Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
1002 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 115 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 | #22601998 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22588252 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22586894 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C71×#1782 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#14556176 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #17986668 | — | 2 | 4 | no | — |
| 7 | #17986656 | — | 2 | 4 | no | — |
| 8 | #17986566 | — | 2 | 2 | no | — |
| 9 | #17986566 | C4 | 1 | 2 | yes | complemented |
| 10 | #17986510 | — | 2 | 2 | no | — |
| 11 | #17986510 | C22 | 1 | 2 | yes | — |
| 12 | #17986477 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×C22⋊(C71⋊C8) | — | 2 | 2 | no | — |
| 14 | C2×C22⋊(C71⋊C8) | C22 | 1 | 2 | yes | — |
| 15 | #17985735 | — | 2 | 4 | no | — |
| 16 | #17985732 | — | 2 | 4 | no | — |
| 17 | C22×#41×C71 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#99×C71 | — | 2 | 2 | no | — |
| 19 | #381×C71 | C22 | 1 | 2 | yes | — |
| 20 | #14556176 | — | 4 | 4 | no | — |
| 21 | C2×#11765866 | — | 2 | 2 | no | — |
| 22 | C2×#11765865 | — | 2 | 2 | no | — |
| 23 | C2×#11765865 | C2×C4 | 1 | 2 | yes | — |
| 24 | C2×Q8×D71 | — | 2 | 2 | no | — |
| 25 | C22×C71⋊C8 | — | 2 | 2 | no | — |
| 26 | #14556124 | — | 2 | 4 | no | — |
| 27 | #14556124 | — | 2 | 2 | no | — |
| 28 | #14556115 | — | 2 | 4 | no | — |
| 29 | #14556115 | — | 2 | 2 | no | — |
| 30 | #14556111 | — | 2 | 2 | no | — |
| 31 | #14556107 | — | 2 | 4 | no | — |
| 32 | #14556107 | D4 | 1 | 4 | yes | — |
| 33 | #14556102 | — | 2 | 4 | no | — |
| 34 | #14556102 | D4 | 1 | 4 | yes | — |
| 35 | #14556099 | — | 2 | 8 | no | — |
| 36 | #14556099 | — | 2 | 4 | no | — |
| 37 | #14556097 | — | 2 | 4 | no | — |
| 38 | #14556097 | — | 2 | 2 | no | — |
| 39 | Q8×Dic71 | — | 2 | 4 | no | — |
| 40 | C4×C22⋊D71 | — | 2 | 8 | no | — |
| 41 | C4×C22⋊D71 | — | 2 | 4 | no | — |
| 42 | #14556087 | — | 2 | 2 | no | — |
| 43 | #14556087 | C2×C4 | 1 | 2 | yes | — |
| 44 | C22⋊(C71⋊C8) | — | 2 | 4 | no | — |
| 45 | C22⋊(C71⋊C8) | — | 4 | 4 | no | — |
| 46 | C2×#41×C71 | — | 2 | 4 | no | — |
| 47 | C2×#41×C71 | — | 4 | 4 | no | — |
| 48 | C22×Q8×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C22×D4×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C22×D4×C71 | D4 | 1 | 2 | yes | — |
| 51 | C23×C4×C71 | D4 | 1 | 2 | yes | — |
| 52 | C23×C4×C71 | C23 | 1 | 1 | yes | characteristic |
| 53 | C2×C22⋊C4×C71 | — | 2 | 2 | no | — |
| 54 | #99×C71 | — | 4 | 4 | no | — |
| 55 | #95×C71 | — | 2 | 4 | no | — |
| 56 | #11765866 | — | 4 | 8 | no | — |
| 57 | #11765865 | — | 4 | 4 | no | — |
| 58 | #11765865 | — | 4 | 4 | no | — |
| 59 | #11765865 | — | 4 | 4 | no | — |
| 60 | C2×Dic142 | — | 2 | 2 | no | — |
| 61 | C2×Dic142 | C22⋊C4 | 1 | 2 | yes | — |
| 62 | Q8×D71 | — | 4 | 8 | no | — |
| 63 | C2×D284 | — | 2 | 2 | no | — |
| 64 | C2×D284 | C22⋊C4 | 1 | 2 | yes | — |
| 65 | C2×C22⋊D71 | — | 2 | 2 | no | — |
| 66 | C2×C22⋊D71 | — | 2 | 8 | no | — |
| 67 | C2×C22⋊D71 | — | 2 | 4 | no | — |
| 68 | C2×C4×D71 | — | 2 | 4 | no | — |
| 69 | C2×C4×D71 | — | 2 | 2 | no | — |
| 70 | C2×C71⋊C8 | — | 4 | 4 | no | — |
| 71 | C2×C71⋊C8 | — | 4 | 2 | no | — |
| 72 | C4⋊Dic71 | — | 2 | 4 | no | — |
| 73 | C4⋊Dic71 | C22⋊C4 | 1 | 4 | yes | — |
| 74 | Dic71⋊C4 | — | 2 | 4 | no | — |
| 75 | C22⋊Dic71 | — | 2 | 8 | no | — |
| 76 | C22⋊Dic71 | — | 2 | 2 | no | — |
| 77 | C22⋊Dic71 | — | 2 | 4 | no | — |
| 78 | (C2×D71)⋊C4 | — | 2 | 4 | no | — |
| 79 | C4×Dic71 | — | 2 | 4 | no | — |
| 80 | C4×Dic71 | — | 2 | 2 | no | — |
| 81 | C24×C71 | C22⋊C4 | 1 | 2 | yes | — |
| 82 | C24×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | #41×C71 | — | 4 | 16 | no | — |
| 84 | C2×Q8×C71 | — | 2 | 4 | no | — |
| 85 | C2×Q8×C71 | — | 2 | 2 | no | — |
| 86 | C2×D4×C71 | — | 2 | 4 | no | — |
| 87 | C2×D4×C71 | — | 2 | 4 | no | — |
| 88 | C2×D4×C71 | — | 2 | 2 | no | — |
| 89 | C2×D4×C71 | — | 4 | 2 | no | — |
| 90 | C2×D4×C71 | — | 4 | 2 | no | — |
| 91 | C22×C4×C71 | — | 2 | 4 | no | — |
| 92 | C22×C4×C71 | — | 2 | 4 | no | — |
| 93 | C22×C4×C71 | — | 4 | 2 | no | — |
| 94 | C22×C4×C71 | — | 4 | 2 | no | — |
| 95 | C22×C4×C71 | — | 4 | 2 | no | — |
| 96 | C22×C4×C71 | C22⋊C4 | 1 | 2 | yes | — |
| 97 | C22×C4×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 98 | C22×C4×C71 | C2×D4 | 1 | 4 | yes | — |
| 99 | C22×C4×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C22⋊C4×C71 | — | 4 | 4 | no | — |
| 101 | C23×C71 | — | 2 | 2 | no | — |
| 102 | C23×C71 | — | 2 | 2 | no | — |
| 103 | C23×C71 | — | 2 | 2 | no | — |
| 104 | C23×C71 | — | 2 | 2 | no | — |
| 105 | C23×C71 | — | 4 | 2 | no | — |
| 106 | C23×C71 | — | 4 | 4 | no | — |
| 107 | C23×C71 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 108 | C23×C71 | #120 | 1 | 2 | yes | — |
| 109 | C22×D71 | — | 2 | 2 | no | — |
| 110 | Q8×C71 | — | 2 | 8 | no | — |
| 111 | D4×C71 | — | 4 | 8 | no | — |
| 112 | D4×C71 | — | 4 | 4 | no | — |
| 113 | C2×C4×C71 | — | 2 | 2 | no | — |
| 114 | C2×C4×C71 | — | 2 | 2 | no | — |
| 115 | C2×C4×C71 | — | 2 | 2 | no | — |
| 116 | C2×C4×C71 | — | 2 | 2 | no | — |
| 117 | C2×C4×C71 | — | 4 | 4 | no | — |
| 118 | C2×C4×C71 | — | 4 | 8 | no | — |
| 119 | C2×C4×C71 | — | 8 | 2 | no | — |
| 120 | C2×C4×C71 | — | 8 | 4 | no | — |
| 121 | C2×C4×C71 | #99 | 1 | 4 | yes | — |
| 122 | C2×C4×C71 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 123 | C2×C4×C71 | #120 | 1 | 2 | yes | — |
| 124 | C22⋊D71 | — | 4 | 4 | no | — |
| 125 | C22⋊D71 | — | 8 | 4 | no | — |
| 126 | C22⋊D71 | — | 8 | 8 | no | — |
| 127 | C2×Dic71 | — | 2 | 4 | no | — |
| 128 | C2×Dic71 | — | 2 | 2 | no | — |
| 129 | D284 | — | 4 | 2 | no | — |
| 130 | D284 | — | 4 | 4 | no | — |
| 131 | C4×D71 | — | 4 | 8 | no | — |
| 132 | C4×D71 | — | 4 | 4 | no | — |
| 133 | Dic142 | — | 4 | 4 | no | — |
| 134 | Dic142 | — | 4 | 2 | no | — |
| 135 | C71⋊C8 | — | 8 | 2 | no | — |
| 136 | C22×C71 | — | 2 | 2 | no | — |
| 137 | C22×C71 | — | 2 | 4 | no | — |
| 138 | C22×C71 | — | 2 | 2 | no | — |
| 139 | C22×C71 | — | 4 | 4 | no | — |
| 140 | C22×C71 | — | 4 | 4 | no | — |
| 141 | C22×C71 | — | 4 | 2 | no | — |
| 142 | C22×C71 | — | 8 | 2 | no | — |
| 143 | C22×C71 | — | 8 | 2 | no | — |
| 144 | C22×C71 | #379 | 1 | 1 | yes | characteristic |
| 145 | C22×C71 | C2×#99 | 1 | 2 | yes | — |
| 146 | C2×D71 | — | 4 | 4 | no | — |
| 147 | C2×D71 | — | 4 | 2 | no | — |
| 148 | C4×C71 | — | 2 | 2 | no | — |
| 149 | C4×C71 | — | 2 | 4 | no | — |
| 150 | C4×C71 | — | 2 | 2 | no | — |
| 151 | C4×C71 | — | 16 | 2 | no | — |
| 152 | Dic71 | — | 4 | 4 | no | — |
| 153 | Dic71 | — | 4 | 2 | no | — |
| 154 | #8464 | — | 71 | 1 | no | maximal, Sylow |
| 155 | C2×C71 | — | 2 | 2 | no | — |
| 156 | C2×C71 | — | 4 | 2 | no | — |
| 157 | C2×C71 | — | 8 | 2 | no | — |
| 158 | C2×C71 | #1781 | 1 | 1 | yes | characteristic |
| 159 | C2×C71 | #1794 | 1 | 2 | yes | — |
| 160 | D71 | — | 8 | 2 | no | — |
| 161 | #3487 | — | 71 | 1 | no | — |
| 162 | #1782 | D71 | 1 | 1 | yes | characteristic, complemented |
| 163 | #1780 | — | 71 | 1 | no | — |
| 164 | #1516 | — | 71 | 4 | no | — |
| 165 | C71 | #8464 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 166 | C2×2−1+4 | — | 71 | 1 | no | — |
| 167 | C22×#41 | C2×D71 | 1 | 1 | yes | characteristic |
| 168 | #550 | — | 142 | 4 | no | — |
| 169 | #548 | — | 142 | 4 | no | — |
| 170 | #535 | — | 71 | 2 | no | — |
| 171 | #535 | — | 142 | 2 | no | — |
| 172 | #525 | — | 71 | 2 | no | — |
| 173 | #525 | — | 142 | 2 | no | — |
| 174 | #519 | — | 71 | 1 | no | — |
| 175 | C2×#99 | — | 2 | 2 | no | — |
| 176 | C2×C22⋊C8 | — | 71 | 2 | no | — |
| 177 | C2×C22⋊C8 | — | 142 | 2 | no | — |
| 178 | #381 | C2×D71 | 1 | 2 | yes | — |
| 179 | #328 | — | 142 | 4 | no | — |
| 180 | #327 | — | 142 | 4 | no | — |
| 181 | 2−1+4 | — | 284 | 4 | no | — |
| 182 | C2×#41 | — | 2 | 4 | no | — |
| 183 | C2×#41 | — | 4 | 4 | no | — |
| 184 | C2×#41 | — | 71 | 2 | no | — |
| 185 | C2×#41 | — | 142 | 2 | no | — |
| 186 | C2×#41 | — | 142 | 2 | no | — |
| 187 | C22×Q8 | — | 142 | 2 | no | — |
| 188 | C22×Q8 | C4×D71 | 1 | 1 | yes | characteristic |
| 189 | C22×D4 | C4×D71 | 1 | 1 | yes | characteristic |
| 190 | C22×D4 | C22⋊D71 | 1 | 2 | yes | — |
| 191 | C23×C4 | D284 | 1 | 2 | yes | — |
| 192 | C23×C4 | C22×D71 | 1 | 1 | yes | characteristic |
| 193 | C22×C8 | — | 142 | 2 | no | — |
| 194 | #127 | — | 142 | 2 | no | — |
| 195 | #127 | — | 142 | 4 | no | — |
| 196 | #124 | — | 142 | 2 | no | — |
| 197 | #124 | — | 142 | 4 | no | — |
| 198 | #123 | — | 142 | 2 | no | — |
| 199 | C22⋊Q8 | — | 71 | 4 | no | — |
| 200 | C22⋊Q8 | — | 142 | 4 | no | — |
| 201 | #121 | — | 71 | 4 | no | — |
| 202 | #121 | — | 142 | 4 | no | — |
| 203 | #121 | — | 142 | 4 | no | — |
| 204 | #121 | — | 142 | 8 | no | — |
| 205 | #120 | — | 142 | 4 | no | — |
| 206 | #120 | — | 142 | 2 | no | — |
| 207 | C4×Q8 | — | 142 | 4 | no | — |
| 208 | C4×D4 | — | 142 | 8 | no | — |
| 209 | C4×D4 | — | 142 | 4 | no | — |
| 210 | #117 | — | 71 | 2 | no | — |
| 211 | #117 | — | 142 | 2 | no | — |
| 212 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 213 | #99 | — | 4 | 4 | no | — |
| 214 | C22⋊C8 | — | 284 | 4 | no | — |
| 215 | C22⋊C8 | — | 142 | 4 | no | — |
| 216 | #95 | — | 2 | 4 | no | — |
| 217 | C24 | (C2×D71)⋊C4 | 1 | 2 | yes | — |
| 218 | C24 | C2×C22⋊D71 | 1 | 1 | yes | characteristic |
| 219 | #41 | — | 4 | 16 | no | — |
| 220 | #41 | — | 284 | 4 | no | — |
| 221 | #41 | — | 284 | 4 | no | — |
| 222 | #41 | — | 284 | 8 | no | — |
| 223 | #41 | — | 284 | 4 | no | — |
| 224 | C2×Q8 | — | 2 | 4 | no | — |
| 225 | C2×Q8 | — | 2 | 2 | no | — |
| 226 | C2×Q8 | — | 284 | 8 | no | — |
| 227 | C2×Q8 | — | 71 | 2 | no | — |
| 228 | C2×Q8 | — | 142 | 2 | no | — |
| 229 | C2×D4 | — | 2 | 4 | no | — |
| 230 | C2×D4 | — | 2 | 4 | no | — |
| 231 | C2×D4 | — | 2 | 2 | no | — |
| 232 | C2×D4 | — | 4 | 2 | no | — |
| 233 | C2×D4 | — | 4 | 2 | no | — |
| 234 | C2×D4 | — | 71 | 2 | no | — |
| 235 | C2×D4 | — | 142 | 2 | no | — |
| 236 | C2×D4 | — | 142 | 2 | no | — |
| 237 | C2×D4 | — | 142 | 8 | no | — |
| 238 | C2×D4 | — | 142 | 4 | no | — |
| 239 | C22×C4 | — | 2 | 4 | no | — |
| 240 | C22×C4 | — | 2 | 4 | no | — |
| 241 | C22×C4 | — | 4 | 2 | no | — |
| 242 | C22×C4 | — | 4 | 2 | no | — |
| 243 | C22×C4 | — | 4 | 2 | no | — |
| 244 | C22×C4 | — | 142 | 4 | no | — |
| 245 | C22×C4 | — | 142 | 2 | no | — |
| 246 | C22×C4 | (C2×D71)⋊C4 | 1 | 2 | yes | — |
| 247 | C22×C4 | C2×C4×D71 | 1 | 1 | yes | characteristic |
| 248 | C22×C4 | D4×D71 | 1 | 4 | yes | — |
| 249 | C22×C4 | C2×D284 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | — | 284 | 4 | no | — |
| 251 | C2×C8 | — | 284 | 2 | no | — |
| 252 | C4⋊C4 | — | 71 | 4 | no | — |
| 253 | C4⋊C4 | — | 142 | 4 | no | — |
| 254 | C4⋊C4 | — | 142 | 4 | no | — |
| 255 | C22⋊C4 | — | 4 | 4 | no | — |
| 256 | C22⋊C4 | — | 142 | 8 | no | — |
| 257 | C22⋊C4 | — | 142 | 2 | no | — |
| 258 | C22⋊C4 | — | 142 | 4 | no | — |
| 259 | C22⋊C4 | — | 142 | 4 | no | — |
| 260 | C42 | — | 142 | 4 | no | — |
| 261 | C42 | — | 142 | 2 | no | — |
| 262 | C23 | — | 2 | 2 | no | — |
| 263 | C23 | — | 2 | 2 | no | — |
| 264 | C23 | — | 2 | 2 | no | — |
| 265 | C23 | — | 2 | 2 | no | — |
| 266 | C23 | — | 4 | 2 | no | — |
| 267 | C23 | — | 4 | 4 | no | — |
| 268 | C23 | — | 142 | 2 | no | — |
| 269 | C23 | C2×(C2×D71)⋊C4 | 1 | 1 | yes | characteristic |
| 270 | C23 | #14556096 | 1 | 2 | yes | — |
| 271 | Q8 | — | 2 | 8 | no | — |
| 272 | Q8 | — | 284 | 2 | no | — |
| 273 | Q8 | — | 284 | 4 | no | — |
| 274 | D4 | — | 4 | 8 | no | — |
| 275 | D4 | — | 4 | 4 | no | — |
| 276 | D4 | — | 284 | 2 | no | — |
| 277 | D4 | — | 284 | 4 | no | — |
| 278 | D4 | — | 284 | 4 | no | — |
| 279 | D4 | — | 568 | 4 | no | — |
| 280 | D4 | — | 568 | 8 | no | — |
| 281 | C2×C4 | — | 2 | 2 | no | — |
| 282 | C2×C4 | — | 2 | 2 | no | — |
| 283 | C2×C4 | — | 2 | 2 | no | — |
| 284 | C2×C4 | — | 2 | 2 | no | — |
| 285 | C2×C4 | — | 4 | 4 | no | — |
| 286 | C2×C4 | — | 4 | 8 | no | — |
| 287 | C2×C4 | — | 8 | 2 | no | — |
| 288 | C2×C4 | — | 8 | 4 | no | — |
| 289 | C2×C4 | — | 284 | 8 | no | — |
| 290 | C2×C4 | — | 284 | 4 | no | — |
| 291 | C2×C4 | — | 142 | 4 | no | — |
| 292 | C2×C4 | — | 142 | 2 | no | — |
| 293 | C2×C4 | #14556042 | 1 | 4 | yes | — |
| 294 | C2×C4 | C22⋊C4×D71 | 1 | 2 | yes | — |
| 295 | C2×C4 | #14556098 | 1 | 2 | yes | — |
| 296 | C8 | — | 568 | 2 | no | — |
| 297 | C22 | — | 2 | 2 | no | — |
| 298 | C22 | — | 2 | 2 | no | — |
| 299 | C22 | — | 2 | 4 | no | — |
| 300 | C22 | — | 4 | 4 | no | — |
| 301 | C22 | — | 4 | 2 | no | — |
| 302 | C22 | — | 4 | 4 | no | — |
| 303 | C22 | — | 8 | 2 | no | — |
| 304 | C22 | — | 8 | 2 | no | — |
| 305 | C22 | — | 284 | 2 | no | — |
| 306 | C22 | — | 284 | 4 | no | — |
| 307 | C22 | #17985854 | 1 | 1 | yes | characteristic |
| 308 | C22 | C2×#14556042 | 1 | 2 | yes | — |
| 309 | C4 | — | 2 | 2 | no | — |
| 310 | C4 | — | 2 | 4 | no | — |
| 311 | C4 | — | 2 | 2 | no | — |
| 312 | C4 | — | 16 | 2 | no | — |
| 313 | C4 | — | 284 | 2 | no | — |
| 314 | C4 | — | 284 | 4 | no | — |
| 315 | C2 | — | 2 | 2 | no | — |
| 316 | C2 | — | 4 | 2 | no | — |
| 317 | C2 | — | 8 | 2 | no | — |
| 318 | C2 | — | 568 | 2 | no | — |
| 319 | C2 | #22588254 | 1 | 1 | yes | characteristic |
| 320 | C2 | #22588306 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22588252 | C2 | semidirect |
| 3. #22586894 | C2 | semidirect |
| 4. C71×#1782 | C2 | semidirect |
| 5. C2×#14556176 | C4 | semidirect |
| 9. #17986566 | C4 | semidirect |
| 12. #17986477 | C4 | semidirect |
| 162. #1782 | D71 | semidirect |
| 165. C71 | #8464 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22601998 | 2 |
| #22588252 | 2 |
| #22586894 | 2 |
| C71×#1782 | 2 |
| #8464 | 71 |