Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
834 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 834 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 | #22602217 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22589180 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22587078 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22586780 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C71×#1526 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #17986734 | — | 2 | 2 | no | — |
| 7 | #17986671 | — | 2 | 2 | no | — |
| 8 | #17986640 | — | 2 | 4 | no | — |
| 9 | #17986605 | — | 2 | 2 | no | — |
| 10 | #17986568 | C4 | 1 | 2 | yes | — |
| 11 | #17986550 | — | 2 | 2 | no | — |
| 12 | C2×#14556125 | C22 | 1 | 1 | yes | characteristic |
| 13 | #17986473 | C4 | 1 | 2 | yes | — |
| 14 | C4×#11765865 | C22 | 1 | 1 | yes | characteristic |
| 15 | #17986023 | C22 | 1 | 2 | yes | — |
| 16 | C2×#14556044 | — | 2 | 4 | no | — |
| 17 | #17985745 | — | 2 | 2 | no | — |
| 18 | #17985731 | — | 2 | 2 | no | — |
| 19 | #17985728 | — | 2 | 2 | no | — |
| 20 | #533×C71 | C22 | 1 | 1 | yes | characteristic |
| 21 | C4×M16×C71 | C22 | 1 | 2 | yes | — |
| 22 | #330×C71 | — | 2 | 2 | no | — |
| 23 | C2×#11765866 | — | 2 | 4 | no | — |
| 24 | C2×#11765865 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C22×D284 | C2×C4 | 1 | 2 | yes | — |
| 26 | C22×D284 | D4 | 1 | 4 | yes | — |
| 27 | C2×#11765849 | — | 2 | 2 | no | — |
| 28 | #14556125 | — | 2 | 2 | no | — |
| 29 | #14556125 | — | 2 | 1 | no | — |
| 30 | #14556125 | — | 2 | 2 | no | — |
| 31 | #14556125 | — | 2 | 1 | no | — |
| 32 | #14556117 | — | 2 | 4 | no | — |
| 33 | #14556116 | — | 2 | 2 | no | — |
| 34 | #14556113 | — | 2 | 8 | no | — |
| 35 | #14556103 | — | 2 | 8 | no | — |
| 36 | #14556103 | — | 2 | 4 | no | — |
| 37 | #14556103 | — | 2 | 4 | no | — |
| 38 | #14556098 | — | 2 | 4 | no | — |
| 39 | C4×Dic142 | D4 | 1 | 2 | yes | — |
| 40 | C4×D284 | — | 2 | 4 | no | — |
| 41 | C4×D284 | D4 | 1 | 2 | yes | — |
| 42 | #14556090 | — | 2 | 4 | no | — |
| 43 | #14556090 | — | 2 | 4 | no | — |
| 44 | C4×C22⋊D71 | — | 2 | 2 | no | — |
| 45 | #14556087 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | #14556085 | — | 2 | 1 | no | — |
| 47 | C22⋊C4×D71 | — | 2 | 4 | no | — |
| 48 | C42×D71 | — | 2 | 1 | no | — |
| 49 | C4⋊(C71⋊C8) | — | 2 | 2 | no | — |
| 50 | C4⋊(C71⋊C8) | — | 2 | 2 | no | — |
| 51 | #14556044 | — | 4 | 8 | no | — |
| 52 | C2×M16×C71 | — | 2 | 2 | no | — |
| 53 | C4⋊Q8×C71 | D4 | 1 | 2 | yes | — |
| 54 | #125×C71 | D4 | 1 | 2 | yes | — |
| 55 | C22⋊Q8×C71 | — | 2 | 2 | no | — |
| 56 | C4×Q8×C71 | — | 2 | 2 | no | — |
| 57 | #117×C71 | C2×C4 | 1 | 2 | yes | — |
| 58 | C2×C42×C71 | C23 | 1 | 1 | yes | characteristic |
| 59 | #97×C71 | — | 2 | 2 | no | — |
| 60 | C4×C8×C71 | — | 2 | 2 | no | — |
| 61 | C23×D71 | C22⋊C4 | 1 | 4 | yes | — |
| 62 | #11765866 | — | 8 | 4 | no | — |
| 63 | #11765866 | — | 8 | 4 | no | — |
| 64 | #11765865 | — | 4 | 2 | no | — |
| 65 | C2×Dic142 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 66 | C2×D284 | — | 2 | 2 | no | — |
| 67 | C2×D284 | — | 2 | 2 | no | — |
| 68 | C2×D284 | — | 2 | 4 | no | — |
| 69 | C2×D284 | — | 2 | 8 | no | — |
| 70 | C2×D284 | — | 2 | 4 | no | — |
| 71 | C2×D284 | — | 4 | 4 | no | — |
| 72 | C2×D284 | — | 4 | 4 | no | — |
| 73 | C2×D284 | — | 4 | 4 | no | — |
| 74 | C2×D284 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 75 | C2×C22⋊D71 | — | 2 | 1 | no | — |
| 76 | C2×C4×D71 | — | 2 | 1 | no | — |
| 77 | C2×C4×D71 | — | 2 | 2 | no | — |
| 78 | C2×C4×D71 | — | 2 | 8 | no | — |
| 79 | #11765849 | — | 4 | 4 | no | — |
| 80 | C2×C71⋊C8 | — | 4 | 2 | no | — |
| 81 | C4⋊Dic71 | C22⋊C4 | 1 | 2 | yes | — |
| 82 | Dic71⋊C4 | — | 2 | 2 | no | — |
| 83 | C22⋊Dic71 | — | 2 | 1 | no | — |
| 84 | (C2×D71)⋊C4 | — | 2 | 8 | no | — |
| 85 | (C2×D71)⋊C4 | — | 2 | 8 | no | — |
| 86 | (C2×D71)⋊C4 | — | 2 | 2 | no | — |
| 87 | C4×Dic71 | — | 2 | 1 | no | — |
| 88 | C4×Dic71 | — | 2 | 2 | no | — |
| 89 | C2×Q8×C71 | — | 2 | 2 | no | — |
| 90 | C22×C4×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C22×C4×C71 | C2×D4 | 1 | 2 | yes | — |
| 92 | M16×C71 | — | 2 | 4 | no | — |
| 93 | M16×C71 | — | 4 | 2 | no | — |
| 94 | C2×C8×C71 | — | 4 | 2 | no | — |
| 95 | C4⋊C4×C71 | — | 2 | 4 | no | — |
| 96 | C4⋊C4×C71 | — | 2 | 2 | no | — |
| 97 | C4⋊C4×C71 | C22⋊C4 | 1 | 4 | yes | — |
| 98 | C22⋊C4×C71 | — | 2 | 2 | no | — |
| 99 | C42×C71 | — | 2 | 2 | no | — |
| 100 | C42×C71 | C2×D4 | 1 | 2 | yes | — |
| 101 | C42×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C42×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C23×C71 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C22×D71 | — | 2 | 4 | no | — |
| 105 | C22×D71 | — | 2 | 1 | no | — |
| 106 | C22×D71 | — | 4 | 8 | no | — |
| 107 | C22×D71 | — | 4 | 4 | no | — |
| 108 | Q8×C71 | — | 4 | 2 | no | — |
| 109 | Q8×C71 | — | 4 | 2 | no | — |
| 110 | C2×C4×C71 | — | 2 | 2 | no | — |
| 111 | C2×C4×C71 | — | 2 | 2 | no | — |
| 112 | C2×C4×C71 | — | 2 | 2 | no | — |
| 113 | C2×C4×C71 | — | 2 | 4 | no | — |
| 114 | C2×C4×C71 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C71 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C71 | #120 | 1 | 2 | yes | — |
| 117 | C2×C4×C71 | #120 | 1 | 2 | yes | — |
| 118 | C22⋊D71 | — | 4 | 2 | no | — |
| 119 | C2×Dic71 | — | 2 | 1 | no | — |
| 120 | C2×Dic71 | — | 2 | 2 | no | — |
| 121 | D284 | — | 2 | 2 | no | — |
| 122 | D284 | — | 4 | 4 | no | — |
| 123 | D284 | — | 4 | 4 | no | — |
| 124 | D284 | — | 4 | 4 | no | — |
| 125 | D284 | — | 4 | 4 | no | — |
| 126 | D284 | — | 8 | 4 | no | — |
| 127 | D284 | — | 8 | 8 | no | — |
| 128 | C4×D71 | — | 4 | 2 | no | — |
| 129 | C4×D71 | — | 4 | 2 | no | — |
| 130 | C4×D71 | — | 4 | 2 | no | — |
| 131 | C4×D71 | — | 8 | 8 | no | — |
| 132 | Dic142 | — | 2 | 2 | no | — |
| 133 | C8×C71 | — | 4 | 4 | no | — |
| 134 | C71⋊C8 | — | 8 | 2 | no | — |
| 135 | C22×C71 | — | 2 | 2 | no | — |
| 136 | C22×C71 | — | 2 | 1 | no | — |
| 137 | C22×C71 | #379 | 1 | 1 | yes | characteristic |
| 138 | C2×D71 | — | 4 | 2 | no | — |
| 139 | C2×D71 | — | 4 | 8 | no | — |
| 140 | C2×D71 | — | 4 | 1 | no | — |
| 141 | C2×D71 | — | 4 | 4 | no | — |
| 142 | C2×D71 | — | 16 | 4 | no | — |
| 143 | C4×C71 | — | 2 | 2 | no | — |
| 144 | C4×C71 | — | 2 | 1 | no | — |
| 145 | C4×C71 | — | 4 | 4 | no | — |
| 146 | C4×C71 | #413 | 1 | 2 | yes | — |
| 147 | C4×C71 | #453 | 1 | 2 | yes | — |
| 148 | C4×C71 | #455 | 1 | 2 | yes | — |
| 149 | Dic71 | — | 4 | 2 | no | — |
| 150 | Dic71 | — | 4 | 1 | no | — |
| 151 | #8472 | — | 71 | 1 | no | maximal, Sylow |
| 152 | C2×C71 | — | 4 | 1 | no | — |
| 153 | C2×C71 | #1785 | 1 | 1 | yes | characteristic |
| 154 | C2×C71 | #1792 | 1 | 2 | yes | — |
| 155 | D71 | — | 8 | 1 | no | — |
| 156 | D71 | — | 8 | 4 | no | — |
| 157 | #3508 | — | 71 | 1 | no | — |
| 158 | #1965 | — | 71 | 2 | no | — |
| 159 | #1561 | — | 71 | 1 | no | — |
| 160 | #1526 | D71 | 1 | 1 | yes | characteristic, complemented |
| 161 | #1489 | — | 71 | 2 | no | — |
| 162 | C71 | #8472 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 163 | #559 | — | 142 | 2 | no | — |
| 164 | #550 | — | 142 | 2 | no | — |
| 165 | #546 | — | 142 | 4 | no | — |
| 166 | #540 | — | 142 | 2 | no | — |
| 167 | #535 | — | 71 | 2 | no | — |
| 168 | #533 | C2×D71 | 1 | 1 | yes | characteristic |
| 169 | #532 | — | 142 | 2 | no | — |
| 170 | C2×#127 | — | 71 | 1 | no | — |
| 171 | #518 | — | 71 | 2 | no | — |
| 172 | C4×#41 | — | 71 | 1 | no | — |
| 173 | #423 | — | 71 | 2 | no | — |
| 174 | C2×#100 | — | 142 | 4 | no | — |
| 175 | C4×M16 | C2×D71 | 1 | 2 | yes | — |
| 176 | #332 | — | 142 | 2 | no | — |
| 177 | #330 | — | 2 | 2 | no | — |
| 178 | Q8⋊C8 | — | 142 | 2 | no | — |
| 179 | D4⋊C8 | — | 142 | 2 | no | — |
| 180 | C2×#41 | — | 71 | 1 | no | — |
| 181 | C2×#41 | — | 142 | 4 | no | — |
| 182 | C22×D4 | — | 71 | 2 | no | — |
| 183 | C22×D4 | — | 71 | 4 | no | — |
| 184 | C2×M16 | — | 2 | 2 | no | — |
| 185 | C2×M16 | — | 142 | 2 | no | — |
| 186 | C4⋊Q8 | D284 | 1 | 2 | yes | — |
| 187 | #127 | — | 142 | 2 | no | — |
| 188 | #127 | — | 142 | 2 | no | — |
| 189 | #127 | — | 142 | 1 | no | — |
| 190 | #127 | — | 142 | 1 | no | — |
| 191 | #125 | C22⋊D71 | 1 | 2 | yes | — |
| 192 | #124 | — | 142 | 2 | no | — |
| 193 | #124 | — | 142 | 4 | no | — |
| 194 | #123 | — | 142 | 8 | no | — |
| 195 | C22⋊Q8 | — | 2 | 2 | no | — |
| 196 | #121 | — | 142 | 4 | no | — |
| 197 | #121 | — | 142 | 8 | no | — |
| 198 | #121 | — | 142 | 4 | no | — |
| 199 | #120 | — | 142 | 4 | no | — |
| 200 | C4×Q8 | — | 2 | 2 | no | — |
| 201 | C4×Q8 | — | 71 | 2 | no | — |
| 202 | C4×D4 | — | 71 | 2 | no | — |
| 203 | C4×D4 | — | 142 | 4 | no | — |
| 204 | C4×D4 | — | 142 | 4 | no | — |
| 205 | C4×D4 | — | 142 | 2 | no | — |
| 206 | C4×D4 | — | 142 | 4 | no | — |
| 207 | #117 | — | 71 | 1 | no | — |
| 208 | #117 | — | 142 | 1 | no | — |
| 209 | #117 | C4×D71 | 1 | 2 | yes | — |
| 210 | C2×C22⋊C4 | — | 142 | 4 | no | — |
| 211 | C2×C42 | — | 142 | 1 | no | — |
| 212 | C2×C42 | C22×D71 | 1 | 1 | yes | characteristic |
| 213 | C4⋊C8 | — | 142 | 2 | no | — |
| 214 | C4⋊C8 | — | 142 | 2 | no | — |
| 215 | #100 | — | 284 | 8 | no | — |
| 216 | #97 | — | 2 | 2 | no | — |
| 217 | C4×C8 | — | 2 | 2 | no | — |
| 218 | C24 | — | 71 | 4 | no | — |
| 219 | #41 | — | 284 | 2 | no | — |
| 220 | #41 | — | 568 | 4 | no | — |
| 221 | #41 | — | 568 | 4 | no | — |
| 222 | C2×Q8 | — | 2 | 2 | no | — |
| 223 | C2×Q8 | — | 71 | 1 | no | — |
| 224 | C2×D4 | — | 284 | 4 | no | — |
| 225 | C2×D4 | — | 284 | 4 | no | — |
| 226 | C2×D4 | — | 284 | 4 | no | — |
| 227 | C2×D4 | — | 71 | 1 | no | — |
| 228 | C2×D4 | — | 142 | 8 | no | — |
| 229 | C2×D4 | — | 142 | 1 | no | — |
| 230 | C2×D4 | — | 142 | 2 | no | — |
| 231 | C2×D4 | — | 142 | 2 | no | — |
| 232 | C2×D4 | — | 142 | 4 | no | — |
| 233 | C2×D4 | — | 142 | 4 | no | — |
| 234 | C22×C4 | — | 142 | 8 | no | — |
| 235 | C22×C4 | — | 142 | 1 | no | — |
| 236 | C22×C4 | — | 142 | 2 | no | — |
| 237 | C22×C4 | C2×C4×D71 | 1 | 1 | yes | characteristic |
| 238 | C22×C4 | D4×D71 | 1 | 2 | yes | — |
| 239 | M16 | — | 2 | 4 | no | — |
| 240 | M16 | — | 4 | 2 | no | — |
| 241 | M16 | — | 284 | 4 | no | — |
| 242 | C2×C8 | — | 4 | 2 | no | — |
| 243 | C2×C8 | — | 284 | 2 | no | — |
| 244 | C4⋊C4 | — | 2 | 4 | no | — |
| 245 | C4⋊C4 | — | 2 | 2 | no | — |
| 246 | C4⋊C4 | — | 71 | 2 | no | — |
| 247 | C4⋊C4 | — | 142 | 2 | no | — |
| 248 | C4⋊C4 | (C2×D71)⋊C4 | 1 | 4 | yes | — |
| 249 | C22⋊C4 | — | 2 | 2 | no | — |
| 250 | C22⋊C4 | — | 142 | 8 | no | — |
| 251 | C22⋊C4 | — | 142 | 8 | no | — |
| 252 | C22⋊C4 | — | 142 | 1 | no | — |
| 253 | C22⋊C4 | — | 142 | 2 | no | — |
| 254 | C42 | — | 2 | 2 | no | — |
| 255 | C42 | — | 142 | 1 | no | — |
| 256 | C42 | — | 142 | 2 | no | — |
| 257 | C42 | D4×D71 | 1 | 2 | yes | — |
| 258 | C42 | C2×C22⋊D71 | 1 | 1 | yes | characteristic |
| 259 | C42 | C2×D284 | 1 | 1 | yes | characteristic |
| 260 | C23 | — | 284 | 4 | no | — |
| 261 | C23 | — | 284 | 8 | no | — |
| 262 | C23 | — | 142 | 4 | no | — |
| 263 | C23 | — | 142 | 1 | no | — |
| 264 | C23 | C22⋊C4×D71 | 1 | 1 | yes | characteristic |
| 265 | Q8 | — | 4 | 2 | no | — |
| 266 | Q8 | — | 4 | 2 | no | — |
| 267 | Q8 | — | 142 | 2 | no | — |
| 268 | D4 | — | 284 | 4 | no | — |
| 269 | D4 | — | 284 | 2 | no | — |
| 270 | D4 | — | 284 | 4 | no | — |
| 271 | D4 | — | 284 | 4 | no | — |
| 272 | D4 | — | 284 | 4 | no | — |
| 273 | D4 | — | 568 | 4 | no | — |
| 274 | D4 | — | 568 | 8 | no | — |
| 275 | D4 | — | 142 | 2 | no | — |
| 276 | C2×C4 | — | 2 | 2 | no | — |
| 277 | C2×C4 | — | 2 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 2 | no | — |
| 279 | C2×C4 | — | 2 | 4 | no | — |
| 280 | C2×C4 | — | 284 | 2 | no | — |
| 281 | C2×C4 | — | 284 | 2 | no | — |
| 282 | C2×C4 | — | 284 | 2 | no | — |
| 283 | C2×C4 | — | 568 | 8 | no | — |
| 284 | C2×C4 | — | 142 | 1 | no | — |
| 285 | C2×C4 | — | 142 | 2 | no | — |
| 286 | C2×C4 | C22⋊C4×D71 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | C2×(C2×D71)⋊C4 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | #14556096 | 1 | 2 | yes | — |
| 289 | C2×C4 | #14556098 | 1 | 2 | yes | — |
| 290 | C8 | — | 4 | 4 | no | — |
| 291 | C8 | — | 568 | 2 | no | — |
| 292 | C22 | — | 2 | 1 | no | — |
| 293 | C22 | — | 2 | 2 | no | — |
| 294 | C22 | — | 284 | 1 | no | — |
| 295 | C22 | — | 284 | 4 | no | — |
| 296 | C22 | — | 284 | 2 | no | — |
| 297 | C22 | — | 284 | 8 | no | — |
| 298 | C22 | — | 1136 | 4 | no | — |
| 299 | C22 | #17985854 | 1 | 1 | yes | characteristic |
| 300 | C4 | — | 2 | 1 | no | — |
| 301 | C4 | — | 2 | 2 | no | — |
| 302 | C4 | — | 4 | 4 | no | — |
| 303 | C4 | — | 284 | 1 | no | — |
| 304 | C4 | — | 284 | 2 | no | — |
| 305 | C4 | #17985982 | 1 | 2 | yes | — |
| 306 | C4 | #17986168 | 1 | 2 | yes | — |
| 307 | C4 | #17986178 | 1 | 2 | yes | — |
| 308 | C2 | — | 4 | 1 | no | — |
| 309 | C2 | — | 568 | 4 | no | — |
| 310 | C2 | — | 568 | 1 | no | — |
| 311 | C2 | #22588267 | 1 | 1 | yes | characteristic |
| 312 | C2 | #22588295 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22589180 | C2 | semidirect |
| 3. #22587078 | C2 | semidirect |
| 4. #22586780 | C2 | semidirect |
| 5. C71×#1526 | C2 | semidirect |
| 160. #1526 | D71 | semidirect |
| 162. C71 | #8472 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22602217 | 2 |
| #22589180 | 2 |
| #22587078 | 2 |
| #22586780 | 2 |
| C71×#1526 | 2 |
| #8472 | 71 |