Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
768 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 85 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 | #22599091 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22597594 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22588251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22586701 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C71×#1293 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#11765865 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#14556156 | — | 2 | 2 | no | — |
| 8 | C2×#14556102 | C22 | 1 | 2 | yes | — |
| 9 | #17986313 | — | 2 | 2 | no | — |
| 10 | #17986235 | — | 2 | 2 | no | — |
| 11 | #17986146 | — | 2 | 4 | no | — |
| 12 | #17986140 | — | 2 | 2 | no | — |
| 13 | #17986135 | — | 2 | 2 | no | — |
| 14 | #17986135 | — | 2 | 2 | no | — |
| 15 | #17986001 | — | 2 | 2 | no | — |
| 16 | C2×C22⋊(C71⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×(C2×D71)⋊C8 | — | 2 | 2 | no | — |
| 18 | #17985722 | — | 2 | 4 | no | — |
| 19 | #407×C71 | C22 | 1 | 2 | yes | — |
| 20 | C2×C22⋊C8×C71 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#11765865 | — | 2 | 2 | no | — |
| 22 | C2×#11765865 | — | 2 | 2 | no | — |
| 23 | C2×#11765865 | — | 4 | 4 | no | — |
| 24 | C22×Dic142 | D4 | 1 | 1 | yes | characteristic |
| 25 | C22×D284 | D4 | 1 | 1 | yes | characteristic |
| 26 | C22×D284 | D4 | 1 | 2 | yes | — |
| 27 | C22×C22⋊D71 | — | 2 | 1 | no | — |
| 28 | C22×C22⋊D71 | — | 2 | 2 | no | — |
| 29 | C22×C4×D71 | — | 2 | 1 | no | — |
| 30 | #14556156 | — | 4 | 8 | no | — |
| 31 | C2×#11765854 | — | 2 | 4 | no | — |
| 32 | C2×D568 | — | 2 | 4 | no | — |
| 33 | C2×D568 | — | 2 | 2 | no | — |
| 34 | C22×C71⋊C8 | — | 2 | 1 | no | — |
| 35 | #14556102 | — | 2 | 2 | no | — |
| 36 | #14556102 | — | 2 | 2 | no | — |
| 37 | #14556102 | — | 2 | 2 | no | — |
| 38 | #14556102 | — | 2 | 2 | no | — |
| 39 | C2×C4⋊Dic71 | D4 | 1 | 2 | yes | — |
| 40 | C2×(C2×D71)⋊C4 | — | 2 | 2 | no | — |
| 41 | #14556062 | — | 2 | 2 | no | — |
| 42 | #14556053 | — | 2 | 4 | no | — |
| 43 | #14556050 | — | 2 | 4 | no | — |
| 44 | #14556050 | — | 2 | 4 | no | — |
| 45 | C22⋊(C71⋊C8) | — | 2 | 2 | no | — |
| 46 | (C2×D71)⋊C8 | — | 4 | 4 | no | — |
| 47 | C23×C4×C71 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×M16×C71 | — | 2 | 2 | no | — |
| 49 | C22×C8×C71 | D4 | 1 | 2 | yes | — |
| 50 | C22⋊C8×C71 | — | 2 | 2 | no | — |
| 51 | C22⋊C8×C71 | — | 2 | 2 | no | — |
| 52 | C22⋊C8×C71 | D4 | 1 | 4 | yes | — |
| 53 | C23×D71 | — | 2 | 1 | no | — |
| 54 | C23×D71 | — | 2 | 2 | no | — |
| 55 | #11765865 | — | 2 | 8 | no | — |
| 56 | #11765865 | — | 4 | 4 | no | — |
| 57 | #11765865 | — | 4 | 4 | no | — |
| 58 | #11765865 | — | 4 | 4 | no | — |
| 59 | C2×Dic142 | — | 2 | 2 | no | — |
| 60 | C2×Dic142 | — | 4 | 2 | no | — |
| 61 | C2×D284 | — | 2 | 2 | no | — |
| 62 | C2×D284 | — | 2 | 2 | no | — |
| 63 | C2×D284 | — | 2 | 2 | no | — |
| 64 | C2×D284 | — | 4 | 2 | no | — |
| 65 | C2×D284 | — | 4 | 4 | no | — |
| 66 | C2×C22⋊D71 | — | 4 | 2 | no | — |
| 67 | C2×C22⋊D71 | — | 4 | 2 | no | — |
| 68 | C2×C22⋊D71 | — | 4 | 2 | no | — |
| 69 | C2×C22⋊D71 | — | 4 | 2 | no | — |
| 70 | C2×C22⋊D71 | — | 4 | 2 | no | — |
| 71 | C2×C22⋊D71 | — | 8 | 4 | no | — |
| 72 | C22×Dic71 | — | 2 | 1 | no | — |
| 73 | C22×Dic71 | — | 2 | 2 | no | — |
| 74 | C2×C4×D71 | — | 2 | 4 | no | — |
| 75 | C2×C4×D71 | — | 4 | 2 | no | — |
| 76 | C2×C4×D71 | — | 4 | 1 | no | — |
| 77 | C2×C4×D71 | — | 4 | 1 | no | — |
| 78 | #11765854 | — | 4 | 8 | no | — |
| 79 | D568 | — | 4 | 8 | no | — |
| 80 | D568 | — | 8 | 2 | no | — |
| 81 | C2×C71⋊C8 | — | 4 | 2 | no | — |
| 82 | C2×C71⋊C8 | — | 4 | 1 | no | — |
| 83 | C4⋊Dic71 | — | 2 | 2 | no | — |
| 84 | C4⋊Dic71 | — | 2 | 2 | no | — |
| 85 | (C2×D71)⋊C4 | — | 4 | 2 | no | — |
| 86 | (C2×D71)⋊C4 | — | 4 | 2 | no | — |
| 87 | C24×C71 | #41 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C71 | — | 4 | 2 | no | — |
| 89 | C22×C4×C71 | C2×D4 | 1 | 2 | yes | — |
| 90 | C22×C4×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C22×C4×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C22×C4×C71 | C2×D4 | 1 | 2 | yes | — |
| 93 | M16×C71 | — | 4 | 4 | no | — |
| 94 | C2×C8×C71 | — | 2 | 2 | no | — |
| 95 | C2×C8×C71 | — | 2 | 4 | no | — |
| 96 | C2×C8×C71 | — | 4 | 2 | no | — |
| 97 | C23×C71 | — | 2 | 2 | no | — |
| 98 | C23×C71 | — | 4 | 1 | no | — |
| 99 | C23×C71 | — | 4 | 1 | no | — |
| 100 | C23×C71 | #121 | 1 | 2 | yes | — |
| 101 | C23×C71 | #121 | 1 | 1 | yes | characteristic |
| 102 | C22×D71 | — | 2 | 2 | no | — |
| 103 | C22×D71 | — | 4 | 1 | no | — |
| 104 | C22×D71 | — | 4 | 2 | no | — |
| 105 | C22×D71 | — | 4 | 1 | no | — |
| 106 | C22×D71 | — | 4 | 2 | no | — |
| 107 | C22×D71 | — | 4 | 2 | no | — |
| 108 | C22×D71 | — | 4 | 4 | no | — |
| 109 | C22×D71 | — | 8 | 2 | no | — |
| 110 | C22×D71 | — | 8 | 2 | no | — |
| 111 | C2×C4×C71 | — | 2 | 2 | no | — |
| 112 | C2×C4×C71 | — | 2 | 4 | no | — |
| 113 | C2×C4×C71 | — | 4 | 1 | no | — |
| 114 | C2×C4×C71 | — | 4 | 2 | no | — |
| 115 | C2×C4×C71 | #120 | 1 | 2 | yes | — |
| 116 | C2×C4×C71 | #120 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C71 | #127 | 1 | 1 | yes | characteristic |
| 118 | C22⋊D71 | — | 4 | 8 | no | — |
| 119 | C22⋊D71 | — | 8 | 4 | no | — |
| 120 | C22⋊D71 | — | 8 | 4 | no | — |
| 121 | C2×Dic71 | — | 2 | 2 | no | — |
| 122 | C2×Dic71 | — | 4 | 1 | no | — |
| 123 | C2×Dic71 | — | 4 | 1 | no | — |
| 124 | C2×Dic71 | — | 4 | 2 | no | — |
| 125 | C2×Dic71 | — | 8 | 2 | no | — |
| 126 | D284 | — | 2 | 4 | no | — |
| 127 | D284 | — | 4 | 2 | no | — |
| 128 | D284 | — | 4 | 4 | no | — |
| 129 | D284 | — | 4 | 4 | no | — |
| 130 | C4×D71 | — | 4 | 4 | no | — |
| 131 | C4×D71 | — | 4 | 4 | no | — |
| 132 | Dic142 | — | 2 | 4 | no | — |
| 133 | Dic142 | — | 4 | 2 | no | — |
| 134 | C8×C71 | — | 4 | 4 | no | — |
| 135 | C8×C71 | — | 4 | 2 | no | — |
| 136 | C71⋊C8 | — | 8 | 1 | no | — |
| 137 | C22×C71 | — | 2 | 1 | no | — |
| 138 | C22×C71 | — | 2 | 1 | no | — |
| 139 | C22×C71 | — | 2 | 2 | no | — |
| 140 | C22×C71 | — | 2 | 2 | no | — |
| 141 | C22×C71 | — | 2 | 2 | no | — |
| 142 | C22×C71 | — | 8 | 2 | no | — |
| 143 | C22×C71 | #392 | 1 | 1 | yes | characteristic |
| 144 | C22×C71 | #472 | 1 | 2 | yes | — |
| 145 | C2×D71 | — | 4 | 4 | no | — |
| 146 | C2×D71 | — | 4 | 2 | no | — |
| 147 | C2×D71 | — | 8 | 2 | no | — |
| 148 | C2×D71 | — | 8 | 1 | no | — |
| 149 | C2×D71 | — | 8 | 1 | no | — |
| 150 | C2×D71 | — | 8 | 2 | no | — |
| 151 | C2×D71 | — | 8 | 2 | no | — |
| 152 | C2×D71 | — | 8 | 2 | no | — |
| 153 | C2×D71 | — | 16 | 4 | no | — |
| 154 | C4×C71 | — | 2 | 1 | no | — |
| 155 | C4×C71 | — | 2 | 2 | no | — |
| 156 | C4×C71 | B(4,𝔽2) | 1 | 2 | yes | — |
| 157 | Dic71 | — | 4 | 2 | no | — |
| 158 | Dic71 | — | 8 | 2 | no | — |
| 159 | #9974 | — | 71 | 1 | no | maximal, Sylow |
| 160 | C2×C71 | — | 2 | 2 | no | — |
| 161 | C2×C71 | — | 4 | 2 | no | — |
| 162 | C2×C71 | #2017 | 1 | 1 | yes | characteristic |
| 163 | C2×C71 | #2018 | 1 | 1 | yes | characteristic |
| 164 | C2×C71 | #2021 | 1 | 1 | yes | characteristic |
| 165 | D71 | — | 8 | 2 | no | — |
| 166 | D71 | — | 16 | 2 | no | — |
| 167 | #3186 | — | 71 | 1 | no | — |
| 168 | #3006 | — | 71 | 2 | no | — |
| 169 | #1780 | — | 71 | 1 | no | — |
| 170 | #1471 | — | 71 | 2 | no | — |
| 171 | #1293 | D71 | 1 | 1 | yes | characteristic, complemented |
| 172 | C71 | #9974 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 173 | C22×#41 | — | 71 | 1 | no | — |
| 174 | C2×AGL(1,ℤ/8) | — | 142 | 2 | no | — |
| 175 | C2×#121 | — | 71 | 2 | no | — |
| 176 | #480 | — | 142 | 2 | no | — |
| 177 | #466 | — | 142 | 2 | no | — |
| 178 | #449 | — | 142 | 4 | no | — |
| 179 | #448 | — | 142 | 2 | no | — |
| 180 | #447 | — | 142 | 2 | no | — |
| 181 | #447 | — | 142 | 2 | no | — |
| 182 | #417 | — | 142 | 2 | no | — |
| 183 | #407 | C2×D71 | 1 | 2 | yes | — |
| 184 | C2×C22⋊C8 | — | 71 | 1 | no | — |
| 185 | C2×C22⋊C8 | — | 142 | 2 | no | — |
| 186 | C2×C22⋊C8 | C2×D71 | 1 | 1 | yes | characteristic |
| 187 | #323 | — | 142 | 4 | no | — |
| 188 | C2×#41 | — | 284 | 4 | no | — |
| 189 | C2×#41 | — | 142 | 2 | no | — |
| 190 | C2×#41 | — | 142 | 2 | no | — |
| 191 | C22×Q8 | — | 71 | 1 | no | — |
| 192 | C22×D4 | — | 71 | 2 | no | — |
| 193 | C22×D4 | — | 71 | 1 | no | — |
| 194 | C22×D4 | — | 142 | 1 | no | — |
| 195 | C22×D4 | — | 142 | 2 | no | — |
| 196 | C23×C4 | — | 142 | 1 | no | — |
| 197 | C23×C4 | C22×D71 | 1 | 1 | yes | characteristic |
| 198 | AGL(1,ℤ/8) | — | 284 | 8 | no | — |
| 199 | C2×SD16 | — | 142 | 4 | no | — |
| 200 | C2×D8 | — | 142 | 4 | no | — |
| 201 | C2×D8 | — | 142 | 2 | no | — |
| 202 | C2×M16 | — | 2 | 2 | no | — |
| 203 | C22×C8 | — | 142 | 1 | no | — |
| 204 | C22×C8 | D284 | 1 | 2 | yes | — |
| 205 | #121 | — | 142 | 2 | no | — |
| 206 | #121 | — | 142 | 2 | no | — |
| 207 | #121 | — | 142 | 2 | no | — |
| 208 | #121 | — | 142 | 2 | no | — |
| 209 | C2×C4⋊C4 | — | 71 | 2 | no | — |
| 210 | C2×C22⋊C4 | — | 142 | 2 | no | — |
| 211 | #107 | — | 142 | 2 | no | — |
| 212 | #103 | — | 142 | 4 | no | — |
| 213 | #102 | — | 142 | 4 | no | — |
| 214 | #102 | — | 142 | 4 | no | — |
| 215 | C22⋊C8 | — | 2 | 2 | no | — |
| 216 | C22⋊C8 | — | 2 | 2 | no | — |
| 217 | C22⋊C8 | — | 284 | 4 | no | — |
| 218 | C22⋊C8 | — | 142 | 2 | no | — |
| 219 | C22⋊C8 | D284 | 1 | 4 | yes | — |
| 220 | C24 | — | 142 | 1 | no | — |
| 221 | C24 | — | 142 | 2 | no | — |
| 222 | C24 | #11765866 | 1 | 1 | yes | characteristic |
| 223 | #41 | — | 284 | 4 | no | — |
| 224 | #41 | — | 284 | 4 | no | — |
| 225 | #41 | — | 284 | 4 | no | — |
| 226 | #41 | — | 142 | 8 | no | — |
| 227 | C2×Q8 | — | 284 | 2 | no | — |
| 228 | C2×Q8 | — | 142 | 2 | no | — |
| 229 | C2×D4 | — | 284 | 4 | no | — |
| 230 | C2×D4 | — | 284 | 2 | no | — |
| 231 | C2×D4 | — | 284 | 2 | no | — |
| 232 | C2×D4 | — | 284 | 2 | no | — |
| 233 | C2×D4 | — | 284 | 2 | no | — |
| 234 | C2×D4 | — | 284 | 2 | no | — |
| 235 | C2×D4 | — | 284 | 2 | no | — |
| 236 | C2×D4 | — | 568 | 4 | no | — |
| 237 | C2×D4 | — | 142 | 2 | no | — |
| 238 | C2×D4 | — | 142 | 2 | no | — |
| 239 | C2×D4 | — | 142 | 2 | no | — |
| 240 | C22×C4 | — | 4 | 2 | no | — |
| 241 | C22×C4 | — | 284 | 2 | no | — |
| 242 | C22×C4 | — | 284 | 1 | no | — |
| 243 | C22×C4 | — | 284 | 1 | no | — |
| 244 | C22×C4 | — | 142 | 4 | no | — |
| 245 | C22×C4 | — | 142 | 1 | no | — |
| 246 | C22×C4 | — | 142 | 2 | no | — |
| 247 | C22×C4 | D4×D71 | 1 | 2 | yes | — |
| 248 | C22×C4 | D4×D71 | 1 | 1 | yes | characteristic |
| 249 | C22×C4 | C2×D284 | 1 | 1 | yes | characteristic |
| 250 | C22×C4 | C2×D284 | 1 | 2 | yes | — |
| 251 | SD16 | — | 284 | 8 | no | — |
| 252 | D8 | — | 284 | 8 | no | — |
| 253 | D8 | — | 568 | 2 | no | — |
| 254 | M16 | — | 4 | 4 | no | — |
| 255 | C2×C8 | — | 2 | 2 | no | — |
| 256 | C2×C8 | — | 2 | 4 | no | — |
| 257 | C2×C8 | — | 4 | 2 | no | — |
| 258 | C2×C8 | — | 284 | 2 | no | — |
| 259 | C2×C8 | — | 284 | 1 | no | — |
| 260 | C4⋊C4 | — | 142 | 2 | no | — |
| 261 | C4⋊C4 | — | 142 | 2 | no | — |
| 262 | C22⋊C4 | — | 284 | 2 | no | — |
| 263 | C22⋊C4 | — | 284 | 2 | no | — |
| 264 | C23 | — | 2 | 2 | no | — |
| 265 | C23 | — | 4 | 1 | no | — |
| 266 | C23 | — | 4 | 1 | no | — |
| 267 | C23 | — | 284 | 1 | no | — |
| 268 | C23 | — | 284 | 2 | no | — |
| 269 | C23 | — | 284 | 4 | no | — |
| 270 | C23 | — | 284 | 1 | no | — |
| 271 | C23 | — | 284 | 2 | no | — |
| 272 | C23 | — | 284 | 2 | no | — |
| 273 | C23 | — | 568 | 2 | no | — |
| 274 | C23 | — | 568 | 2 | no | — |
| 275 | C23 | — | 142 | 2 | no | — |
| 276 | C23 | #14556103 | 1 | 2 | yes | — |
| 277 | C23 | #14556103 | 1 | 1 | yes | characteristic |
| 278 | Q8 | — | 284 | 2 | no | — |
| 279 | Q8 | — | 142 | 4 | no | — |
| 280 | D4 | — | 284 | 2 | no | — |
| 281 | D4 | — | 284 | 4 | no | — |
| 282 | D4 | — | 284 | 8 | no | — |
| 283 | D4 | — | 284 | 4 | no | — |
| 284 | D4 | — | 568 | 4 | no | — |
| 285 | D4 | — | 568 | 4 | no | — |
| 286 | D4 | — | 142 | 4 | no | — |
| 287 | C2×C4 | — | 2 | 2 | no | — |
| 288 | C2×C4 | — | 2 | 4 | no | — |
| 289 | C2×C4 | — | 4 | 1 | no | — |
| 290 | C2×C4 | — | 4 | 2 | no | — |
| 291 | C2×C4 | — | 284 | 4 | no | — |
| 292 | C2×C4 | — | 284 | 1 | no | — |
| 293 | C2×C4 | — | 284 | 1 | no | — |
| 294 | C2×C4 | — | 284 | 4 | no | — |
| 295 | C2×C4 | — | 284 | 2 | no | — |
| 296 | C2×C4 | — | 568 | 2 | no | — |
| 297 | C2×C4 | — | 142 | 2 | no | — |
| 298 | C2×C4 | #14556098 | 1 | 2 | yes | — |
| 299 | C2×C4 | #14556098 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #14556125 | 1 | 1 | yes | characteristic |
| 301 | C8 | — | 4 | 4 | no | — |
| 302 | C8 | — | 4 | 2 | no | — |
| 303 | C8 | — | 568 | 1 | no | — |
| 304 | C22 | — | 2 | 1 | no | — |
| 305 | C22 | — | 2 | 1 | no | — |
| 306 | C22 | — | 2 | 2 | no | — |
| 307 | C22 | — | 2 | 2 | no | — |
| 308 | C22 | — | 2 | 2 | no | — |
| 309 | C22 | — | 8 | 2 | no | — |
| 310 | C22 | — | 284 | 4 | no | — |
| 311 | C22 | — | 284 | 2 | no | — |
| 312 | C22 | — | 568 | 2 | no | — |
| 313 | C22 | — | 568 | 1 | no | — |
| 314 | C22 | — | 568 | 1 | no | — |
| 315 | C22 | — | 568 | 2 | no | — |
| 316 | C22 | — | 568 | 2 | no | — |
| 317 | C22 | — | 568 | 2 | no | — |
| 318 | C22 | — | 1136 | 4 | no | — |
| 319 | C22 | #17985909 | 1 | 1 | yes | characteristic |
| 320 | C22 | #17986270 | 1 | 2 | yes | — |
| 321 | C4 | — | 2 | 2 | no | — |
| 322 | C4 | — | 2 | 1 | no | — |
| 323 | C4 | — | 284 | 2 | no | — |
| 324 | C4 | — | 568 | 2 | no | — |
| 325 | C4 | #17986187 | 1 | 2 | yes | — |
| 326 | C2 | — | 2 | 2 | no | — |
| 327 | C2 | — | 4 | 2 | no | — |
| 328 | C2 | — | 568 | 2 | no | — |
| 329 | C2 | — | 1136 | 2 | no | — |
| 330 | C2 | #22589445 | 1 | 1 | yes | characteristic |
| 331 | C2 | #22589450 | 1 | 1 | yes | characteristic |
| 332 | C2 | #22589467 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22599091 | C2 | semidirect |
| 2. #22597594 | C2 | semidirect |
| 3. #22588251 | C2 | semidirect |
| 4. #22586701 | C2 | semidirect |
| 5. C71×#1293 | C2 | semidirect |
| 171. #1293 | D71 | semidirect |
| 172. C71 | #9974 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22599091 | 2 |
| #22597594 | 2 |
| #22588251 | 2 |
| #22586701 | 2 |
| C71×#1293 | 2 |
| #9974 | 71 |