Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
3958 conjugacy classes of subgroups; only the 387 normal subgroups are shown.
This group has 3958 conjugacy classes of subgroups — too many to draw. Only its 387 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 | C22×#41×D71 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#17985961 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #22588252 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22588251 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C71×#1780 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#11765866 | C4 | 1 | 1 | yes | characteristic |
| 7 | C22×#11765865 | C22 | 1 | 3 | yes | — |
| 8 | C22×#11765864 | C4 | 1 | 3 | yes | — |
| 9 | C2×#41×D71 | — | 2 | 24 | no | — |
| 10 | C22×Q8×D71 | C4 | 1 | 1 | yes | characteristic |
| 11 | C22×D4×D71 | C4 | 1 | 3 | yes | — |
| 12 | C23×C4×D71 | C22 | 1 | 3 | yes | — |
| 13 | C22×#11765848 | C22 | 1 | 4 | yes | — |
| 14 | #17985961 | — | 2 | 24 | no | — |
| 15 | C2×C22⋊(C71⋊C8) | C22 | 1 | 6 | yes | complemented |
| 16 | C2×(C2×D71)⋊C8 | C22 | 1 | 12 | yes | complemented |
| 17 | C22×#41×C71 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×C22⋊C8×C71 | C22 | 1 | 6 | yes | complemented |
| 19 | C24×D71 | C2×C4 | 1 | 3 | yes | — |
| 20 | C2×#11765866 | — | 2 | 12 | no | — |
| 21 | C2×#11765865 | — | 2 | 24 | no | — |
| 22 | C2×#11765865 | — | 2 | 12 | no | — |
| 23 | C2×#11765864 | — | 2 | 24 | no | — |
| 24 | C2×#11765864 | — | 2 | 12 | no | — |
| 25 | #41×D71 | — | 2 | 128 | no | — |
| 26 | C22×Dic142 | C2×C4 | 1 | 3 | yes | — |
| 27 | C2×Q8×D71 | — | 2 | 12 | no | — |
| 28 | C22×D284 | C2×C4 | 1 | 3 | yes | — |
| 29 | C22×C22⋊D71 | C2×C4 | 1 | 6 | yes | — |
| 30 | C2×D4×D71 | — | 2 | 24 | no | — |
| 31 | C2×D4×D71 | — | 2 | 12 | no | — |
| 32 | C23×Dic71 | C2×C4 | 1 | 3 | yes | — |
| 33 | C22×C4×D71 | — | 2 | 24 | no | — |
| 34 | C22×C4×D71 | C2×C4 | 1 | 3 | yes | — |
| 35 | C22×C4×D71 | C2×C4 | 1 | 3 | yes | — |
| 36 | C22×C4×D71 | D4 | 1 | 24 | yes | — |
| 37 | C22×C4×D71 | C23 | 1 | 3 | yes | — |
| 38 | C22×C4×D71 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×#11765848 | — | 2 | 24 | no | — |
| 40 | C22×C71⋊C8 | C23 | 1 | 4 | yes | — |
| 41 | C22⋊(C71⋊C8) | — | 2 | 12 | no | — |
| 42 | (C2×D71)⋊C8 | — | 2 | 24 | no | — |
| 43 | C2×#41×C71 | — | 2 | 12 | no | — |
| 44 | C22×Q8×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C22×D4×C71 | C2×C4 | 1 | 3 | yes | — |
| 46 | C23×C4×C71 | C23 | 1 | 3 | yes | — |
| 47 | C22×C8×C71 | C23 | 1 | 4 | yes | — |
| 48 | C22⋊C8×C71 | — | 2 | 12 | no | — |
| 49 | C23×D71 | — | 2 | 12 | no | — |
| 50 | C23×D71 | — | 4 | 12 | no | — |
| 51 | C23×D71 | C22⋊C4 | 1 | 12 | yes | — |
| 52 | C23×D71 | C22×C4 | 1 | 3 | yes | — |
| 53 | C23×D71 | C22×C4 | 1 | 1 | yes | characteristic |
| 54 | #11765866 | — | 2 | 32 | no | — |
| 55 | #11765865 | — | 2 | 96 | no | — |
| 56 | #11765864 | — | 2 | 96 | no | — |
| 57 | C2×Dic142 | — | 2 | 6 | no | — |
| 58 | C2×Dic142 | — | 2 | 12 | no | — |
| 59 | Q8×D71 | — | 2 | 32 | no | — |
| 60 | C2×D284 | — | 2 | 12 | no | — |
| 61 | C2×D284 | — | 2 | 6 | no | — |
| 62 | C2×C22⋊D71 | — | 2 | 12 | no | — |
| 63 | C2×C22⋊D71 | — | 2 | 12 | no | — |
| 64 | C2×C22⋊D71 | — | 2 | 12 | no | — |
| 65 | D4×D71 | — | 2 | 96 | no | — |
| 66 | C22×Dic71 | — | 2 | 12 | no | — |
| 67 | C22×Dic71 | C22⋊C4 | 1 | 12 | yes | — |
| 68 | C22×Dic71 | C22×C4 | 1 | 3 | yes | — |
| 69 | C22×Dic71 | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | C2×C4×D71 | — | 2 | 12 | no | — |
| 71 | C2×C4×D71 | — | 2 | 12 | no | — |
| 72 | C2×C4×D71 | — | 2 | 12 | no | — |
| 73 | C2×C4×D71 | — | 2 | 96 | no | — |
| 74 | C2×C4×D71 | C22⋊C4 | 1 | 12 | yes | — |
| 75 | C2×C4×D71 | C22⋊C4 | 1 | 12 | yes | — |
| 76 | C2×C4×D71 | C2×D4 | 1 | 12 | yes | — |
| 77 | C2×C4×D71 | C2×D4 | 1 | 12 | yes | — |
| 78 | #11765848 | — | 4 | 16 | no | — |
| 79 | C2×C71⋊C8 | — | 2 | 12 | no | — |
| 80 | C24×C71 | C22×C4 | 1 | 3 | yes | — |
| 81 | #41×C71 | — | 2 | 32 | no | — |
| 82 | C2×Q8×C71 | — | 2 | 6 | no | — |
| 83 | C2×D4×C71 | — | 2 | 12 | no | — |
| 84 | C2×D4×C71 | — | 2 | 6 | no | — |
| 85 | C22×C4×C71 | — | 2 | 12 | no | — |
| 86 | C22×C4×C71 | C22×C4 | 1 | 3 | yes | — |
| 87 | C22×C4×C71 | C2×D4 | 1 | 12 | yes | — |
| 88 | C22×C4×C71 | C24 | 1 | 1 | yes | characteristic |
| 89 | C2×C8×C71 | — | 2 | 12 | no | — |
| 90 | C23×C71 | — | 2 | 6 | no | — |
| 91 | C23×C71 | — | 2 | 12 | no | — |
| 92 | C23×C71 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 93 | C23×C71 | C23×C4 | 1 | 1 | yes | characteristic |
| 94 | C22×D71 | — | 2 | 4 | no | — |
| 95 | C22×D71 | — | 2 | 6 | no | — |
| 96 | C22×D71 | — | 2 | 12 | no | — |
| 97 | C22×D71 | — | 2 | 24 | no | — |
| 98 | C22×D71 | — | 4 | 24 | no | — |
| 99 | C22×D71 | — | 4 | 48 | no | — |
| 100 | C22×D71 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 101 | C22×D71 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 102 | Q8×C71 | — | 2 | 8 | no | — |
| 103 | D4×C71 | — | 2 | 24 | no | — |
| 104 | C2×C4×C71 | — | 2 | 6 | no | — |
| 105 | C2×C4×C71 | — | 2 | 24 | no | — |
| 106 | C2×C4×C71 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 107 | C2×C4×C71 | C22×D4 | 1 | 6 | yes | — |
| 108 | C22⋊D71 | — | 2 | 48 | no | — |
| 109 | C2×Dic71 | — | 2 | 6 | no | — |
| 110 | C2×Dic71 | — | 2 | 24 | no | — |
| 111 | C2×Dic71 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 112 | C2×Dic71 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 113 | D284 | — | 2 | 24 | no | — |
| 114 | C4×D71 | — | 2 | 24 | no | — |
| 115 | C4×D71 | — | 2 | 24 | no | — |
| 116 | C4×D71 | — | 2 | 24 | no | — |
| 117 | C4×D71 | #120 | 1 | 16 | yes | — |
| 118 | Dic142 | — | 2 | 24 | no | — |
| 119 | C8×C71 | — | 4 | 4 | no | — |
| 120 | C71⋊C8 | — | 4 | 4 | no | — |
| 121 | C22×C71 | — | 2 | 12 | no | — |
| 122 | C22×C71 | — | 2 | 6 | no | — |
| 123 | C22×C71 | — | 4 | 12 | no | — |
| 124 | C22×C71 | C22×C22⋊C4 | 1 | 3 | yes | — |
| 125 | C22×C71 | #568 | 1 | 4 | yes | — |
| 126 | C2×D71 | — | 2 | 12 | no | — |
| 127 | C2×D71 | — | 2 | 12 | no | — |
| 128 | C2×D71 | — | 2 | 6 | no | — |
| 129 | C2×D71 | — | 4 | 48 | no | — |
| 130 | C2×D71 | — | 4 | 12 | no | — |
| 131 | C2×D71 | #379 | 1 | 4 | yes | — |
| 132 | C4×C71 | — | 2 | 6 | no | — |
| 133 | C4×C71 | C2×#120 | 1 | 4 | yes | — |
| 134 | Dic71 | — | 2 | 6 | no | — |
| 135 | Dic71 | #379 | 1 | 4 | yes | — |
| 136 | #20580 | — | 71 | 1 | no | maximal, Sylow |
| 137 | C2×C71 | — | 4 | 6 | no | — |
| 138 | C2×C71 | C2×#379 | 1 | 1 | yes | characteristic |
| 139 | C2×C71 | #2879 | 1 | 6 | yes | — |
| 140 | D71 | — | 2 | 4 | no | — |
| 141 | D71 | — | 4 | 6 | no | — |
| 142 | C23×#41 | — | 71 | 1 | no | — |
| 143 | C2×#407 | — | 71 | 6 | no | — |
| 144 | #1780 | — | 71 | 6 | no | — |
| 145 | #1780 | — | 71 | 1 | no | — |
| 146 | #1780 | D71 | 1 | 1 | yes | characteristic, complemented |
| 147 | C71 | #20580 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 148 | C22×#41 | — | 71 | 1 | no | — |
| 149 | C22×#41 | — | 71 | 3 | no | — |
| 150 | C22×#41 | — | 71 | 3 | no | — |
| 151 | C22×#41 | — | 142 | 24 | no | — |
| 152 | C22×#41 | C2×D71 | 1 | 1 | yes | characteristic |
| 153 | C23×Q8 | — | 71 | 1 | no | — |
| 154 | C23×D4 | — | 71 | 3 | no | — |
| 155 | C24×C4 | — | 71 | 3 | no | — |
| 156 | C22×M16 | — | 71 | 4 | no | — |
| 157 | #407 | — | 142 | 24 | no | — |
| 158 | C2×C22⋊C8 | — | 71 | 6 | no | — |
| 159 | C2×C22⋊C8 | — | 71 | 12 | no | — |
| 160 | C2×C22⋊C8 | C2×D71 | 1 | 6 | yes | complemented |
| 161 | C25 | — | 71 | 3 | no | — |
| 162 | C2×#41 | — | 2 | 12 | no | — |
| 163 | C2×#41 | — | 142 | 24 | no | — |
| 164 | C2×#41 | — | 142 | 24 | no | — |
| 165 | C2×#41 | — | 142 | 12 | no | — |
| 166 | C2×#41 | — | 142 | 12 | no | — |
| 167 | C2×#41 | — | 142 | 128 | no | — |
| 168 | C2×#41 | — | 142 | 12 | no | — |
| 169 | C22×Q8 | — | 71 | 3 | no | — |
| 170 | C22×Q8 | — | 142 | 12 | no | — |
| 171 | C22×Q8 | C4×D71 | 1 | 1 | yes | characteristic |
| 172 | C22×D4 | — | 71 | 6 | no | — |
| 173 | C22×D4 | — | 71 | 3 | no | — |
| 174 | C22×D4 | — | 142 | 12 | no | — |
| 175 | C22×D4 | — | 142 | 24 | no | — |
| 176 | C22×D4 | C4×D71 | 1 | 3 | yes | — |
| 177 | C23×C4 | — | 71 | 1 | no | — |
| 178 | C23×C4 | — | 71 | 3 | no | — |
| 179 | C23×C4 | — | 71 | 3 | no | — |
| 180 | C23×C4 | — | 71 | 3 | no | — |
| 181 | C23×C4 | — | 71 | 24 | no | — |
| 182 | C23×C4 | — | 71 | 3 | no | — |
| 183 | C23×C4 | — | 142 | 24 | no | — |
| 184 | C23×C4 | C22×D71 | 1 | 3 | yes | — |
| 185 | C2×M16 | — | 142 | 24 | no | — |
| 186 | C22×C8 | — | 71 | 4 | no | — |
| 187 | C22×C8 | C22×D71 | 1 | 4 | yes | — |
| 188 | C22⋊C8 | — | 2 | 12 | no | — |
| 189 | C22⋊C8 | — | 142 | 24 | no | — |
| 190 | C22⋊C8 | — | 142 | 12 | no | — |
| 191 | C24 | — | 284 | 12 | no | — |
| 192 | C24 | — | 71 | 12 | no | — |
| 193 | C24 | — | 71 | 1 | no | — |
| 194 | C24 | — | 71 | 3 | no | — |
| 195 | C24 | — | 142 | 12 | no | — |
| 196 | C24 | C2×C4×D71 | 1 | 3 | yes | — |
| 197 | #41 | — | 2 | 32 | no | — |
| 198 | #41 | — | 142 | 96 | no | — |
| 199 | #41 | — | 142 | 96 | no | — |
| 200 | #41 | — | 142 | 32 | no | — |
| 201 | C2×Q8 | — | 2 | 6 | no | — |
| 202 | C2×Q8 | — | 142 | 12 | no | — |
| 203 | C2×Q8 | — | 142 | 32 | no | — |
| 204 | C2×Q8 | — | 142 | 6 | no | — |
| 205 | C2×D4 | — | 2 | 12 | no | — |
| 206 | C2×D4 | — | 2 | 6 | no | — |
| 207 | C2×D4 | — | 142 | 12 | no | — |
| 208 | C2×D4 | — | 142 | 12 | no | — |
| 209 | C2×D4 | — | 142 | 96 | no | — |
| 210 | C2×D4 | — | 142 | 12 | no | — |
| 211 | C2×D4 | — | 142 | 12 | no | — |
| 212 | C2×D4 | — | 142 | 6 | no | — |
| 213 | C22×C4 | — | 2 | 12 | no | — |
| 214 | C22×C4 | — | 71 | 1 | no | — |
| 215 | C22×C4 | — | 71 | 12 | no | — |
| 216 | C22×C4 | — | 71 | 3 | no | — |
| 217 | C22×C4 | — | 71 | 12 | no | — |
| 218 | C22×C4 | — | 71 | 12 | no | — |
| 219 | C22×C4 | — | 71 | 12 | no | — |
| 220 | C22×C4 | — | 71 | 12 | no | — |
| 221 | C22×C4 | — | 142 | 12 | no | — |
| 222 | C22×C4 | — | 142 | 96 | no | — |
| 223 | C22×C4 | — | 142 | 12 | no | — |
| 224 | C22×C4 | — | 142 | 12 | no | — |
| 225 | C22×C4 | — | 142 | 12 | no | — |
| 226 | C22×C4 | C2×C4×D71 | 1 | 3 | yes | — |
| 227 | C22×C4 | D4×D71 | 1 | 12 | yes | — |
| 228 | C22×C4 | C23×D71 | 1 | 1 | yes | characteristic |
| 229 | M16 | — | 284 | 16 | no | — |
| 230 | C2×C8 | — | 2 | 12 | no | — |
| 231 | C2×C8 | — | 142 | 12 | no | — |
| 232 | C23 | — | 2 | 6 | no | — |
| 233 | C23 | — | 2 | 12 | no | — |
| 234 | C23 | — | 284 | 24 | no | — |
| 235 | C23 | — | 284 | 48 | no | — |
| 236 | C23 | — | 71 | 6 | no | — |
| 237 | C23 | — | 71 | 6 | no | — |
| 238 | C23 | — | 142 | 24 | no | — |
| 239 | C23 | — | 142 | 6 | no | — |
| 240 | C23 | — | 142 | 12 | no | — |
| 241 | C23 | — | 142 | 4 | no | — |
| 242 | C23 | C22⋊C4×D71 | 1 | 6 | yes | — |
| 243 | C23 | C22×C4×D71 | 1 | 1 | yes | characteristic |
| 244 | Q8 | — | 2 | 8 | no | — |
| 245 | Q8 | — | 142 | 24 | no | — |
| 246 | D4 | — | 2 | 24 | no | — |
| 247 | D4 | — | 142 | 24 | no | — |
| 248 | D4 | — | 142 | 48 | no | — |
| 249 | C2×C4 | — | 2 | 6 | no | — |
| 250 | C2×C4 | — | 2 | 24 | no | — |
| 251 | C2×C4 | — | 71 | 6 | no | — |
| 252 | C2×C4 | — | 71 | 16 | no | — |
| 253 | C2×C4 | — | 71 | 6 | no | — |
| 254 | C2×C4 | — | 142 | 24 | no | — |
| 255 | C2×C4 | — | 142 | 24 | no | — |
| 256 | C2×C4 | — | 142 | 24 | no | — |
| 257 | C2×C4 | — | 142 | 6 | no | — |
| 258 | C2×C4 | — | 142 | 24 | no | — |
| 259 | C2×C4 | C22⋊C4×D71 | 1 | 6 | yes | — |
| 260 | C2×C4 | C2×D4×D71 | 1 | 6 | yes | — |
| 261 | C8 | — | 4 | 4 | no | — |
| 262 | C8 | — | 284 | 4 | no | — |
| 263 | C22 | — | 2 | 12 | no | — |
| 264 | C22 | — | 2 | 6 | no | — |
| 265 | C22 | — | 4 | 12 | no | — |
| 266 | C22 | — | 284 | 48 | no | — |
| 267 | C22 | — | 284 | 12 | no | — |
| 268 | C22 | — | 71 | 4 | no | — |
| 269 | C22 | — | 142 | 12 | no | — |
| 270 | C22 | — | 142 | 6 | no | — |
| 271 | C22 | — | 142 | 12 | no | — |
| 272 | C22 | C2×C22⋊C4×D71 | 1 | 3 | yes | — |
| 273 | C22 | #17986764 | 1 | 4 | yes | — |
| 274 | C4 | — | 2 | 6 | no | — |
| 275 | C4 | — | 71 | 4 | no | — |
| 276 | C4 | — | 142 | 6 | no | — |
| 277 | C4 | #120×D71 | 1 | 4 | yes | — |
| 278 | C2 | — | 4 | 6 | no | — |
| 279 | C2 | — | 284 | 6 | no | — |
| 280 | C2 | — | 142 | 4 | no | — |
| 281 | C2 | #379×D71 | 1 | 1 | yes | characteristic |
| 282 | C2 | #22596782 | 1 | 6 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#17985961 | C2 | semidirect |
| 3. #22588252 | C2 | semidirect |
| 4. #22588251 | C2 | semidirect |
| 5. C71×#1780 | C2 | semidirect |
| 15. C2×C22⋊(C71⋊C8) | C22 | semidirect |
| 16. C2×(C2×D71)⋊C8 | C22 | semidirect |
| 18. C2×C22⋊C8×C71 | C22 | semidirect |
| 146. #1780 | D71 | semidirect |
| 147. C71 | #20580 | semidirect |
| 160. C2×C22⋊C8 | C2×D71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41×D71 | 2 |
| C2×#17985961 | 2 |
| #22588252 | 2 |
| #22588251 | 2 |
| C71×#1780 | 2 |
| #20580 | 71 |