← #30866465
Subgroups — #30866465
Order 17152.
658 conjugacy classes of subgroups.
The subgroup lattice has 262 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16863457 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22568773 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22560984 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #22560413 | C2 | 1 | 4 | yes | maximal |
| 5 | #22560383 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22560380 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #22560349 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #22560346 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C67×#2921 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C22×#11761612 | C22 | 1 | 2 | yes | — |
| 11 | #16863457 | C4 | 1 | 8 | yes | — |
| 12 | C2×#14523978 | C22 | 1 | 4 | yes | — |
| 13 | C2×#14523978 | C22 | 1 | 4 | yes | — |
| 14 | #16863423 | — | 2 | 4 | no | — |
| 15 | #16863420 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16863420 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16863420 | C22 | 1 | 2 | yes | — |
| 18 | C2×C4⋊(C67⋊C8) | C22 | 1 | 2 | yes | — |
| 19 | C2×C22⋊(C67⋊C8) | C22 | 1 | 2 | yes | — |
| 20 | C2×#14523954 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C4×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 22 | #16863160 | C22 | 1 | 4 | yes | — |
| 23 | #16863160 | C22 | 1 | 4 | yes | — |
| 24 | C2×#117×C67 | C22 | 1 | 1 | yes | characteristic |
| 25 | #433×C67 | — | 2 | 4 | no | — |
| 26 | #423×C67 | C22 | 1 | 1 | yes | characteristic |
| 27 | #423×C67 | C22 | 1 | 1 | yes | characteristic |
| 28 | #407×C67 | C22 | 1 | 2 | yes | — |
| 29 | C4×M16×C67 | C22 | 1 | 2 | yes | — |
| 30 | C2×#11761612 | — | 2 | 2 | no | — |
| 31 | C2×#11761612 | — | 4 | 4 | no | — |
| 32 | C2×#11761612 | C23 | 1 | 2 | yes | — |
| 33 | C2×#11761612 | C23 | 1 | 2 | yes | — |
| 34 | C22×C67⋊C8 | — | 2 | 2 | no | — |
| 35 | C22×C67⋊C8 | C23 | 1 | 2 | yes | — |
| 36 | C22×C67⋊C8 | C23 | 1 | 2 | yes | — |
| 37 | #14523978 | — | 4 | 4 | no | — |
| 38 | #14523978 | — | 4 | 4 | no | — |
| 39 | C4⋊(C67⋊C8) | — | 2 | 2 | no | — |
| 40 | C4⋊(C67⋊C8) | — | 2 | 2 | no | — |
| 41 | C4⋊(C67⋊C8) | — | 2 | 4 | no | — |
| 42 | C4⋊(C67⋊C8) | C2×C4 | 1 | 8 | yes | — |
| 43 | C22⋊(C67⋊C8) | C2×C4 | 1 | 8 | yes | — |
| 44 | #14523954 | C2×C4 | 1 | 4 | yes | — |
| 45 | C4×C67⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 46 | C23×C4×C67 | C23 | 1 | 1 | yes | characteristic |
| 47 | C2×M16×C67 | C23 | 1 | 2 | yes | — |
| 48 | C2×M16×C67 | C23 | 1 | 2 | yes | — |
| 49 | #117×C67 | C2×C4 | 1 | 4 | yes | — |
| 50 | #117×C67 | D4 | 1 | 4 | yes | — |
| 51 | C2×C4⋊C4×C67 | Q8 | 1 | 2 | yes | — |
| 52 | C2×C22⋊C4×C67 | Q8 | 1 | 2 | yes | — |
| 53 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 55 | C4⋊C8×C67 | — | 2 | 2 | no | — |
| 56 | C4⋊C8×C67 | — | 2 | 2 | no | — |
| 57 | C22⋊C8×C67 | — | 2 | 4 | no | — |
| 58 | #97×C67 | — | 2 | 2 | no | — |
| 59 | C4×C8×C67 | — | 2 | 2 | no | — |
| 60 | #11761612 | — | 2 | 4 | no | — |
| 61 | #11761612 | — | 2 | 4 | no | — |
| 62 | #11761612 | — | 4 | 4 | no | — |
| 63 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 64 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 65 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 66 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 67 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 68 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 69 | C2×C67⋊C8 | — | 4 | 4 | no | — |
| 70 | C2×C67⋊C8 | C22×C4 | 1 | 4 | yes | — |
| 71 | C2×C67⋊C8 | C22×C4 | 1 | 4 | yes | — |
| 72 | C24×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 73 | C22×C4×C67 | — | 2 | 2 | no | — |
| 74 | C22×C4×C67 | — | 2 | 2 | no | — |
| 75 | C22×C4×C67 | C22×C4 | 1 | 2 | yes | — |
| 76 | C22×C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 77 | C22×C4×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 78 | C22×C4×C67 | C2×Q8 | 1 | 2 | yes | — |
| 79 | C22×C4×C67 | C2×Q8 | 1 | 2 | yes | — |
| 80 | C22×C4×C67 | C24 | 1 | 1 | yes | characteristic |
| 81 | M16×C67 | — | 2 | 4 | no | — |
| 82 | M16×C67 | — | 2 | 4 | no | — |
| 83 | C2×C8×C67 | — | 2 | 2 | no | — |
| 84 | C2×C8×C67 | — | 2 | 2 | no | — |
| 85 | C4⋊C4×C67 | C4⋊C4 | 1 | 8 | yes | — |
| 86 | C22⋊C4×C67 | C4⋊C4 | 1 | 8 | yes | — |
| 87 | C42×C67 | C22×C4 | 1 | 2 | yes | — |
| 88 | C42×C67 | C22×C4 | 1 | 2 | yes | — |
| 89 | C42×C67 | C2×D4 | 1 | 2 | yes | — |
| 90 | C42×C67 | C2×D4 | 1 | 2 | yes | — |
| 91 | C23×C67 | — | 2 | 2 | no | — |
| 92 | C23×C67 | — | 4 | 2 | no | — |
| 93 | C23×C67 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 94 | C23×C67 | C22×Q8 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C67 | — | 2 | 2 | no | — |
| 96 | C2×C4×C67 | — | 2 | 4 | no | — |
| 97 | C2×C4×C67 | — | 2 | 2 | no | — |
| 98 | C2×C4×C67 | — | 2 | 2 | no | — |
| 99 | C2×C4×C67 | — | 4 | 2 | no | — |
| 100 | C2×C4×C67 | — | 4 | 2 | no | — |
| 101 | C2×C4×C67 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 102 | C2×C4×C67 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 103 | C2×C4×C67 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 104 | C2×C4×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C67 | H(5,𝔽2) | 1 | 2 | yes | — |
| 107 | C2×C4×C67 | 2−1+4 | 1 | 2 | yes | — |
| 108 | C8×C67 | — | 4 | 2 | no | — |
| 109 | C8×C67 | — | 4 | 2 | no | — |
| 110 | C67⋊C8 | — | 2 | 4 | no | — |
| 111 | C67⋊C8 | — | 2 | 4 | no | — |
| 112 | C67⋊C8 | — | 4 | 2 | no | — |
| 113 | C67⋊C8 | — | 4 | 2 | no | — |
| 114 | C22×C67 | — | 2 | 2 | no | — |
| 115 | C22×C67 | — | 2 | 2 | no | — |
| 116 | C22×C67 | — | 2 | 4 | no | — |
| 117 | C22×C67 | — | 8 | 2 | no | — |
| 118 | C22×C67 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 119 | C22×C67 | #544 | 1 | 2 | yes | — |
| 120 | C4×C67 | — | 2 | 2 | no | — |
| 121 | C4×C67 | — | 2 | 4 | no | — |
| 122 | C4×C67 | — | 2 | 4 | no | — |
| 123 | C4×C67 | #520 | 1 | 2 | yes | — |
| 124 | C4×C67 | #535 | 1 | 1 | yes | characteristic |
| 125 | C4×C67 | #536 | 1 | 1 | yes | characteristic |
| 126 | #21995 | — | 67 | 1 | no | maximal, Sylow |
| 127 | C2×C67 | — | 2 | 2 | no | — |
| 128 | C2×C67 | — | 4 | 2 | no | — |
| 129 | C2×C67 | #2317 | 1 | 1 | yes | characteristic |
| 130 | C2×C67 | #2916 | 1 | 2 | yes | — |
| 131 | #2921 | D67 | 1 | 1 | yes | characteristic |
| 132 | C2×#433 | — | 67 | 1 | no | — |
| 133 | #2905 | — | 67 | 1 | no | — |
| 134 | #1950 | — | 67 | 4 | no | — |
| 135 | #1855 | — | 67 | 4 | no | — |
| 136 | #1849 | — | 67 | 1 | no | — |
| 137 | #1848 | — | 67 | 1 | no | — |
| 138 | #1838 | — | 67 | 1 | no | — |
| 139 | #1837 | — | 67 | 1 | no | — |
| 140 | C67 | #21995 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 141 | C22×M16 | — | 67 | 2 | no | — |
| 142 | C2×#117 | C2×D67 | 1 | 1 | yes | characteristic |
| 143 | #433 | — | 2 | 4 | no | — |
| 144 | #433 | — | 67 | 8 | no | — |
| 145 | C2×#108 | — | 67 | 4 | no | — |
| 146 | C2×#108 | — | 67 | 4 | no | — |
| 147 | #424 | — | 134 | 4 | no | — |
| 148 | #423 | — | 67 | 1 | no | — |
| 149 | #423 | — | 67 | 1 | no | — |
| 150 | #423 | — | 67 | 2 | no | — |
| 151 | #423 | C2×D67 | 1 | 1 | yes | characteristic |
| 152 | #423 | C2×D67 | 1 | 1 | yes | characteristic |
| 153 | C2×C4⋊C8 | — | 67 | 2 | no | — |
| 154 | #407 | C2×D67 | 1 | 2 | yes | — |
| 155 | C2×C22⋊C8 | — | 67 | 2 | no | — |
| 156 | C4×M16 | C2×D67 | 1 | 2 | yes | — |
| 157 | C2×#97 | — | 67 | 1 | no | — |
| 158 | C2×C4×C8 | — | 67 | 1 | no | — |
| 159 | #341 | — | 67 | 4 | no | — |
| 160 | #341 | — | 67 | 4 | no | — |
| 161 | C23×C4 | C22×D67 | 1 | 1 | yes | characteristic |
| 162 | C2×M16 | — | 268 | 4 | no | — |
| 163 | C2×M16 | — | 67 | 2 | no | — |
| 164 | C2×M16 | — | 67 | 2 | no | — |
| 165 | C2×M16 | — | 134 | 2 | no | — |
| 166 | C2×M16 | C22×D67 | 1 | 2 | yes | — |
| 167 | C2×M16 | C22×D67 | 1 | 2 | yes | — |
| 168 | C22×C8 | — | 67 | 2 | no | — |
| 169 | C22×C8 | — | 67 | 2 | no | — |
| 170 | C22×C8 | — | 134 | 2 | no | — |
| 171 | #117 | C4×D67 | 1 | 4 | yes | — |
| 172 | #117 | C22⋊D67 | 1 | 4 | yes | — |
| 173 | C2×C4⋊C4 | Dic134 | 1 | 2 | yes | — |
| 174 | C2×C22⋊C4 | Dic134 | 1 | 2 | yes | — |
| 175 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 176 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 177 | #108 | — | 268 | 4 | no | — |
| 178 | #108 | — | 268 | 4 | no | — |
| 179 | C4⋊C8 | — | 2 | 2 | no | — |
| 180 | C4⋊C8 | — | 2 | 2 | no | — |
| 181 | C4⋊C8 | — | 67 | 8 | no | — |
| 182 | C4⋊C8 | — | 134 | 4 | no | — |
| 183 | C4⋊C8 | — | 134 | 2 | no | — |
| 184 | C4⋊C8 | — | 134 | 2 | no | — |
| 185 | C22⋊C8 | — | 2 | 4 | no | — |
| 186 | C22⋊C8 | — | 67 | 8 | no | — |
| 187 | #97 | — | 2 | 2 | no | — |
| 188 | #97 | — | 67 | 4 | no | — |
| 189 | C4×C8 | — | 2 | 2 | no | — |
| 190 | C4×C8 | — | 67 | 4 | no | — |
| 191 | C24 | C2×Dic134 | 1 | 1 | yes | characteristic |
| 192 | C22×C4 | — | 2 | 2 | no | — |
| 193 | C22×C4 | — | 2 | 2 | no | — |
| 194 | C22×C4 | C2×C4×D67 | 1 | 2 | yes | — |
| 195 | C22×C4 | C2×C22⋊D67 | 1 | 2 | yes | — |
| 196 | C22×C4 | C2×Dic134 | 1 | 1 | yes | characteristic |
| 197 | C22×C4 | C2×Dic134 | 1 | 2 | yes | — |
| 198 | C22×C4 | C2×Dic134 | 1 | 2 | yes | — |
| 199 | C22×C4 | C23×D67 | 1 | 1 | yes | characteristic |
| 200 | M16 | — | 2 | 4 | no | — |
| 201 | M16 | — | 2 | 4 | no | — |
| 202 | M16 | — | 268 | 4 | no | — |
| 203 | M16 | — | 134 | 4 | no | — |
| 204 | M16 | — | 134 | 4 | no | — |
| 205 | C2×C8 | — | 2 | 2 | no | — |
| 206 | C2×C8 | — | 2 | 2 | no | — |
| 207 | C2×C8 | — | 268 | 4 | no | — |
| 208 | C2×C8 | — | 67 | 4 | no | — |
| 209 | C2×C8 | — | 67 | 4 | no | — |
| 210 | C2×C8 | — | 134 | 2 | no | — |
| 211 | C2×C8 | — | 134 | 2 | no | — |
| 212 | C2×C8 | — | 134 | 2 | no | — |
| 213 | C2×C8 | — | 134 | 2 | no | — |
| 214 | C2×C8 | — | 134 | 2 | no | — |
| 215 | C2×C8 | — | 134 | 2 | no | — |
| 216 | C4⋊C4 | Dic67⋊C4 | 1 | 8 | yes | — |
| 217 | C22⋊C4 | Dic67⋊C4 | 1 | 8 | yes | — |
| 218 | C42 | C2×C4×D67 | 1 | 2 | yes | — |
| 219 | C42 | C2×C4×D67 | 1 | 2 | yes | — |
| 220 | C42 | C2×C22⋊D67 | 1 | 2 | yes | — |
| 221 | C42 | C2×C22⋊D67 | 1 | 2 | yes | — |
| 222 | C23 | — | 2 | 2 | no | — |
| 223 | C23 | — | 4 | 2 | no | — |
| 224 | C23 | C2×Dic67⋊C4 | 1 | 2 | yes | — |
| 225 | C23 | C22×Dic134 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | — | 2 | 2 | no | — |
| 227 | C2×C4 | — | 2 | 4 | no | — |
| 228 | C2×C4 | — | 2 | 2 | no | — |
| 229 | C2×C4 | — | 2 | 2 | no | — |
| 230 | C2×C4 | — | 4 | 2 | no | — |
| 231 | C2×C4 | — | 4 | 2 | no | — |
| 232 | C2×C4 | C2×Dic67⋊C4 | 1 | 4 | yes | — |
| 233 | C2×C4 | C2×Dic67⋊C4 | 1 | 4 | yes | — |
| 234 | C2×C4 | C2×Dic67⋊C4 | 1 | 2 | yes | — |
| 235 | C2×C4 | C22×C4×D67 | 1 | 1 | yes | characteristic |
| 236 | C2×C4 | C22×C22⋊D67 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | #14524089 | 1 | 2 | yes | — |
| 238 | C2×C4 | #14524090 | 1 | 2 | yes | — |
| 239 | C8 | — | 4 | 2 | no | — |
| 240 | C8 | — | 4 | 2 | no | — |
| 241 | C8 | — | 268 | 2 | no | — |
| 242 | C8 | — | 268 | 2 | no | — |
| 243 | C8 | — | 134 | 4 | no | — |
| 244 | C8 | — | 134 | 4 | no | — |
| 245 | C22 | — | 2 | 2 | no | — |
| 246 | C22 | — | 2 | 2 | no | — |
| 247 | C22 | — | 2 | 4 | no | — |
| 248 | C22 | — | 8 | 2 | no | — |
| 249 | C22 | C22×Dic67⋊C4 | 1 | 1 | yes | characteristic |
| 250 | C22 | #16864024 | 1 | 2 | yes | — |
| 251 | C4 | — | 2 | 2 | no | — |
| 252 | C4 | — | 2 | 4 | no | — |
| 253 | C4 | — | 2 | 4 | no | — |
| 254 | C4 | #16863876 | 1 | 2 | yes | — |
| 255 | C4 | #16863960 | 1 | 1 | yes | characteristic |
| 256 | C4 | #16863967 | 1 | 1 | yes | characteristic |
| 257 | C2 | — | 2 | 2 | no | — |
| 258 | C2 | — | 4 | 2 | no | — |
| 259 | C2 | #22562829 | 1 | 1 | yes | characteristic |
| 260 | C2 | #22568818 | 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