← #30825050
Subgroups — #30825050
Order 17152.
572 conjugacy classes of subgroups.
The subgroup lattice has 328 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22569328 | C2 | 1 | 2 | yes | maximal |
| 2 | #22568851 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22558996 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22558991 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22558988 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #22558982 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #22558977 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #22558968 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C67×#1568 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C4×#11761628 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16863524 | — | 2 | 2 | no | — |
| 12 | #16863471 | — | 2 | 2 | no | — |
| 13 | #16863470 | — | 2 | 2 | no | — |
| 14 | #16863469 | — | 2 | 2 | no | — |
| 15 | #16863455 | C4 | 1 | 4 | yes | — |
| 16 | #16863452 | C22 | 1 | 2 | yes | — |
| 17 | #16863448 | C22 | 1 | 2 | yes | — |
| 18 | #16863447 | C22 | 1 | 2 | yes | — |
| 19 | #16863447 | C22 | 1 | 2 | yes | — |
| 20 | #16863422 | C4 | 1 | 2 | yes | — |
| 21 | #16863420 | C22 | 1 | 2 | yes | — |
| 22 | #16863418 | C22 | 1 | 2 | yes | — |
| 23 | C4⋊C8×D67 | C22 | 1 | 2 | yes | — |
| 24 | #16863362 | C4 | 1 | 2 | yes | — |
| 25 | C4×#11761612 | C22 | 1 | 1 | yes | characteristic |
| 26 | #16863149 | C22 | 1 | 2 | yes | — |
| 27 | #16863149 | C22 | 1 | 1 | yes | characteristic |
| 28 | #16863149 | C22 | 1 | 1 | yes | characteristic |
| 29 | #16863148 | C22 | 1 | 2 | yes | — |
| 30 | #16863148 | C22 | 1 | 2 | yes | — |
| 31 | #16863146 | C22 | 1 | 2 | yes | — |
| 32 | #16863146 | C22 | 1 | 2 | yes | — |
| 33 | #432×C67 | C22 | 1 | 2 | yes | — |
| 34 | C2×C4×C8×C67 | C22 | 1 | 1 | yes | characteristic |
| 35 | #335×C67 | C22 | 1 | 1 | yes | characteristic |
| 36 | #335×C67 | C22 | 1 | 1 | yes | characteristic |
| 37 | #334×C67 | C22 | 1 | 2 | yes | — |
| 38 | C2×#11761628 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C2×#11761612 | — | 2 | 2 | no | — |
| 40 | C2×#11761612 | C2×C4 | 1 | 2 | yes | — |
| 41 | C2×#11761611 | — | 2 | 2 | no | — |
| 42 | C2×C8×D67 | — | 2 | 2 | no | — |
| 43 | C4×Dic134 | D4 | 1 | 1 | yes | characteristic |
| 44 | C4×Dic134 | Q8 | 1 | 1 | yes | characteristic |
| 45 | C4×D268 | D4 | 1 | 1 | yes | characteristic |
| 46 | C4×D268 | Q8 | 1 | 1 | yes | characteristic |
| 47 | C4×C22⋊D67 | D4 | 1 | 2 | yes | — |
| 48 | C4×C22⋊D67 | Q8 | 1 | 2 | yes | — |
| 49 | #14524002 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | #14524000 | C2×C4 | 1 | 2 | yes | — |
| 51 | C42×D67 | C23 | 1 | 2 | yes | — |
| 52 | C4⋊(C67⋊C8) | — | 2 | 2 | no | — |
| 53 | C4⋊(C67⋊C8) | C23 | 1 | 2 | yes | — |
| 54 | C4⋊(C67⋊C8) | C23 | 1 | 2 | yes | — |
| 55 | Dic67⋊C8 | — | 2 | 2 | no | — |
| 56 | Dic67⋊C8 | — | 2 | 2 | no | — |
| 57 | Dic67⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 58 | Dic67⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 59 | (C2×D67)⋊C8 | — | 2 | 2 | no | — |
| 60 | (C2×D67)⋊C8 | — | 2 | 2 | no | — |
| 61 | (C2×D67)⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 62 | (C2×D67)⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 63 | #14523954 | C2×C4 | 1 | 2 | yes | — |
| 64 | #14523953 | — | 2 | 2 | no | — |
| 65 | C4×C67⋊C8 | C23 | 1 | 2 | yes | — |
| 66 | C8×Dic67 | — | 2 | 2 | no | — |
| 67 | C22×C8×C67 | C2×C4 | 1 | 2 | yes | — |
| 68 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 69 | C4⋊C8×C67 | C23 | 1 | 2 | yes | — |
| 70 | C4⋊C8×C67 | C23 | 1 | 2 | yes | — |
| 71 | C22⋊C8×C67 | — | 2 | 2 | no | — |
| 72 | #97×C67 | — | 2 | 2 | no | — |
| 73 | C4×C8×C67 | C2×C4 | 1 | 2 | yes | — |
| 74 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 75 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 76 | #11761628 | — | 4 | 2 | no | — |
| 77 | C2×Dic134 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 78 | C2×D268 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 79 | C2×C22⋊D67 | C4⋊C4 | 1 | 2 | yes | — |
| 80 | C2×C4×D67 | C22×C4 | 1 | 2 | yes | — |
| 81 | C2×C4×D67 | C2×D4 | 1 | 2 | yes | — |
| 82 | C2×C4×D67 | C2×Q8 | 1 | 2 | yes | — |
| 83 | #11761612 | — | 4 | 2 | no | — |
| 84 | #11761612 | — | 8 | 2 | no | — |
| 85 | #11761611 | — | 8 | 2 | no | — |
| 86 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 87 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 88 | C2×C67⋊C8 | C22×C4 | 1 | 4 | yes | — |
| 89 | C8×D67 | — | 8 | 2 | no | — |
| 90 | C4⋊Dic67 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 91 | C4⋊Dic67 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 92 | Dic67⋊C4 | C4⋊C4 | 1 | 2 | yes | — |
| 93 | Dic67⋊C4 | C4⋊C4 | 1 | 2 | yes | — |
| 94 | C22⋊Dic67 | C4⋊C4 | 1 | 2 | yes | — |
| 95 | (C2×D67)⋊C4 | C4⋊C4 | 1 | 2 | yes | — |
| 96 | (C2×D67)⋊C4 | C4⋊C4 | 1 | 2 | yes | — |
| 97 | C4×Dic67 | C22×C4 | 1 | 2 | yes | — |
| 98 | C4×Dic67 | C2×D4 | 1 | 2 | yes | — |
| 99 | C4×Dic67 | C2×Q8 | 1 | 2 | yes | — |
| 100 | C22×C4×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 101 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C22×C4×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 103 | C2×C8×C67 | — | 2 | 2 | no | — |
| 104 | C2×C8×C67 | — | 2 | 2 | no | — |
| 105 | C2×C8×C67 | — | 4 | 2 | no | — |
| 106 | C2×C8×C67 | C22×C4 | 1 | 2 | yes | — |
| 107 | C2×C8×C67 | C22×C4 | 1 | 2 | yes | — |
| 108 | C42×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 109 | C42×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C42×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 111 | C42×C67 | C24 | 1 | 1 | yes | characteristic |
| 112 | C23×C67 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 113 | C22×D67 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 114 | C2×C4×C67 | — | 2 | 2 | no | — |
| 115 | C2×C4×C67 | — | 2 | 1 | no | — |
| 116 | C2×C4×C67 | — | 2 | 1 | no | — |
| 117 | C2×C4×C67 | — | 4 | 1 | no | — |
| 118 | C2×C4×C67 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C67 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C67 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C67 | C22×Q8 | 1 | 1 | yes | characteristic |
| 124 | C22⋊D67 | — | 4 | 2 | no | — |
| 125 | C2×Dic67 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 126 | C2×Dic67 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 127 | C2×Dic67 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 128 | D268 | — | 4 | 1 | no | — |
| 129 | C4×D67 | — | 2 | 4 | no | — |
| 130 | C4×D67 | — | 2 | 2 | no | — |
| 131 | C4×D67 | — | 2 | 2 | no | — |
| 132 | C4×D67 | — | 4 | 2 | no | — |
| 133 | Dic134 | — | 4 | 1 | no | — |
| 134 | C8×C67 | — | 2 | 2 | no | — |
| 135 | C8×C67 | — | 2 | 2 | no | — |
| 136 | C8×C67 | — | 4 | 2 | no | — |
| 137 | C8×C67 | — | 4 | 2 | no | — |
| 138 | C67⋊C8 | — | 2 | 4 | no | — |
| 139 | C67⋊C8 | — | 4 | 2 | no | — |
| 140 | C67⋊C8 | — | 4 | 2 | no | — |
| 141 | C22×C67 | — | 2 | 1 | no | — |
| 142 | C22×C67 | — | 2 | 1 | no | — |
| 143 | C22×C67 | — | 2 | 1 | no | — |
| 144 | C22×C67 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 145 | C2×D67 | — | 2 | 2 | no | — |
| 146 | C2×D67 | — | 2 | 2 | no | — |
| 147 | C2×D67 | — | 2 | 2 | no | — |
| 148 | C4×C67 | — | 2 | 1 | no | — |
| 149 | C4×C67 | — | 2 | 1 | no | — |
| 150 | C4×C67 | — | 2 | 1 | no | — |
| 151 | C4×C67 | — | 2 | 1 | no | — |
| 152 | C4×C67 | #568 | 1 | 2 | yes | — |
| 153 | C4×C67 | #576 | 1 | 1 | yes | characteristic |
| 154 | C4×C67 | #578 | 1 | 1 | yes | characteristic |
| 155 | Dic67 | — | 2 | 2 | no | — |
| 156 | Dic67 | — | 2 | 2 | no | — |
| 157 | Dic67 | — | 2 | 2 | no | — |
| 158 | #17825 | — | 67 | 1 | no | maximal, Sylow |
| 159 | C2×C67 | — | 4 | 1 | no | — |
| 160 | C2×C67 | #2905 | 1 | 1 | yes | characteristic |
| 161 | C2×C67 | #2913 | 1 | 1 | yes | characteristic |
| 162 | C2×C67 | #2916 | 1 | 1 | yes | characteristic |
| 163 | D67 | — | 4 | 2 | no | — |
| 164 | #2989 | — | 67 | 2 | no | — |
| 165 | #2922 | — | 67 | 1 | no | — |
| 166 | #1574 | — | 67 | 2 | no | — |
| 167 | #1573 | — | 67 | 2 | no | — |
| 168 | #1572 | — | 67 | 2 | no | — |
| 169 | #1571 | — | 67 | 2 | no | — |
| 170 | #1569 | — | 67 | 2 | no | — |
| 171 | #1568 | D67 | 1 | 1 | yes | characteristic, complemented |
| 172 | #1567 | — | 67 | 1 | no | — |
| 173 | C67 | #17825 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 174 | C4×#41 | — | 67 | 1 | no | — |
| 175 | #446 | — | 134 | 2 | no | — |
| 176 | #436 | — | 134 | 2 | no | — |
| 177 | #435 | — | 134 | 2 | no | — |
| 178 | #435 | — | 134 | 2 | no | — |
| 179 | #433 | — | 67 | 4 | no | — |
| 180 | #432 | — | 67 | 2 | no | — |
| 181 | #432 | C2×D67 | 1 | 2 | yes | — |
| 182 | #431 | — | 67 | 2 | no | — |
| 183 | #431 | — | 67 | 2 | no | — |
| 184 | #431 | — | 67 | 2 | no | — |
| 185 | #424 | — | 67 | 2 | no | — |
| 186 | #423 | — | 67 | 2 | no | — |
| 187 | #423 | — | 67 | 2 | no | — |
| 188 | C2×C4⋊C8 | — | 67 | 2 | no | — |
| 189 | #408 | — | 67 | 2 | no | — |
| 190 | C4×M16 | — | 67 | 1 | no | — |
| 191 | C2×C4×C8 | C2×D67 | 1 | 1 | yes | characteristic |
| 192 | #335 | — | 67 | 2 | no | — |
| 193 | #335 | — | 67 | 2 | no | — |
| 194 | #335 | — | 67 | 2 | no | — |
| 195 | #335 | — | 67 | 1 | no | — |
| 196 | #335 | — | 67 | 1 | no | — |
| 197 | #335 | C2×D67 | 1 | 1 | yes | characteristic |
| 198 | #335 | C2×D67 | 1 | 1 | yes | characteristic |
| 199 | #334 | — | 67 | 2 | no | — |
| 200 | #334 | — | 67 | 2 | no | — |
| 201 | #334 | C2×D67 | 1 | 2 | yes | — |
| 202 | C2×#41 | — | 67 | 1 | no | — |
| 203 | C2×M16 | — | 67 | 2 | no | — |
| 204 | C2×M16 | — | 134 | 2 | no | — |
| 205 | C2×M16 | — | 134 | 2 | no | — |
| 206 | C22×C8 | — | 134 | 2 | no | — |
| 207 | C22×C8 | C4×D67 | 1 | 2 | yes | — |
| 208 | C4×Q8 | — | 67 | 1 | no | — |
| 209 | C4×Q8 | — | 67 | 1 | no | — |
| 210 | C4×D4 | — | 67 | 2 | no | — |
| 211 | C4×D4 | — | 67 | 2 | no | — |
| 212 | C4×D4 | — | 67 | 1 | no | — |
| 213 | C4×D4 | — | 67 | 1 | no | — |
| 214 | #117 | — | 67 | 1 | no | — |
| 215 | #117 | — | 67 | 2 | no | — |
| 216 | C2×C42 | — | 67 | 2 | no | — |
| 217 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 218 | C4⋊C8 | — | 67 | 4 | no | — |
| 219 | C4⋊C8 | — | 67 | 4 | no | — |
| 220 | C4⋊C8 | — | 67 | 2 | no | — |
| 221 | C4⋊C8 | — | 67 | 2 | no | — |
| 222 | C4⋊C8 | — | 134 | 2 | no | — |
| 223 | C4⋊C8 | — | 134 | 2 | no | — |
| 224 | C4⋊C8 | — | 134 | 2 | no | — |
| 225 | C4⋊C8 | C22×D67 | 1 | 2 | yes | — |
| 226 | C4⋊C8 | C22×D67 | 1 | 2 | yes | — |
| 227 | C22⋊C8 | — | 2 | 2 | no | — |
| 228 | C22⋊C8 | — | 67 | 4 | no | — |
| 229 | C22⋊C8 | — | 67 | 4 | no | — |
| 230 | C22⋊C8 | — | 134 | 2 | no | — |
| 231 | C22⋊C8 | — | 134 | 2 | no | — |
| 232 | #97 | — | 2 | 2 | no | — |
| 233 | #97 | — | 67 | 2 | no | — |
| 234 | #97 | — | 134 | 2 | no | — |
| 235 | C4×C8 | — | 67 | 2 | no | — |
| 236 | C4×C8 | — | 134 | 2 | no | — |
| 237 | C4×C8 | C4×D67 | 1 | 2 | yes | — |
| 238 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 239 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 240 | #41 | — | 268 | 2 | no | — |
| 241 | C2×Q8 | — | 67 | 1 | no | — |
| 242 | C2×D4 | — | 67 | 2 | no | — |
| 243 | C2×D4 | — | 67 | 1 | no | — |
| 244 | C22×C4 | — | 67 | 2 | no | — |
| 245 | C22×C4 | — | 67 | 2 | no | — |
| 246 | C22×C4 | — | 67 | 2 | no | — |
| 247 | C22×C4 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 248 | C22×C4 | D4×D67 | 1 | 1 | yes | characteristic |
| 249 | C22×C4 | Q8×D67 | 1 | 1 | yes | characteristic |
| 250 | M16 | — | 268 | 2 | no | — |
| 251 | M16 | — | 536 | 2 | no | — |
| 252 | M16 | — | 536 | 2 | no | — |
| 253 | C2×C8 | — | 2 | 2 | no | — |
| 254 | C2×C8 | — | 2 | 2 | no | — |
| 255 | C2×C8 | — | 4 | 2 | no | — |
| 256 | C2×C8 | — | 536 | 2 | no | — |
| 257 | C2×C8 | — | 67 | 4 | no | — |
| 258 | C2×C8 | — | 134 | 2 | no | — |
| 259 | C2×C8 | — | 134 | 2 | no | — |
| 260 | C2×C8 | C2×C4×D67 | 1 | 2 | yes | — |
| 261 | C2×C8 | C2×C4×D67 | 1 | 2 | yes | — |
| 262 | C4⋊C4 | — | 67 | 2 | no | — |
| 263 | C4⋊C4 | — | 67 | 2 | no | — |
| 264 | C4⋊C4 | — | 67 | 1 | no | — |
| 265 | C4⋊C4 | — | 67 | 1 | no | — |
| 266 | C22⋊C4 | — | 67 | 2 | no | — |
| 267 | C22⋊C4 | — | 67 | 2 | no | — |
| 268 | C22⋊C4 | — | 67 | 2 | no | — |
| 269 | C42 | — | 67 | 2 | no | — |
| 270 | C42 | — | 67 | 2 | no | — |
| 271 | C42 | — | 67 | 2 | no | — |
| 272 | C42 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 273 | C42 | D4×D67 | 1 | 1 | yes | characteristic |
| 274 | C42 | Q8×D67 | 1 | 1 | yes | characteristic |
| 275 | C42 | C23×D67 | 1 | 1 | yes | characteristic |
| 276 | C23 | — | 67 | 2 | no | — |
| 277 | C23 | C4⋊C4×D67 | 1 | 1 | yes | characteristic |
| 278 | Q8 | — | 268 | 1 | no | — |
| 279 | D4 | — | 268 | 1 | no | — |
| 280 | D4 | — | 268 | 2 | no | — |
| 281 | C2×C4 | — | 2 | 2 | no | — |
| 282 | C2×C4 | — | 2 | 1 | no | — |
| 283 | C2×C4 | — | 2 | 1 | no | — |
| 284 | C2×C4 | — | 4 | 1 | no | — |
| 285 | C2×C4 | — | 268 | 2 | no | — |
| 286 | C2×C4 | — | 67 | 2 | no | — |
| 287 | C2×C4 | — | 67 | 2 | no | — |
| 288 | C2×C4 | — | 67 | 2 | no | — |
| 289 | C2×C4 | — | 134 | 4 | no | — |
| 290 | C2×C4 | — | 134 | 2 | no | — |
| 291 | C2×C4 | — | 134 | 2 | no | — |
| 292 | C2×C4 | C4⋊C4×D67 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | C4⋊C4×D67 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C4⋊C4×D67 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | C22×C4×D67 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | C2×D4×D67 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | C2×Q8×D67 | 1 | 1 | yes | characteristic |
| 298 | C8 | — | 2 | 2 | no | — |
| 299 | C8 | — | 2 | 2 | no | — |
| 300 | C8 | — | 4 | 2 | no | — |
| 301 | C8 | — | 4 | 2 | no | — |
| 302 | C8 | — | 268 | 2 | no | — |
| 303 | C8 | — | 268 | 2 | no | — |
| 304 | C8 | — | 134 | 4 | no | — |
| 305 | C22 | — | 2 | 1 | no | — |
| 306 | C22 | — | 2 | 1 | no | — |
| 307 | C22 | — | 2 | 1 | no | — |
| 308 | C22 | — | 134 | 2 | no | — |
| 309 | C22 | — | 134 | 2 | no | — |
| 310 | C22 | — | 134 | 2 | no | — |
| 311 | C22 | C2×C4⋊C4×D67 | 1 | 1 | yes | characteristic |
| 312 | C4 | — | 2 | 1 | no | — |
| 313 | C4 | — | 2 | 1 | no | — |
| 314 | C4 | — | 2 | 1 | no | — |
| 315 | C4 | — | 2 | 1 | no | — |
| 316 | C4 | — | 134 | 2 | no | — |
| 317 | C4 | — | 134 | 2 | no | — |
| 318 | C4 | — | 134 | 2 | no | — |
| 319 | C4 | #16864162 | 1 | 2 | yes | — |
| 320 | C4 | #16864198 | 1 | 1 | yes | characteristic |
| 321 | C4 | #16864210 | 1 | 1 | yes | characteristic |
| 322 | C2 | — | 4 | 1 | no | — |
| 323 | C2 | — | 268 | 2 | no | — |
| 324 | C2 | #22568771 | 1 | 1 | yes | characteristic |
| 325 | C2 | #22568806 | 1 | 1 | yes | characteristic |
| 326 | C2 | #22568817 | 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