← Q8×#508 #7623916
Subgroups — Q8×#508 #7623916
Order 512.
1252 conjugacy classes of subgroups.
The subgroup lattice has 287 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | Q8×#135 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #33950 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | Q8×Q32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #33948 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #33947 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | Q8×SD32 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | Q8×D16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #33726 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #33626 | C2 | 1 | 6 | yes | maximal, complemented |
| 10 | C2×Q8×C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | C4×#508 | C2 | 1 | 3 | yes | maximal |
| 12 | Q8×#41 | — | 2 | 2 | no | — |
| 13 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 14 | #3383 | C22 | 1 | 6 | yes | — |
| 15 | Q8×Q16 | C22 | 1 | 2 | yes | — |
| 16 | #3381 | — | 2 | 6 | no | — |
| 17 | #3380 | C22 | 1 | 6 | yes | — |
| 18 | Q8×SD16 | — | 2 | 2 | no | — |
| 19 | Q8×D8 | C22 | 1 | 2 | yes | — |
| 20 | #3160 | C22 | 1 | 6 | yes | — |
| 21 | #3076 | — | 2 | 6 | no | — |
| 22 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 23 | C4×#135 | C22 | 1 | 6 | yes | — |
| 24 | #2254 | C22 | 1 | 6 | yes | — |
| 25 | #2253 | C22 | 1 | 6 | yes | — |
| 26 | #2253 | C22 | 1 | 6 | yes | — |
| 27 | #2252 | C22 | 1 | 3 | yes | — |
| 28 | #2252 | C22 | 1 | 3 | yes | — |
| 29 | #2229 | C22 | 1 | 12 | yes | — |
| 30 | #2228 | C22 | 1 | 12 | yes | — |
| 31 | #2227 | C22 | 1 | 6 | yes | — |
| 32 | #2226 | C22 | 1 | 6 | yes | — |
| 33 | #2225 | C22 | 1 | 6 | yes | — |
| 34 | #2224 | C22 | 1 | 6 | yes | — |
| 35 | Q8×C16 | C22 | 1 | 1 | yes | characteristic |
| 36 | Q8×C16 | C22 | 1 | 1 | yes | characteristic |
| 37 | Q8×C16 | C22 | 1 | 3 | yes | — |
| 38 | Q8×C16 | C22 | 1 | 3 | yes | — |
| 39 | C4×Q32 | C22 | 1 | 3 | yes | — |
| 40 | C4×Q32 | C22 | 1 | 3 | yes | — |
| 41 | C4×SD32 | C22 | 1 | 6 | yes | — |
| 42 | C4×SD32 | C22 | 1 | 6 | yes | — |
| 43 | C4×D16 | C22 | 1 | 3 | yes | — |
| 44 | C4×D16 | C22 | 1 | 3 | yes | — |
| 45 | #2158 | C22 | 1 | 6 | yes | — |
| 46 | #2158 | C22 | 1 | 3 | yes | — |
| 47 | C2×C4⋊C16 | C22 | 1 | 3 | yes | — |
| 48 | #2138 | C22 | 1 | 6 | yes | — |
| 49 | C2×C4×C16 | C22 | 1 | 3 | yes | — |
| 50 | C2×#135 | C23 | 1 | 2 | yes | — |
| 51 | Q82 | — | 2 | 2 | no | — |
| 52 | #557 | — | 2 | 6 | no | — |
| 53 | #554 | — | 2 | 6 | no | — |
| 54 | #554 | — | 4 | 6 | no | — |
| 55 | D4×Q8 | — | 2 | 2 | no | — |
| 56 | D4×Q8 | — | 4 | 2 | no | — |
| 57 | #533 | — | 2 | 6 | no | — |
| 58 | #533 | — | 4 | 6 | no | — |
| 59 | C4×#41 | — | 2 | 6 | no | — |
| 60 | C2×C4×Q8 | — | 4 | 2 | no | — |
| 61 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 62 | #508 | Q8 | 1 | 8 | yes | direct product factor |
| 63 | C2×Q32 | C23 | 1 | 1 | yes | characteristic |
| 64 | C2×SD32 | C23 | 1 | 2 | yes | — |
| 65 | C2×D16 | C23 | 1 | 1 | yes | characteristic |
| 66 | C22×C16 | C23 | 1 | 3 | yes | — |
| 67 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 68 | #500 | C23 | 1 | 6 | yes | — |
| 69 | #500 | C23 | 1 | 6 | yes | — |
| 70 | #499 | — | 2 | 6 | no | — |
| 71 | #499 | C23 | 1 | 12 | yes | — |
| 72 | #498 | — | 2 | 6 | no | — |
| 73 | #479 | — | 2 | 12 | no | — |
| 74 | #478 | — | 2 | 12 | no | — |
| 75 | #477 | — | 2 | 6 | no | — |
| 76 | #476 | — | 2 | 6 | no | — |
| 77 | #475 | — | 2 | 6 | no | — |
| 78 | #474 | — | 2 | 6 | no | — |
| 79 | C8×Q8 | D4 | 1 | 1 | yes | characteristic |
| 80 | C8×Q8 | D4 | 1 | 3 | yes | — |
| 81 | C8×Q8 | C23 | 1 | 1 | yes | characteristic |
| 82 | C8×Q8 | C23 | 1 | 3 | yes | — |
| 83 | C4×Q16 | C23 | 1 | 6 | yes | — |
| 84 | C4×Q16 | C23 | 1 | 6 | yes | — |
| 85 | C4×SD16 | — | 2 | 6 | no | — |
| 86 | C4×SD16 | — | 2 | 6 | no | — |
| 87 | C4×D8 | C23 | 1 | 6 | yes | — |
| 88 | C4×D8 | C23 | 1 | 6 | yes | — |
| 89 | #427 | — | 2 | 6 | no | — |
| 90 | #427 | C23 | 1 | 6 | yes | — |
| 91 | C2×C4⋊C8 | D4 | 1 | 3 | yes | — |
| 92 | #416 | — | 2 | 6 | no | — |
| 93 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 94 | #367 | C23 | 1 | 6 | yes | — |
| 95 | #367 | C23 | 1 | 12 | yes | — |
| 96 | #366 | C23 | 1 | 3 | yes | — |
| 97 | #366 | C23 | 1 | 3 | yes | — |
| 98 | #366 | C23 | 1 | 6 | yes | — |
| 99 | #366 | C23 | 1 | 6 | yes | — |
| 100 | C4⋊C16 | C23 | 1 | 3 | yes | — |
| 101 | C4⋊C16 | C23 | 1 | 3 | yes | — |
| 102 | C4⋊C16 | C23 | 1 | 6 | yes | — |
| 103 | #358 | C23 | 1 | 6 | yes | — |
| 104 | #358 | C23 | 1 | 6 | yes | — |
| 105 | #357 | C23 | 1 | 6 | yes | — |
| 106 | #357 | C23 | 1 | 6 | yes | — |
| 107 | C4×C16 | C23 | 1 | 3 | yes | — |
| 108 | C4×C16 | C23 | 1 | 3 | yes | — |
| 109 | C4×C16 | C23 | 1 | 3 | yes | — |
| 110 | C4×C16 | C23 | 1 | 3 | yes | — |
| 111 | C2×#41 | — | 2 | 2 | no | — |
| 112 | C22×Q8 | — | 4 | 2 | no | — |
| 113 | C22×Q8 | D8 | 1 | 1 | yes | characteristic |
| 114 | #135 | — | 2 | 4 | no | — |
| 115 | #135 | C2×Q8 | 1 | 8 | yes | complemented |
| 116 | C2×Q16 | C24 | 1 | 2 | yes | — |
| 117 | C2×SD16 | — | 2 | 2 | no | — |
| 118 | C2×D8 | C24 | 1 | 2 | yes | — |
| 119 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 120 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 121 | C4⋊Q8 | — | 2 | 6 | no | — |
| 122 | C4⋊Q8 | — | 2 | 6 | no | — |
| 123 | C4⋊Q8 | — | 4 | 6 | no | — |
| 124 | C4⋊Q8 | — | 4 | 6 | no | — |
| 125 | #125 | — | 2 | 12 | no | — |
| 126 | #125 | — | 4 | 12 | no | — |
| 127 | C22⋊Q8 | — | 4 | 6 | no | — |
| 128 | C22⋊Q8 | — | 4 | 6 | no | — |
| 129 | C22⋊Q8 | — | 4 | 6 | no | — |
| 130 | C22⋊Q8 | — | 4 | 6 | no | — |
| 131 | C22⋊Q8 | — | 4 | 12 | no | — |
| 132 | C4×Q8 | — | 2 | 6 | no | — |
| 133 | C4×Q8 | — | 2 | 6 | no | — |
| 134 | C4×Q8 | — | 4 | 2 | no | — |
| 135 | C4×Q8 | — | 4 | 6 | no | — |
| 136 | C4×Q8 | — | 4 | 6 | no | — |
| 137 | C4×Q8 | — | 4 | 6 | no | — |
| 138 | C4×Q8 | D8 | 1 | 1 | yes | characteristic |
| 139 | C4×Q8 | D8 | 1 | 3 | yes | — |
| 140 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 141 | C4×Q8 | C2×D4 | 1 | 3 | yes | — |
| 142 | C4×D4 | — | 2 | 6 | no | — |
| 143 | C4×D4 | — | 2 | 6 | no | — |
| 144 | C4×D4 | — | 4 | 6 | no | — |
| 145 | C4×D4 | — | 4 | 6 | no | — |
| 146 | #117 | — | 2 | 6 | no | — |
| 147 | #117 | — | 4 | 6 | no | — |
| 148 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 149 | C2×C4⋊C4 | D8 | 1 | 3 | yes | — |
| 150 | C2×C42 | — | 4 | 6 | no | — |
| 151 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 152 | Q32 | C2×Q8 | 1 | 4 | yes | complemented |
| 153 | SD32 | C2×Q8 | 1 | 8 | yes | complemented |
| 154 | D16 | C2×Q8 | 1 | 4 | yes | complemented |
| 155 | C2×C16 | — | 2 | 6 | no | — |
| 156 | C2×C16 | C2×Q8 | 1 | 4 | yes | complemented |
| 157 | C2×C16 | C24 | 1 | 3 | yes | — |
| 158 | C2×C16 | C24 | 1 | 3 | yes | — |
| 159 | C2×C16 | C24 | 1 | 1 | yes | characteristic |
| 160 | C2×C16 | C24 | 1 | 1 | yes | characteristic |
| 161 | #107 | — | 2 | 6 | no | — |
| 162 | #107 | — | 2 | 6 | no | — |
| 163 | #107 | C24 | 1 | 6 | yes | — |
| 164 | #107 | C24 | 1 | 6 | yes | — |
| 165 | #106 | — | 2 | 12 | no | — |
| 166 | #106 | — | 2 | 6 | no | — |
| 167 | C4⋊C8 | C2×D4 | 1 | 3 | yes | — |
| 168 | C4⋊C8 | C2×D4 | 1 | 3 | yes | — |
| 169 | C4⋊C8 | C2×D4 | 1 | 6 | yes | — |
| 170 | #103 | — | 2 | 6 | no | — |
| 171 | #103 | — | 2 | 6 | no | — |
| 172 | #102 | — | 2 | 6 | no | — |
| 173 | #102 | — | 2 | 6 | no | — |
| 174 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 175 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 176 | C4×C8 | C24 | 1 | 3 | yes | — |
| 177 | C4×C8 | C24 | 1 | 3 | yes | — |
| 178 | #41 | — | 2 | 8 | no | — |
| 179 | #41 | — | 4 | 4 | no | — |
| 180 | C2×Q8 | — | 2 | 4 | no | — |
| 181 | C2×Q8 | — | 2 | 2 | no | — |
| 182 | C2×Q8 | — | 4 | 6 | no | — |
| 183 | C2×Q8 | — | 4 | 6 | no | — |
| 184 | C2×Q8 | — | 8 | 8 | no | — |
| 185 | C2×Q8 | C2×D8 | 1 | 3 | yes | — |
| 186 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 187 | C2×D4 | — | 2 | 2 | no | — |
| 188 | C2×D4 | — | 4 | 2 | no | — |
| 189 | C22×C4 | — | 4 | 2 | no | — |
| 190 | C22×C4 | — | 4 | 6 | no | — |
| 191 | C22×C4 | — | 4 | 6 | no | — |
| 192 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 193 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 194 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 195 | Q16 | — | 2 | 2 | no | — |
| 196 | Q16 | C22×Q8 | 1 | 4 | yes | — |
| 197 | SD16 | — | 2 | 8 | no | — |
| 198 | D8 | — | 2 | 2 | no | — |
| 199 | D8 | C22×Q8 | 1 | 4 | yes | — |
| 200 | C2×C8 | — | 2 | 6 | no | — |
| 201 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 202 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 203 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 204 | C2×C8 | C22×Q8 | 1 | 2 | yes | — |
| 205 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 206 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 207 | C4⋊C4 | — | 2 | 6 | no | — |
| 208 | C4⋊C4 | — | 2 | 6 | no | — |
| 209 | C4⋊C4 | — | 4 | 12 | no | — |
| 210 | C4⋊C4 | — | 4 | 6 | no | — |
| 211 | C4⋊C4 | — | 4 | 6 | no | — |
| 212 | C4⋊C4 | — | 4 | 6 | no | — |
| 213 | C4⋊C4 | — | 4 | 12 | no | — |
| 214 | C4⋊C4 | — | 4 | 6 | no | — |
| 215 | C4⋊C4 | C2×D8 | 1 | 3 | yes | — |
| 216 | C4⋊C4 | C2×D8 | 1 | 6 | yes | — |
| 217 | C4⋊C4 | C2×D8 | 1 | 3 | yes | — |
| 218 | C22⋊C4 | — | 4 | 6 | no | — |
| 219 | C22⋊C4 | — | 4 | 6 | no | — |
| 220 | C22⋊C4 | — | 4 | 6 | no | — |
| 221 | C42 | — | 4 | 6 | no | — |
| 222 | C42 | — | 4 | 6 | no | — |
| 223 | C42 | — | 4 | 6 | no | — |
| 224 | C42 | C2×D8 | 1 | 3 | yes | — |
| 225 | C42 | C2×D8 | 1 | 3 | yes | — |
| 226 | C42 | C22×D4 | 1 | 3 | yes | — |
| 227 | C42 | C22×D4 | 1 | 3 | yes | — |
| 228 | C16 | — | 2 | 3 | no | — |
| 229 | C16 | — | 2 | 3 | no | — |
| 230 | C16 | C22×Q8 | 1 | 2 | yes | — |
| 231 | C16 | C22×Q8 | 1 | 2 | yes | — |
| 232 | C23 | — | 4 | 2 | no | — |
| 233 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 234 | Q8 | — | 2 | 6 | no | — |
| 235 | Q8 | — | 2 | 4 | no | — |
| 236 | Q8 | — | 4 | 2 | no | — |
| 237 | Q8 | — | 8 | 12 | no | — |
| 238 | Q8 | — | 16 | 6 | no | — |
| 239 | Q8 | #508 | 1 | 4 | yes | direct product factor |
| 240 | D4 | — | 2 | 4 | no | — |
| 241 | D4 | — | 4 | 8 | no | — |
| 242 | D4 | — | 4 | 2 | no | — |
| 243 | C2×C4 | — | 2 | 6 | no | — |
| 244 | C2×C4 | — | 4 | 3 | no | — |
| 245 | C2×C4 | — | 4 | 2 | no | — |
| 246 | C2×C4 | — | 4 | 8 | no | — |
| 247 | C2×C4 | — | 4 | 6 | no | — |
| 248 | C2×C4 | — | 4 | 6 | no | — |
| 249 | C2×C4 | — | 8 | 12 | no | — |
| 250 | C2×C4 | — | 16 | 6 | no | — |
| 251 | C2×C4 | D4×Q8 | 1 | 2 | yes | — |
| 252 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 253 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 254 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 255 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 256 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | #578 | 1 | 2 | yes | — |
| 258 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 259 | C8 | — | 2 | 3 | no | — |
| 260 | C8 | — | 2 | 3 | no | — |
| 261 | C8 | D4×Q8 | 1 | 2 | yes | — |
| 262 | C8 | C23×Q8 | 1 | 1 | yes | characteristic |
| 263 | C8 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 264 | C22 | — | 2 | 1 | no | — |
| 265 | C22 | — | 2 | 1 | no | — |
| 266 | C22 | — | 4 | 4 | no | — |
| 267 | C22 | — | 8 | 2 | no | — |
| 268 | C22 | — | 8 | 2 | no | — |
| 269 | C22 | Q8×D8 | 1 | 2 | yes | — |
| 270 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 271 | C4 | — | 2 | 3 | no | — |
| 272 | C4 | — | 2 | 3 | no | — |
| 273 | C4 | — | 2 | 3 | no | — |
| 274 | C4 | — | 4 | 4 | no | — |
| 275 | C4 | — | 8 | 6 | no | — |
| 276 | C4 | — | 8 | 6 | no | — |
| 277 | C4 | Q8×D8 | 1 | 2 | yes | — |
| 278 | C4 | C2×#508 | 1 | 6 | yes | — |
| 279 | C4 | C2×D4×Q8 | 1 | 1 | yes | characteristic |
| 280 | C4 | C2×#578 | 1 | 1 | yes | characteristic |
| 281 | C2 | — | 2 | 2 | no | — |
| 282 | C2 | — | 8 | 4 | no | — |
| 283 | C2 | C2×Q8×D8 | 1 | 1 | yes | characteristic |
| 284 | C2 | C22×#508 | 1 | 1 | yes | characteristic |
| 285 | C2 | #62635 | 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