Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
417 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 76 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 | #33594 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #29154 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19596 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13259 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11956 | 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 | #3414 | — | 2 | 1 | no | — |
| 9 | #3167 | — | 2 | 2 | no | — |
| 10 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3147 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 13 | #2214 | — | 2 | 1 | no | — |
| 14 | #2213 | — | 2 | 1 | no | — |
| 15 | #2212 | — | 2 | 2 | no | — |
| 16 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 18 | #2148 | — | 2 | 1 | no | — |
| 19 | #2145 | — | 2 | 1 | no | — |
| 20 | #2143 | — | 2 | 1 | no | — |
| 21 | #2031 | — | 2 | 2 | no | — |
| 22 | #1976 | — | 2 | 1 | no | — |
| 23 | #1833 | — | 2 | 1 | no | — |
| 24 | #1814 | — | 2 | 1 | no | — |
| 25 | #1812 | — | 2 | 1 | no | — |
| 26 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 27 | #1708 | — | 2 | 1 | no | — |
| 28 | #1707 | — | 2 | 1 | no | — |
| 29 | #1706 | — | 2 | 1 | no | — |
| 30 | #1705 | — | 2 | 1 | no | — |
| 31 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1366 | — | 2 | 2 | no | — |
| 33 | #1335 | — | 2 | 1 | no | — |
| 34 | #1335 | — | 2 | 1 | no | — |
| 35 | C2×#135 | — | 2 | 1 | no | — |
| 36 | C2×#135 | — | 2 | 1 | no | — |
| 37 | C2×#135 | D4 | 1 | 1 | yes | characteristic |
| 38 | C2×#131 | — | 2 | 1 | no | — |
| 39 | #533 | D4 | 1 | 2 | yes | — |
| 40 | #532 | — | 2 | 1 | no | — |
| 41 | #510 | — | 2 | 2 | no | — |
| 42 | #510 | — | 4 | 1 | no | — |
| 43 | #509 | — | 2 | 2 | no | — |
| 44 | #509 | — | 4 | 1 | no | — |
| 45 | #508 | — | 4 | 2 | no | — |
| 46 | C2×M32 | — | 2 | 1 | no | — |
| 47 | C2×M32 | — | 2 | 1 | no | — |
| 48 | #500 | — | 2 | 1 | no | — |
| 49 | #499 | — | 2 | 1 | no | — |
| 50 | #499 | — | 2 | 1 | no | — |
| 51 | #498 | — | 2 | 2 | no | — |
| 52 | #498 | — | 2 | 1 | no | — |
| 53 | #495 | — | 2 | 1 | no | — |
| 54 | #495 | — | 2 | 1 | no | — |
| 55 | #494 | D4 | 1 | 1 | yes | characteristic |
| 56 | #493 | D4 | 1 | 1 | yes | characteristic |
| 57 | #492 | — | 2 | 2 | no | — |
| 58 | #492 | — | 2 | 1 | no | — |
| 59 | #487 | — | 2 | 2 | no | — |
| 60 | #486 | — | 2 | 2 | no | — |
| 61 | #465 | — | 4 | 4 | no | — |
| 62 | #443 | — | 2 | 1 | no | — |
| 63 | #443 | — | 2 | 1 | no | — |
| 64 | #443 | — | 2 | 1 | no | — |
| 65 | #443 | — | 2 | 1 | no | — |
| 66 | C4×SD16 | — | 4 | 4 | no | — |
| 67 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 68 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 69 | #427 | — | 2 | 1 | no | — |
| 70 | #427 | — | 2 | 1 | no | — |
| 71 | #427 | D4 | 1 | 1 | yes | characteristic |
| 72 | #421 | — | 2 | 2 | no | — |
| 73 | C2×#104 | — | 2 | 1 | no | — |
| 74 | #410 | — | 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 | — |
| 82 | #362 | — | 2 | 2 | no | — |
| 83 | #361 | — | 2 | 2 | no | — |
| 84 | #360 | — | 4 | 2 | no | — |
| 85 | #359 | — | 4 | 2 | no | — |
| 86 | #344 | — | 2 | 2 | no | — |
| 87 | #339 | — | 2 | 1 | no | — |
| 88 | H(3,ℤ/4) | — | 2 | 2 | no | — |
| 89 | C82 | — | 2 | 1 | no | — |
| 90 | C82 | D4 | 1 | 1 | yes | characteristic |
| 91 | C82 | D4 | 1 | 1 | yes | characteristic |
| 92 | C2×#41 | — | 2 | 1 | no | — |
| 93 | C2×#41 | — | 4 | 1 | no | — |
| 94 | #135 | — | 2 | 1 | no | — |
| 95 | #135 | — | 2 | 1 | no | — |
| 96 | #135 | — | 2 | 1 | no | — |
| 97 | #135 | — | 2 | 1 | no | — |
| 98 | #135 | — | 4 | 1 | no | — |
| 99 | #135 | — | 4 | 1 | no | — |
| 100 | #135 | — | 4 | 1 | no | — |
| 101 | #135 | — | 4 | 1 | no | — |
| 102 | #135 | — | 4 | 1 | no | — |
| 103 | #135 | — | 4 | 1 | no | — |
| 104 | #135 | — | 4 | 1 | no | — |
| 105 | #135 | — | 4 | 1 | no | — |
| 106 | C2×Q16 | — | 2 | 1 | no | — |
| 107 | C2×Q16 | — | 2 | 1 | no | — |
| 108 | C2×Q16 | SD16 | 1 | 1 | yes | characteristic, complemented |
| 109 | C2×SD16 | — | 2 | 1 | no | — |
| 110 | C2×SD16 | — | 4 | 4 | no | — |
| 111 | C2×SD16 | — | 4 | 1 | no | — |
| 112 | C2×SD16 | — | 4 | 1 | no | — |
| 113 | C2×D8 | — | 2 | 1 | no | — |
| 114 | C2×D8 | — | 2 | 1 | no | — |
| 115 | C2×D8 | SD16 | 1 | 1 | yes | characteristic, complemented |
| 116 | #131 | — | 4 | 1 | no | — |
| 117 | #131 | — | 4 | 1 | no | — |
| 118 | #131 | — | 4 | 1 | no | — |
| 119 | #131 | — | 4 | 1 | no | — |
| 120 | C2×M16 | — | 2 | 1 | no | — |
| 121 | C2×M16 | — | 2 | 1 | no | — |
| 122 | C2×M16 | — | 4 | 1 | no | — |
| 123 | C22×C8 | — | 2 | 2 | no | — |
| 124 | C22×C8 | — | 4 | 1 | no | — |
| 125 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 126 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 127 | C4⋊Q8 | — | 2 | 1 | no | — |
| 128 | C4⋊Q8 | — | 2 | 2 | no | — |
| 129 | #127 | — | 2 | 1 | no | — |
| 130 | #125 | — | 2 | 2 | no | — |
| 131 | #124 | — | 4 | 1 | no | — |
| 132 | C22⋊Q8 | — | 4 | 2 | no | — |
| 133 | #121 | — | 4 | 2 | no | — |
| 134 | C4×Q8 | — | 4 | 2 | no | — |
| 135 | C4×D4 | — | 4 | 2 | no | — |
| 136 | #117 | — | 2 | 2 | no | — |
| 137 | #117 | — | 2 | 1 | no | — |
| 138 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 139 | Q32 | — | 4 | 2 | no | — |
| 140 | SD32 | — | 4 | 2 | no | — |
| 141 | SD32 | — | 4 | 2 | no | — |
| 142 | D16 | — | 4 | 2 | no | — |
| 143 | M32 | — | 2 | 2 | no | — |
| 144 | M32 | — | 4 | 1 | no | — |
| 145 | M32 | — | 4 | 1 | no | — |
| 146 | M32 | — | 4 | 1 | no | — |
| 147 | C2×C16 | — | 4 | 1 | no | — |
| 148 | C2×C16 | — | 4 | 1 | no | — |
| 149 | #108 | — | 2 | 1 | no | — |
| 150 | #108 | — | 2 | 1 | no | — |
| 151 | #108 | — | 2 | 1 | no | — |
| 152 | #108 | — | 2 | 1 | no | — |
| 153 | #107 | — | 2 | 1 | no | — |
| 154 | #107 | — | 4 | 1 | no | — |
| 155 | #107 | — | 4 | 1 | no | — |
| 156 | #106 | — | 2 | 1 | no | — |
| 157 | #106 | — | 4 | 4 | no | — |
| 158 | #106 | — | 4 | 1 | no | — |
| 159 | #106 | — | 4 | 1 | no | — |
| 160 | #104 | — | 4 | 1 | no | — |
| 161 | #104 | — | 4 | 1 | no | — |
| 162 | #104 | — | 8 | 2 | no | — |
| 163 | #103 | — | 4 | 4 | no | — |
| 164 | #102 | — | 4 | 4 | no | — |
| 165 | #99 | — | 8 | 2 | no | — |
| 166 | #97 | — | 4 | 1 | no | — |
| 167 | C4×C8 | — | 2 | 2 | no | — |
| 168 | C4×C8 | — | 2 | 1 | no | — |
| 169 | C4×C8 | — | 2 | 1 | no | — |
| 170 | C4×C8 | — | 4 | 1 | no | — |
| 171 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 172 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 173 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 174 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 175 | #41 | — | 4 | 1 | no | — |
| 176 | #41 | — | 4 | 1 | no | — |
| 177 | #41 | — | 4 | 1 | no | — |
| 178 | #41 | — | 4 | 1 | no | — |
| 179 | #41 | — | 8 | 1 | no | — |
| 180 | #41 | — | 8 | 1 | no | — |
| 181 | #41 | — | 8 | 1 | no | — |
| 182 | #41 | — | 8 | 1 | no | — |
| 183 | C2×Q8 | — | 2 | 1 | no | — |
| 184 | C2×Q8 | — | 4 | 2 | no | — |
| 185 | C2×Q8 | — | 4 | 1 | no | — |
| 186 | C2×D4 | — | 2 | 1 | no | — |
| 187 | C2×D4 | — | 4 | 2 | no | — |
| 188 | C2×D4 | — | 4 | 1 | no | — |
| 189 | C2×D4 | — | 4 | 1 | no | — |
| 190 | C2×D4 | — | 8 | 1 | no | — |
| 191 | C22×C4 | — | 2 | 1 | no | — |
| 192 | C22×C4 | — | 4 | 1 | no | — |
| 193 | C22×C4 | — | 8 | 1 | no | — |
| 194 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 195 | Q16 | — | 2 | 1 | no | — |
| 196 | Q16 | — | 2 | 1 | no | — |
| 197 | Q16 | — | 4 | 1 | no | — |
| 198 | Q16 | — | 4 | 1 | no | — |
| 199 | Q16 | — | 4 | 1 | no | — |
| 200 | Q16 | — | 4 | 1 | no | — |
| 201 | SD16 | — | 4 | 1 | no | — |
| 202 | SD16 | — | 4 | 1 | no | — |
| 203 | SD16 | — | 8 | 8 | no | — |
| 204 | SD16 | — | 8 | 1 | no | — |
| 205 | SD16 | — | 8 | 1 | no | — |
| 206 | SD16 | — | 8 | 1 | no | — |
| 207 | SD16 | — | 8 | 1 | no | — |
| 208 | D8 | — | 2 | 1 | no | — |
| 209 | D8 | — | 2 | 1 | no | — |
| 210 | D8 | — | 4 | 1 | no | — |
| 211 | D8 | — | 4 | 1 | no | — |
| 212 | D8 | — | 4 | 1 | no | — |
| 213 | D8 | — | 4 | 1 | no | — |
| 214 | M16 | — | 4 | 1 | no | — |
| 215 | M16 | — | 4 | 1 | no | — |
| 216 | M16 | — | 4 | 1 | no | — |
| 217 | M16 | — | 4 | 1 | no | — |
| 218 | M16 | — | 8 | 1 | no | — |
| 219 | M16 | — | 8 | 1 | no | — |
| 220 | C2×C8 | — | 2 | 2 | no | — |
| 221 | C2×C8 | — | 2 | 2 | no | — |
| 222 | C2×C8 | — | 4 | 4 | no | — |
| 223 | C2×C8 | — | 4 | 1 | no | — |
| 224 | C2×C8 | — | 4 | 1 | no | — |
| 225 | C2×C8 | — | 8 | 1 | no | — |
| 226 | C2×C8 | — | 8 | 1 | no | — |
| 227 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 228 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 229 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 230 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 231 | C2×C8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 232 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 234 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | #137 | 1 | 1 | yes | characteristic |
| 239 | C4⋊C4 | — | 4 | 2 | no | — |
| 240 | C4⋊C4 | — | 4 | 2 | no | — |
| 241 | C4⋊C4 | — | 4 | 2 | no | — |
| 242 | C4⋊C4 | — | 4 | 1 | no | — |
| 243 | C4⋊C4 | — | 8 | 2 | no | — |
| 244 | C22⋊C4 | — | 4 | 1 | no | — |
| 245 | C22⋊C4 | — | 4 | 2 | no | — |
| 246 | C22⋊C4 | — | 8 | 2 | no | — |
| 247 | C42 | — | 2 | 1 | no | — |
| 248 | C42 | — | 4 | 1 | no | — |
| 249 | C42 | — | 4 | 2 | no | — |
| 250 | C42 | #120 | 1 | 1 | yes | characteristic |
| 251 | C42 | #121 | 1 | 1 | yes | characteristic |
| 252 | C16 | — | 4 | 2 | no | — |
| 253 | C16 | — | 8 | 1 | no | — |
| 254 | C23 | — | 4 | 1 | no | — |
| 255 | C23 | — | 8 | 1 | no | — |
| 256 | C23 | #453 | 1 | 1 | yes | characteristic |
| 257 | Q8 | — | 4 | 1 | no | — |
| 258 | Q8 | — | 4 | 1 | no | — |
| 259 | Q8 | — | 8 | 4 | no | — |
| 260 | Q8 | — | 8 | 1 | no | — |
| 261 | Q8 | — | 8 | 1 | no | — |
| 262 | D4 | — | 4 | 1 | no | — |
| 263 | D4 | — | 4 | 1 | no | — |
| 264 | D4 | — | 8 | 4 | no | — |
| 265 | D4 | — | 8 | 1 | no | — |
| 266 | D4 | — | 8 | 1 | no | — |
| 267 | D4 | — | 8 | 1 | no | — |
| 268 | D4 | — | 8 | 1 | no | — |
| 269 | D4 | — | 16 | 1 | no | — |
| 270 | D4 | — | 16 | 1 | no | — |
| 271 | C2×C4 | — | 2 | 1 | no | — |
| 272 | C2×C4 | — | 2 | 1 | no | — |
| 273 | C2×C4 | — | 4 | 2 | no | — |
| 274 | C2×C4 | — | 4 | 1 | no | — |
| 275 | C2×C4 | — | 8 | 1 | no | — |
| 276 | C2×C4 | — | 8 | 1 | no | — |
| 277 | C2×C4 | — | 8 | 1 | no | — |
| 278 | C2×C4 | — | 8 | 2 | no | — |
| 279 | C2×C4 | — | 8 | 1 | no | — |
| 280 | C2×C4 | — | 16 | 1 | no | — |
| 281 | C2×C4 | — | 16 | 1 | no | — |
| 282 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | #448 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #449 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | #449 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #449 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | #453 | 1 | 1 | yes | characteristic |
| 288 | C8 | — | 2 | 4 | no | — |
| 289 | C8 | — | 4 | 2 | no | — |
| 290 | C8 | — | 8 | 1 | no | — |
| 291 | C8 | — | 8 | 1 | no | — |
| 292 | C8 | #450 | 1 | 1 | yes | characteristic |
| 293 | C8 | #452 | 1 | 1 | yes | characteristic |
| 294 | C8 | #463 | 1 | 1 | yes | characteristic |
| 295 | C8 | #463 | 1 | 1 | yes | characteristic |
| 296 | C8 | #465 | 1 | 1 | yes | characteristic |
| 297 | C8 | #468 | 1 | 1 | yes | characteristic |
| 298 | C8 | #495 | 1 | 1 | yes | characteristic |
| 299 | C8 | #496 | 1 | 1 | yes | characteristic |
| 300 | C22 | — | 4 | 1 | no | — |
| 301 | C22 | — | 8 | 1 | no | — |
| 302 | C22 | — | 16 | 1 | no | — |
| 303 | C22 | — | 16 | 2 | no | — |
| 304 | C22 | — | 16 | 1 | no | — |
| 305 | C22 | #1622 | 1 | 1 | yes | characteristic |
| 306 | C22 | #1624 | 1 | 1 | yes | characteristic |
| 307 | C22 | #2002 | 1 | 1 | yes | characteristic |
| 308 | C4 | — | 2 | 2 | no | — |
| 309 | C4 | — | 4 | 1 | no | — |
| 310 | C4 | — | 8 | 1 | no | — |
| 311 | C4 | — | 16 | 1 | no | — |
| 312 | C4 | — | 16 | 2 | no | — |
| 313 | C4 | — | 16 | 1 | no | — |
| 314 | C4 | #1622 | 1 | 1 | yes | characteristic |
| 315 | C4 | #1624 | 1 | 1 | yes | characteristic |
| 316 | C4 | #2000 | 1 | 1 | yes | characteristic |
| 317 | C4 | #2012 | 1 | 1 | yes | characteristic |
| 318 | C2 | — | 2 | 1 | no | — |
| 319 | C2 | — | 2 | 1 | no | — |
| 320 | C2 | — | 2 | 1 | no | — |
| 321 | C2 | — | 16 | 1 | no | — |
| 322 | C2 | — | 32 | 1 | no | — |
| 323 | C2 | #10048 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1624 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10048 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ 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 |
|---|---|---|
| 2. #29154 | C2 | semidirect |
| 3. #19596 | C2 | semidirect |
| 4. #13259 | C2 | semidirect |
| 5. #11956 | C2 | semidirect |
| 6. C2×#1335 | C2 | semidirect |
| 7. #7179 | C2 | semidirect |
| 108. C2×Q16 | SD16 | semidirect |
| 115. C2×D8 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33594 | 2 |
| #29154 | 2 |
| #19596 | 2 |
| #13259 | 2 |
| #11956 | 2 |
| C2×#1335 | 2 |
| #7179 | 2 |