Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
537 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 537 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#2214 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #28504 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#1709 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13262 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#1357 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#1335 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7179 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#510 | — | 2 | 1 | no | — |
| 9 | #3155 | — | 2 | 1 | no | — |
| 10 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#486 | — | 2 | 2 | no | — |
| 13 | C2×#472 | — | 2 | 1 | no | — |
| 14 | #3058 | — | 2 | 1 | no | — |
| 15 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2214 | — | 2 | 2 | no | — |
| 17 | #2214 | — | 2 | 2 | no | — |
| 18 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#362 | — | 2 | 1 | no | — |
| 21 | C2×#362 | — | 2 | 1 | no | — |
| 22 | #2141 | — | 2 | 1 | no | — |
| 23 | #2002 | — | 2 | 2 | no | — |
| 24 | #1978 | — | 2 | 1 | no | — |
| 25 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 26 | #1709 | — | 2 | 2 | no | — |
| 27 | #1709 | — | 2 | 2 | no | — |
| 28 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1357 | — | 2 | 2 | no | — |
| 30 | #1335 | — | 2 | 2 | no | — |
| 31 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 32 | C2×#135 | D4 | 1 | 1 | yes | characteristic |
| 33 | C22×Q16 | — | 2 | 1 | no | — |
| 34 | C22×Q16 | — | 2 | 1 | no | — |
| 35 | C22×D8 | — | 2 | 1 | no | — |
| 36 | C22×D8 | — | 2 | 1 | no | — |
| 37 | C2×#131 | — | 2 | 1 | no | — |
| 38 | #535 | — | 2 | 2 | no | — |
| 39 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 40 | C2×#127 | D4 | 1 | 2 | yes | — |
| 41 | #510 | — | 2 | 4 | no | — |
| 42 | #510 | — | 4 | 2 | no | — |
| 43 | C2×Q32 | — | 4 | 1 | no | — |
| 44 | C2×SD32 | — | 4 | 1 | no | — |
| 45 | C2×M32 | — | 2 | 1 | no | — |
| 46 | C2×M32 | — | 2 | 1 | no | — |
| 47 | #496 | — | 4 | 2 | no | — |
| 48 | #495 | — | 4 | 1 | no | — |
| 49 | #494 | — | 2 | 1 | no | — |
| 50 | #494 | — | 2 | 1 | no | — |
| 51 | #494 | — | 2 | 1 | no | — |
| 52 | #494 | D4 | 1 | 2 | yes | complemented |
| 53 | #493 | — | 2 | 1 | no | — |
| 54 | #493 | — | 2 | 1 | no | — |
| 55 | #493 | — | 2 | 1 | no | — |
| 56 | #493 | — | 2 | 1 | no | — |
| 57 | #493 | — | 4 | 1 | no | — |
| 58 | #486 | — | 2 | 8 | no | — |
| 59 | #486 | — | 4 | 4 | no | — |
| 60 | #472 | — | 4 | 2 | no | — |
| 61 | #472 | — | 4 | 2 | no | — |
| 62 | #469 | — | 4 | 2 | no | — |
| 63 | #466 | — | 4 | 2 | no | — |
| 64 | #443 | — | 2 | 1 | no | — |
| 65 | #443 | — | 2 | 1 | no | — |
| 66 | #443 | — | 2 | 1 | no | — |
| 67 | #443 | — | 2 | 1 | no | — |
| 68 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 69 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 70 | #427 | D4 | 1 | 1 | yes | characteristic |
| 71 | C2×#104 | — | 2 | 1 | no | — |
| 72 | #418 | — | 2 | 2 | no | — |
| 73 | C2×#102 | — | 4 | 4 | no | — |
| 74 | C2×#100 | — | 2 | 2 | no | — |
| 75 | #405 | — | 2 | 1 | no | — |
| 76 | C2×C4×C8 | — | 2 | 1 | no | — |
| 77 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 78 | #364 | — | 2 | 1 | no | — |
| 79 | #364 | — | 2 | 1 | no | — |
| 80 | #364 | — | 2 | 1 | no | — |
| 81 | #364 | D4 | 1 | 2 | yes | complemented |
| 82 | #362 | — | 2 | 4 | no | — |
| 83 | #362 | — | 4 | 2 | no | — |
| 84 | #358 | — | 4 | 2 | no | — |
| 85 | #339 | — | 2 | 1 | no | — |
| 86 | C82 | — | 2 | 1 | no | — |
| 87 | C82 | D4 | 1 | 2 | yes | complemented |
| 88 | C2×#41 | — | 2 | 1 | no | — |
| 89 | C22×Q8 | — | 4 | 1 | no | — |
| 90 | C22×D4 | — | 2 | 2 | no | — |
| 91 | C22×D4 | — | 4 | 2 | no | — |
| 92 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 93 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 94 | #135 | — | 2 | 1 | no | — |
| 95 | #135 | — | 2 | 1 | no | — |
| 96 | #135 | — | 2 | 1 | no | — |
| 97 | #135 | — | 2 | 1 | no | — |
| 98 | C2×Q16 | — | 2 | 2 | no | — |
| 99 | C2×Q16 | — | 2 | 2 | no | — |
| 100 | C2×Q16 | — | 4 | 1 | no | — |
| 101 | C2×Q16 | — | 4 | 1 | no | — |
| 102 | C2×Q16 | — | 4 | 1 | no | — |
| 103 | C2×Q16 | — | 4 | 1 | no | — |
| 104 | C2×Q16 | — | 4 | 1 | no | — |
| 105 | C2×Q16 | — | 4 | 1 | no | — |
| 106 | C2×Q16 | — | 4 | 1 | no | — |
| 107 | C2×Q16 | — | 4 | 1 | no | — |
| 108 | C2×Q16 | — | 4 | 1 | no | — |
| 109 | C2×Q16 | — | 4 | 1 | no | — |
| 110 | C2×Q16 | D8 | 1 | 1 | yes | characteristic |
| 111 | C2×SD16 | — | 2 | 1 | no | — |
| 112 | C2×SD16 | — | 4 | 2 | no | — |
| 113 | C2×D8 | — | 4 | 2 | no | — |
| 114 | C2×D8 | — | 4 | 1 | no | — |
| 115 | C2×D8 | — | 4 | 1 | no | — |
| 116 | C2×D8 | — | 4 | 1 | no | — |
| 117 | C2×D8 | — | 4 | 1 | no | — |
| 118 | C2×D8 | — | 4 | 1 | no | — |
| 119 | C2×D8 | — | 4 | 1 | no | — |
| 120 | C2×D8 | — | 4 | 1 | no | — |
| 121 | C2×D8 | — | 4 | 1 | no | — |
| 122 | C2×D8 | — | 4 | 1 | no | — |
| 123 | C2×D8 | — | 4 | 1 | no | — |
| 124 | C2×D8 | — | 4 | 1 | no | — |
| 125 | C2×D8 | — | 4 | 1 | no | — |
| 126 | C2×D8 | D8 | 1 | 1 | yes | characteristic |
| 127 | #131 | — | 4 | 1 | no | — |
| 128 | #131 | — | 4 | 1 | no | — |
| 129 | #131 | — | 4 | 1 | no | — |
| 130 | #131 | — | 4 | 1 | no | — |
| 131 | C2×M16 | — | 2 | 1 | no | — |
| 132 | C2×M16 | — | 2 | 1 | no | — |
| 133 | C2×M16 | — | 4 | 1 | no | — |
| 134 | C22×C8 | — | 2 | 2 | no | — |
| 135 | C22×C8 | — | 4 | 1 | no | — |
| 136 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 137 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 138 | C4⋊Q8 | — | 2 | 2 | no | — |
| 139 | C4⋊Q8 | — | 4 | 2 | no | — |
| 140 | C4⋊Q8 | — | 4 | 1 | no | — |
| 141 | #127 | — | 2 | 2 | no | — |
| 142 | #127 | — | 2 | 2 | no | — |
| 143 | #127 | — | 4 | 2 | no | — |
| 144 | #127 | — | 4 | 2 | no | — |
| 145 | #124 | — | 4 | 2 | no | — |
| 146 | #121 | — | 4 | 4 | no | — |
| 147 | #120 | — | 8 | 2 | no | — |
| 148 | #117 | — | 2 | 1 | no | — |
| 149 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 150 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 151 | Q32 | — | 4 | 4 | no | — |
| 152 | SD32 | — | 4 | 4 | no | — |
| 153 | M32 | — | 2 | 2 | no | — |
| 154 | M32 | — | 4 | 1 | no | — |
| 155 | M32 | — | 4 | 1 | no | — |
| 156 | M32 | — | 4 | 1 | no | — |
| 157 | C2×C16 | — | 4 | 1 | no | — |
| 158 | C2×C16 | — | 4 | 1 | no | — |
| 159 | #108 | — | 2 | 1 | no | — |
| 160 | #108 | — | 2 | 1 | no | — |
| 161 | #108 | — | 2 | 1 | no | — |
| 162 | #108 | D8 | 1 | 2 | yes | — |
| 163 | #107 | D8 | 1 | 2 | yes | — |
| 164 | #106 | — | 2 | 1 | no | — |
| 165 | #104 | — | 4 | 1 | no | — |
| 166 | #104 | — | 4 | 1 | no | — |
| 167 | #102 | — | 4 | 4 | no | — |
| 168 | #102 | — | 4 | 16 | no | — |
| 169 | #100 | — | 4 | 4 | no | — |
| 170 | #97 | — | 4 | 1 | no | — |
| 171 | C4×C8 | — | 2 | 2 | no | — |
| 172 | C4×C8 | — | 2 | 2 | no | — |
| 173 | C4×C8 | — | 4 | 1 | no | — |
| 174 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 175 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 176 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 177 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 178 | C24 | — | 4 | 2 | no | — |
| 179 | #41 | — | 4 | 1 | no | — |
| 180 | #41 | — | 4 | 1 | no | — |
| 181 | #41 | — | 4 | 1 | no | — |
| 182 | #41 | — | 4 | 1 | no | — |
| 183 | C2×Q8 | — | 2 | 1 | no | — |
| 184 | C2×Q8 | — | 4 | 2 | no | — |
| 185 | C2×Q8 | — | 8 | 1 | no | — |
| 186 | C2×Q8 | — | 8 | 1 | no | — |
| 187 | C2×Q8 | — | 8 | 1 | no | — |
| 188 | C2×Q8 | — | 8 | 1 | no | — |
| 189 | C2×Q8 | — | 8 | 1 | no | — |
| 190 | C2×D4 | — | 2 | 1 | no | — |
| 191 | C2×D4 | — | 4 | 4 | no | — |
| 192 | C2×D4 | — | 4 | 1 | no | — |
| 193 | C2×D4 | — | 4 | 2 | no | — |
| 194 | C2×D4 | — | 4 | 2 | no | — |
| 195 | C2×D4 | — | 4 | 2 | no | — |
| 196 | C2×D4 | — | 4 | 2 | no | — |
| 197 | C2×D4 | — | 4 | 2 | no | — |
| 198 | C2×D4 | — | 4 | 2 | no | — |
| 199 | C2×D4 | — | 8 | 8 | no | — |
| 200 | C2×D4 | — | 8 | 2 | no | — |
| 201 | C2×D4 | — | 8 | 4 | no | — |
| 202 | C22×C4 | — | 2 | 1 | no | — |
| 203 | C22×C4 | — | 4 | 1 | no | — |
| 204 | C22×C4 | — | 8 | 1 | no | — |
| 205 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 206 | Q16 | — | 2 | 1 | no | — |
| 207 | Q16 | — | 2 | 1 | no | — |
| 208 | Q16 | — | 4 | 2 | no | — |
| 209 | Q16 | — | 4 | 2 | no | — |
| 210 | Q16 | — | 4 | 2 | no | — |
| 211 | Q16 | — | 4 | 2 | no | — |
| 212 | Q16 | — | 8 | 1 | no | — |
| 213 | Q16 | — | 8 | 1 | no | — |
| 214 | Q16 | — | 8 | 1 | no | — |
| 215 | Q16 | — | 8 | 1 | no | — |
| 216 | SD16 | — | 4 | 1 | no | — |
| 217 | SD16 | — | 4 | 1 | no | — |
| 218 | SD16 | — | 8 | 4 | no | — |
| 219 | D8 | — | 2 | 1 | no | — |
| 220 | D8 | — | 2 | 1 | no | — |
| 221 | D8 | — | 8 | 4 | no | — |
| 222 | D8 | — | 8 | 1 | no | — |
| 223 | D8 | — | 8 | 1 | no | — |
| 224 | D8 | — | 8 | 1 | no | — |
| 225 | D8 | — | 8 | 1 | no | — |
| 226 | D8 | — | 8 | 1 | no | — |
| 227 | D8 | — | 8 | 1 | no | — |
| 228 | D8 | — | 8 | 1 | no | — |
| 229 | D8 | — | 8 | 1 | no | — |
| 230 | M16 | — | 2 | 2 | no | — |
| 231 | M16 | — | 4 | 1 | no | — |
| 232 | M16 | — | 4 | 1 | no | — |
| 233 | M16 | — | 4 | 1 | no | — |
| 234 | M16 | — | 8 | 1 | no | — |
| 235 | M16 | — | 8 | 1 | no | — |
| 236 | C2×C8 | — | 2 | 4 | no | — |
| 237 | C2×C8 | — | 4 | 4 | no | — |
| 238 | C2×C8 | — | 4 | 1 | no | — |
| 239 | C2×C8 | — | 4 | 1 | no | — |
| 240 | C2×C8 | — | 8 | 1 | no | — |
| 241 | C2×C8 | — | 8 | 1 | no | — |
| 242 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 243 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 244 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 245 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 246 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 247 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 252 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 253 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 254 | C4⋊C4 | — | 2 | 2 | no | — |
| 255 | C4⋊C4 | — | 4 | 4 | no | — |
| 256 | C4⋊C4 | — | 8 | 2 | no | — |
| 257 | C22⋊C4 | — | 4 | 1 | no | — |
| 258 | C22⋊C4 | — | 8 | 4 | no | — |
| 259 | C42 | — | 2 | 1 | no | — |
| 260 | C42 | — | 4 | 1 | no | — |
| 261 | C42 | #120 | 1 | 1 | yes | characteristic |
| 262 | C42 | #121 | 1 | 1 | yes | characteristic |
| 263 | C16 | — | 4 | 2 | no | — |
| 264 | C16 | — | 8 | 1 | no | — |
| 265 | C23 | — | 4 | 1 | no | — |
| 266 | C23 | — | 8 | 2 | no | — |
| 267 | C23 | — | 8 | 2 | no | — |
| 268 | C23 | — | 8 | 2 | no | — |
| 269 | C23 | — | 16 | 4 | no | — |
| 270 | C23 | #453 | 1 | 1 | yes | characteristic |
| 271 | Q8 | — | 4 | 1 | no | — |
| 272 | Q8 | — | 4 | 1 | no | — |
| 273 | Q8 | — | 8 | 2 | no | — |
| 274 | Q8 | — | 8 | 2 | no | — |
| 275 | Q8 | — | 16 | 1 | no | — |
| 276 | Q8 | — | 16 | 1 | no | — |
| 277 | D4 | — | 4 | 1 | no | — |
| 278 | D4 | — | 4 | 1 | no | — |
| 279 | D4 | — | 8 | 8 | no | — |
| 280 | D4 | — | 8 | 1 | no | — |
| 281 | D4 | — | 8 | 1 | no | — |
| 282 | D4 | — | 8 | 2 | no | — |
| 283 | D4 | — | 8 | 2 | no | — |
| 284 | D4 | — | 8 | 2 | no | — |
| 285 | D4 | — | 8 | 2 | no | — |
| 286 | D4 | — | 16 | 4 | no | — |
| 287 | D4 | — | 32 | 4 | no | — |
| 288 | C2×C4 | — | 2 | 4 | no | — |
| 289 | C2×C4 | — | 2 | 1 | no | — |
| 290 | C2×C4 | — | 2 | 1 | no | — |
| 291 | C2×C4 | — | 4 | 2 | no | — |
| 292 | C2×C4 | — | 4 | 1 | no | — |
| 293 | C2×C4 | — | 8 | 1 | no | — |
| 294 | C2×C4 | — | 8 | 1 | no | — |
| 295 | C2×C4 | — | 8 | 2 | no | — |
| 296 | C2×C4 | — | 16 | 1 | no | — |
| 297 | C2×C4 | — | 16 | 1 | no | — |
| 298 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | #449 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | #453 | 1 | 1 | yes | characteristic |
| 304 | C8 | — | 4 | 4 | no | — |
| 305 | C8 | — | 4 | 2 | no | — |
| 306 | C8 | — | 8 | 1 | no | — |
| 307 | C8 | #447 | 1 | 1 | yes | characteristic |
| 308 | C8 | #449 | 1 | 1 | yes | characteristic |
| 309 | C8 | #459 | 1 | 1 | yes | characteristic |
| 310 | C8 | #459 | 1 | 1 | yes | characteristic |
| 311 | C8 | #466 | 1 | 1 | yes | characteristic |
| 312 | C8 | #469 | 1 | 1 | yes | characteristic |
| 313 | C8 | #493 | 1 | 1 | yes | characteristic |
| 314 | C8 | #496 | 1 | 1 | yes | characteristic |
| 315 | C22 | — | 2 | 2 | no | — |
| 316 | C22 | — | 8 | 2 | no | — |
| 317 | C22 | — | 8 | 1 | no | — |
| 318 | C22 | — | 16 | 4 | no | — |
| 319 | C22 | — | 16 | 4 | no | — |
| 320 | C22 | — | 16 | 2 | no | — |
| 321 | C22 | — | 32 | 2 | no | — |
| 322 | C22 | #1619 | 1 | 1 | yes | characteristic |
| 323 | C22 | #1621 | 1 | 1 | yes | characteristic |
| 324 | C22 | #2002 | 1 | 1 | yes | characteristic |
| 325 | C4 | — | 2 | 2 | no | — |
| 326 | C4 | — | 2 | 2 | no | — |
| 327 | C4 | — | 8 | 2 | no | — |
| 328 | C4 | — | 8 | 1 | no | — |
| 329 | C4 | — | 16 | 2 | no | — |
| 330 | C4 | #1619 | 1 | 1 | yes | characteristic |
| 331 | C4 | #1621 | 1 | 1 | yes | characteristic |
| 332 | C4 | #1999 | 1 | 1 | yes | characteristic |
| 333 | C4 | #2011 | 1 | 1 | yes | characteristic |
| 334 | C2 | — | 2 | 1 | no | — |
| 335 | C2 | — | 2 | 1 | no | — |
| 336 | C2 | — | 16 | 1 | no | — |
| 337 | C2 | — | 32 | 2 | no | — |
| 338 | C2 | #10046 | 1 | 1 | yes | characteristic |
| 339 | C2 | #12312 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28504 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2214 | C2 | semidirect |
| 3. C2×#1709 | C2 | semidirect |
| 4. #13262 | C2 | semidirect |
| 5. C2×#1357 | C2 | semidirect |
| 6. C2×#1335 | C2 | semidirect |
| 7. #7179 | C2 | semidirect |
| 52. #494 | D4 | semidirect |
| 81. #364 | D4 | semidirect |
| 87. C82 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2214 | 2 |
| #28504 | 2 |
| C2×#1709 | 2 |
| #13262 | 2 |
| C2×#1357 | 2 |
| C2×#1335 | 2 |
| #7179 | 2 |