← #32150694
Subgroups — #32150694
Order 18176.
646 conjugacy classes of subgroups.
The subgroup lattice has 256 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#17986415 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #22597244 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22590183 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #22590178 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #22590106 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22589967 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C71×#2108 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #17986799 | C4 | 1 | 1 | yes | characteristic |
| 9 | C2×#14556134 | C22 | 1 | 2 | yes | — |
| 10 | #17986415 | — | 2 | 2 | no | — |
| 11 | #17986415 | — | 2 | 2 | no | — |
| 12 | #17986415 | C22 | 1 | 4 | yes | complemented |
| 13 | #17986415 | C22 | 1 | 4 | yes | complemented |
| 14 | C2×#14556068 | C22 | 1 | 2 | yes | — |
| 15 | C2×#14556067 | — | 2 | 2 | no | — |
| 16 | C2×C16×D71 | — | 2 | 2 | no | — |
| 17 | #17986122 | C4 | 1 | 4 | yes | — |
| 18 | #17986117 | C22 | 1 | 4 | yes | — |
| 19 | #17986048 | C4 | 1 | 1 | yes | characteristic |
| 20 | #17986017 | C4 | 1 | 2 | yes | — |
| 21 | #17985820 | C22 | 1 | 4 | yes | complemented |
| 22 | #17985782 | C22 | 1 | 8 | yes | complemented |
| 23 | C2×M32×C71 | C22 | 1 | 1 | yes | characteristic |
| 24 | C22×C16×C71 | C22 | 1 | 1 | yes | characteristic |
| 25 | #405×C71 | C22 | 1 | 1 | yes | characteristic |
| 26 | #346×C71 | C22 | 1 | 2 | yes | complemented |
| 27 | C4×C16×C71 | C22 | 1 | 2 | yes | complemented |
| 28 | C2×#11765865 | C2×C4 | 1 | 2 | yes | — |
| 29 | #14556161 | C8 | 1 | 4 | yes | — |
| 30 | #14556156 | C8 | 1 | 4 | yes | — |
| 31 | #14556150 | C2×C4 | 1 | 4 | yes | — |
| 32 | #14556134 | — | 2 | 2 | no | — |
| 33 | #14556134 | — | 2 | 2 | no | — |
| 34 | #14556134 | C2×C4 | 1 | 4 | yes | — |
| 35 | #14556134 | C23 | 1 | 4 | yes | — |
| 36 | C2×#11765849 | C2×C4 | 1 | 2 | yes | — |
| 37 | C2×#11765848 | — | 2 | 2 | no | — |
| 38 | C2×C8×D71 | — | 2 | 2 | no | — |
| 39 | #14556068 | — | 2 | 2 | no | — |
| 40 | #14556068 | C23 | 1 | 4 | yes | — |
| 41 | #14556067 | — | 2 | 4 | no | — |
| 42 | #14556067 | — | 2 | 4 | no | — |
| 43 | C2×C71⋊C16 | — | 2 | 2 | no | — |
| 44 | C16×D71 | — | 2 | 4 | no | — |
| 45 | C16×D71 | — | 2 | 4 | no | — |
| 46 | #14556064 | C2×C4 | 1 | 4 | yes | — |
| 47 | #14556056 | C2×C4 | 1 | 8 | yes | — |
| 48 | C2×M16×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C22×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 50 | #117×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | M32×C71 | — | 2 | 2 | no | — |
| 52 | C2×C16×C71 | D4 | 1 | 2 | yes | — |
| 53 | C2×C16×C71 | D4 | 1 | 2 | yes | — |
| 54 | C2×C16×C71 | C23 | 1 | 2 | yes | — |
| 55 | C2×C16×C71 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×C16×C71 | C23 | 1 | 1 | yes | characteristic |
| 57 | #97×C71 | C2×C4 | 1 | 2 | yes | — |
| 58 | C4×C8×C71 | C23 | 1 | 2 | yes | — |
| 59 | #11765865 | — | 2 | 2 | no | — |
| 60 | #11765865 | — | 2 | 2 | no | — |
| 61 | #11765865 | C2×C8 | 1 | 4 | yes | — |
| 62 | #11765865 | C22×C4 | 1 | 4 | yes | — |
| 63 | C2×Dic142 | C2×C8 | 1 | 2 | yes | — |
| 64 | C2×D284 | C2×C8 | 1 | 2 | yes | — |
| 65 | C2×C22⋊D71 | — | 2 | 2 | no | — |
| 66 | C2×C4×D71 | — | 2 | 2 | no | — |
| 67 | Dic284 | C2×C8 | 1 | 4 | yes | — |
| 68 | #11765854 | C2×C8 | 1 | 8 | yes | — |
| 69 | D568 | C2×C8 | 1 | 4 | yes | — |
| 70 | #11765849 | — | 2 | 2 | no | — |
| 71 | #11765849 | C22×C4 | 1 | 4 | yes | — |
| 72 | #11765848 | — | 2 | 4 | no | — |
| 73 | #11765848 | — | 2 | 4 | no | — |
| 74 | C2×C71⋊C8 | — | 2 | 2 | no | — |
| 75 | C8×D71 | — | 2 | 4 | no | — |
| 76 | C8×D71 | — | 2 | 4 | no | — |
| 77 | C71⋊C16 | — | 2 | 4 | no | — |
| 78 | C22×C4×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | M16×C71 | C2×C8 | 1 | 4 | yes | — |
| 80 | C2×C8×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 81 | C2×C8×C71 | C22×C4 | 1 | 2 | yes | — |
| 82 | C2×C8×C71 | C2×D4 | 1 | 2 | yes | — |
| 83 | C2×C8×C71 | #41 | 1 | 2 | yes | — |
| 84 | C2×C8×C71 | C24 | 1 | 1 | yes | characteristic |
| 85 | C4⋊C4×C71 | — | 2 | 1 | no | — |
| 86 | C22⋊C4×C71 | — | 2 | 1 | no | — |
| 87 | C42×C71 | C22×C4 | 1 | 2 | yes | — |
| 88 | C16×C71 | — | 2 | 2 | no | — |
| 89 | C16×C71 | C2×D4 | 1 | 2 | yes | — |
| 90 | C16×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C16×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C23×C71 | #131 | 1 | 1 | yes | characteristic |
| 93 | C22×D71 | — | 2 | 2 | no | — |
| 94 | C2×C4×C71 | — | 2 | 2 | no | — |
| 95 | C2×C4×C71 | C4×D4 | 1 | 2 | yes | — |
| 96 | C2×C4×C71 | C22×C8 | 1 | 2 | yes | — |
| 97 | C2×C4×C71 | #131 | 1 | 1 | yes | characteristic |
| 98 | C2×C4×C71 | C23×C4 | 1 | 1 | yes | characteristic |
| 99 | C22⋊D71 | — | 2 | 4 | no | — |
| 100 | C22⋊D71 | — | 2 | 4 | no | — |
| 101 | C2×Dic71 | — | 2 | 2 | no | — |
| 102 | D284 | — | 2 | 2 | no | — |
| 103 | D284 | C22×C8 | 1 | 4 | yes | — |
| 104 | C4×D71 | — | 2 | 4 | no | — |
| 105 | C4×D71 | — | 2 | 4 | no | — |
| 106 | Dic142 | — | 2 | 2 | no | — |
| 107 | Dic142 | C22×C8 | 1 | 4 | yes | — |
| 108 | C8×C71 | C4×D4 | 1 | 2 | yes | — |
| 109 | C8×C71 | C22×C8 | 1 | 4 | yes | — |
| 110 | C8×C71 | C22×D4 | 1 | 1 | yes | characteristic |
| 111 | C8×C71 | C2×#41 | 1 | 1 | yes | characteristic |
| 112 | C71⋊C8 | — | 2 | 4 | no | — |
| 113 | C22×C71 | — | 4 | 1 | no | — |
| 114 | C22×C71 | C8×D4 | 1 | 2 | yes | — |
| 115 | C22×C71 | C2×#131 | 1 | 1 | yes | characteristic |
| 116 | C2×D71 | — | 2 | 4 | no | — |
| 117 | C2×D71 | — | 4 | 4 | no | — |
| 118 | C4×C71 | — | 4 | 2 | no | — |
| 119 | C4×C71 | C8×D4 | 1 | 2 | yes | — |
| 120 | C4×C71 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 121 | C4×C71 | C23×C8 | 1 | 1 | yes | characteristic |
| 122 | Dic71 | — | 2 | 4 | no | — |
| 123 | #33441 | — | 71 | 1 | no | maximal, Sylow |
| 124 | C2×C71 | — | 2 | 1 | no | — |
| 125 | C2×C71 | — | 2 | 2 | no | — |
| 126 | C2×C71 | C2×C8×D4 | 1 | 1 | yes | characteristic |
| 127 | D71 | — | 4 | 4 | no | — |
| 128 | C2×#504 | — | 71 | 2 | no | — |
| 129 | #2955 | — | 71 | 1 | no | — |
| 130 | #2171 | — | 71 | 4 | no | — |
| 131 | #2170 | — | 71 | 4 | no | — |
| 132 | #2153 | — | 71 | 1 | no | — |
| 133 | #2117 | — | 71 | 2 | no | — |
| 134 | #2108 | D71 | 1 | 1 | yes | characteristic, complemented |
| 135 | C71 | #33441 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 136 | #575 | — | 71 | 1 | no | — |
| 137 | C2×#131 | — | 71 | 2 | no | — |
| 138 | #504 | — | 71 | 4 | no | — |
| 139 | #504 | — | 71 | 4 | no | — |
| 140 | #504 | — | 142 | 2 | no | — |
| 141 | #504 | — | 142 | 2 | no | — |
| 142 | C2×M32 | — | 71 | 2 | no | — |
| 143 | C2×M32 | — | 142 | 2 | no | — |
| 144 | C2×M32 | C2×D71 | 1 | 1 | yes | characteristic |
| 145 | C22×C16 | — | 142 | 2 | no | — |
| 146 | C22×C16 | C2×D71 | 1 | 1 | yes | characteristic |
| 147 | #444 | — | 71 | 4 | no | — |
| 148 | #443 | — | 71 | 4 | no | — |
| 149 | #430 | — | 71 | 1 | no | — |
| 150 | #421 | — | 71 | 2 | no | — |
| 151 | #405 | C2×D71 | 1 | 1 | yes | characteristic |
| 152 | #364 | — | 71 | 4 | no | — |
| 153 | #350 | — | 71 | 8 | no | — |
| 154 | #346 | C2×D71 | 1 | 2 | yes | complemented |
| 155 | C4×C16 | C2×D71 | 1 | 2 | yes | complemented |
| 156 | C2×#41 | — | 71 | 2 | no | — |
| 157 | #137 | — | 71 | 4 | no | — |
| 158 | AGL(1,ℤ/8) | — | 71 | 4 | no | — |
| 159 | #135 | — | 71 | 4 | no | — |
| 160 | #131 | — | 71 | 4 | no | — |
| 161 | #131 | — | 71 | 4 | no | — |
| 162 | #131 | — | 142 | 2 | no | — |
| 163 | #131 | — | 142 | 2 | no | — |
| 164 | C2×M16 | — | 71 | 2 | no | — |
| 165 | C2×M16 | — | 142 | 2 | no | — |
| 166 | C2×M16 | C4×D71 | 1 | 1 | yes | characteristic |
| 167 | C22×C8 | — | 142 | 2 | no | — |
| 168 | C22×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 169 | #117 | C4×D71 | 1 | 1 | yes | characteristic |
| 170 | M32 | — | 2 | 2 | no | — |
| 171 | M32 | — | 71 | 4 | no | — |
| 172 | M32 | — | 142 | 2 | no | — |
| 173 | M32 | — | 142 | 4 | no | — |
| 174 | M32 | — | 142 | 4 | no | — |
| 175 | C2×C16 | — | 142 | 2 | no | — |
| 176 | C2×C16 | — | 142 | 4 | no | — |
| 177 | C2×C16 | — | 142 | 4 | no | — |
| 178 | C2×C16 | D284 | 1 | 2 | yes | — |
| 179 | C2×C16 | D284 | 1 | 2 | yes | — |
| 180 | C2×C16 | C22×D71 | 1 | 2 | yes | — |
| 181 | C2×C16 | C22×D71 | 1 | 1 | yes | characteristic |
| 182 | C2×C16 | C22×D71 | 1 | 1 | yes | characteristic |
| 183 | #108 | — | 71 | 4 | no | — |
| 184 | #104 | — | 71 | 8 | no | — |
| 185 | #97 | C4×D71 | 1 | 2 | yes | — |
| 186 | C4×C8 | C22×D71 | 1 | 2 | yes | — |
| 187 | #41 | — | 71 | 4 | no | — |
| 188 | #41 | — | 71 | 4 | no | — |
| 189 | #41 | — | 142 | 2 | no | — |
| 190 | #41 | — | 142 | 2 | no | — |
| 191 | C2×Q8 | — | 71 | 2 | no | — |
| 192 | C2×D4 | — | 71 | 2 | no | — |
| 193 | C2×D4 | — | 142 | 2 | no | — |
| 194 | C22×C4 | — | 142 | 2 | no | — |
| 195 | C22×C4 | C2×C4×D71 | 1 | 1 | yes | characteristic |
| 196 | Q16 | — | 71 | 4 | no | — |
| 197 | SD16 | — | 71 | 8 | no | — |
| 198 | D8 | — | 71 | 4 | no | — |
| 199 | M16 | — | 71 | 4 | no | — |
| 200 | M16 | — | 142 | 2 | no | — |
| 201 | M16 | — | 142 | 4 | no | — |
| 202 | M16 | — | 142 | 4 | no | — |
| 203 | M16 | C8×D71 | 1 | 4 | yes | — |
| 204 | C2×C8 | — | 142 | 2 | no | — |
| 205 | C2×C8 | — | 142 | 4 | no | — |
| 206 | C2×C8 | — | 142 | 4 | no | — |
| 207 | C2×C8 | C2×C4×D71 | 1 | 1 | yes | characteristic |
| 208 | C2×C8 | C2×C4×D71 | 1 | 2 | yes | — |
| 209 | C2×C8 | C2×D284 | 1 | 2 | yes | — |
| 210 | C2×C8 | #11765865 | 1 | 2 | yes | — |
| 211 | C2×C8 | C23×D71 | 1 | 1 | yes | characteristic |
| 212 | C4⋊C4 | — | 2 | 1 | no | — |
| 213 | C22⋊C4 | — | 2 | 1 | no | — |
| 214 | C42 | C2×C4×D71 | 1 | 2 | yes | — |
| 215 | C16 | — | 2 | 2 | no | — |
| 216 | C16 | — | 142 | 4 | no | — |
| 217 | C16 | C2×D284 | 1 | 2 | yes | — |
| 218 | C16 | C2×D284 | 1 | 1 | yes | characteristic |
| 219 | C16 | C2×D284 | 1 | 1 | yes | characteristic |
| 220 | C23 | — | 142 | 2 | no | — |
| 221 | C23 | #14556134 | 1 | 1 | yes | characteristic |
| 222 | Q8 | — | 71 | 4 | no | — |
| 223 | Q8 | — | 142 | 2 | no | — |
| 224 | D4 | — | 71 | 4 | no | — |
| 225 | D4 | — | 142 | 4 | no | — |
| 226 | D4 | — | 142 | 4 | no | — |
| 227 | D4 | — | 142 | 2 | no | — |
| 228 | C2×C4 | — | 2 | 2 | no | — |
| 229 | C2×C4 | — | 142 | 2 | no | — |
| 230 | C2×C4 | — | 142 | 4 | no | — |
| 231 | C2×C4 | — | 142 | 4 | no | — |
| 232 | C2×C4 | C4×D284 | 1 | 2 | yes | — |
| 233 | C2×C4 | C2×C8×D71 | 1 | 2 | yes | — |
| 234 | C2×C4 | #14556134 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | C22×C4×D71 | 1 | 1 | yes | characteristic |
| 236 | C8 | — | 142 | 4 | no | — |
| 237 | C8 | C4×D284 | 1 | 2 | yes | — |
| 238 | C8 | C2×C8×D71 | 1 | 4 | yes | — |
| 239 | C8 | C22×D284 | 1 | 1 | yes | characteristic |
| 240 | C8 | C2×#11765865 | 1 | 1 | yes | characteristic |
| 241 | C22 | — | 4 | 1 | no | — |
| 242 | C22 | — | 284 | 4 | no | — |
| 243 | C22 | — | 142 | 4 | no | — |
| 244 | C22 | C8×D284 | 1 | 2 | yes | — |
| 245 | C22 | C2×#14556134 | 1 | 1 | yes | characteristic |
| 246 | C4 | — | 4 | 2 | no | — |
| 247 | C4 | — | 142 | 4 | no | — |
| 248 | C4 | C8×D284 | 1 | 2 | yes | — |
| 249 | C4 | C2×C4×D284 | 1 | 1 | yes | characteristic |
| 250 | C4 | C22×C8×D71 | 1 | 1 | yes | characteristic |
| 251 | C2 | — | 2 | 1 | no | — |
| 252 | C2 | — | 2 | 2 | no | — |
| 253 | C2 | — | 284 | 4 | no | — |
| 254 | C2 | C2×C8×D284 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta