← #30856465
Subgroups — #30856465
Order 17152.
770 conjugacy classes of subgroups.
The subgroup lattice has 270 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22568838 | C2 | 1 | 2 | yes | maximal |
| 2 | C22×#14523953 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22560382 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22560348 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22560247 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #22560245 | C2 | 1 | 2 | yes | maximal |
| 7 | #22560117 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #22560114 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C67×#1810 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C23×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×C4×Dic67 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16863450 | — | 2 | 4 | no | — |
| 13 | #16863446 | — | 2 | 4 | no | — |
| 14 | C2×#14523979 | C22 | 1 | 2 | yes | — |
| 15 | C2×#14523978 | C22 | 1 | 4 | yes | — |
| 16 | #16863419 | C22 | 1 | 4 | yes | — |
| 17 | #16863360 | C22 | 1 | 2 | yes | — |
| 18 | M16×Dic67 | C22 | 1 | 4 | yes | — |
| 19 | C2×#14523953 | C4 | 1 | 8 | yes | — |
| 20 | C2×#14523953 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#14523953 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#14523953 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#14523953 | C22 | 1 | 1 | yes | characteristic |
| 24 | #16863161 | C22 | 1 | 2 | yes | — |
| 25 | #16863160 | C22 | 1 | 4 | yes | — |
| 26 | C23×C8×C67 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#108×C67 | C22 | 1 | 2 | yes | — |
| 28 | #407×C67 | C22 | 1 | 2 | yes | — |
| 29 | #341×C67 | C22 | 1 | 2 | yes | — |
| 30 | C23×Dic67 | Q8 | 1 | 2 | yes | — |
| 31 | C2×#11761612 | C23 | 1 | 4 | yes | — |
| 32 | C22×C67⋊C8 | — | 2 | 4 | no | — |
| 33 | C22×C67⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 34 | C22×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 35 | C22×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×C4×Dic67 | C2×C4 | 1 | 4 | yes | — |
| 37 | C2×C4×Dic67 | D4 | 1 | 4 | yes | — |
| 38 | C2×C4×Dic67 | Q8 | 1 | 2 | yes | — |
| 39 | C2×C4×Dic67 | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×C4×Dic67 | C23 | 1 | 1 | yes | characteristic |
| 41 | #14523979 | — | 2 | 4 | no | — |
| 42 | #14523978 | — | 2 | 8 | no | — |
| 43 | Dic67⋊C8 | — | 2 | 8 | no | — |
| 44 | C22⋊(C67⋊C8) | — | 2 | 4 | no | — |
| 45 | #14523953 | — | 2 | 4 | no | — |
| 46 | #14523953 | C2×C4 | 1 | 4 | yes | — |
| 47 | #14523953 | C2×C4 | 1 | 4 | yes | — |
| 48 | #14523953 | C2×C4 | 1 | 4 | yes | — |
| 49 | #14523953 | C2×C4 | 1 | 4 | yes | — |
| 50 | C8×Dic67 | — | 2 | 4 | no | — |
| 51 | C23×C4×C67 | C23 | 1 | 1 | yes | characteristic |
| 52 | C2×M16×C67 | C23 | 1 | 4 | yes | — |
| 53 | C22×C8×C67 | — | 2 | 4 | no | — |
| 54 | C22×C8×C67 | C2×C4 | 1 | 4 | yes | — |
| 55 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 56 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 57 | #108×C67 | — | 2 | 4 | no | — |
| 58 | C22⋊C8×C67 | — | 2 | 4 | no | — |
| 59 | C22×Dic67 | — | 4 | 4 | no | — |
| 60 | C22×Dic67 | C4⋊C4 | 1 | 4 | yes | — |
| 61 | C22×Dic67 | C4⋊C4 | 1 | 4 | yes | — |
| 62 | C22×Dic67 | C2×Q8 | 1 | 2 | yes | — |
| 63 | C22×Dic67 | C2×Q8 | 1 | 2 | yes | — |
| 64 | #11761612 | — | 2 | 4 | no | — |
| 65 | #11761612 | — | 2 | 4 | no | — |
| 66 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 67 | C2×C67⋊C8 | — | 2 | 8 | no | — |
| 68 | C2×C67⋊C8 | — | 2 | 4 | no | — |
| 69 | C2×C67⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 70 | C2×C67⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 71 | C2×C67⋊C8 | #41 | 1 | 4 | yes | — |
| 72 | C4×Dic67 | C4⋊C4 | 1 | 8 | yes | — |
| 73 | C4×Dic67 | C22×C4 | 1 | 2 | yes | — |
| 74 | C4×Dic67 | C22×C4 | 1 | 2 | yes | — |
| 75 | C4×Dic67 | C2×D4 | 1 | 2 | yes | — |
| 76 | C4×Dic67 | C2×D4 | 1 | 2 | yes | — |
| 77 | C24×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 78 | C22×C4×C67 | — | 2 | 2 | no | — |
| 79 | C22×C4×C67 | — | 2 | 2 | no | — |
| 80 | C22×C4×C67 | C22×C4 | 1 | 2 | yes | — |
| 81 | C22×C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 82 | C22×C4×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 83 | C22×C4×C67 | C24 | 1 | 1 | yes | characteristic |
| 84 | M16×C67 | — | 2 | 4 | no | — |
| 85 | M16×C67 | — | 2 | 4 | no | — |
| 86 | C2×C8×C67 | — | 2 | 2 | no | — |
| 87 | C2×C8×C67 | — | 2 | 8 | no | — |
| 88 | C2×C8×C67 | — | 2 | 4 | no | — |
| 89 | C2×C8×C67 | C22×C4 | 1 | 2 | yes | — |
| 90 | C2×C8×C67 | C22×C4 | 1 | 2 | yes | — |
| 91 | C2×C8×C67 | #41 | 1 | 4 | yes | — |
| 92 | C23×C67 | — | 2 | 2 | no | — |
| 93 | C23×C67 | — | 2 | 4 | no | — |
| 94 | C23×C67 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 95 | C23×C67 | C22×Q8 | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C67 | — | 2 | 2 | no | — |
| 97 | C2×C4×C67 | — | 2 | 2 | no | — |
| 98 | C2×C4×C67 | — | 2 | 2 | no | — |
| 99 | C2×C4×C67 | — | 2 | 4 | no | — |
| 100 | C2×C4×C67 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 101 | C2×C4×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 102 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 103 | C2×C4×C67 | C2×#41 | 1 | 2 | yes | — |
| 104 | C2×C4×C67 | C2×#41 | 1 | 2 | yes | — |
| 105 | C2×Dic67 | — | 2 | 8 | no | — |
| 106 | C2×Dic67 | — | 4 | 4 | no | — |
| 107 | C2×Dic67 | — | 4 | 2 | no | — |
| 108 | C2×Dic67 | — | 4 | 2 | no | — |
| 109 | C2×Dic67 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 110 | C2×Dic67 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 111 | C8×C67 | — | 2 | 2 | no | — |
| 112 | C8×C67 | — | 4 | 4 | no | — |
| 113 | C8×C67 | #117 | 1 | 4 | yes | — |
| 114 | C67⋊C8 | — | 2 | 2 | no | — |
| 115 | C67⋊C8 | — | 4 | 4 | no | — |
| 116 | C67⋊C8 | #117 | 1 | 4 | yes | — |
| 117 | C22×C67 | — | 2 | 8 | no | — |
| 118 | C22×C67 | — | 2 | 2 | no | — |
| 119 | C22×C67 | — | 2 | 2 | no | — |
| 120 | C22×C67 | — | 2 | 2 | no | — |
| 121 | C22×C67 | #430 | 1 | 4 | yes | — |
| 122 | C22×C67 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 123 | C22×C67 | #533 | 1 | 2 | yes | — |
| 124 | C4×C67 | — | 2 | 2 | no | — |
| 125 | C4×C67 | C2×#117 | 1 | 1 | yes | characteristic |
| 126 | C4×C67 | C2×#117 | 1 | 1 | yes | characteristic |
| 127 | C4×C67 | #532 | 1 | 2 | yes | — |
| 128 | Dic67 | — | 2 | 4 | no | — |
| 129 | Dic67 | — | 2 | 4 | no | — |
| 130 | #20830 | — | 67 | 1 | no | maximal, Sylow |
| 131 | C2×C67 | — | 2 | 2 | no | — |
| 132 | C2×C67 | — | 2 | 4 | no | — |
| 133 | C2×C67 | #2285 | 1 | 1 | yes | characteristic |
| 134 | C2×C67 | C2×#430 | 1 | 2 | yes | — |
| 135 | #2920 | — | 67 | 2 | no | — |
| 136 | C22×#97 | — | 67 | 1 | no | — |
| 137 | #1849 | — | 67 | 2 | no | — |
| 138 | #1838 | — | 67 | 2 | no | — |
| 139 | #1810 | — | 67 | 2 | no | — |
| 140 | #1810 | — | 67 | 1 | no | — |
| 141 | #1810 | D67 | 1 | 1 | yes | characteristic |
| 142 | #1778 | — | 67 | 2 | no | — |
| 143 | #1778 | — | 67 | 2 | no | — |
| 144 | C67 | #20830 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 145 | C23×C8 | — | 67 | 1 | no | — |
| 146 | C23×C8 | C2×D67 | 1 | 1 | yes | characteristic |
| 147 | C22×C42 | — | 67 | 1 | no | — |
| 148 | #432 | — | 134 | 4 | no | — |
| 149 | #431 | — | 134 | 4 | no | — |
| 150 | C2×#108 | — | 67 | 4 | no | — |
| 151 | C2×#108 | — | 67 | 2 | no | — |
| 152 | C2×#108 | C2×D67 | 1 | 2 | yes | — |
| 153 | #423 | — | 67 | 4 | no | — |
| 154 | #407 | — | 67 | 2 | no | — |
| 155 | #407 | C2×D67 | 1 | 2 | yes | — |
| 156 | C4×M16 | — | 67 | 4 | no | — |
| 157 | C2×#97 | — | 67 | 8 | no | — |
| 158 | C2×#97 | — | 67 | 1 | no | — |
| 159 | C2×#97 | — | 67 | 1 | no | — |
| 160 | C2×#97 | — | 67 | 1 | no | — |
| 161 | C2×#97 | — | 67 | 1 | no | — |
| 162 | #341 | — | 67 | 4 | no | — |
| 163 | #341 | — | 67 | 2 | no | — |
| 164 | #341 | C2×D67 | 1 | 2 | yes | — |
| 165 | C23×C4 | — | 67 | 2 | no | — |
| 166 | C23×C4 | C22×D67 | 1 | 1 | yes | characteristic |
| 167 | C2×M16 | — | 67 | 4 | no | — |
| 168 | C2×M16 | C22×D67 | 1 | 4 | yes | — |
| 169 | C22×C8 | — | 2 | 4 | no | — |
| 170 | C22×C8 | — | 67 | 1 | no | — |
| 171 | C22×C8 | — | 67 | 4 | no | — |
| 172 | C22×C8 | — | 67 | 1 | no | — |
| 173 | C22×C8 | — | 134 | 4 | no | — |
| 174 | C22×C8 | C4×D67 | 1 | 4 | yes | — |
| 175 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 176 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 177 | C2×C42 | — | 67 | 4 | no | — |
| 178 | C2×C42 | — | 67 | 4 | no | — |
| 179 | C2×C42 | — | 67 | 2 | no | — |
| 180 | C2×C42 | — | 67 | 1 | no | — |
| 181 | C2×C42 | — | 67 | 1 | no | — |
| 182 | #108 | — | 2 | 4 | no | — |
| 183 | #108 | — | 134 | 8 | no | — |
| 184 | #108 | — | 134 | 4 | no | — |
| 185 | C4⋊C8 | — | 134 | 8 | no | — |
| 186 | C22⋊C8 | — | 2 | 4 | no | — |
| 187 | C22⋊C8 | — | 134 | 4 | no | — |
| 188 | #97 | — | 67 | 4 | no | — |
| 189 | #97 | — | 67 | 4 | no | — |
| 190 | #97 | — | 67 | 4 | no | — |
| 191 | #97 | — | 67 | 4 | no | — |
| 192 | #97 | — | 134 | 4 | no | — |
| 193 | C4×C8 | — | 134 | 4 | no | — |
| 194 | C24 | Q8×D67 | 1 | 1 | yes | characteristic |
| 195 | C22×C4 | — | 2 | 2 | no | — |
| 196 | C22×C4 | — | 2 | 2 | no | — |
| 197 | C22×C4 | — | 268 | 4 | no | — |
| 198 | C22×C4 | — | 67 | 4 | no | — |
| 199 | C22×C4 | — | 67 | 2 | no | — |
| 200 | C22×C4 | — | 67 | 2 | no | — |
| 201 | C22×C4 | — | 67 | 4 | no | — |
| 202 | C22×C4 | C2×C4×D67 | 1 | 2 | yes | — |
| 203 | C22×C4 | D4×D67 | 1 | 2 | yes | — |
| 204 | C22×C4 | Q8×D67 | 1 | 1 | yes | characteristic |
| 205 | C22×C4 | C23×D67 | 1 | 1 | yes | characteristic |
| 206 | M16 | — | 2 | 4 | no | — |
| 207 | M16 | — | 2 | 4 | no | — |
| 208 | M16 | — | 134 | 4 | no | — |
| 209 | M16 | — | 134 | 4 | no | — |
| 210 | C2×C8 | — | 2 | 2 | no | — |
| 211 | C2×C8 | — | 2 | 8 | no | — |
| 212 | C2×C8 | — | 2 | 4 | no | — |
| 213 | C2×C8 | — | 67 | 2 | no | — |
| 214 | C2×C8 | — | 67 | 4 | no | — |
| 215 | C2×C8 | — | 67 | 2 | no | — |
| 216 | C2×C8 | — | 134 | 4 | no | — |
| 217 | C2×C8 | — | 134 | 8 | no | — |
| 218 | C2×C8 | — | 134 | 2 | no | — |
| 219 | C2×C8 | C2×C4×D67 | 1 | 2 | yes | — |
| 220 | C2×C8 | C2×C4×D67 | 1 | 2 | yes | — |
| 221 | C2×C8 | #11761627 | 1 | 4 | yes | — |
| 222 | C42 | — | 67 | 8 | no | — |
| 223 | C42 | — | 67 | 2 | no | — |
| 224 | C42 | — | 67 | 2 | no | — |
| 225 | C42 | — | 67 | 2 | no | — |
| 226 | C42 | — | 67 | 2 | no | — |
| 227 | C23 | — | 2 | 2 | no | — |
| 228 | C23 | — | 2 | 4 | no | — |
| 229 | C23 | C4⋊C4×D67 | 1 | 2 | yes | — |
| 230 | C23 | C2×Q8×D67 | 1 | 1 | yes | characteristic |
| 231 | C2×C4 | — | 2 | 2 | no | — |
| 232 | C2×C4 | — | 2 | 2 | no | — |
| 233 | C2×C4 | — | 2 | 2 | no | — |
| 234 | C2×C4 | — | 2 | 4 | no | — |
| 235 | C2×C4 | — | 268 | 4 | no | — |
| 236 | C2×C4 | — | 268 | 2 | no | — |
| 237 | C2×C4 | — | 268 | 2 | no | — |
| 238 | C2×C4 | — | 67 | 4 | no | — |
| 239 | C2×C4 | — | 67 | 4 | no | — |
| 240 | C2×C4 | — | 134 | 8 | no | — |
| 241 | C2×C4 | C4⋊C4×D67 | 1 | 2 | yes | — |
| 242 | C2×C4 | C22×C4×D67 | 1 | 1 | yes | characteristic |
| 243 | C2×C4 | C2×D4×D67 | 1 | 1 | yes | characteristic |
| 244 | C2×C4 | #41×D67 | 1 | 2 | yes | — |
| 245 | C2×C4 | C2×#11761627 | 1 | 2 | yes | — |
| 246 | C8 | — | 2 | 2 | no | — |
| 247 | C8 | — | 4 | 4 | no | — |
| 248 | C8 | — | 268 | 4 | no | — |
| 249 | C8 | — | 67 | 4 | no | — |
| 250 | C8 | — | 134 | 2 | no | — |
| 251 | C8 | #14523999 | 1 | 4 | yes | — |
| 252 | C22 | — | 2 | 8 | no | — |
| 253 | C22 | — | 2 | 2 | no | — |
| 254 | C22 | — | 2 | 2 | no | — |
| 255 | C22 | — | 2 | 2 | no | — |
| 256 | C22 | #16863443 | 1 | 4 | yes | — |
| 257 | C22 | C2×C4⋊C4×D67 | 1 | 1 | yes | characteristic |
| 258 | C22 | #16863948 | 1 | 2 | yes | — |
| 259 | C4 | — | 2 | 2 | no | — |
| 260 | C4 | — | 134 | 4 | no | — |
| 261 | C4 | — | 134 | 4 | no | — |
| 262 | C4 | #117×D67 | 1 | 1 | yes | characteristic |
| 263 | C4 | C2×#14523999 | 1 | 1 | yes | characteristic |
| 264 | C4 | #16863941 | 1 | 2 | yes | — |
| 265 | C2 | — | 2 | 2 | no | — |
| 266 | C2 | — | 2 | 4 | no | — |
| 267 | C2 | #22562611 | 1 | 1 | yes | characteristic |
| 268 | C2 | C2×#16863443 | 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