← #508×Dic67 #31009114
Subgroups — #508×Dic67 #31009114
Order 17152.
690 conjugacy classes of subgroups.
The subgroup lattice has 258 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #135×Dic67 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | Q32×Dic67 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | SD32×Dic67 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | D16×Dic67 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22561993 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22561916 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×C16×Dic67 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#508×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #41×Dic67 | — | 2 | 2 | no | — |
| 10 | C22×C67⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 11 | Q16×Dic67 | C22 | 1 | 4 | yes | — |
| 12 | SD16×Dic67 | — | 2 | 4 | no | — |
| 13 | D8×Dic67 | C22 | 1 | 4 | yes | — |
| 14 | #16863434 | C22 | 1 | 2 | yes | — |
| 15 | #16863392 | — | 2 | 2 | no | — |
| 16 | C2×C8×Dic67 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16863224 | C22 | 1 | 2 | yes | — |
| 18 | #16863222 | C22 | 1 | 2 | yes | — |
| 19 | #16863200 | C22 | 1 | 4 | yes | — |
| 20 | #16863197 | C22 | 1 | 4 | yes | — |
| 21 | C16×Dic67 | C22 | 1 | 2 | yes | — |
| 22 | C16×Dic67 | C22 | 1 | 2 | yes | — |
| 23 | C2×#135×C67 | C22 | 1 | 2 | yes | — |
| 24 | #508×C67 | C4 | 1 | 8 | yes | complemented |
| 25 | C2×Q32×C67 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×SD32×C67 | C22 | 1 | 2 | yes | — |
| 27 | C2×D16×C67 | C22 | 1 | 1 | yes | characteristic |
| 28 | C22×C16×C67 | C22 | 1 | 1 | yes | characteristic |
| 29 | C22×C67⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 30 | Q8×Dic67 | — | 2 | 4 | no | — |
| 31 | D4×Dic67 | — | 2 | 4 | no | — |
| 32 | D4×Dic67 | — | 4 | 4 | no | — |
| 33 | #14524002 | — | 2 | 2 | no | — |
| 34 | #14524002 | — | 4 | 2 | no | — |
| 35 | C2×C4×Dic67 | — | 4 | 2 | no | — |
| 36 | C2×C4×Dic67 | D4 | 1 | 1 | yes | characteristic |
| 37 | C2×C67⋊C16 | — | 2 | 2 | no | — |
| 38 | C2×C67⋊C16 | C23 | 1 | 2 | yes | — |
| 39 | #14523977 | C23 | 1 | 4 | yes | — |
| 40 | #14523975 | — | 2 | 2 | no | — |
| 41 | #14523967 | — | 2 | 4 | no | — |
| 42 | #14523964 | — | 2 | 4 | no | — |
| 43 | C8×Dic67 | D4 | 1 | 2 | yes | — |
| 44 | C8×Dic67 | C23 | 1 | 2 | yes | — |
| 45 | C2×#41×C67 | — | 2 | 2 | no | — |
| 46 | #135×C67 | — | 2 | 4 | no | — |
| 47 | #135×C67 | C2×C4 | 1 | 8 | yes | complemented |
| 48 | C2×Q16×C67 | C23 | 1 | 2 | yes | — |
| 49 | C2×SD16×C67 | — | 2 | 2 | no | — |
| 50 | C2×D8×C67 | C23 | 1 | 2 | yes | — |
| 51 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 52 | Q32×C67 | C2×C4 | 1 | 4 | yes | complemented |
| 53 | SD32×C67 | C2×C4 | 1 | 8 | yes | complemented |
| 54 | D16×C67 | C2×C4 | 1 | 4 | yes | complemented |
| 55 | C2×C16×C67 | C2×C4 | 1 | 4 | yes | complemented |
| 56 | C2×C16×C67 | C23 | 1 | 1 | yes | characteristic |
| 57 | C2×C16×C67 | C23 | 1 | 1 | yes | characteristic |
| 58 | C22×Dic67 | — | 4 | 4 | no | — |
| 59 | C22×Dic67 | D8 | 1 | 2 | yes | — |
| 60 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 61 | C2×C67⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 62 | C4⋊Dic67 | — | 2 | 4 | no | — |
| 63 | C4⋊Dic67 | — | 4 | 4 | no | — |
| 64 | C22⋊Dic67 | — | 4 | 2 | no | — |
| 65 | C22⋊Dic67 | — | 4 | 4 | no | — |
| 66 | C4×Dic67 | — | 4 | 2 | no | — |
| 67 | C4×Dic67 | — | 4 | 4 | no | — |
| 68 | C4×Dic67 | D8 | 1 | 1 | yes | characteristic |
| 69 | C4×Dic67 | D8 | 1 | 1 | yes | characteristic |
| 70 | C4×Dic67 | C2×D4 | 1 | 2 | yes | — |
| 71 | C67⋊C16 | — | 2 | 2 | no | — |
| 72 | #41×C67 | — | 2 | 8 | no | — |
| 73 | #41×C67 | — | 4 | 4 | no | — |
| 74 | C2×Q8×C67 | — | 2 | 2 | no | — |
| 75 | C2×D4×C67 | — | 2 | 2 | no | — |
| 76 | C2×D4×C67 | — | 4 | 2 | no | — |
| 77 | C22×C4×C67 | — | 4 | 2 | no | — |
| 78 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | Q16×C67 | — | 2 | 2 | no | — |
| 80 | Q16×C67 | C22×C4 | 1 | 4 | yes | — |
| 81 | SD16×C67 | — | 2 | 8 | no | — |
| 82 | D8×C67 | — | 2 | 2 | no | — |
| 83 | D8×C67 | C22×C4 | 1 | 4 | yes | — |
| 84 | C2×C8×C67 | C22×C4 | 1 | 2 | yes | — |
| 85 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C67 | #41 | 1 | 2 | yes | — |
| 87 | C2×C8×C67 | C24 | 1 | 1 | yes | characteristic |
| 88 | C16×C67 | C22×C4 | 1 | 2 | yes | — |
| 89 | C16×C67 | C22×C4 | 1 | 2 | yes | — |
| 90 | C23×C67 | — | 4 | 2 | no | — |
| 91 | C23×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 92 | Q8×C67 | — | 2 | 4 | no | — |
| 93 | Q8×C67 | — | 4 | 2 | no | — |
| 94 | D4×C67 | — | 2 | 4 | no | — |
| 95 | D4×C67 | — | 4 | 8 | no | — |
| 96 | D4×C67 | — | 4 | 2 | no | — |
| 97 | C2×C4×C67 | — | 4 | 2 | no | — |
| 98 | C2×C4×C67 | — | 4 | 8 | no | — |
| 99 | C2×C4×C67 | C4×D4 | 1 | 2 | yes | — |
| 100 | C2×C4×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 101 | C2×C4×C67 | #135 | 1 | 2 | yes | — |
| 102 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 103 | C2×Dic67 | — | 2 | 4 | no | — |
| 104 | C2×Dic67 | — | 4 | 4 | no | — |
| 105 | C2×Dic67 | — | 8 | 8 | no | — |
| 106 | C2×Dic67 | C2×D8 | 1 | 2 | yes | — |
| 107 | C2×Dic67 | C2×D8 | 1 | 2 | yes | — |
| 108 | C8×C67 | C4×D4 | 1 | 2 | yes | — |
| 109 | C8×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 110 | C8×C67 | C2×#41 | 1 | 1 | yes | characteristic |
| 111 | C67⋊C8 | — | 2 | 2 | no | — |
| 112 | C22×C67 | — | 2 | 2 | no | — |
| 113 | C22×C67 | — | 4 | 4 | no | — |
| 114 | C22×C67 | — | 8 | 4 | no | — |
| 115 | C22×C67 | C4×D8 | 1 | 2 | yes | — |
| 116 | C22×C67 | C22×D8 | 1 | 1 | yes | characteristic |
| 117 | C4×C67 | — | 4 | 4 | no | — |
| 118 | C4×C67 | C4×D8 | 1 | 2 | yes | — |
| 119 | C4×C67 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 120 | C4×C67 | C2×#135 | 1 | 1 | yes | characteristic |
| 121 | Dic67 | — | 2 | 2 | no | — |
| 122 | Dic67 | — | 8 | 4 | no | — |
| 123 | Dic67 | #508 | 1 | 4 | yes | direct product factor |
| 124 | C4×#508 | — | 67 | 1 | no | maximal, Sylow |
| 125 | C2×C67 | — | 2 | 2 | no | — |
| 126 | C2×C67 | — | 8 | 4 | no | — |
| 127 | C2×C67 | C2×C4×D8 | 1 | 1 | yes | characteristic |
| 128 | C2×C67 | C2×#508 | 1 | 2 | yes | — |
| 129 | C2×#508 | D67 | 1 | 1 | yes | characteristic |
| 130 | C4×#135 | — | 67 | 2 | no | — |
| 131 | C4×Q32 | — | 67 | 2 | no | — |
| 132 | C4×SD32 | — | 67 | 4 | no | — |
| 133 | C4×D16 | — | 67 | 2 | no | — |
| 134 | #2158 | — | 67 | 1 | no | — |
| 135 | #2138 | — | 67 | 2 | no | — |
| 136 | C2×C4×C16 | — | 67 | 1 | no | — |
| 137 | C67 | C4×#508 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 138 | C2×#135 | C2×D67 | 1 | 2 | yes | — |
| 139 | C4×#41 | — | 134 | 2 | no | — |
| 140 | #508 | Dic67 | 1 | 8 | yes | direct product factor |
| 141 | C2×Q32 | C2×D67 | 1 | 1 | yes | characteristic |
| 142 | C2×SD32 | C2×D67 | 1 | 2 | yes | — |
| 143 | C2×D16 | C2×D67 | 1 | 1 | yes | characteristic |
| 144 | C22×C16 | — | 67 | 1 | no | — |
| 145 | C22×C16 | C2×D67 | 1 | 1 | yes | characteristic |
| 146 | C4×Q16 | — | 67 | 4 | no | — |
| 147 | C4×SD16 | — | 134 | 4 | no | — |
| 148 | C4×D8 | — | 67 | 4 | no | — |
| 149 | #427 | — | 67 | 2 | no | — |
| 150 | #416 | — | 134 | 2 | no | — |
| 151 | C2×C4×C8 | — | 67 | 1 | no | — |
| 152 | #367 | — | 67 | 2 | no | — |
| 153 | #366 | — | 67 | 2 | no | — |
| 154 | #358 | — | 67 | 4 | no | — |
| 155 | #357 | — | 67 | 4 | no | — |
| 156 | C4×C16 | — | 67 | 2 | no | — |
| 157 | C4×C16 | — | 67 | 2 | no | — |
| 158 | C2×#41 | — | 2 | 2 | no | — |
| 159 | #135 | — | 2 | 4 | no | — |
| 160 | #135 | C2×Dic67 | 1 | 8 | yes | complemented |
| 161 | C2×Q16 | C22×D67 | 1 | 2 | yes | — |
| 162 | C2×SD16 | — | 2 | 2 | no | — |
| 163 | C2×D8 | C22×D67 | 1 | 2 | yes | — |
| 164 | C22×C8 | — | 67 | 1 | no | — |
| 165 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 166 | C4×Q8 | — | 134 | 4 | no | — |
| 167 | C4×D4 | — | 268 | 4 | no | — |
| 168 | C4×D4 | — | 134 | 4 | no | — |
| 169 | #117 | — | 268 | 2 | no | — |
| 170 | #117 | — | 134 | 2 | no | — |
| 171 | C2×C42 | — | 268 | 2 | no | — |
| 172 | C2×C42 | — | 67 | 1 | no | — |
| 173 | Q32 | C2×Dic67 | 1 | 4 | yes | complemented |
| 174 | SD32 | C2×Dic67 | 1 | 8 | yes | complemented |
| 175 | D16 | C2×Dic67 | 1 | 4 | yes | complemented |
| 176 | C2×C16 | — | 67 | 2 | no | — |
| 177 | C2×C16 | — | 134 | 2 | no | — |
| 178 | C2×C16 | C2×Dic67 | 1 | 4 | yes | complemented |
| 179 | C2×C16 | C22×D67 | 1 | 1 | yes | characteristic |
| 180 | C2×C16 | C22×D67 | 1 | 1 | yes | characteristic |
| 181 | #107 | — | 67 | 4 | no | — |
| 182 | #106 | — | 134 | 2 | no | — |
| 183 | #103 | — | 134 | 4 | no | — |
| 184 | #102 | — | 134 | 4 | no | — |
| 185 | C4×C8 | — | 67 | 2 | no | — |
| 186 | C4×C8 | — | 67 | 2 | no | — |
| 187 | #41 | — | 2 | 8 | no | — |
| 188 | #41 | — | 4 | 4 | no | — |
| 189 | C2×Q8 | — | 2 | 2 | no | — |
| 190 | C2×D4 | — | 2 | 2 | no | — |
| 191 | C2×D4 | — | 4 | 2 | no | — |
| 192 | C22×C4 | — | 4 | 2 | no | — |
| 193 | C22×C4 | — | 268 | 4 | no | — |
| 194 | C22×C4 | — | 67 | 2 | no | — |
| 195 | C22×C4 | D4×D67 | 1 | 1 | yes | characteristic |
| 196 | Q16 | — | 2 | 2 | no | — |
| 197 | Q16 | C22×Dic67 | 1 | 4 | yes | — |
| 198 | SD16 | — | 2 | 8 | no | — |
| 199 | D8 | — | 2 | 2 | no | — |
| 200 | D8 | C22×Dic67 | 1 | 4 | yes | — |
| 201 | C2×C8 | — | 67 | 2 | no | — |
| 202 | C2×C8 | — | 134 | 2 | no | — |
| 203 | C2×C8 | C22×Dic67 | 1 | 2 | yes | — |
| 204 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 205 | C2×C8 | #11761627 | 1 | 2 | yes | — |
| 206 | C2×C8 | C23×D67 | 1 | 1 | yes | characteristic |
| 207 | C4⋊C4 | — | 268 | 4 | no | — |
| 208 | C4⋊C4 | — | 134 | 4 | no | — |
| 209 | C22⋊C4 | — | 268 | 2 | no | — |
| 210 | C22⋊C4 | — | 268 | 4 | no | — |
| 211 | C42 | — | 268 | 4 | no | — |
| 212 | C42 | — | 268 | 2 | no | — |
| 213 | C42 | — | 67 | 1 | no | — |
| 214 | C42 | — | 67 | 1 | no | — |
| 215 | C42 | — | 67 | 2 | no | — |
| 216 | C16 | — | 134 | 2 | no | — |
| 217 | C16 | C22×Dic67 | 1 | 2 | yes | — |
| 218 | C16 | C22×Dic67 | 1 | 2 | yes | — |
| 219 | C23 | — | 4 | 2 | no | — |
| 220 | C23 | D8×D67 | 1 | 1 | yes | characteristic |
| 221 | Q8 | — | 2 | 4 | no | — |
| 222 | Q8 | — | 4 | 2 | no | — |
| 223 | D4 | — | 2 | 4 | no | — |
| 224 | D4 | — | 4 | 8 | no | — |
| 225 | D4 | — | 4 | 2 | no | — |
| 226 | C2×C4 | — | 4 | 2 | no | — |
| 227 | C2×C4 | — | 4 | 8 | no | — |
| 228 | C2×C4 | — | 268 | 4 | no | — |
| 229 | C2×C4 | — | 536 | 8 | no | — |
| 230 | C2×C4 | — | 67 | 2 | no | — |
| 231 | C2×C4 | — | 67 | 2 | no | — |
| 232 | C2×C4 | — | 134 | 4 | no | — |
| 233 | C2×C4 | D4×Dic67 | 1 | 2 | yes | — |
| 234 | C2×C4 | D8×D67 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | #14524062 | 1 | 2 | yes | — |
| 236 | C2×C4 | C2×D4×D67 | 1 | 1 | yes | characteristic |
| 237 | C8 | — | 134 | 2 | no | — |
| 238 | C8 | D4×Dic67 | 1 | 2 | yes | — |
| 239 | C8 | C23×Dic67 | 1 | 1 | yes | characteristic |
| 240 | C8 | C2×#11761627 | 1 | 1 | yes | characteristic |
| 241 | C22 | — | 2 | 2 | no | — |
| 242 | C22 | — | 4 | 4 | no | — |
| 243 | C22 | — | 8 | 4 | no | — |
| 244 | C22 | D8×Dic67 | 1 | 2 | yes | — |
| 245 | C22 | C2×D8×D67 | 1 | 1 | yes | characteristic |
| 246 | C4 | — | 4 | 4 | no | — |
| 247 | C4 | — | 536 | 4 | no | — |
| 248 | C4 | — | 67 | 4 | no | — |
| 249 | C4 | — | 134 | 2 | no | — |
| 250 | C4 | D8×Dic67 | 1 | 2 | yes | — |
| 251 | C4 | C2×D4×Dic67 | 1 | 1 | yes | characteristic |
| 252 | C4 | C2×#14524062 | 1 | 1 | yes | characteristic |
| 253 | C2 | — | 2 | 2 | no | — |
| 254 | C2 | — | 8 | 4 | no | — |
| 255 | C2 | C2×D8×Dic67 | 1 | 1 | yes | characteristic |
| 256 | C2 | #508×D67 | 1 | 2 | yes | — |
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