Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
906 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 139 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 | #22603030 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22599359 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #22587642 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22587635 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #22587632 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #22586970 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C71×#1658 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #17986747 | C22 | 1 | 3 | yes | complemented |
| 9 | #17986702 | — | 2 | 3 | no | — |
| 10 | #17986681 | — | 2 | 3 | no | — |
| 11 | #17986605 | — | 2 | 3 | no | — |
| 12 | #17986597 | — | 2 | 3 | no | — |
| 13 | #127×D71 | C22 | 1 | 1 | yes | characteristic |
| 14 | #17986534 | C22 | 1 | 3 | yes | — |
| 15 | #17986388 | C22 | 1 | 3 | yes | — |
| 16 | #17986386 | C22 | 1 | 3 | yes | — |
| 17 | #17986365 | C22 | 1 | 3 | yes | — |
| 18 | #17986198 | — | 2 | 3 | no | — |
| 19 | #17986193 | — | 2 | 3 | no | — |
| 20 | #17986151 | — | 2 | 3 | no | — |
| 21 | #17986134 | — | 2 | 3 | no | — |
| 22 | #17986054 | C22 | 1 | 3 | yes | — |
| 23 | #17985741 | C22 | 1 | 3 | yes | complemented |
| 24 | #17985738 | C22 | 1 | 6 | yes | complemented |
| 25 | #564×C71 | C22 | 1 | 1 | yes | characteristic, complemented |
| 26 | #490×C71 | C22 | 1 | 3 | yes | — |
| 27 | #330×C71 | C22 | 1 | 3 | yes | complemented |
| 28 | C2×D4×D71 | — | 2 | 3 | no | — |
| 29 | C2×#11765854 | — | 2 | 3 | no | — |
| 30 | C2×#11765853 | — | 2 | 3 | no | — |
| 31 | C2×D568 | — | 2 | 3 | no | — |
| 32 | C2×#11765850 | — | 2 | 3 | no | — |
| 33 | #14556125 | — | 2 | 1 | no | — |
| 34 | #14556125 | C23 | 1 | 1 | yes | characteristic |
| 35 | #14556124 | — | 2 | 3 | no | — |
| 36 | #14556123 | — | 2 | 3 | no | — |
| 37 | #14556123 | — | 2 | 3 | no | — |
| 38 | #14556122 | — | 2 | 1 | no | — |
| 39 | #14556118 | — | 2 | 3 | no | — |
| 40 | #14556116 | C23 | 1 | 3 | yes | — |
| 41 | #14556113 | — | 2 | 3 | no | — |
| 42 | #14556113 | — | 2 | 3 | no | — |
| 43 | #14556113 | — | 2 | 3 | no | — |
| 44 | #14556113 | — | 2 | 3 | no | — |
| 45 | #14556113 | — | 2 | 3 | no | — |
| 46 | #14556108 | D4 | 1 | 3 | yes | — |
| 47 | #14556106 | — | 2 | 3 | no | — |
| 48 | #14556106 | — | 2 | 3 | no | — |
| 49 | #14556103 | — | 2 | 3 | no | — |
| 50 | #14556103 | — | 2 | 3 | no | — |
| 51 | #14556103 | — | 2 | 3 | no | — |
| 52 | #14556103 | — | 2 | 3 | no | — |
| 53 | #14556103 | D4 | 1 | 3 | yes | — |
| 54 | #14556093 | D4 | 1 | 3 | yes | — |
| 55 | C4×D284 | — | 2 | 3 | no | — |
| 56 | #14556090 | — | 2 | 3 | no | — |
| 57 | #14556090 | — | 2 | 3 | no | — |
| 58 | #14556090 | D4 | 1 | 3 | yes | — |
| 59 | #14556086 | — | 2 | 3 | no | — |
| 60 | #14556085 | — | 2 | 3 | no | — |
| 61 | C4⋊C4×D71 | — | 2 | 3 | no | — |
| 62 | C42×D71 | C23 | 1 | 1 | yes | characteristic |
| 63 | Dic71⋊C8 | — | 2 | 3 | no | — |
| 64 | #14556048 | — | 2 | 3 | no | — |
| 65 | #14556048 | — | 2 | 3 | no | — |
| 66 | (C2×D71)⋊C8 | — | 2 | 3 | no | — |
| 67 | #14556039 | C23 | 1 | 3 | yes | — |
| 68 | #127×C71 | C23 | 1 | 1 | yes | characteristic |
| 69 | #125×C71 | — | 2 | 3 | no | — |
| 70 | #125×C71 | — | 2 | 3 | no | — |
| 71 | #125×C71 | C23 | 1 | 3 | yes | — |
| 72 | #102×C71 | — | 2 | 3 | no | — |
| 73 | #102×C71 | — | 2 | 3 | no | — |
| 74 | #97×C71 | C23 | 1 | 3 | yes | — |
| 75 | C23×D71 | — | 4 | 1 | no | — |
| 76 | C2×Dic142 | C2×D4 | 1 | 3 | yes | — |
| 77 | C2×D284 | — | 2 | 3 | no | — |
| 78 | C2×D284 | — | 2 | 3 | no | — |
| 79 | C2×D284 | — | 2 | 3 | no | — |
| 80 | C2×D284 | — | 2 | 3 | no | — |
| 81 | C2×D284 | C2×D4 | 1 | 3 | yes | — |
| 82 | C2×C22⋊D71 | — | 4 | 3 | no | — |
| 83 | D4×D71 | — | 8 | 3 | no | — |
| 84 | D4×D71 | — | 8 | 3 | no | — |
| 85 | C2×C4×D71 | — | 2 | 3 | no | — |
| 86 | C2×C4×D71 | — | 2 | 3 | no | — |
| 87 | C2×C4×D71 | — | 2 | 3 | no | — |
| 88 | C2×C4×D71 | C2×D4 | 1 | 3 | yes | — |
| 89 | #11765854 | — | 8 | 3 | no | — |
| 90 | #11765853 | — | 8 | 3 | no | — |
| 91 | D568 | — | 8 | 3 | no | — |
| 92 | #11765850 | — | 8 | 3 | no | — |
| 93 | C2×C71⋊C8 | — | 2 | 3 | no | — |
| 94 | C4⋊Dic71 | — | 2 | 3 | no | — |
| 95 | Dic71⋊C4 | — | 2 | 3 | no | — |
| 96 | Dic71⋊C4 | — | 2 | 3 | no | — |
| 97 | Dic71⋊C4 | — | 2 | 3 | no | — |
| 98 | (C2×D71)⋊C4 | — | 2 | 3 | no | — |
| 99 | (C2×D71)⋊C4 | — | 2 | 3 | no | — |
| 100 | (C2×D71)⋊C4 | — | 2 | 3 | no | — |
| 101 | (C2×D71)⋊C4 | — | 2 | 3 | no | — |
| 102 | (C2×D71)⋊C4 | — | 2 | 3 | no | — |
| 103 | (C2×D71)⋊C4 | — | 2 | 3 | no | — |
| 104 | (C2×D71)⋊C4 | — | 2 | 3 | no | — |
| 105 | C4×Dic71 | — | 2 | 3 | no | — |
| 106 | C4×Dic71 | C2×D4 | 1 | 3 | yes | — |
| 107 | C2×D4×C71 | — | 2 | 3 | no | — |
| 108 | C2×C8×C71 | — | 2 | 3 | no | — |
| 109 | C4⋊C4×C71 | — | 2 | 3 | no | — |
| 110 | C4⋊C4×C71 | — | 2 | 3 | no | — |
| 111 | C4⋊C4×C71 | — | 2 | 3 | no | — |
| 112 | C4⋊C4×C71 | — | 2 | 3 | no | — |
| 113 | C4⋊C4×C71 | C2×D4 | 1 | 3 | yes | — |
| 114 | C4⋊C4×C71 | C2×D4 | 1 | 3 | yes | — |
| 115 | C42×C71 | — | 2 | 1 | no | — |
| 116 | C42×C71 | — | 2 | 1 | no | — |
| 117 | C42×C71 | C24 | 1 | 1 | yes | characteristic |
| 118 | C23×C71 | — | 4 | 1 | no | — |
| 119 | C22×D71 | — | 2 | 3 | no | — |
| 120 | C22×D71 | — | 4 | 1 | no | — |
| 121 | C22×D71 | — | 8 | 6 | no | — |
| 122 | C22×D71 | #120 | 1 | 1 | yes | characteristic |
| 123 | D4×C71 | — | 4 | 3 | no | — |
| 124 | D4×C71 | — | 4 | 3 | no | — |
| 125 | C2×C4×C71 | — | 2 | 3 | no | — |
| 126 | C2×C4×C71 | — | 2 | 3 | no | — |
| 127 | C2×C4×C71 | C22×D4 | 1 | 3 | yes | — |
| 128 | C22⋊D71 | — | 8 | 6 | no | — |
| 129 | C2×Dic71 | — | 2 | 3 | no | — |
| 130 | C2×Dic71 | #120 | 1 | 3 | yes | — |
| 131 | D284 | — | 4 | 3 | no | — |
| 132 | D284 | — | 4 | 3 | no | — |
| 133 | D284 | — | 4 | 3 | no | — |
| 134 | D284 | — | 8 | 3 | no | — |
| 135 | D284 | — | 8 | 3 | no | — |
| 136 | D284 | — | 8 | 3 | no | — |
| 137 | C4×D71 | — | 4 | 3 | no | — |
| 138 | C4×D71 | — | 8 | 3 | no | — |
| 139 | C4×D71 | — | 8 | 3 | no | — |
| 140 | C4×D71 | — | 8 | 3 | no | — |
| 141 | Dic142 | — | 4 | 3 | no | — |
| 142 | C8×C71 | — | 4 | 3 | no | — |
| 143 | C71⋊C8 | — | 4 | 3 | no | — |
| 144 | C22×C71 | — | 8 | 3 | no | — |
| 145 | C22×C71 | C2×#120 | 1 | 1 | yes | characteristic |
| 146 | C2×D71 | — | 2 | 3 | no | — |
| 147 | C2×D71 | — | 4 | 3 | no | — |
| 148 | C2×D71 | — | 4 | 3 | no | — |
| 149 | C2×D71 | — | 4 | 3 | no | — |
| 150 | C2×D71 | — | 8 | 3 | no | — |
| 151 | C2×D71 | — | 16 | 4 | no | — |
| 152 | C4×C71 | — | 2 | 3 | no | — |
| 153 | C4×C71 | — | 4 | 3 | no | — |
| 154 | C4×C71 | — | 4 | 3 | no | — |
| 155 | Dic71 | — | 2 | 3 | no | — |
| 156 | Dic71 | — | 4 | 3 | no | — |
| 157 | #19096 | — | 71 | 1 | no | maximal, Sylow |
| 158 | C2×C71 | — | 16 | 1 | no | — |
| 159 | C2×C71 | #3021 | 1 | 3 | yes | — |
| 160 | D71 | — | 4 | 1 | no | — |
| 161 | D71 | — | 8 | 3 | no | — |
| 162 | D71 | — | 16 | 1 | no | — |
| 163 | #3567 | — | 71 | 1 | no | — |
| 164 | #3217 | — | 71 | 3 | no | — |
| 165 | #1658 | — | 71 | 1 | no | — |
| 166 | #1658 | D71 | 1 | 1 | yes | characteristic, complemented |
| 167 | #1656 | — | 71 | 3 | no | — |
| 168 | #1656 | — | 71 | 3 | no | — |
| 169 | #1535 | — | 71 | 3 | no | — |
| 170 | C71 | #19096 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 171 | #564 | C2×D71 | 1 | 1 | yes | characteristic, complemented |
| 172 | #562 | — | 71 | 3 | no | — |
| 173 | #553 | — | 142 | 3 | no | — |
| 174 | #552 | — | 142 | 3 | no | — |
| 175 | #540 | — | 142 | 3 | no | — |
| 176 | #539 | — | 142 | 3 | no | — |
| 177 | C2×#127 | — | 71 | 1 | no | — |
| 178 | #529 | — | 71 | 3 | no | — |
| 179 | #496 | — | 71 | 3 | no | — |
| 180 | #496 | — | 71 | 3 | no | — |
| 181 | #490 | — | 71 | 3 | no | — |
| 182 | #490 | C2×D71 | 1 | 3 | yes | — |
| 183 | #460 | — | 142 | 3 | no | — |
| 184 | #459 | — | 142 | 3 | no | — |
| 185 | #450 | — | 142 | 3 | no | — |
| 186 | #447 | — | 142 | 3 | no | — |
| 187 | #432 | — | 71 | 3 | no | — |
| 188 | #330 | — | 71 | 3 | no | — |
| 189 | #330 | C2×D71 | 1 | 3 | yes | complemented |
| 190 | #329 | — | 71 | 6 | no | — |
| 191 | C22×D4 | — | 142 | 3 | no | — |
| 192 | C2×SD16 | — | 142 | 3 | no | — |
| 193 | C2×SD16 | — | 142 | 3 | no | — |
| 194 | C2×D8 | — | 142 | 3 | no | — |
| 195 | C2×D8 | — | 142 | 3 | no | — |
| 196 | #127 | — | 71 | 1 | no | — |
| 197 | #127 | — | 142 | 1 | no | — |
| 198 | #127 | — | 142 | 3 | no | — |
| 199 | #127 | C22×D71 | 1 | 1 | yes | characteristic |
| 200 | #126 | — | 142 | 1 | no | — |
| 201 | #126 | — | 142 | 3 | no | — |
| 202 | #126 | — | 142 | 3 | no | — |
| 203 | #125 | — | 2 | 3 | no | — |
| 204 | #125 | — | 2 | 3 | no | — |
| 205 | #125 | — | 142 | 3 | no | — |
| 206 | #125 | C22×D71 | 1 | 3 | yes | — |
| 207 | #124 | — | 71 | 3 | no | — |
| 208 | #123 | — | 142 | 3 | no | — |
| 209 | #123 | — | 142 | 3 | no | — |
| 210 | #123 | — | 142 | 3 | no | — |
| 211 | #123 | — | 142 | 3 | no | — |
| 212 | #123 | — | 142 | 3 | no | — |
| 213 | C22⋊Q8 | — | 71 | 3 | no | — |
| 214 | C22⋊Q8 | — | 142 | 3 | no | — |
| 215 | C22⋊Q8 | — | 142 | 3 | no | — |
| 216 | #121 | — | 71 | 3 | no | — |
| 217 | #121 | — | 142 | 3 | no | — |
| 218 | #121 | — | 142 | 3 | no | — |
| 219 | #121 | — | 142 | 3 | no | — |
| 220 | #121 | — | 142 | 3 | no | — |
| 221 | C4×Q8 | — | 71 | 3 | no | — |
| 222 | C4×D4 | — | 71 | 3 | no | — |
| 223 | C4×D4 | — | 142 | 3 | no | — |
| 224 | C4×D4 | — | 142 | 3 | no | — |
| 225 | C4×D4 | — | 142 | 3 | no | — |
| 226 | #117 | — | 142 | 3 | no | — |
| 227 | #117 | — | 142 | 3 | no | — |
| 228 | C2×C4⋊C4 | — | 142 | 3 | no | — |
| 229 | C2×C42 | — | 71 | 1 | no | — |
| 230 | C4⋊C8 | — | 142 | 3 | no | — |
| 231 | #102 | — | 2 | 3 | no | — |
| 232 | #102 | — | 2 | 3 | no | — |
| 233 | #102 | — | 142 | 3 | no | — |
| 234 | #102 | — | 142 | 3 | no | — |
| 235 | C22⋊C8 | — | 142 | 3 | no | — |
| 236 | #97 | — | 71 | 3 | no | — |
| 237 | #97 | C22×D71 | 1 | 3 | yes | — |
| 238 | C24 | — | 284 | 1 | no | — |
| 239 | C2×Q8 | — | 71 | 3 | no | — |
| 240 | C2×D4 | — | 2 | 3 | no | — |
| 241 | C2×D4 | — | 284 | 3 | no | — |
| 242 | C2×D4 | — | 568 | 3 | no | — |
| 243 | C2×D4 | — | 568 | 3 | no | — |
| 244 | C2×D4 | — | 71 | 3 | no | — |
| 245 | C2×D4 | — | 142 | 3 | no | — |
| 246 | C2×D4 | — | 142 | 3 | no | — |
| 247 | C2×D4 | — | 142 | 3 | no | — |
| 248 | C2×D4 | — | 142 | 3 | no | — |
| 249 | C22×C4 | — | 71 | 3 | no | — |
| 250 | C22×C4 | — | 142 | 3 | no | — |
| 251 | C22×C4 | — | 142 | 3 | no | — |
| 252 | C22×C4 | — | 142 | 3 | no | — |
| 253 | SD16 | — | 568 | 3 | no | — |
| 254 | SD16 | — | 568 | 3 | no | — |
| 255 | D8 | — | 568 | 3 | no | — |
| 256 | D8 | — | 568 | 3 | no | — |
| 257 | C2×C8 | — | 2 | 3 | no | — |
| 258 | C2×C8 | — | 142 | 3 | no | — |
| 259 | C4⋊C4 | — | 2 | 3 | no | — |
| 260 | C4⋊C4 | — | 2 | 3 | no | — |
| 261 | C4⋊C4 | — | 2 | 3 | no | — |
| 262 | C4⋊C4 | — | 2 | 3 | no | — |
| 263 | C4⋊C4 | — | 142 | 3 | no | — |
| 264 | C4⋊C4 | — | 142 | 3 | no | — |
| 265 | C4⋊C4 | — | 142 | 3 | no | — |
| 266 | C4⋊C4 | — | 142 | 3 | no | — |
| 267 | C4⋊C4 | D4×D71 | 1 | 3 | yes | — |
| 268 | C4⋊C4 | D4×D71 | 1 | 3 | yes | — |
| 269 | C22⋊C4 | — | 142 | 3 | no | — |
| 270 | C22⋊C4 | — | 142 | 3 | no | — |
| 271 | C22⋊C4 | — | 142 | 3 | no | — |
| 272 | C22⋊C4 | — | 142 | 3 | no | — |
| 273 | C22⋊C4 | — | 142 | 3 | no | — |
| 274 | C22⋊C4 | — | 142 | 3 | no | — |
| 275 | C22⋊C4 | — | 142 | 3 | no | — |
| 276 | C42 | — | 2 | 1 | no | — |
| 277 | C42 | — | 2 | 1 | no | — |
| 278 | C42 | — | 71 | 3 | no | — |
| 279 | C42 | — | 142 | 3 | no | — |
| 280 | C42 | C23×D71 | 1 | 1 | yes | characteristic |
| 281 | C23 | — | 4 | 1 | no | — |
| 282 | C23 | — | 284 | 1 | no | — |
| 283 | C23 | — | 568 | 6 | no | — |
| 284 | C23 | — | 71 | 1 | no | — |
| 285 | C23 | — | 142 | 3 | no | — |
| 286 | Q8 | — | 284 | 3 | no | — |
| 287 | D4 | — | 4 | 3 | no | — |
| 288 | D4 | — | 4 | 3 | no | — |
| 289 | D4 | — | 284 | 3 | no | — |
| 290 | D4 | — | 284 | 3 | no | — |
| 291 | D4 | — | 284 | 3 | no | — |
| 292 | D4 | — | 568 | 3 | no | — |
| 293 | D4 | — | 568 | 3 | no | — |
| 294 | D4 | — | 568 | 3 | no | — |
| 295 | D4 | — | 568 | 6 | no | — |
| 296 | C2×C4 | — | 2 | 3 | no | — |
| 297 | C2×C4 | — | 2 | 3 | no | — |
| 298 | C2×C4 | — | 284 | 3 | no | — |
| 299 | C2×C4 | — | 568 | 3 | no | — |
| 300 | C2×C4 | — | 568 | 3 | no | — |
| 301 | C2×C4 | — | 568 | 3 | no | — |
| 302 | C2×C4 | — | 71 | 3 | no | — |
| 303 | C2×C4 | — | 142 | 3 | no | — |
| 304 | C2×C4 | C2×D4×D71 | 1 | 3 | yes | — |
| 305 | C8 | — | 4 | 3 | no | — |
| 306 | C8 | — | 284 | 3 | no | — |
| 307 | C22 | — | 8 | 3 | no | — |
| 308 | C22 | — | 284 | 3 | no | — |
| 309 | C22 | — | 284 | 3 | no | — |
| 310 | C22 | — | 284 | 3 | no | — |
| 311 | C22 | — | 568 | 3 | no | — |
| 312 | C22 | — | 1136 | 4 | no | — |
| 313 | C22 | — | 142 | 3 | no | — |
| 314 | C22 | #120×D71 | 1 | 1 | yes | characteristic |
| 315 | C4 | — | 2 | 3 | no | — |
| 316 | C4 | — | 4 | 3 | no | — |
| 317 | C4 | — | 4 | 3 | no | — |
| 318 | C4 | — | 284 | 3 | no | — |
| 319 | C4 | — | 142 | 3 | no | — |
| 320 | C2 | — | 16 | 1 | no | — |
| 321 | C2 | — | 284 | 1 | no | — |
| 322 | C2 | — | 568 | 3 | no | — |
| 323 | C2 | — | 1136 | 1 | no | — |
| 324 | C2 | #22597710 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22603030 | C2 | semidirect |
| 2. #22599359 | C2 | semidirect |
| 3. #22587642 | C2 | semidirect |
| 4. #22587635 | C2 | semidirect |
| 5. #22587632 | C2 | semidirect |
| 6. #22586970 | C2 | semidirect |
| 7. C71×#1658 | C2 | semidirect |
| 8. #17986747 | C22 | semidirect |
| 23. #17985741 | C22 | semidirect |
| 24. #17985738 | C22 | semidirect |
| 25. #564×C71 | C22 | semidirect |
| 27. #330×C71 | C22 | semidirect |
| 166. #1658 | D71 | semidirect |
| 170. C71 | #19096 | semidirect |
| 171. #564 | C2×D71 | semidirect |
| 189. #330 | C2×D71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22603030 | 2 |
| #22599359 | 2 |
| #22587642 | 2 |
| #22587635 | 2 |
| #22587632 | 2 |
| #22586970 | 2 |
| C71×#1658 | 2 |
| #19096 | 71 |