← D8×Q16×C67 #30699225
Subgroups — D8×Q16×C67 #30699225
Order 17152.
690 conjugacy classes of subgroups.
The subgroup lattice has 276 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | Q8×D8×C67 | C2 | 1 | 2 | yes | maximal |
| 2 | D4×Q16×C67 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C67×#1713 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C67×#1674 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C67×#1669 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C67×#1649 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C8×Q16×C67 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C8×D8×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C22×Q16×C67 | — | 2 | 2 | no | — |
| 10 | D4×Q8×C67 | C22 | 1 | 4 | yes | — |
| 11 | #500×C67 | — | 2 | 2 | no | — |
| 12 | #500×C67 | C22 | 1 | 2 | yes | — |
| 13 | #500×C67 | C22 | 1 | 2 | yes | — |
| 14 | #494×C67 | C22 | 1 | 2 | yes | — |
| 15 | #474×C67 | — | 2 | 4 | no | — |
| 16 | #474×C67 | C22 | 1 | 4 | yes | — |
| 17 | #467×C67 | — | 2 | 2 | no | — |
| 18 | #462×C67 | C22 | 1 | 4 | yes | — |
| 19 | #451×C67 | — | 2 | 4 | no | — |
| 20 | C8×Q8×C67 | C22 | 1 | 2 | yes | — |
| 21 | C4×Q16×C67 | C22 | 1 | 1 | yes | characteristic |
| 22 | C4×D8×C67 | — | 2 | 2 | no | — |
| 23 | C4×D8×C67 | C22 | 1 | 1 | yes | characteristic |
| 24 | C8×D4×C67 | C22 | 1 | 2 | yes | — |
| 25 | #335×C67 | C22 | 1 | 1 | yes | characteristic |
| 26 | #335×C67 | C22 | 1 | 1 | yes | characteristic |
| 27 | #332×C67 | C22 | 1 | 4 | yes | — |
| 28 | Q8⋊C8×C67 | C22 | 1 | 2 | yes | — |
| 29 | D4⋊C8×C67 | C22 | 1 | 2 | yes | — |
| 30 | C82×C67 | C22 | 1 | 1 | yes | characteristic |
| 31 | C22×Q8×C67 | — | 2 | 4 | no | — |
| 32 | C2×Q16×C67 | — | 2 | 4 | no | — |
| 33 | C2×Q16×C67 | — | 2 | 2 | no | — |
| 34 | C2×Q16×C67 | — | 2 | 2 | no | — |
| 35 | C2×Q16×C67 | — | 4 | 8 | no | — |
| 36 | C2×Q16×C67 | D4 | 1 | 1 | yes | characteristic |
| 37 | C2×D8×C67 | D4 | 1 | 1 | yes | characteristic |
| 38 | C22×C8×C67 | — | 2 | 2 | no | — |
| 39 | C4⋊Q8×C67 | — | 2 | 4 | no | — |
| 40 | C4⋊Q8×C67 | C23 | 1 | 4 | yes | — |
| 41 | C22⋊Q8×C67 | — | 2 | 4 | no | — |
| 42 | C22⋊Q8×C67 | — | 4 | 4 | no | — |
| 43 | C4×Q8×C67 | C23 | 1 | 2 | yes | — |
| 44 | C4×D4×C67 | — | 2 | 4 | no | — |
| 45 | C4×D4×C67 | C23 | 1 | 2 | yes | — |
| 46 | #107×C67 | — | 2 | 2 | no | — |
| 47 | #107×C67 | — | 2 | 2 | no | — |
| 48 | #107×C67 | — | 2 | 4 | no | — |
| 49 | #107×C67 | — | 4 | 2 | no | — |
| 50 | #107×C67 | D4 | 1 | 1 | yes | characteristic |
| 51 | #107×C67 | D4 | 1 | 1 | yes | characteristic |
| 52 | C4⋊C8×C67 | — | 2 | 2 | no | — |
| 53 | C4⋊C8×C67 | C23 | 1 | 2 | yes | — |
| 54 | C4⋊C8×C67 | C23 | 1 | 2 | yes | — |
| 55 | #103×C67 | — | 2 | 4 | no | — |
| 56 | #103×C67 | D4 | 1 | 2 | yes | — |
| 57 | #102×C67 | — | 2 | 4 | no | — |
| 58 | #102×C67 | D4 | 1 | 2 | yes | — |
| 59 | C22⋊C8×C67 | — | 2 | 2 | no | — |
| 60 | C4×C8×C67 | — | 2 | 2 | no | — |
| 61 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 62 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 63 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 64 | C2×Q8×C67 | — | 2 | 4 | no | — |
| 65 | C2×Q8×C67 | — | 4 | 4 | no | — |
| 66 | C2×Q8×C67 | — | 4 | 8 | no | — |
| 67 | C2×Q8×C67 | — | 8 | 4 | no | — |
| 68 | C2×Q8×C67 | C2×D4 | 1 | 2 | yes | — |
| 69 | C2×D4×C67 | C2×D4 | 1 | 2 | yes | — |
| 70 | C22×C4×C67 | — | 2 | 2 | no | — |
| 71 | C22×C4×C67 | — | 4 | 4 | no | — |
| 72 | Q16×C67 | — | 4 | 8 | no | — |
| 73 | Q16×C67 | — | 4 | 4 | no | — |
| 74 | Q16×C67 | — | 8 | 2 | no | — |
| 75 | Q16×C67 | D8 | 1 | 4 | yes | direct product factor |
| 76 | D8×C67 | Q16 | 1 | 4 | yes | direct product factor |
| 77 | C2×C8×C67 | — | 2 | 1 | no | — |
| 78 | C2×C8×C67 | — | 2 | 2 | no | — |
| 79 | C2×C8×C67 | — | 2 | 2 | no | — |
| 80 | C2×C8×C67 | — | 4 | 4 | no | — |
| 81 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 86 | C4⋊C4×C67 | — | 2 | 4 | no | — |
| 87 | C4⋊C4×C67 | — | 2 | 4 | no | — |
| 88 | C4⋊C4×C67 | — | 4 | 4 | no | — |
| 89 | C4⋊C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 90 | C4⋊C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 91 | C22⋊C4×C67 | — | 2 | 2 | no | — |
| 92 | C22⋊C4×C67 | — | 4 | 4 | no | — |
| 93 | C42×C67 | — | 2 | 2 | no | — |
| 94 | C42×C67 | C24 | 1 | 1 | yes | characteristic |
| 95 | C23×C67 | — | 2 | 2 | no | — |
| 96 | Q8×C67 | — | 2 | 2 | no | — |
| 97 | Q8×C67 | — | 4 | 4 | no | — |
| 98 | Q8×C67 | — | 4 | 4 | no | — |
| 99 | Q8×C67 | — | 8 | 8 | no | — |
| 100 | Q8×C67 | C2×D8 | 1 | 4 | yes | — |
| 101 | D4×C67 | — | 2 | 2 | no | — |
| 102 | D4×C67 | C2×Q16 | 1 | 4 | yes | complemented |
| 103 | C2×C4×C67 | — | 2 | 2 | no | — |
| 104 | C2×C4×C67 | — | 2 | 2 | no | — |
| 105 | C2×C4×C67 | — | 2 | 2 | no | — |
| 106 | C2×C4×C67 | — | 4 | 2 | no | — |
| 107 | C2×C4×C67 | — | 4 | 2 | no | — |
| 108 | C2×C4×C67 | — | 4 | 4 | no | — |
| 109 | C2×C4×C67 | — | 8 | 8 | no | — |
| 110 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 111 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C67 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 113 | C8×C67 | — | 2 | 2 | no | — |
| 114 | C8×C67 | — | 2 | 1 | no | — |
| 115 | C8×C67 | — | 2 | 1 | no | — |
| 116 | C8×C67 | — | 4 | 2 | no | — |
| 117 | C8×C67 | — | 4 | 2 | no | — |
| 118 | C8×C67 | C2×D8 | 1 | 2 | yes | — |
| 119 | C8×C67 | C2×Q16 | 1 | 2 | yes | complemented |
| 120 | C22×C67 | — | 2 | 4 | no | — |
| 121 | C22×C67 | — | 4 | 2 | no | — |
| 122 | C22×C67 | — | 4 | 2 | no | — |
| 123 | C22×C67 | D42 | 1 | 1 | yes | characteristic |
| 124 | C4×C67 | — | 2 | 4 | no | — |
| 125 | C4×C67 | — | 2 | 1 | no | — |
| 126 | C4×C67 | — | 4 | 2 | no | — |
| 127 | C4×C67 | — | 4 | 2 | no | — |
| 128 | C4×C67 | — | 8 | 4 | no | — |
| 129 | C4×C67 | C22×D8 | 1 | 1 | yes | characteristic |
| 130 | C4×C67 | C22×Q16 | 1 | 1 | yes | characteristic |
| 131 | C4×C67 | #576 | 1 | 1 | yes | characteristic |
| 132 | C4×C67 | #578 | 1 | 1 | yes | characteristic |
| 133 | D8×Q16 | C67 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 134 | C2×C67 | — | 4 | 4 | no | — |
| 135 | C2×C67 | D4×D8 | 1 | 1 | yes | characteristic |
| 136 | C2×C67 | D4×Q16 | 1 | 1 | yes | characteristic |
| 137 | C2×C67 | #3293 | 1 | 1 | yes | characteristic |
| 138 | Q8×D8 | C2×C67 | 1 | 2 | yes | — |
| 139 | D4×Q16 | C2×C67 | 1 | 2 | yes | complemented |
| 140 | #1713 | C2×C67 | 1 | 1 | yes | characteristic, complemented |
| 141 | #1674 | C2×C67 | 1 | 2 | yes | complemented |
| 142 | #1669 | C2×C67 | 1 | 2 | yes | complemented |
| 143 | #1649 | C2×C67 | 1 | 4 | yes | complemented |
| 144 | C8×Q16 | C2×C67 | 1 | 1 | yes | characteristic, complemented |
| 145 | C8×D8 | C2×C67 | 1 | 1 | yes | characteristic |
| 146 | C67 | D8×Q16 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 147 | C22×Q16 | — | 2 | 2 | no | — |
| 148 | D4×Q8 | C22×C67 | 1 | 4 | yes | — |
| 149 | #500 | — | 2 | 2 | no | — |
| 150 | #500 | C22×C67 | 1 | 2 | yes | — |
| 151 | #500 | C22×C67 | 1 | 2 | yes | — |
| 152 | #494 | C22×C67 | 1 | 2 | yes | — |
| 153 | #474 | — | 2 | 4 | no | — |
| 154 | #474 | C22×C67 | 1 | 4 | yes | — |
| 155 | #467 | — | 2 | 2 | no | — |
| 156 | #462 | C22×C67 | 1 | 4 | yes | — |
| 157 | #451 | — | 2 | 4 | no | — |
| 158 | C8×Q8 | C22×C67 | 1 | 2 | yes | — |
| 159 | C4×Q16 | C22×C67 | 1 | 1 | yes | characteristic |
| 160 | C4×D8 | — | 2 | 2 | no | — |
| 161 | C4×D8 | C22×C67 | 1 | 1 | yes | characteristic |
| 162 | C8×D4 | C22×C67 | 1 | 2 | yes | — |
| 163 | #335 | C22×C67 | 1 | 1 | yes | characteristic |
| 164 | #335 | C22×C67 | 1 | 1 | yes | characteristic |
| 165 | #332 | C22×C67 | 1 | 4 | yes | — |
| 166 | Q8⋊C8 | C22×C67 | 1 | 2 | yes | — |
| 167 | D4⋊C8 | C22×C67 | 1 | 2 | yes | — |
| 168 | C82 | C22×C67 | 1 | 1 | yes | characteristic |
| 169 | C22×Q8 | — | 2 | 4 | no | — |
| 170 | C2×Q16 | — | 2 | 4 | no | — |
| 171 | C2×Q16 | — | 2 | 2 | no | — |
| 172 | C2×Q16 | — | 2 | 2 | no | — |
| 173 | C2×Q16 | — | 4 | 8 | no | — |
| 174 | C2×Q16 | D4×C67 | 1 | 1 | yes | characteristic |
| 175 | C2×D8 | D4×C67 | 1 | 1 | yes | characteristic |
| 176 | C22×C8 | — | 2 | 2 | no | — |
| 177 | C4⋊Q8 | — | 2 | 4 | no | — |
| 178 | C4⋊Q8 | C23×C67 | 1 | 4 | yes | — |
| 179 | C22⋊Q8 | — | 2 | 4 | no | — |
| 180 | C22⋊Q8 | — | 4 | 4 | no | — |
| 181 | C4×Q8 | C23×C67 | 1 | 2 | yes | — |
| 182 | C4×D4 | — | 2 | 4 | no | — |
| 183 | C4×D4 | C23×C67 | 1 | 2 | yes | — |
| 184 | #107 | — | 2 | 2 | no | — |
| 185 | #107 | — | 2 | 2 | no | — |
| 186 | #107 | — | 2 | 4 | no | — |
| 187 | #107 | — | 4 | 2 | no | — |
| 188 | #107 | D4×C67 | 1 | 1 | yes | characteristic |
| 189 | #107 | D4×C67 | 1 | 1 | yes | characteristic |
| 190 | C4⋊C8 | — | 2 | 2 | no | — |
| 191 | C4⋊C8 | C23×C67 | 1 | 2 | yes | — |
| 192 | C4⋊C8 | C23×C67 | 1 | 2 | yes | — |
| 193 | #103 | — | 2 | 4 | no | — |
| 194 | #103 | D4×C67 | 1 | 2 | yes | — |
| 195 | #102 | — | 2 | 4 | no | — |
| 196 | #102 | D4×C67 | 1 | 2 | yes | — |
| 197 | C22⋊C8 | — | 2 | 2 | no | — |
| 198 | C4×C8 | — | 2 | 2 | no | — |
| 199 | C4×C8 | C23×C67 | 1 | 1 | yes | characteristic |
| 200 | C4×C8 | C23×C67 | 1 | 1 | yes | characteristic |
| 201 | C4×C8 | C23×C67 | 1 | 1 | yes | characteristic |
| 202 | C2×Q8 | — | 2 | 4 | no | — |
| 203 | C2×Q8 | — | 4 | 4 | no | — |
| 204 | C2×Q8 | — | 4 | 8 | no | — |
| 205 | C2×Q8 | — | 8 | 4 | no | — |
| 206 | C2×Q8 | C2×D4×C67 | 1 | 2 | yes | — |
| 207 | C2×D4 | C2×D4×C67 | 1 | 2 | yes | — |
| 208 | C22×C4 | — | 2 | 2 | no | — |
| 209 | C22×C4 | — | 4 | 4 | no | — |
| 210 | Q16 | — | 4 | 8 | no | — |
| 211 | Q16 | — | 4 | 4 | no | — |
| 212 | Q16 | — | 8 | 2 | no | — |
| 213 | Q16 | D8×C67 | 1 | 4 | yes | direct product factor |
| 214 | D8 | Q16×C67 | 1 | 4 | yes | direct product factor |
| 215 | C2×C8 | — | 2 | 1 | no | — |
| 216 | C2×C8 | — | 2 | 2 | no | — |
| 217 | C2×C8 | — | 2 | 2 | no | — |
| 218 | C2×C8 | — | 4 | 4 | no | — |
| 219 | C2×C8 | C2×D4×C67 | 1 | 1 | yes | characteristic |
| 220 | C2×C8 | C2×D4×C67 | 1 | 1 | yes | characteristic |
| 221 | C2×C8 | C2×D4×C67 | 1 | 1 | yes | characteristic |
| 222 | C2×C8 | C2×D4×C67 | 1 | 1 | yes | characteristic |
| 223 | C4⋊C4 | — | 2 | 2 | no | — |
| 224 | C4⋊C4 | — | 2 | 4 | no | — |
| 225 | C4⋊C4 | — | 2 | 4 | no | — |
| 226 | C4⋊C4 | — | 4 | 4 | no | — |
| 227 | C4⋊C4 | C2×D4×C67 | 1 | 2 | yes | — |
| 228 | C4⋊C4 | C2×D4×C67 | 1 | 2 | yes | — |
| 229 | C22⋊C4 | — | 2 | 2 | no | — |
| 230 | C22⋊C4 | — | 4 | 4 | no | — |
| 231 | C42 | — | 2 | 2 | no | — |
| 232 | C42 | C24×C67 | 1 | 1 | yes | characteristic |
| 233 | C23 | — | 2 | 2 | no | — |
| 234 | Q8 | — | 2 | 2 | no | — |
| 235 | Q8 | — | 4 | 4 | no | — |
| 236 | Q8 | — | 4 | 4 | no | — |
| 237 | Q8 | — | 8 | 8 | no | — |
| 238 | Q8 | C2×D8×C67 | 1 | 4 | yes | — |
| 239 | D4 | — | 2 | 2 | no | — |
| 240 | D4 | C2×Q16×C67 | 1 | 4 | yes | complemented |
| 241 | C2×C4 | — | 2 | 2 | no | — |
| 242 | C2×C4 | — | 2 | 2 | no | — |
| 243 | C2×C4 | — | 2 | 2 | no | — |
| 244 | C2×C4 | — | 4 | 2 | no | — |
| 245 | C2×C4 | — | 4 | 2 | no | — |
| 246 | C2×C4 | — | 4 | 4 | no | — |
| 247 | C2×C4 | — | 8 | 8 | no | — |
| 248 | C2×C4 | C22×D4×C67 | 1 | 1 | yes | characteristic |
| 249 | C2×C4 | C22×D4×C67 | 1 | 1 | yes | characteristic |
| 250 | C2×C4 | H(5,𝔽2)×C67 | 1 | 1 | yes | characteristic |
| 251 | C8 | — | 2 | 2 | no | — |
| 252 | C8 | — | 2 | 1 | no | — |
| 253 | C8 | — | 2 | 1 | no | — |
| 254 | C8 | — | 4 | 2 | no | — |
| 255 | C8 | — | 4 | 2 | no | — |
| 256 | C8 | C2×D8×C67 | 1 | 2 | yes | — |
| 257 | C8 | C2×Q16×C67 | 1 | 2 | yes | complemented |
| 258 | C22 | — | 2 | 4 | no | — |
| 259 | C22 | — | 4 | 2 | no | — |
| 260 | C22 | — | 4 | 2 | no | — |
| 261 | C22 | D42×C67 | 1 | 1 | yes | characteristic |
| 262 | C4 | — | 2 | 4 | no | — |
| 263 | C4 | — | 2 | 1 | no | — |
| 264 | C4 | — | 4 | 2 | no | — |
| 265 | C4 | — | 4 | 2 | no | — |
| 266 | C4 | — | 8 | 4 | no | — |
| 267 | C4 | C22×D8×C67 | 1 | 1 | yes | characteristic |
| 268 | C4 | C22×Q16×C67 | 1 | 1 | yes | characteristic |
| 269 | C4 | #576×C67 | 1 | 1 | yes | characteristic |
| 270 | C4 | #578×C67 | 1 | 1 | yes | characteristic |
| 271 | C2 | — | 4 | 4 | no | — |
| 272 | C2 | D4×D8×C67 | 1 | 1 | yes | characteristic |
| 273 | C2 | D4×Q16×C67 | 1 | 1 | yes | characteristic |
| 274 | C2 | C67×#3293 | 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