← #30774202
Subgroups — #30774202
Order 17152.
510 conjugacy classes of subgroups.
The subgroup lattice has 244 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16863222 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#16863197 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×C16×Dic67 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22561076 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C22×D16×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C22×C67⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#14523977 | C22 | 1 | 2 | yes | — |
| 8 | C2×#14523964 | — | 2 | 2 | no | — |
| 9 | C2×C8×Dic67 | C22 | 1 | 1 | yes | characteristic |
| 10 | #16863299 | — | 2 | 2 | no | — |
| 11 | #16863222 | — | 2 | 2 | no | — |
| 12 | #16863197 | — | 2 | 4 | no | — |
| 13 | C16×Dic67 | — | 2 | 2 | no | — |
| 14 | C22×D8×C67 | C22 | 1 | 2 | yes | — |
| 15 | C2×D16×C67 | — | 2 | 2 | no | — |
| 16 | C2×D16×C67 | — | 2 | 2 | no | — |
| 17 | C2×D16×C67 | C4 | 1 | 4 | yes | complemented |
| 18 | C22×C16×C67 | C22 | 1 | 1 | yes | characteristic |
| 19 | C22×C67⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 20 | C2×C4⋊Dic67 | — | 2 | 2 | no | — |
| 21 | C2×C22⋊Dic67 | — | 4 | 4 | no | — |
| 22 | C2×C4×Dic67 | D4 | 1 | 1 | yes | characteristic |
| 23 | C2×C67⋊C16 | — | 2 | 1 | no | — |
| 24 | C2×C67⋊C16 | — | 2 | 1 | no | — |
| 25 | C2×C67⋊C16 | D4 | 1 | 2 | yes | — |
| 26 | #14523977 | — | 2 | 2 | no | — |
| 27 | #14523977 | — | 2 | 2 | no | — |
| 28 | #14523964 | — | 4 | 4 | no | — |
| 29 | C8×Dic67 | — | 2 | 1 | no | — |
| 30 | C8×Dic67 | — | 2 | 1 | no | — |
| 31 | C22×D4×C67 | — | 2 | 2 | no | — |
| 32 | C2×D8×C67 | — | 2 | 2 | no | — |
| 33 | C2×D8×C67 | — | 2 | 2 | no | — |
| 34 | C2×D8×C67 | — | 2 | 2 | no | — |
| 35 | C2×D8×C67 | — | 2 | 2 | no | — |
| 36 | C2×D8×C67 | — | 2 | 2 | no | — |
| 37 | C2×D8×C67 | C2×C4 | 1 | 4 | yes | — |
| 38 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 39 | D16×C67 | — | 2 | 8 | no | — |
| 40 | C2×C16×C67 | C2×C4 | 1 | 2 | yes | — |
| 41 | C2×C16×C67 | D4 | 1 | 2 | yes | — |
| 42 | C2×C16×C67 | D4 | 1 | 2 | yes | — |
| 43 | C22×Dic67 | — | 4 | 2 | no | — |
| 44 | C22×Dic67 | D8 | 1 | 2 | yes | — |
| 45 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 46 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 47 | C2×C67⋊C8 | D8 | 1 | 2 | yes | — |
| 48 | C4⋊Dic67 | — | 4 | 2 | no | — |
| 49 | C4⋊Dic67 | — | 4 | 2 | no | — |
| 50 | C22⋊Dic67 | — | 8 | 8 | no | — |
| 51 | C4×Dic67 | — | 2 | 1 | no | — |
| 52 | C4×Dic67 | — | 2 | 1 | no | — |
| 53 | C67⋊C16 | — | 2 | 2 | no | — |
| 54 | C24×C67 | — | 4 | 2 | no | — |
| 55 | C2×D4×C67 | — | 2 | 4 | no | — |
| 56 | C2×D4×C67 | — | 4 | 2 | no | — |
| 57 | C2×D4×C67 | — | 4 | 2 | no | — |
| 58 | C2×D4×C67 | — | 4 | 2 | no | — |
| 59 | C2×D4×C67 | — | 4 | 2 | no | — |
| 60 | C2×D4×C67 | — | 4 | 2 | no | — |
| 61 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | D8×C67 | — | 2 | 4 | no | — |
| 63 | D8×C67 | — | 2 | 4 | no | — |
| 64 | D8×C67 | — | 4 | 2 | no | — |
| 65 | D8×C67 | — | 4 | 2 | no | — |
| 66 | C2×C8×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 67 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 71 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 72 | C16×C67 | C22⋊C4 | 1 | 4 | yes | complemented |
| 73 | C23×C67 | — | 4 | 4 | no | — |
| 74 | C23×C67 | — | 8 | 2 | no | — |
| 75 | C23×C67 | — | 8 | 2 | no | — |
| 76 | C23×C67 | — | 8 | 4 | no | — |
| 77 | C23×C67 | — | 8 | 4 | no | — |
| 78 | C23×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 79 | D4×C67 | — | 4 | 4 | no | — |
| 80 | D4×C67 | — | 4 | 4 | no | — |
| 81 | D4×C67 | — | 8 | 2 | no | — |
| 82 | D4×C67 | — | 8 | 2 | no | — |
| 83 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C4×C67 | #121 | 1 | 1 | yes | characteristic |
| 85 | C2×C4×C67 | #127 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C67 | #135 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C67 | #135 | 1 | 1 | yes | characteristic |
| 89 | C2×Dic67 | — | 2 | 2 | no | — |
| 90 | C2×Dic67 | — | 2 | 2 | no | — |
| 91 | C2×Dic67 | — | 8 | 2 | no | — |
| 92 | C2×Dic67 | — | 8 | 4 | no | — |
| 93 | C2×Dic67 | D16 | 1 | 4 | yes | complemented |
| 94 | C8×C67 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 95 | C8×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 96 | C8×C67 | #121 | 1 | 1 | yes | characteristic |
| 97 | C8×C67 | #124 | 1 | 1 | yes | characteristic |
| 98 | C67⋊C8 | — | 2 | 2 | no | — |
| 99 | C22×C67 | — | 8 | 8 | no | — |
| 100 | C22×C67 | — | 8 | 4 | no | — |
| 101 | C22×C67 | — | 16 | 4 | no | — |
| 102 | C22×C67 | — | 16 | 2 | no | — |
| 103 | C22×C67 | — | 16 | 2 | no | — |
| 104 | C22×C67 | C4×D8 | 1 | 1 | yes | characteristic |
| 105 | C22×C67 | #466 | 1 | 1 | yes | characteristic |
| 106 | C22×C67 | #493 | 1 | 1 | yes | characteristic |
| 107 | C22×C67 | C2×D16 | 1 | 2 | yes | — |
| 108 | C22×C67 | #508 | 1 | 2 | yes | — |
| 109 | C4×C67 | #390 | 1 | 1 | yes | characteristic |
| 110 | C4×C67 | C4×D8 | 1 | 1 | yes | characteristic |
| 111 | C4×C67 | #466 | 1 | 1 | yes | characteristic |
| 112 | C4×C67 | #495 | 1 | 1 | yes | characteristic |
| 113 | Dic67 | — | 2 | 4 | no | — |
| 114 | Dic67 | — | 16 | 2 | no | — |
| 115 | #13235 | — | 67 | 1 | no | maximal, Sylow |
| 116 | C2×C67 | — | 16 | 4 | no | — |
| 117 | C2×C67 | #1970 | 1 | 1 | yes | characteristic |
| 118 | C2×C67 | C4×D16 | 1 | 2 | yes | — |
| 119 | C2×C67 | #2215 | 1 | 2 | yes | — |
| 120 | C2×C67 | #2246 | 1 | 1 | yes | characteristic |
| 121 | C2×C67 | #2249 | 1 | 1 | yes | characteristic |
| 122 | C22×D16 | D67 | 1 | 1 | yes | characteristic |
| 123 | C2×#366 | — | 67 | 1 | no | — |
| 124 | C2×#357 | — | 67 | 2 | no | — |
| 125 | C2×C4×C16 | — | 67 | 1 | no | — |
| 126 | #1970 | — | 67 | 2 | no | — |
| 127 | C67 | #13235 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 128 | C22×D8 | C2×D67 | 1 | 2 | yes | — |
| 129 | C2×D16 | — | 2 | 2 | no | — |
| 130 | C2×D16 | — | 2 | 2 | no | — |
| 131 | C2×D16 | Dic67 | 1 | 4 | yes | complemented |
| 132 | C22×C16 | — | 67 | 1 | no | — |
| 133 | C22×C16 | C2×D67 | 1 | 1 | yes | characteristic |
| 134 | C2×#107 | — | 67 | 2 | no | — |
| 135 | C2×#102 | — | 134 | 2 | no | — |
| 136 | C2×C4×C8 | — | 67 | 1 | no | — |
| 137 | #390 | — | 134 | 2 | no | — |
| 138 | #366 | — | 134 | 2 | no | — |
| 139 | #357 | — | 134 | 4 | no | — |
| 140 | C4×C16 | — | 134 | 2 | no | — |
| 141 | C22×D4 | — | 2 | 2 | no | — |
| 142 | C2×D8 | — | 2 | 2 | no | — |
| 143 | C2×D8 | — | 2 | 2 | no | — |
| 144 | C2×D8 | — | 2 | 2 | no | — |
| 145 | C2×D8 | — | 2 | 2 | no | — |
| 146 | C2×D8 | — | 2 | 2 | no | — |
| 147 | C2×D8 | C2×Dic67 | 1 | 4 | yes | — |
| 148 | C22×C8 | — | 67 | 1 | no | — |
| 149 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 150 | C2×C4⋊C4 | — | 134 | 2 | no | — |
| 151 | C2×C22⋊C4 | — | 268 | 4 | no | — |
| 152 | C2×C42 | — | 67 | 1 | no | — |
| 153 | D16 | — | 2 | 8 | no | — |
| 154 | C2×C16 | — | 67 | 2 | no | — |
| 155 | C2×C16 | — | 134 | 1 | no | — |
| 156 | C2×C16 | — | 134 | 1 | no | — |
| 157 | C2×C16 | C2×Dic67 | 1 | 2 | yes | — |
| 158 | C2×C16 | C22⋊D67 | 1 | 2 | yes | — |
| 159 | C2×C16 | C22⋊D67 | 1 | 2 | yes | — |
| 160 | #107 | — | 134 | 2 | no | — |
| 161 | #107 | — | 134 | 2 | no | — |
| 162 | #102 | — | 268 | 4 | no | — |
| 163 | C4×C8 | — | 134 | 1 | no | — |
| 164 | C4×C8 | — | 134 | 1 | no | — |
| 165 | C24 | — | 4 | 2 | no | — |
| 166 | C2×D4 | — | 2 | 4 | no | — |
| 167 | C2×D4 | — | 4 | 2 | no | — |
| 168 | C2×D4 | — | 4 | 2 | no | — |
| 169 | C2×D4 | — | 4 | 2 | no | — |
| 170 | C2×D4 | — | 4 | 2 | no | — |
| 171 | C2×D4 | — | 4 | 2 | no | — |
| 172 | C22×C4 | — | 268 | 2 | no | — |
| 173 | C22×C4 | — | 67 | 2 | no | — |
| 174 | C22×C4 | D4×D67 | 1 | 1 | yes | characteristic |
| 175 | D8 | — | 2 | 4 | no | — |
| 176 | D8 | — | 2 | 4 | no | — |
| 177 | D8 | — | 4 | 2 | no | — |
| 178 | D8 | — | 4 | 2 | no | — |
| 179 | C2×C8 | — | 67 | 2 | no | — |
| 180 | C2×C8 | — | 134 | 1 | no | — |
| 181 | C2×C8 | — | 134 | 1 | no | — |
| 182 | C2×C8 | C22×Dic67 | 1 | 1 | yes | characteristic |
| 183 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 184 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 185 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 186 | C2×C8 | #11761627 | 1 | 1 | yes | characteristic |
| 187 | C2×C8 | #11761627 | 1 | 1 | yes | characteristic |
| 188 | C4⋊C4 | — | 268 | 2 | no | — |
| 189 | C4⋊C4 | — | 268 | 2 | no | — |
| 190 | C22⋊C4 | — | 536 | 8 | no | — |
| 191 | C42 | — | 134 | 1 | no | — |
| 192 | C42 | — | 134 | 1 | no | — |
| 193 | C16 | — | 134 | 2 | no | — |
| 194 | C16 | C22⋊Dic67 | 1 | 4 | yes | complemented |
| 195 | C23 | — | 4 | 4 | no | — |
| 196 | C23 | — | 8 | 2 | no | — |
| 197 | C23 | — | 8 | 4 | no | — |
| 198 | C23 | — | 8 | 4 | no | — |
| 199 | C23 | — | 8 | 2 | no | — |
| 200 | C23 | D8×D67 | 1 | 1 | yes | characteristic |
| 201 | D4 | — | 4 | 4 | no | — |
| 202 | D4 | — | 4 | 4 | no | — |
| 203 | D4 | — | 8 | 2 | no | — |
| 204 | D4 | — | 8 | 2 | no | — |
| 205 | C2×C4 | — | 536 | 2 | no | — |
| 206 | C2×C4 | — | 536 | 4 | no | — |
| 207 | C2×C4 | — | 67 | 4 | no | — |
| 208 | C2×C4 | — | 134 | 2 | no | — |
| 209 | C2×C4 | — | 134 | 2 | no | — |
| 210 | C2×C4 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | #14524014 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | #14524039 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | D8×D67 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | #14524062 | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | #14524062 | 1 | 1 | yes | characteristic |
| 216 | C8 | — | 134 | 2 | no | — |
| 217 | C8 | C2×C22⋊Dic67 | 1 | 1 | yes | characteristic |
| 218 | C8 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 219 | C8 | #14524014 | 1 | 1 | yes | characteristic |
| 220 | C8 | #14524030 | 1 | 1 | yes | characteristic |
| 221 | C22 | — | 8 | 8 | no | — |
| 222 | C22 | — | 8 | 4 | no | — |
| 223 | C22 | — | 16 | 2 | no | — |
| 224 | C22 | — | 16 | 2 | no | — |
| 225 | C22 | — | 16 | 4 | no | — |
| 226 | C22 | D8×Dic67 | 1 | 1 | yes | characteristic |
| 227 | C22 | #16863629 | 1 | 1 | yes | characteristic |
| 228 | C22 | #16863771 | 1 | 1 | yes | characteristic |
| 229 | C22 | D16×D67 | 1 | 2 | yes | — |
| 230 | C22 | #16863825 | 1 | 2 | yes | — |
| 231 | C4 | — | 1072 | 2 | no | — |
| 232 | C4 | — | 134 | 4 | no | — |
| 233 | C4 | #16863299 | 1 | 1 | yes | characteristic |
| 234 | C4 | D8×Dic67 | 1 | 1 | yes | characteristic |
| 235 | C4 | #16863629 | 1 | 1 | yes | characteristic |
| 236 | C4 | #16863779 | 1 | 1 | yes | characteristic |
| 237 | C2 | — | 16 | 4 | no | — |
| 238 | C2 | #22561076 | 1 | 1 | yes | characteristic |
| 239 | C2 | D16×Dic67 | 1 | 2 | yes | — |
| 240 | C2 | #22562292 | 1 | 2 | yes | — |
| 241 | C2 | #22562457 | 1 | 1 | yes | characteristic |
| 242 | C2 | #22562467 | 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