Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
525 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 124 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 | #21709 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #21295 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13540 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13532 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13416 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13340 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #13137 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #13027 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #13024 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #12944 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #12897 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3414 | C22 | 1 | 2 | yes | — |
| 13 | #3150 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#444 | — | 2 | 1 | no | — |
| 15 | C2×#444 | C22 | 1 | 2 | yes | — |
| 16 | C4×#135 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2931 | — | 2 | 1 | no | — |
| 18 | #2923 | — | 2 | 1 | no | — |
| 19 | #2161 | C22 | 1 | 2 | yes | — |
| 20 | #2148 | C22 | 1 | 2 | yes | — |
| 21 | #2145 | C22 | 1 | 2 | yes | — |
| 22 | #2143 | C22 | 1 | 2 | yes | — |
| 23 | #2143 | C22 | 1 | 2 | yes | — |
| 24 | #1975 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1975 | C22 | 1 | 1 | yes | characteristic |
| 26 | #1974 | — | 2 | 1 | no | — |
| 27 | #1974 | C22 | 1 | 2 | yes | — |
| 28 | #1963 | — | 2 | 1 | no | — |
| 29 | #1946 | C22 | 1 | 1 | yes | characteristic |
| 30 | #1946 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1937 | — | 2 | 1 | no | — |
| 32 | #1937 | — | 2 | 1 | no | — |
| 33 | #1849 | C22 | 1 | 1 | yes | characteristic |
| 34 | C4×#108 | C22 | 1 | 1 | yes | characteristic |
| 35 | #1759 | — | 2 | 1 | no | — |
| 36 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 37 | #1389 | C22 | 1 | 2 | yes | — |
| 38 | #1388 | C22 | 1 | 2 | yes | — |
| 39 | #1377 | C22 | 1 | 2 | yes | — |
| 40 | #1376 | C22 | 1 | 2 | yes | — |
| 41 | #1375 | C22 | 1 | 1 | yes | characteristic |
| 42 | #1375 | C22 | 1 | 1 | yes | characteristic |
| 43 | C2×#135 | — | 2 | 1 | no | — |
| 44 | C2×#135 | C23 | 1 | 2 | yes | — |
| 45 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×#131 | — | 2 | 2 | no | — |
| 47 | #532 | — | 2 | 1 | no | — |
| 48 | C4×#41 | — | 2 | 1 | no | — |
| 49 | #510 | — | 2 | 2 | no | — |
| 50 | #510 | — | 2 | 2 | no | — |
| 51 | #509 | — | 2 | 2 | no | — |
| 52 | #509 | — | 2 | 2 | no | — |
| 53 | #508 | — | 4 | 2 | no | — |
| 54 | C2×M32 | C23 | 1 | 2 | yes | — |
| 55 | C2×M32 | C23 | 1 | 2 | yes | — |
| 56 | #497 | — | 2 | 2 | no | — |
| 57 | #496 | — | 2 | 2 | no | — |
| 58 | #444 | — | 4 | 2 | no | — |
| 59 | #444 | — | 4 | 2 | no | — |
| 60 | #444 | — | 4 | 2 | no | — |
| 61 | #444 | — | 4 | 2 | no | — |
| 62 | C4×Q16 | — | 2 | 1 | no | — |
| 63 | C4×SD16 | — | 2 | 2 | no | — |
| 64 | C4×D8 | — | 2 | 1 | no | — |
| 65 | #436 | — | 4 | 2 | no | — |
| 66 | C8×D4 | — | 4 | 2 | no | — |
| 67 | #433 | — | 4 | 2 | no | — |
| 68 | #432 | — | 4 | 2 | no | — |
| 69 | C2×#108 | — | 2 | 1 | no | — |
| 70 | C2×#108 | C23 | 1 | 2 | yes | — |
| 71 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 72 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 73 | #427 | C23 | 1 | 1 | yes | characteristic |
| 74 | #424 | — | 2 | 1 | no | — |
| 75 | #424 | — | 2 | 1 | no | — |
| 76 | #423 | — | 2 | 1 | no | — |
| 77 | C2×#104 | — | 2 | 2 | no | — |
| 78 | C2×#104 | — | 2 | 2 | no | — |
| 79 | C2×#104 | — | 2 | 1 | no | — |
| 80 | C2×#104 | — | 2 | 1 | no | — |
| 81 | #416 | — | 2 | 1 | no | — |
| 82 | #408 | — | 2 | 1 | no | — |
| 83 | #408 | — | 2 | 1 | no | — |
| 84 | #408 | — | 2 | 2 | no | — |
| 85 | C4×M16 | — | 2 | 1 | no | — |
| 86 | C2×#97 | D4 | 1 | 1 | yes | characteristic |
| 87 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 88 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 92 | #376 | D4 | 1 | 1 | yes | characteristic |
| 93 | #368 | — | 4 | 2 | no | — |
| 94 | #362 | — | 2 | 2 | no | — |
| 95 | #361 | — | 2 | 2 | no | — |
| 96 | #360 | — | 2 | 4 | no | — |
| 97 | #359 | — | 4 | 2 | no | — |
| 98 | #359 | — | 4 | 2 | no | — |
| 99 | #341 | — | 2 | 1 | no | — |
| 100 | #339 | — | 2 | 1 | no | — |
| 101 | #339 | — | 2 | 1 | no | — |
| 102 | C2×#41 | — | 2 | 1 | no | — |
| 103 | C2×#41 | — | 2 | 2 | no | — |
| 104 | #135 | — | 2 | 2 | no | — |
| 105 | #135 | — | 2 | 2 | no | — |
| 106 | #135 | — | 2 | 2 | no | — |
| 107 | #135 | — | 2 | 2 | no | — |
| 108 | #135 | — | 2 | 2 | no | — |
| 109 | #135 | — | 2 | 2 | no | — |
| 110 | #135 | — | 4 | 2 | no | — |
| 111 | #135 | — | 4 | 2 | no | — |
| 112 | C2×Q16 | — | 2 | 1 | no | — |
| 113 | C2×Q16 | #41 | 1 | 2 | yes | — |
| 114 | C2×Q16 | #41 | 1 | 1 | yes | characteristic |
| 115 | C2×SD16 | — | 2 | 2 | no | — |
| 116 | C2×SD16 | — | 2 | 1 | no | — |
| 117 | C2×SD16 | — | 2 | 2 | no | — |
| 118 | C2×D8 | — | 2 | 1 | no | — |
| 119 | C2×D8 | #41 | 1 | 2 | yes | — |
| 120 | C2×D8 | #41 | 1 | 1 | yes | characteristic |
| 121 | #131 | — | 4 | 4 | no | — |
| 122 | #131 | — | 4 | 4 | no | — |
| 123 | C2×M16 | — | 2 | 2 | no | — |
| 124 | C2×M16 | — | 2 | 1 | no | — |
| 125 | C2×M16 | — | 2 | 1 | no | — |
| 126 | C2×M16 | — | 4 | 2 | no | — |
| 127 | C22×C8 | — | 4 | 2 | no | — |
| 128 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 129 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 130 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 131 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 132 | C4⋊Q8 | — | 2 | 1 | no | — |
| 133 | #127 | — | 2 | 1 | no | — |
| 134 | #124 | — | 2 | 2 | no | — |
| 135 | #121 | — | 4 | 2 | no | — |
| 136 | C4×Q8 | — | 2 | 2 | no | — |
| 137 | C4×D4 | — | 2 | 2 | no | — |
| 138 | C4×D4 | — | 4 | 2 | no | — |
| 139 | C4×D4 | — | 4 | 2 | no | — |
| 140 | #117 | — | 2 | 1 | no | — |
| 141 | #117 | — | 4 | 1 | no | — |
| 142 | C2×C42 | — | 4 | 1 | no | — |
| 143 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 144 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 145 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 146 | Q32 | — | 4 | 2 | no | — |
| 147 | SD32 | — | 4 | 2 | no | — |
| 148 | SD32 | — | 4 | 2 | no | — |
| 149 | D16 | — | 4 | 2 | no | — |
| 150 | M32 | — | 2 | 2 | no | — |
| 151 | M32 | — | 2 | 2 | no | — |
| 152 | M32 | — | 2 | 2 | no | — |
| 153 | M32 | — | 2 | 2 | no | — |
| 154 | C2×C16 | — | 2 | 2 | no | — |
| 155 | C2×C16 | — | 2 | 2 | no | — |
| 156 | #108 | — | 2 | 2 | no | — |
| 157 | #108 | — | 2 | 2 | no | — |
| 158 | #108 | — | 2 | 2 | no | — |
| 159 | #108 | — | 2 | 2 | no | — |
| 160 | #108 | — | 4 | 2 | no | — |
| 161 | #107 | — | 2 | 1 | no | — |
| 162 | #106 | #41 | 1 | 2 | yes | — |
| 163 | C4⋊C8 | — | 4 | 2 | no | — |
| 164 | C4⋊C8 | — | 4 | 2 | no | — |
| 165 | C4⋊C8 | — | 4 | 2 | no | — |
| 166 | #104 | — | 4 | 2 | no | — |
| 167 | #104 | — | 4 | 4 | no | — |
| 168 | #104 | — | 4 | 2 | no | — |
| 169 | #104 | — | 4 | 2 | no | — |
| 170 | #104 | — | 4 | 2 | no | — |
| 171 | #103 | — | 2 | 2 | no | — |
| 172 | #102 | — | 2 | 2 | no | — |
| 173 | C22⋊C8 | — | 4 | 4 | no | — |
| 174 | C22⋊C8 | — | 4 | 2 | no | — |
| 175 | C22⋊C8 | — | 4 | 2 | no | — |
| 176 | #97 | — | 2 | 2 | no | — |
| 177 | #97 | — | 2 | 2 | no | — |
| 178 | #97 | — | 4 | 1 | no | — |
| 179 | #97 | D8 | 1 | 2 | yes | — |
| 180 | #97 | D8 | 1 | 2 | yes | — |
| 181 | C4×C8 | — | 2 | 1 | no | — |
| 182 | C4×C8 | — | 2 | 1 | no | — |
| 183 | C4×C8 | — | 2 | 2 | no | — |
| 184 | C4×C8 | — | 4 | 1 | no | — |
| 185 | C4×C8 | — | 4 | 1 | no | — |
| 186 | #95 | — | 4 | 1 | no | — |
| 187 | #41 | — | 4 | 1 | no | — |
| 188 | #41 | — | 4 | 2 | no | — |
| 189 | #41 | — | 4 | 2 | no | — |
| 190 | #41 | — | 4 | 1 | no | — |
| 191 | #41 | — | 4 | 2 | no | — |
| 192 | #41 | — | 4 | 1 | no | — |
| 193 | #41 | — | 4 | 1 | no | — |
| 194 | #41 | — | 4 | 2 | no | — |
| 195 | C2×Q8 | — | 2 | 1 | no | — |
| 196 | C2×Q8 | — | 2 | 2 | no | — |
| 197 | C2×D4 | — | 2 | 1 | no | — |
| 198 | C2×D4 | — | 2 | 2 | no | — |
| 199 | C2×D4 | — | 4 | 2 | no | — |
| 200 | C2×D4 | — | 4 | 1 | no | — |
| 201 | C2×D4 | — | 4 | 2 | no | — |
| 202 | C22×C4 | — | 2 | 1 | no | — |
| 203 | C22×C4 | — | 2 | 1 | no | — |
| 204 | C22×C4 | — | 2 | 1 | no | — |
| 205 | C22×C4 | — | 4 | 2 | no | — |
| 206 | C22×C4 | — | 4 | 1 | no | — |
| 207 | C22×C4 | — | 4 | 2 | no | — |
| 208 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 209 | Q16 | — | 2 | 2 | no | — |
| 210 | Q16 | — | 2 | 2 | no | — |
| 211 | Q16 | — | 2 | 2 | no | — |
| 212 | Q16 | — | 4 | 2 | no | — |
| 213 | SD16 | — | 4 | 4 | no | — |
| 214 | SD16 | — | 4 | 2 | no | — |
| 215 | SD16 | — | 4 | 2 | no | — |
| 216 | SD16 | — | 4 | 2 | no | — |
| 217 | D8 | — | 2 | 2 | no | — |
| 218 | D8 | — | 2 | 2 | no | — |
| 219 | D8 | — | 2 | 2 | no | — |
| 220 | D8 | — | 4 | 2 | no | — |
| 221 | M16 | — | 4 | 1 | no | — |
| 222 | M16 | — | 4 | 2 | no | — |
| 223 | M16 | — | 4 | 1 | no | — |
| 224 | M16 | — | 4 | 2 | no | — |
| 225 | M16 | — | 4 | 1 | no | — |
| 226 | M16 | — | 4 | 1 | no | — |
| 227 | M16 | — | 8 | 4 | no | — |
| 228 | C2×C8 | — | 2 | 1 | no | — |
| 229 | C2×C8 | — | 2 | 1 | no | — |
| 230 | C2×C8 | — | 2 | 1 | no | — |
| 231 | C2×C8 | — | 2 | 1 | no | — |
| 232 | C2×C8 | — | 2 | 1 | no | — |
| 233 | C2×C8 | — | 2 | 1 | no | — |
| 234 | C2×C8 | — | 2 | 1 | no | — |
| 235 | C2×C8 | — | 4 | 1 | no | — |
| 236 | C2×C8 | — | 4 | 1 | no | — |
| 237 | C2×C8 | — | 4 | 2 | no | — |
| 238 | C2×C8 | — | 8 | 4 | no | — |
| 239 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 240 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 241 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 242 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 243 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 244 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 245 | C4⋊C4 | — | 2 | 2 | no | — |
| 246 | C4⋊C4 | — | 4 | 2 | no | — |
| 247 | C4⋊C4 | — | 4 | 2 | no | — |
| 248 | C22⋊C4 | — | 4 | 2 | no | — |
| 249 | C22⋊C4 | — | 4 | 4 | no | — |
| 250 | C22⋊C4 | — | 4 | 1 | no | — |
| 251 | C42 | — | 2 | 1 | no | — |
| 252 | C42 | — | 2 | 1 | no | — |
| 253 | C42 | — | 2 | 1 | no | — |
| 254 | C42 | — | 2 | 1 | no | — |
| 255 | C42 | — | 4 | 2 | no | — |
| 256 | C42 | — | 4 | 1 | no | — |
| 257 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 258 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 259 | C42 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 260 | C16 | — | 4 | 2 | no | — |
| 261 | C16 | — | 4 | 2 | no | — |
| 262 | C23 | — | 4 | 1 | no | — |
| 263 | C23 | — | 4 | 2 | no | — |
| 264 | C23 | #575 | 1 | 1 | yes | characteristic |
| 265 | Q8 | — | 4 | 2 | no | — |
| 266 | Q8 | — | 4 | 2 | no | — |
| 267 | Q8 | — | 4 | 1 | no | — |
| 268 | Q8 | — | 4 | 1 | no | — |
| 269 | D4 | — | 4 | 2 | no | — |
| 270 | D4 | — | 4 | 1 | no | — |
| 271 | D4 | — | 4 | 2 | no | — |
| 272 | D4 | — | 4 | 1 | no | — |
| 273 | D4 | — | 8 | 4 | no | — |
| 274 | D4 | — | 8 | 2 | no | — |
| 275 | D4 | — | 8 | 1 | no | — |
| 276 | D4 | — | 8 | 1 | no | — |
| 277 | D4 | — | 8 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 1 | no | — |
| 279 | C2×C4 | — | 2 | 1 | no | — |
| 280 | C2×C4 | — | 2 | 2 | no | — |
| 281 | C2×C4 | — | 4 | 2 | no | — |
| 282 | C2×C4 | — | 4 | 1 | no | — |
| 283 | C2×C4 | — | 4 | 2 | no | — |
| 284 | C2×C4 | — | 4 | 1 | no | — |
| 285 | C2×C4 | — | 8 | 1 | no | — |
| 286 | C2×C4 | — | 8 | 1 | no | — |
| 287 | C2×C4 | — | 8 | 1 | no | — |
| 288 | C2×C4 | — | 8 | 2 | no | — |
| 289 | C2×C4 | — | 8 | 2 | no | — |
| 290 | C2×C4 | — | 8 | 1 | no | — |
| 291 | C2×C4 | — | 8 | 1 | no | — |
| 292 | C2×C4 | — | 16 | 2 | no | — |
| 293 | C2×C4 | #525 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | #525 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #575 | 1 | 1 | yes | characteristic |
| 299 | C8 | — | 2 | 2 | no | — |
| 300 | C8 | — | 2 | 2 | no | — |
| 301 | C8 | — | 4 | 1 | no | — |
| 302 | C8 | — | 8 | 1 | no | — |
| 303 | C8 | — | 8 | 2 | no | — |
| 304 | C8 | — | 8 | 1 | no | — |
| 305 | C8 | #551 | 1 | 2 | yes | — |
| 306 | C8 | #553 | 1 | 2 | yes | — |
| 307 | C22 | — | 4 | 1 | no | — |
| 308 | C22 | — | 8 | 2 | no | — |
| 309 | C22 | — | 8 | 4 | no | — |
| 310 | C22 | — | 8 | 1 | no | — |
| 311 | C22 | — | 8 | 2 | no | — |
| 312 | C22 | #3100 | 1 | 1 | yes | characteristic |
| 313 | C22 | #3114 | 1 | 1 | yes | characteristic |
| 314 | C22 | #3219 | 1 | 1 | yes | characteristic |
| 315 | C4 | — | 4 | 1 | no | — |
| 316 | C4 | — | 4 | 1 | no | — |
| 317 | C4 | — | 4 | 2 | no | — |
| 318 | C4 | — | 8 | 1 | no | — |
| 319 | C4 | — | 8 | 2 | no | — |
| 320 | C4 | — | 8 | 1 | no | — |
| 321 | C4 | — | 8 | 2 | no | — |
| 322 | C4 | #2598 | 1 | 1 | yes | characteristic |
| 323 | C4 | #3100 | 1 | 1 | yes | characteristic |
| 324 | C4 | #3115 | 1 | 1 | yes | characteristic |
| 325 | C4 | #3205 | 1 | 1 | yes | characteristic |
| 326 | C2 | — | 2 | 1 | no | — |
| 327 | C2 | — | 2 | 1 | no | — |
| 328 | C2 | — | 2 | 1 | no | — |
| 329 | C2 | — | 16 | 1 | no | — |
| 330 | C2 | — | 16 | 2 | no | — |
| 331 | C2 | #29364 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3205 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #29364 |
| prime radical | #13340 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21709 | C2 | semidirect |
| 2. #21295 | C2 | semidirect |
| 3. #13540 | C2 | semidirect |
| 4. #13532 | C2 | semidirect |
| 5. #13416 | C2 | semidirect |
| 7. #13137 | C2 | semidirect |
| 8. #13027 | C2 | semidirect |
| 9. #13024 | C2 | semidirect |
| 10. #12944 | C2 | semidirect |
| 11. #12897 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21709 | 2 |
| #21295 | 2 |
| #13540 | 2 |
| #13532 | 2 |
| #13416 | 2 |
| #13340 | 2 |
| #13137 | 2 |
| #13027 | 2 |
| #13024 | 2 |
| #12944 | 2 |
| #12897 | 2 |