1 | no | — |
| 237 | C2×C3 | — | 6 | 6 | no | — |
| 238 | C2×C3 | — | 6 | 3 | no | — |
| 239 | C2×C3 | — | 54 | 2 | no | — |
| 240 | C2×C3 | — | 54 | 1 | no | — |
| 241 | C2×C3 | — | 54 | 1 | no | — |
| 242 | D3 | — | 18 | 3 | no | — |
| 243 | D3 | — | 18 | 1 | no | — |
| 244 | D3 | — | 108 | 1 | no | — |
| 245 | C22 | — | 27 | 1 | no | — |
| 246 | C22 | C3×#4183 | 1 | 1 | yes | characteristic, complemented |
| 247 | C4 | — | 27 | 1 | no | — |
| 248 | C3 | — | 2 | 1 | no | — |
| 249 | C3 | — | 3 | 3 | no | — |
| 250 | C3 | — | 3 | 2 | no | — |
| 251 | C3 | — | 3 | 1 | no | — |
| 252 | C3 | — | 6 | 3 | no | — |
| 253 | C3 | — | 36 | 1 | no | — |
| 254 | C3 | — | 72 | 1 | no | — |
| 255 | C3 | #10619762 | 1 | 1 | yes | characteristic, direct product factor |
| 256 | C3 | C3×#6258 | 1 | 1 | yes | characteristic |
| 257 | C2 | — | 3 | 1 | no | — |
| 258 | C2 | — | 54 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #10619762 |
| commutator, cosocle, hypocenter | #66368 | C2×C3 |
| Frattini | C32 | C3×C3⋊S4 |
| Fitting | C22×C34 | D3 |
| socle | C22×C32 | #272 |
| LCS | G ▷ #66368 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#66368 | C2 | semidirect |
| 5. #10619762 | C3 | direct (N × H) |
| 7. C22×C34 | D3 | semidirect |
| 10. C3×C22⋊H(3,𝔽3) | D3 | semidirect |
| 11. C3×#1094 | D3 | semidirect |
| 15. #66368 | C2×C3 | semidirect |
| 38. C22×C33 | C3×D3 | semidirect |
| 53. C22⋊H(3,𝔽3) | C3×D3 | semidirect |
| 56. #1094 | C3×D3 | semidirect |
| 58. C34 | S4 | semidirect |
| 139. C33 | C3×S4 | semidirect |
| 197. C22×C3 | #4183 | semidirect |
| 246. C22 | C3×#4183 | semidirect |
| 255. C3 | #10619762 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#66368 | 2 |
| C32×#638 | 3 |
| C3×(C3⋊S4)⋊C3 | 3 |
| C3×((C22⋊C9)⋊C2)⋊C3 | 3 |
| #10619762 | 3 |
| C3×#4183 | 4 |