← #7622513
Subgroups — #7622513
Order 512.
3913 conjugacy classes of subgroups.
The subgroup lattice has 323 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#3584 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#3407 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#2916 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #33393 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #33392 | C2 | 1 | 8 | yes | maximal, complemented |
| 6 | C2×#2148 | C2 | 1 | 12 | yes | maximal, complemented |
| 7 | C2×#585 | — | 2 | 1 | no | — |
| 8 | #3584 | — | 2 | 4 | no | — |
| 9 | #3584 | — | 2 | 4 | no | — |
| 10 | #3584 | C4 | 1 | 8 | yes | — |
| 11 | #3584 | C4 | 1 | 8 | yes | — |
| 12 | C2×#578 | C22 | 1 | 4 | yes | — |
| 13 | C2×#577 | — | 2 | 4 | no | — |
| 14 | C2×#576 | C22 | 1 | 4 | yes | — |
| 15 | C2×#575 | — | 2 | 3 | no | — |
| 16 | C22×#135 | C22 | 1 | 6 | yes | — |
| 17 | C2×#568 | — | 2 | 1 | no | — |
| 18 | C22×#131 | C22 | 1 | 1 | yes | characteristic |
| 19 | #3407 | — | 2 | 4 | no | — |
| 20 | #3407 | C22 | 1 | 8 | yes | complemented |
| 21 | C2×#504 | — | 2 | 4 | no | — |
| 22 | C22×M32 | C22 | 1 | 6 | yes | complemented |
| 23 | #2916 | — | 2 | 4 | no | — |
| 24 | #2916 | C22 | 1 | 8 | yes | complemented |
| 25 | C2×#430 | — | 2 | 3 | no | — |
| 26 | C22×#108 | C22 | 1 | 6 | yes | complemented |
| 27 | #2148 | C22 | 1 | 16 | yes | complemented |
| 28 | #2148 | C22 | 1 | 48 | yes | complemented |
| 29 | C2×#362 | C22 | 1 | 24 | yes | complemented |
| 30 | C2×#361 | C22 | 1 | 24 | yes | complemented |
| 31 | #585 | — | 2 | 8 | no | — |
| 32 | #585 | — | 2 | 8 | no | — |
| 33 | #585 | — | 4 | 4 | no | — |
| 34 | #585 | — | 4 | 4 | no | — |
| 35 | C2×2−1+4 | — | 2 | 4 | no | — |
| 36 | C2×2−1+4 | — | 4 | 1 | no | — |
| 37 | C2×H(5,𝔽2) | — | 2 | 4 | no | — |
| 38 | C2×H(5,𝔽2) | — | 4 | 3 | no | — |
| 39 | C22×#41 | — | 2 | 6 | no | — |
| 40 | C22×#41 | — | 4 | 4 | no | — |
| 41 | C22×#41 | D4 | 1 | 1 | yes | characteristic |
| 42 | #578 | — | 2 | 16 | no | — |
| 43 | #578 | C2×C4 | 1 | 32 | yes | — |
| 44 | #577 | — | 2 | 32 | no | — |
| 45 | #577 | — | 2 | 32 | no | — |
| 46 | #576 | — | 2 | 16 | no | — |
| 47 | #576 | C2×C4 | 1 | 32 | yes | — |
| 48 | #575 | — | 2 | 24 | no | — |
| 49 | #575 | — | 2 | 24 | no | — |
| 50 | C2×#137 | — | 2 | 12 | no | — |
| 51 | C2×AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 52 | C2×#135 | — | 2 | 12 | no | — |
| 53 | C2×#135 | — | 2 | 12 | no | — |
| 54 | C2×#135 | — | 2 | 6 | no | — |
| 55 | C2×#135 | — | 2 | 6 | no | — |
| 56 | C2×#135 | C2×C4 | 1 | 24 | yes | — |
| 57 | C2×#135 | C2×C4 | 1 | 24 | yes | — |
| 58 | C2×#135 | C23 | 1 | 4 | yes | — |
| 59 | C2×#135 | C23 | 1 | 12 | yes | — |
| 60 | C22×Q16 | C23 | 1 | 6 | yes | — |
| 61 | C22×SD16 | — | 2 | 6 | no | — |
| 62 | C22×D8 | C23 | 1 | 6 | yes | — |
| 63 | #568 | — | 2 | 8 | no | — |
| 64 | #568 | — | 4 | 4 | no | — |
| 65 | C2×#131 | — | 4 | 4 | no | — |
| 66 | C2×#131 | C2×C4 | 1 | 4 | yes | — |
| 67 | C2×#131 | C2×C4 | 1 | 4 | yes | — |
| 68 | C2×#131 | D4 | 1 | 4 | yes | — |
| 69 | C2×#131 | D4 | 1 | 4 | yes | — |
| 70 | C2×#131 | D4 | 1 | 4 | yes | — |
| 71 | C2×#131 | C23 | 1 | 4 | yes | — |
| 72 | C22×M16 | — | 2 | 6 | no | — |
| 73 | C22×M16 | D4 | 1 | 3 | yes | — |
| 74 | C23×C8 | C23 | 1 | 3 | yes | — |
| 75 | #504 | — | 2 | 32 | no | — |
| 76 | C2×M32 | — | 2 | 6 | no | — |
| 77 | C2×M32 | — | 2 | 12 | no | — |
| 78 | C2×M32 | C23 | 1 | 4 | yes | — |
| 79 | C2×M32 | C23 | 1 | 24 | yes | — |
| 80 | C22×C16 | — | 2 | 6 | no | — |
| 81 | #430 | — | 2 | 24 | no | — |
| 82 | C2×#108 | — | 2 | 6 | no | — |
| 83 | C2×#108 | — | 2 | 12 | no | — |
| 84 | C2×#108 | C23 | 1 | 24 | yes | — |
| 85 | C2×#108 | C23 | 1 | 4 | yes | — |
| 86 | #362 | C23 | 1 | 32 | yes | — |
| 87 | #361 | C23 | 1 | 32 | yes | — |
| 88 | 2−1+4 | — | 2 | 32 | no | — |
| 89 | 2−1+4 | — | 4 | 16 | no | — |
| 90 | 2−1+4 | — | 4 | 8 | no | — |
| 91 | 2−1+4 | — | 4 | 8 | no | — |
| 92 | H(5,𝔽2) | — | 2 | 32 | no | — |
| 93 | H(5,𝔽2) | — | 4 | 16 | no | — |
| 94 | H(5,𝔽2) | — | 4 | 24 | no | — |
| 95 | H(5,𝔽2) | — | 4 | 24 | no | — |
| 96 | C2×#41 | — | 2 | 24 | no | — |
| 97 | C2×#41 | — | 2 | 24 | no | — |
| 98 | C2×#41 | — | 2 | 4 | no | — |
| 99 | C2×#41 | — | 2 | 12 | no | — |
| 100 | C2×#41 | — | 4 | 32 | no | — |
| 101 | C2×#41 | — | 4 | 32 | no | — |
| 102 | C2×#41 | — | 4 | 4 | no | — |
| 103 | C2×#41 | — | 4 | 12 | no | — |
| 104 | C2×#41 | — | 4 | 12 | no | — |
| 105 | C2×#41 | — | 4 | 12 | no | — |
| 106 | C2×#41 | — | 4 | 6 | no | — |
| 107 | C2×#41 | — | 4 | 6 | no | — |
| 108 | C2×#41 | C22⋊C4 | 1 | 4 | yes | — |
| 109 | C2×#41 | C22⋊C4 | 1 | 4 | yes | — |
| 110 | C2×#41 | D8 | 1 | 4 | yes | — |
| 111 | C2×#41 | D8 | 1 | 4 | yes | — |
| 112 | C2×#41 | SD16 | 1 | 4 | yes | — |
| 113 | C2×#41 | C2×D4 | 1 | 4 | yes | — |
| 114 | C22×Q8 | — | 2 | 6 | no | — |
| 115 | C22×Q8 | — | 4 | 4 | no | — |
| 116 | C22×Q8 | SD16 | 1 | 1 | yes | characteristic |
| 117 | C22×D4 | — | 2 | 6 | no | — |
| 118 | C22×D4 | — | 4 | 6 | no | — |
| 119 | C22×D4 | — | 4 | 12 | no | — |
| 120 | C22×D4 | SD16 | 1 | 3 | yes | — |
| 121 | C23×C4 | — | 4 | 6 | no | — |
| 122 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 123 | #137 | — | 2 | 96 | no | — |
| 124 | AGL(1,ℤ/8) | — | 2 | 96 | no | — |
| 125 | #135 | — | 2 | 24 | no | — |
| 126 | #135 | — | 2 | 24 | no | — |
| 127 | #135 | — | 2 | 24 | no | — |
| 128 | #135 | — | 2 | 24 | no | — |
| 129 | #135 | — | 2 | 4 | no | — |
| 130 | #135 | — | 2 | 4 | no | — |
| 131 | #135 | — | 2 | 24 | no | — |
| 132 | #135 | C22×C4 | 1 | 8 | yes | — |
| 133 | #135 | C22×C4 | 1 | 8 | yes | — |
| 134 | #135 | C22×C4 | 1 | 48 | yes | — |
| 135 | C2×Q16 | — | 2 | 6 | no | — |
| 136 | C2×Q16 | — | 2 | 12 | no | — |
| 137 | C2×Q16 | C22×C4 | 1 | 24 | yes | — |
| 138 | C2×Q16 | C24 | 1 | 4 | yes | — |
| 139 | C2×SD16 | — | 2 | 24 | no | — |
| 140 | C2×SD16 | — | 2 | 24 | no | — |
| 141 | C2×SD16 | — | 2 | 4 | no | — |
| 142 | C2×SD16 | — | 2 | 12 | no | — |
| 143 | C2×D8 | — | 2 | 6 | no | — |
| 144 | C2×D8 | — | 2 | 12 | no | — |
| 145 | C2×D8 | C22×C4 | 1 | 24 | yes | — |
| 146 | C2×D8 | C24 | 1 | 4 | yes | — |
| 147 | #131 | — | 4 | 32 | no | — |
| 148 | #131 | C22⋊C4 | 1 | 16 | yes | — |
| 149 | #131 | C22⋊C4 | 1 | 16 | yes | — |
| 150 | #131 | C22×C4 | 1 | 16 | yes | — |
| 151 | #131 | C2×D4 | 1 | 16 | yes | — |
| 152 | C2×M16 | — | 2 | 4 | no | — |
| 153 | C2×M16 | — | 2 | 24 | no | — |
| 154 | C2×M16 | — | 4 | 6 | no | — |
| 155 | C2×M16 | — | 4 | 12 | no | — |
| 156 | C2×M16 | C22⋊C4 | 1 | 12 | yes | — |
| 157 | C2×M16 | C22⋊C4 | 1 | 12 | yes | — |
| 158 | C2×M16 | C2×D4 | 1 | 12 | yes | — |
| 159 | C22×C8 | — | 4 | 6 | no | — |
| 160 | C22×C8 | C22×C4 | 1 | 6 | yes | — |
| 161 | C22×C8 | C22×C4 | 1 | 6 | yes | — |
| 162 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 163 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 164 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 165 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 166 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 167 | C22×C8 | C24 | 1 | 6 | yes | — |
| 168 | M32 | — | 2 | 4 | no | — |
| 169 | M32 | — | 2 | 24 | no | — |
| 170 | M32 | C24 | 1 | 8 | yes | — |
| 171 | C2×C16 | — | 2 | 4 | no | — |
| 172 | C2×C16 | — | 2 | 24 | no | — |
| 173 | #108 | — | 2 | 4 | no | — |
| 174 | #108 | — | 2 | 24 | no | — |
| 175 | #108 | C24 | 1 | 8 | yes | — |
| 176 | C24 | — | 4 | 6 | no | — |
| 177 | C24 | C2×SD16 | 1 | 3 | yes | — |
| 178 | #41 | — | 2 | 8 | no | — |
| 179 | #41 | — | 2 | 8 | no | — |
| 180 | #41 | — | 2 | 48 | no | — |
| 181 | #41 | — | 4 | 32 | no | — |
| 182 | #41 | — | 4 | 96 | no | — |
| 183 | #41 | — | 4 | 4 | no | — |
| 184 | #41 | — | 4 | 4 | no | — |
| 185 | #41 | — | 4 | 24 | no | — |
| 186 | #41 | — | 4 | 24 | no | — |
| 187 | #41 | — | 4 | 24 | no | — |
| 188 | #41 | — | 4 | 24 | no | — |
| 189 | #41 | — | 4 | 24 | no | — |
| 190 | #41 | #102 | 1 | 16 | yes | — |
| 191 | #41 | #102 | 1 | 16 | yes | — |
| 192 | #41 | C2×C22⋊C4 | 1 | 16 | yes | — |
| 193 | #41 | C2×D8 | 1 | 16 | yes | — |
| 194 | C2×Q8 | — | 2 | 24 | no | — |
| 195 | C2×Q8 | — | 2 | 4 | no | — |
| 196 | C2×Q8 | — | 4 | 32 | no | — |
| 197 | C2×Q8 | — | 4 | 12 | no | — |
| 198 | C2×Q8 | — | 4 | 6 | no | — |
| 199 | C2×Q8 | #102 | 1 | 4 | yes | — |
| 200 | C2×Q8 | #102 | 1 | 4 | yes | — |
| 201 | C2×Q8 | C2×SD16 | 1 | 4 | yes | — |
| 202 | C2×D4 | — | 2 | 24 | no | — |
| 203 | C2×D4 | — | 2 | 4 | no | — |
| 204 | C2×D4 | — | 4 | 12 | no | — |
| 205 | C2×D4 | — | 4 | 4 | no | — |
| 206 | C2×D4 | — | 4 | 24 | no | — |
| 207 | C2×D4 | — | 4 | 24 | no | — |
| 208 | C2×D4 | — | 4 | 96 | no | — |
| 209 | C2×D4 | — | 4 | 12 | no | — |
| 210 | C2×D4 | — | 4 | 6 | no | — |
| 211 | C2×D4 | #102 | 1 | 12 | yes | — |
| 212 | C2×D4 | #102 | 1 | 12 | yes | — |
| 213 | C2×D4 | C2×SD16 | 1 | 12 | yes | — |
| 214 | C22×C4 | — | 4 | 6 | no | — |
| 215 | C22×C4 | — | 4 | 4 | no | — |
| 216 | C22×C4 | — | 4 | 12 | no | — |
| 217 | C22×C4 | — | 4 | 24 | no | — |
| 218 | C22×C4 | — | 4 | 24 | no | — |
| 219 | C22×C4 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 220 | C22×C4 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 221 | C22×C4 | C2×D8 | 1 | 6 | yes | — |
| 222 | C22×C4 | C2×D8 | 1 | 6 | yes | — |
| 223 | C22×C4 | C2×SD16 | 1 | 6 | yes | — |
| 224 | C22×C4 | C2×SD16 | 1 | 3 | yes | — |
| 225 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 226 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 227 | Q16 | — | 2 | 24 | no | — |
| 228 | Q16 | — | 2 | 4 | no | — |
| 229 | Q16 | C23×C4 | 1 | 8 | yes | — |
| 230 | SD16 | — | 2 | 8 | no | — |
| 231 | SD16 | — | 2 | 8 | no | — |
| 232 | SD16 | — | 2 | 48 | no | — |
| 233 | D8 | — | 2 | 24 | no | — |
| 234 | D8 | — | 2 | 4 | no | — |
| 235 | D8 | C23×C4 | 1 | 8 | yes | — |
| 236 | M16 | — | 2 | 8 | no | — |
| 237 | M16 | — | 4 | 4 | no | — |
| 238 | M16 | — | 4 | 24 | no | — |
| 239 | M16 | C2×C22⋊C4 | 1 | 48 | yes | — |
| 240 | C2×C8 | — | 4 | 4 | no | — |
| 241 | C2×C8 | — | 4 | 24 | no | — |
| 242 | C2×C8 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 243 | C2×C8 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 244 | C2×C8 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 245 | C2×C8 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 246 | C2×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 247 | C2×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | C23×C4 | 1 | 12 | yes | — |
| 249 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | C22×D4 | 1 | 6 | yes | — |
| 251 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 252 | C2×C8 | C22×D4 | 1 | 12 | yes | — |
| 253 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 254 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 255 | C16 | — | 2 | 8 | no | — |
| 256 | C23 | — | 4 | 4 | no | — |
| 257 | C23 | — | 4 | 24 | no | — |
| 258 | C23 | — | 8 | 3 | no | — |
| 259 | C23 | — | 16 | 12 | no | — |
| 260 | C23 | C2×#102 | 1 | 6 | yes | — |
| 261 | C23 | C2×#102 | 1 | 6 | yes | — |
| 262 | C23 | C22×SD16 | 1 | 6 | yes | — |
| 263 | C23 | C22×SD16 | 1 | 1 | yes | characteristic |
| 264 | Q8 | — | 2 | 8 | no | — |
| 265 | Q8 | — | 4 | 24 | no | — |
| 266 | Q8 | — | 4 | 4 | no | — |
| 267 | Q8 | C2×#102 | 1 | 16 | yes | — |
| 268 | D4 | — | 2 | 8 | no | — |
| 269 | D4 | — | 4 | 48 | no | — |
| 270 | D4 | — | 4 | 8 | no | — |
| 271 | D4 | — | 4 | 8 | no | — |
| 272 | D4 | — | 4 | 24 | no | — |
| 273 | D4 | — | 4 | 4 | no | — |
| 274 | D4 | C2×#102 | 1 | 48 | yes | — |
| 275 | C2×C4 | — | 4 | 24 | no | — |
| 276 | C2×C4 | — | 4 | 4 | no | — |
| 277 | C2×C4 | — | 4 | 8 | no | — |
| 278 | C2×C4 | — | 4 | 48 | no | — |
| 279 | C2×C4 | — | 4 | 8 | no | — |
| 280 | C2×C4 | C2×#102 | 1 | 12 | yes | — |
| 281 | C2×C4 | C2×#102 | 1 | 6 | yes | — |
| 282 | C2×C4 | C2×#102 | 1 | 6 | yes | — |
| 283 | C2×C4 | C2×#102 | 1 | 12 | yes | — |
| 284 | C2×C4 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | C22×C22⋊C4 | 1 | 12 | yes | — |
| 286 | C2×C4 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | C22×D8 | 1 | 12 | yes | — |
| 288 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | C22×SD16 | 1 | 6 | yes | — |
| 291 | C2×C4 | C22×SD16 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 293 | C8 | — | 4 | 8 | no | — |
| 294 | C8 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 295 | C8 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 296 | C8 | C22×C22⋊C4 | 1 | 12 | yes | — |
| 297 | C8 | C24×C4 | 1 | 1 | yes | characteristic |
| 298 | C8 | C23×D4 | 1 | 1 | yes | characteristic |
| 299 | C22 | — | 4 | 1 | no | — |
| 300 | C22 | — | 4 | 6 | no | — |
| 301 | C22 | — | 4 | 6 | no | — |
| 302 | C22 | — | 4 | 6 | no | — |
| 303 | C22 | — | 4 | 8 | no | — |
| 304 | C22 | — | 8 | 8 | no | — |
| 305 | C22 | — | 16 | 24 | no | — |
| 306 | C22 | C22×#102 | 1 | 12 | yes | — |
| 307 | C22 | C22×#102 | 1 | 1 | yes | characteristic |
| 308 | C22 | C22×#102 | 1 | 1 | yes | characteristic |
| 309 | C22 | C23×SD16 | 1 | 1 | yes | characteristic |
| 310 | C4 | — | 4 | 8 | no | — |
| 311 | C4 | C22×#102 | 1 | 1 | yes | characteristic |
| 312 | C4 | C22×#102 | 1 | 12 | yes | — |
| 313 | C4 | C22×#102 | 1 | 1 | yes | characteristic |
| 314 | C4 | C23×C22⋊C4 | 1 | 1 | yes | characteristic |
| 315 | C4 | C23×D8 | 1 | 1 | yes | characteristic |
| 316 | C2 | — | 2 | 1 | no | — |
| 317 | C2 | — | 2 | 1 | no | — |
| 318 | C2 | — | 2 | 1 | no | — |
| 319 | C2 | — | 2 | 12 | no | — |
| 320 | C2 | — | 8 | 8 | no | — |
| 321 | C2 | C23×#102 | 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