← #14553764
Subgroups — #14553764
Order 2187.
572 conjugacy classes of subgroups.
The subgroup lattice has 300 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #10624906 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10624903 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10624903 | C3 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10624886 | C3 | 1 | 1 | yes | maximal, characteristic |
| 5 | #10624674 | C3 | 1 | 3 | yes | maximal, complemented |
| 6 | #10624674 | C3 | 1 | 3 | yes | maximal, complemented |
| 7 | #10624674 | C3 | 1 | 3 | yes | maximal, complemented |
| 8 | #10624655 | C3 | 1 | 3 | yes | maximal, complemented |
| 9 | #10624655 | C3 | 1 | 3 | yes | maximal, complemented |
| 10 | #10624649 | C3 | 1 | 3 | yes | maximal, complemented |
| 11 | #10624633 | C3 | 1 | 1 | yes | maximal, characteristic |
| 12 | #10624633 | C3 | 1 | 1 | yes | maximal, characteristic |
| 13 | #10624626 | C3 | 1 | 3 | yes | maximal, complemented |
| 14 | #10624618 | C3 | 1 | 3 | yes | maximal, complemented |
| 15 | #10624618 | C3 | 1 | 3 | yes | maximal, complemented |
| 16 | #10624616 | C3 | 1 | 1 | yes | maximal, characteristic |
| 17 | #10624614 | C3 | 1 | 1 | yes | maximal, characteristic |
| 18 | #10624614 | C3 | 1 | 1 | yes | maximal, characteristic |
| 19 | #10624604 | C3 | 1 | 1 | yes | maximal, characteristic |
| 20 | #10624604 | C3 | 1 | 1 | yes | maximal, characteristic |
| 21 | #10624597 | C3 | 1 | 1 | yes | maximal, characteristic |
| 22 | C3×#6858 | C3 | 1 | 1 | yes | maximal, characteristic |
| 23 | C32×3−1+2 | C32 | 1 | 1 | yes | characteristic |
| 24 | #6899 | — | 3 | 3 | no | — |
| 25 | #6899 | — | 3 | 3 | no | — |
| 26 | #6898 | — | 3 | 3 | no | — |
| 27 | #6897 | — | 3 | 3 | no | — |
| 28 | #6897 | — | 3 | 3 | no | — |
| 29 | #6897 | — | 3 | 3 | no | — |
| 30 | #6895 | — | 3 | 1 | no | — |
| 31 | #6895 | — | 3 | 1 | no | — |
| 32 | #6895 | — | 3 | 1 | no | — |
| 33 | #6895 | — | 3 | 1 | no | — |
| 34 | #6895 | — | 3 | 1 | no | — |
| 35 | #6895 | C32 | 1 | 3 | yes | — |
| 36 | #6895 | C32 | 1 | 3 | yes | — |
| 37 | #6894 | — | 3 | 3 | no | — |
| 38 | #6894 | — | 3 | 3 | no | — |
| 39 | #6894 | — | 3 | 1 | no | — |
| 40 | #6894 | — | 3 | 1 | no | — |
| 41 | #6894 | — | 3 | 1 | no | — |
| 42 | #6894 | — | 3 | 1 | no | — |
| 43 | #6894 | C32 | 1 | 3 | yes | — |
| 44 | #6894 | C32 | 1 | 3 | yes | — |
| 45 | #6894 | C32 | 1 | 3 | yes | — |
| 46 | #6894 | C32 | 1 | 3 | yes | — |
| 47 | #6894 | C32 | 1 | 3 | yes | — |
| 48 | #6894 | C32 | 1 | 3 | yes | — |
| 49 | #6894 | C32 | 1 | 3 | yes | — |
| 50 | #6891 | C32 | 1 | 1 | yes | characteristic |
| 51 | #6891 | C32 | 1 | 1 | yes | characteristic |
| 52 | #6891 | C32 | 1 | 1 | yes | characteristic |
| 53 | #6891 | C32 | 1 | 1 | yes | characteristic |
| 54 | #6891 | C32 | 1 | 1 | yes | characteristic |
| 55 | #6891 | C32 | 1 | 1 | yes | characteristic |
| 56 | C9×3−1+2 | — | 3 | 1 | no | — |
| 57 | C9×3−1+2 | — | 3 | 1 | no | — |
| 58 | C9×3−1+2 | C32 | 1 | 3 | yes | — |
| 59 | C9×3−1+2 | C32 | 1 | 3 | yes | — |
| 60 | #6887 | — | 3 | 1 | no | — |
| 61 | #6887 | C32 | 1 | 3 | yes | — |
| 62 | C3×C32⋊C9 | C32 | 1 | 1 | yes | characteristic |
| 63 | C3×C32⋊C9 | C32 | 1 | 1 | yes | characteristic |
| 64 | C3×C32⋊C9 | C32 | 1 | 1 | yes | characteristic |
| 65 | C3×C32⋊C9 | C32 | 1 | 1 | yes | characteristic |
| 66 | C3×C32⋊C9 | C32 | 1 | 1 | yes | characteristic |
| 67 | C3×C32⋊C9 | C32 | 1 | 1 | yes | characteristic |
| 68 | #6861 | C32 | 1 | 3 | yes | — |
| 69 | #6861 | C32 | 1 | 3 | yes | — |
| 70 | #6861 | C32 | 1 | 3 | yes | — |
| 71 | #6861 | C32 | 1 | 3 | yes | — |
| 72 | #6861 | C32 | 1 | 3 | yes | — |
| 73 | #6861 | C32 | 1 | 3 | yes | — |
| 74 | #6861 | C32 | 1 | 3 | yes | — |
| 75 | #6861 | C32 | 1 | 3 | yes | — |
| 76 | #6861 | C32 | 1 | 3 | yes | — |
| 77 | #6860 | C32 | 1 | 3 | yes | — |
| 78 | #6860 | C32 | 1 | 3 | yes | — |
| 79 | #6860 | C32 | 1 | 3 | yes | — |
| 80 | #6860 | C32 | 1 | 3 | yes | — |
| 81 | #6860 | C32 | 1 | 3 | yes | — |
| 82 | #6860 | C32 | 1 | 3 | yes | — |
| 83 | #6860 | C32 | 1 | 3 | yes | — |
| 84 | #6860 | C32 | 1 | 3 | yes | — |
| 85 | #6860 | C32 | 1 | 3 | yes | — |
| 86 | #6858 | C32 | 1 | 1 | yes | characteristic |
| 87 | #6858 | C32 | 1 | 1 | yes | characteristic |
| 88 | #6858 | C32 | 1 | 1 | yes | characteristic |
| 89 | #6858 | C32 | 1 | 1 | yes | characteristic |
| 90 | #6858 | C32 | 1 | 1 | yes | characteristic |
| 91 | #6858 | C32 | 1 | 1 | yes | characteristic |
| 92 | #6858 | C32 | 1 | 1 | yes | characteristic |
| 93 | #6858 | C32 | 1 | 1 | yes | characteristic |
| 94 | #6858 | C32 | 1 | 1 | yes | characteristic |
| 95 | #6858 | C32 | 1 | 3 | yes | — |
| 96 | #6858 | C32 | 1 | 3 | yes | — |
| 97 | #6858 | C32 | 1 | 3 | yes | — |
| 98 | #6858 | C32 | 1 | 3 | yes | — |
| 99 | #6858 | C32 | 1 | 3 | yes | — |
| 100 | #6858 | C32 | 1 | 3 | yes | — |
| 101 | C34 | C33 | 1 | 1 | yes | characteristic |
| 102 | C3×3−1+2 | — | 3 | 1 | no | — |
| 103 | C3×3−1+2 | — | 3 | 1 | no | — |
| 104 | C3×3−1+2 | — | 3 | 1 | no | — |
| 105 | C3×3−1+2 | — | 3 | 1 | no | — |
| 106 | C3×3−1+2 | — | 3 | 1 | no | — |
| 107 | C3×3−1+2 | — | 3 | 1 | no | — |
| 108 | C3×3−1+2 | — | 9 | 3 | no | — |
| 109 | C3×3−1+2 | C33 | 1 | 1 | yes | characteristic |
| 110 | C3×3−1+2 | C33 | 1 | 1 | yes | characteristic |
| 111 | C3×3−1+2 | C33 | 1 | 1 | yes | characteristic |
| 112 | C3×3−1+2 | C33 | 1 | 3 | yes | — |
| 113 | C3×3−1+2 | C33 | 1 | 3 | yes | — |
| 114 | C32×C9 | — | 3 | 1 | no | — |
| 115 | C32×C9 | — | 3 | 1 | no | — |
| 116 | C32×C9 | — | 3 | 1 | no | — |
| 117 | C32×C9 | — | 3 | 1 | no | — |
| 118 | C32×C9 | — | 3 | 1 | no | — |
| 119 | C32×C9 | — | 3 | 1 | no | — |
| 120 | C32×C9 | C33 | 1 | 3 | yes | — |
| 121 | C9⋊C9 | — | 3 | 3 | no | — |
| 122 | C9⋊C9 | — | 3 | 3 | no | — |
| 123 | C9⋊C9 | — | 3 | 3 | no | — |
| 124 | C9⋊C9 | — | 3 | 3 | no | — |
| 125 | C9⋊C9 | — | 3 | 3 | no | — |
| 126 | C9⋊C9 | — | 3 | 3 | no | — |
| 127 | C9⋊C9 | — | 3 | 3 | no | — |
| 128 | C9⋊C9 | — | 3 | 3 | no | — |
| 129 | C9⋊C9 | — | 3 | 3 | no | — |
| 130 | C9⋊C9 | — | 3 | 3 | no | — |
| 131 | C9⋊C9 | — | 3 | 3 | no | — |
| 132 | C9⋊C9 | — | 3 | 3 | no | — |
| 133 | C9⋊C9 | — | 3 | 3 | no | — |
| 134 | C9⋊C9 | — | 3 | 3 | no | — |
| 135 | C9⋊C9 | — | 3 | 3 | no | — |
| 136 | C9⋊C9 | — | 3 | 3 | no | — |
| 137 | C9⋊C9 | — | 3 | 3 | no | — |
| 138 | C9⋊C9 | — | 3 | 3 | no | — |
| 139 | C9⋊C9 | — | 3 | 3 | no | — |
| 140 | C9⋊C9 | — | 3 | 3 | no | — |
| 141 | C9⋊C9 | — | 3 | 3 | no | — |
| 142 | C9⋊C9 | — | 3 | 3 | no | — |
| 143 | C9⋊C9 | — | 3 | 3 | no | — |
| 144 | C9⋊C9 | — | 3 | 3 | no | — |
| 145 | C9⋊C9 | — | 3 | 3 | no | — |
| 146 | C9⋊C9 | — | 3 | 3 | no | — |
| 147 | C9⋊C9 | — | 3 | 3 | no | — |
| 148 | C9⋊C9 | — | 3 | 3 | no | — |
| 149 | C9⋊C9 | — | 3 | 3 | no | — |
| 150 | C9⋊C9 | — | 3 | 3 | no | — |
| 151 | C9⋊C9 | — | 3 | 3 | no | — |
| 152 | C9⋊C9 | — | 3 | 3 | no | — |
| 153 | C9⋊C9 | — | 3 | 3 | no | — |
| 154 | C32⋊C9 | — | 3 | 1 | no | — |
| 155 | C32⋊C9 | — | 3 | 1 | no | — |
| 156 | C32⋊C9 | — | 3 | 1 | no | — |
| 157 | C32⋊C9 | — | 3 | 1 | no | — |
| 158 | C32⋊C9 | — | 3 | 1 | no | — |
| 159 | C32⋊C9 | — | 3 | 1 | no | — |
| 160 | C32⋊C9 | — | 3 | 1 | no | — |
| 161 | C32⋊C9 | — | 3 | 1 | no | — |
| 162 | C32⋊C9 | — | 3 | 1 | no | — |
| 163 | C32⋊C9 | — | 3 | 1 | no | — |
| 164 | C32⋊C9 | — | 3 | 1 | no | — |
| 165 | C32⋊C9 | — | 3 | 1 | no | — |
| 166 | C32⋊C9 | — | 3 | 1 | no | — |
| 167 | C32⋊C9 | — | 3 | 1 | no | — |
| 168 | C32⋊C9 | — | 3 | 1 | no | — |
| 169 | C32⋊C9 | — | 3 | 1 | no | — |
| 170 | C32⋊C9 | — | 3 | 1 | no | — |
| 171 | C32⋊C9 | — | 3 | 1 | no | — |
| 172 | C32⋊C9 | — | 3 | 1 | no | — |
| 173 | C32⋊C9 | — | 3 | 1 | no | — |
| 174 | C32⋊C9 | — | 3 | 1 | no | — |
| 175 | C32⋊C9 | — | 3 | 1 | no | — |
| 176 | C32⋊C9 | — | 3 | 1 | no | — |
| 177 | C32⋊C9 | — | 3 | 1 | no | — |
| 178 | C32⋊C9 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 179 | C32⋊C9 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 180 | C32⋊C9 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 181 | C32⋊C9 | H(3,𝔽3) | 1 | 3 | yes | — |
| 182 | C32⋊C9 | H(3,𝔽3) | 1 | 3 | yes | — |
| 183 | C32⋊C9 | C33 | 1 | 1 | yes | characteristic |
| 184 | C32⋊C9 | C33 | 1 | 1 | yes | characteristic |
| 185 | C32⋊C9 | C33 | 1 | 1 | yes | characteristic |
| 186 | C32⋊C9 | C33 | 1 | 1 | yes | characteristic |
| 187 | C32⋊C9 | C33 | 1 | 1 | yes | characteristic |
| 188 | C32⋊C9 | C33 | 1 | 1 | yes | characteristic |
| 189 | C32⋊C9 | C33 | 1 | 3 | yes | — |
| 190 | C32⋊C9 | C33 | 1 | 1 | yes | characteristic |
| 191 | C32⋊C9 | C33 | 1 | 1 | yes | characteristic |
| 192 | C32⋊C9 | C33 | 1 | 1 | yes | characteristic |
| 193 | C32⋊C9 | C33 | 1 | 3 | yes | — |
| 194 | C32⋊C9 | C33 | 1 | 3 | yes | — |
| 195 | C32⋊C9 | C33 | 1 | 3 | yes | — |
| 196 | C32⋊C9 | C33 | 1 | 3 | yes | — |
| 197 | C32⋊C9 | C33 | 1 | 3 | yes | — |
| 198 | C92 | — | 3 | 3 | no | — |
| 199 | C92 | — | 3 | 3 | no | — |
| 200 | C92 | — | 3 | 3 | no | — |
| 201 | C33 | — | 3 | 3 | no | — |
| 202 | C33 | — | 9 | 1 | no | — |
| 203 | C33 | — | 9 | 1 | no | — |
| 204 | C33 | — | 9 | 1 | no | — |
| 205 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 206 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 207 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 208 | C33 | C34 | 1 | 1 | yes | characteristic |
| 209 | 3−1+2 | — | 9 | 3 | no | — |
| 210 | 3−1+2 | — | 9 | 3 | no | — |
| 211 | 3−1+2 | — | 9 | 3 | no | — |
| 212 | 3−1+2 | — | 9 | 3 | no | — |
| 213 | 3−1+2 | — | 9 | 1 | no | — |
| 214 | 3−1+2 | — | 9 | 1 | no | — |
| 215 | 3−1+2 | — | 9 | 1 | no | — |
| 216 | 3−1+2 | — | 9 | 3 | no | — |
| 217 | 3−1+2 | — | 9 | 3 | no | — |
| 218 | 3−1+2 | — | 27 | 1 | no | — |
| 219 | 3−1+2 | — | 27 | 1 | no | — |
| 220 | C3×C9 | — | 3 | 1 | no | — |
| 221 | C3×C9 | — | 3 | 1 | no | — |
| 222 | C3×C9 | — | 3 | 1 | no | — |
| 223 | C3×C9 | — | 3 | 3 | no | — |
| 224 | C3×C9 | — | 3 | 3 | no | — |
| 225 | C3×C9 | — | 3 | 3 | no | — |
| 226 | C3×C9 | — | 3 | 3 | no | — |
| 227 | C3×C9 | — | 3 | 3 | no | — |
| 228 | C3×C9 | — | 3 | 3 | no | — |
| 229 | C3×C9 | — | 3 | 1 | no | — |
| 230 | C3×C9 | — | 3 | 1 | no | — |
| 231 | C3×C9 | — | 3 | 3 | no | — |
| 232 | C3×C9 | — | 3 | 3 | no | — |
| 233 | C3×C9 | — | 3 | 1 | no | — |
| 234 | C3×C9 | — | 3 | 1 | no | — |
| 235 | C3×C9 | — | 3 | 1 | no | — |
| 236 | C3×C9 | — | 3 | 1 | no | — |
| 237 | C3×C9 | — | 3 | 1 | no | — |
| 238 | C3×C9 | — | 3 | 1 | no | — |
| 239 | C3×C9 | — | 3 | 1 | no | — |
| 240 | C3×C9 | — | 9 | 3 | no | — |
| 241 | C3×C9 | — | 9 | 3 | no | — |
| 242 | C3×C9 | — | 9 | 3 | no | — |
| 243 | C3×C9 | — | 27 | 1 | no | — |
| 244 | C3×C9 | — | 27 | 1 | no | — |
| 245 | C3×C9 | — | 27 | 1 | no | — |
| 246 | C3×C9 | — | 27 | 1 | no | — |
| 247 | C3×C9 | C3×H(3,𝔽3) | 1 | 3 | yes | — |
| 248 | C3×C9 | C3×H(3,𝔽3) | 1 | 3 | yes | — |
| 249 | C3×C9 | C3×H(3,𝔽3) | 1 | 3 | yes | — |
| 250 | C32 | — | 3 | 1 | no | — |
| 251 | C32 | — | 3 | 1 | no | — |
| 252 | C32 | — | 3 | 1 | no | — |
| 253 | C32 | — | 3 | 1 | no | — |
| 254 | C32 | — | 3 | 1 | no | — |
| 255 | C32 | — | 3 | 1 | no | — |
| 256 | C32 | — | 3 | 1 | no | — |
| 257 | C32 | — | 3 | 1 | no | — |
| 258 | C32 | — | 3 | 1 | no | — |
| 259 | C32 | — | 3 | 1 | no | — |
| 260 | C32 | — | 3 | 1 | no | — |
| 261 | C32 | — | 3 | 1 | no | — |
| 262 | C32 | — | 3 | 1 | no | — |
| 263 | C32 | — | 3 | 1 | no | — |
| 264 | C32 | — | 3 | 1 | no | — |
| 265 | C32 | — | 27 | 3 | no | — |
| 266 | C32 | #6913 | 1 | 3 | yes | — |
| 267 | C32 | C32×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 268 | C9 | — | 3 | 3 | no | — |
| 269 | C9 | — | 3 | 3 | no | — |
| 270 | C9 | — | 3 | 3 | no | — |
| 271 | C9 | — | 9 | 1 | no | — |
| 272 | C9 | — | 9 | 1 | no | — |
| 273 | C9 | — | 9 | 1 | no | — |
| 274 | C9 | — | 9 | 1 | no | — |
| 275 | C9 | — | 9 | 1 | no | — |
| 276 | C9 | — | 9 | 1 | no | — |
| 277 | C9 | — | 9 | 3 | no | — |
| 278 | C9 | — | 9 | 3 | no | — |
| 279 | C9 | — | 9 | 1 | no | — |
| 280 | C9 | — | 9 | 1 | no | — |
| 281 | C9 | — | 9 | 3 | no | — |
| 282 | C9 | — | 9 | 3 | no | — |
| 283 | C9 | — | 9 | 1 | no | — |
| 284 | C9 | — | 9 | 1 | no | — |
| 285 | C9 | — | 9 | 1 | no | — |
| 286 | C9 | — | 9 | 1 | no | — |
| 287 | C9 | — | 9 | 3 | no | — |
| 288 | C9 | — | 9 | 3 | no | — |
| 289 | C9 | — | 9 | 3 | no | — |
| 290 | C9 | — | 9 | 3 | no | — |
| 291 | C3 | — | 3 | 3 | no | — |
| 292 | C3 | — | 9 | 1 | no | — |
| 293 | C3 | — | 9 | 1 | no | — |
| 294 | C3 | — | 9 | 1 | no | — |
| 295 | C3 | C3×#6913 | 1 | 1 | yes | characteristic |
| 296 | C3 | #10624933 | 1 | 1 | yes | characteristic |
| 297 | C3 | #10624935 | 1 | 1 | yes | characteristic |
| 298 | C3 | #10624935 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta