Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
512 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 118 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 | #28497 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #27693 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13413 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13404 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13402 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13117 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13068 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #13007 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #12982 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3407 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3151 | — | 2 | 1 | no | — |
| 12 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#472 | C22 | 1 | 2 | yes | — |
| 14 | C2×#472 | C22 | 1 | 2 | yes | — |
| 15 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2916 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2155 | C22 | 1 | 2 | yes | — |
| 18 | C2×#361 | C22 | 1 | 2 | yes | — |
| 19 | C2×#361 | C22 | 1 | 2 | yes | — |
| 20 | #2140 | C22 | 1 | 2 | yes | — |
| 21 | #2140 | C22 | 1 | 2 | yes | — |
| 22 | #2117 | C22 | 1 | 2 | yes | — |
| 23 | #2115 | C22 | 1 | 1 | yes | characteristic |
| 24 | #2102 | — | 2 | 1 | no | — |
| 25 | #2098 | — | 2 | 1 | no | — |
| 26 | #2018 | — | 2 | 1 | no | — |
| 27 | #2018 | — | 2 | 1 | no | — |
| 28 | #1965 | — | 2 | 1 | no | — |
| 29 | #1945 | C22 | 1 | 2 | yes | — |
| 30 | #1884 | — | 2 | 1 | no | — |
| 31 | #1868 | — | 2 | 1 | no | — |
| 32 | #1814 | C22 | 1 | 2 | yes | — |
| 33 | #1812 | C22 | 1 | 1 | yes | characteristic |
| 34 | C4×#108 | C22 | 1 | 1 | yes | characteristic |
| 35 | #1389 | C22 | 1 | 2 | yes | — |
| 36 | #1388 | C22 | 1 | 2 | yes | — |
| 37 | #1385 | C22 | 1 | 2 | yes | — |
| 38 | #1376 | C22 | 1 | 2 | yes | — |
| 39 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 40 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 41 | C22×D8 | — | 2 | 1 | no | — |
| 42 | C22×D8 | C23 | 1 | 2 | yes | — |
| 43 | #568 | — | 2 | 1 | no | — |
| 44 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 46 | #532 | D4 | 1 | 1 | yes | characteristic |
| 47 | C2×#127 | — | 2 | 1 | no | — |
| 48 | #504 | — | 4 | 2 | no | — |
| 49 | C2×M32 | — | 4 | 1 | no | — |
| 50 | C2×M32 | C23 | 1 | 2 | yes | — |
| 51 | C2×M32 | C23 | 1 | 2 | yes | — |
| 52 | #496 | — | 4 | 1 | no | — |
| 53 | #496 | — | 4 | 1 | no | — |
| 54 | #493 | — | 4 | 1 | no | — |
| 55 | #493 | — | 4 | 1 | no | — |
| 56 | #493 | — | 4 | 1 | no | — |
| 57 | #492 | — | 4 | 1 | no | — |
| 58 | #472 | — | 2 | 4 | no | — |
| 59 | #472 | — | 2 | 4 | no | — |
| 60 | #472 | — | 4 | 2 | no | — |
| 61 | #472 | — | 4 | 2 | no | — |
| 62 | #436 | — | 4 | 1 | no | — |
| 63 | C8×D4 | — | 4 | 1 | no | — |
| 64 | #430 | — | 2 | 1 | no | — |
| 65 | #430 | — | 4 | 1 | no | — |
| 66 | C2×#108 | — | 4 | 1 | no | — |
| 67 | C2×#108 | C23 | 1 | 2 | yes | — |
| 68 | C2×#108 | C23 | 1 | 2 | yes | — |
| 69 | #427 | C23 | 1 | 2 | yes | — |
| 70 | #424 | — | 2 | 2 | no | — |
| 71 | #423 | D4 | 1 | 1 | yes | characteristic |
| 72 | #421 | — | 2 | 2 | no | — |
| 73 | C2×#100 | — | 2 | 2 | no | — |
| 74 | C2×#100 | — | 2 | 2 | no | — |
| 75 | #410 | — | 2 | 1 | no | — |
| 76 | #408 | D4 | 1 | 1 | yes | characteristic |
| 77 | #408 | D4 | 1 | 1 | yes | characteristic |
| 78 | C4×M16 | — | 2 | 1 | no | — |
| 79 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 80 | #365 | — | 2 | 4 | no | — |
| 81 | #361 | — | 4 | 2 | no | — |
| 82 | #361 | — | 4 | 2 | no | — |
| 83 | #357 | — | 4 | 2 | no | — |
| 84 | #357 | — | 4 | 2 | no | — |
| 85 | #350 | — | 4 | 2 | no | — |
| 86 | #349 | — | 4 | 1 | no | — |
| 87 | #344 | — | 2 | 2 | no | — |
| 88 | #341 | — | 2 | 1 | no | — |
| 89 | #339 | — | 2 | 2 | no | — |
| 90 | H(3,ℤ/4) | — | 2 | 1 | no | — |
| 91 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C22×D4 | — | 2 | 2 | no | — |
| 94 | C22×D4 | — | 4 | 2 | no | — |
| 95 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 96 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 97 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 98 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 99 | C2×SD16 | — | 4 | 2 | no | — |
| 100 | C2×SD16 | — | 4 | 2 | no | — |
| 101 | C2×D8 | — | 2 | 2 | no | — |
| 102 | C2×D8 | — | 2 | 2 | no | — |
| 103 | C2×D8 | — | 2 | 2 | no | — |
| 104 | C2×D8 | — | 2 | 2 | no | — |
| 105 | C2×D8 | — | 2 | 2 | no | — |
| 106 | C2×D8 | — | 2 | 2 | no | — |
| 107 | C2×D8 | — | 4 | 2 | no | — |
| 108 | C2×D8 | — | 4 | 2 | no | — |
| 109 | C2×D8 | — | 4 | 2 | no | — |
| 110 | C2×D8 | — | 8 | 2 | no | — |
| 111 | #131 | — | 2 | 2 | no | — |
| 112 | #131 | — | 2 | 2 | no | — |
| 113 | #131 | — | 2 | 2 | no | — |
| 114 | #131 | — | 2 | 2 | no | — |
| 115 | #131 | — | 8 | 2 | no | — |
| 116 | C2×M16 | — | 2 | 1 | no | — |
| 117 | C2×M16 | — | 2 | 1 | no | — |
| 118 | C2×M16 | — | 2 | 2 | no | — |
| 119 | C2×M16 | — | 2 | 2 | no | — |
| 120 | C2×M16 | — | 4 | 2 | no | — |
| 121 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 122 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 123 | C22×C8 | — | 2 | 1 | no | — |
| 124 | C22×C8 | — | 2 | 1 | no | — |
| 125 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 126 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 127 | C4⋊Q8 | SD16 | 1 | 1 | yes | characteristic |
| 128 | #127 | — | 4 | 2 | no | — |
| 129 | #127 | — | 4 | 1 | no | — |
| 130 | #127 | — | 4 | 1 | no | — |
| 131 | #127 | SD16 | 1 | 1 | yes | characteristic |
| 132 | #124 | SD16 | 1 | 1 | yes | characteristic |
| 133 | #124 | SD16 | 1 | 1 | yes | characteristic |
| 134 | #121 | — | 4 | 1 | no | — |
| 135 | C4×D4 | — | 4 | 1 | no | — |
| 136 | #117 | — | 2 | 2 | no | — |
| 137 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 138 | M32 | — | 2 | 2 | no | — |
| 139 | M32 | — | 2 | 2 | no | — |
| 140 | M32 | — | 2 | 2 | no | — |
| 141 | M32 | — | 2 | 2 | no | — |
| 142 | M32 | — | 4 | 2 | no | — |
| 143 | C2×C16 | — | 2 | 2 | no | — |
| 144 | C2×C16 | — | 2 | 2 | no | — |
| 145 | C2×C16 | — | 4 | 2 | no | — |
| 146 | #108 | — | 2 | 2 | no | — |
| 147 | #108 | — | 2 | 2 | no | — |
| 148 | #108 | — | 2 | 2 | no | — |
| 149 | #108 | — | 4 | 2 | no | — |
| 150 | #108 | #41 | 1 | 4 | yes | — |
| 151 | #107 | — | 2 | 2 | no | — |
| 152 | #106 | — | 2 | 2 | no | — |
| 153 | C4⋊C8 | — | 4 | 1 | no | — |
| 154 | C4⋊C8 | — | 8 | 2 | no | — |
| 155 | #104 | — | 8 | 2 | no | — |
| 156 | #100 | — | 4 | 4 | no | — |
| 157 | #100 | — | 4 | 4 | no | — |
| 158 | #99 | — | 4 | 2 | no | — |
| 159 | C22⋊C8 | — | 4 | 1 | no | — |
| 160 | C22⋊C8 | — | 4 | 1 | no | — |
| 161 | #97 | — | 4 | 1 | no | — |
| 162 | C4×C8 | — | 2 | 1 | no | — |
| 163 | C4×C8 | — | 2 | 1 | no | — |
| 164 | C4×C8 | — | 4 | 1 | no | — |
| 165 | C24 | — | 4 | 2 | no | — |
| 166 | #41 | — | 2 | 2 | no | — |
| 167 | #41 | — | 2 | 2 | no | — |
| 168 | #41 | — | 2 | 2 | no | — |
| 169 | #41 | — | 2 | 2 | no | — |
| 170 | C2×Q8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 171 | C2×Q8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 172 | C2×D4 | — | 2 | 1 | no | — |
| 173 | C2×D4 | — | 2 | 1 | no | — |
| 174 | C2×D4 | — | 4 | 1 | no | — |
| 175 | C2×D4 | — | 4 | 2 | no | — |
| 176 | C2×D4 | — | 4 | 2 | no | — |
| 177 | C2×D4 | — | 4 | 2 | no | — |
| 178 | C2×D4 | — | 4 | 2 | no | — |
| 179 | C2×D4 | — | 4 | 2 | no | — |
| 180 | C2×D4 | — | 4 | 2 | no | — |
| 181 | C2×D4 | — | 8 | 2 | no | — |
| 182 | C2×D4 | — | 8 | 2 | no | — |
| 183 | C2×D4 | — | 16 | 4 | no | — |
| 184 | C2×D4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 185 | C2×D4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 186 | C22×C4 | — | 2 | 1 | no | — |
| 187 | C22×C4 | — | 2 | 1 | no | — |
| 188 | C22×C4 | — | 2 | 1 | no | — |
| 189 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 190 | SD16 | — | 8 | 4 | no | — |
| 191 | SD16 | — | 8 | 4 | no | — |
| 192 | D8 | — | 4 | 2 | no | — |
| 193 | D8 | — | 4 | 2 | no | — |
| 194 | D8 | — | 4 | 2 | no | — |
| 195 | D8 | — | 4 | 2 | no | — |
| 196 | D8 | — | 8 | 4 | no | — |
| 197 | D8 | — | 8 | 4 | no | — |
| 198 | D8 | — | 8 | 4 | no | — |
| 199 | M16 | — | 2 | 4 | no | — |
| 200 | M16 | — | 2 | 2 | no | — |
| 201 | M16 | — | 2 | 2 | no | — |
| 202 | M16 | — | 4 | 2 | no | — |
| 203 | M16 | — | 4 | 2 | no | — |
| 204 | M16 | — | 4 | 2 | no | — |
| 205 | M16 | — | 4 | 2 | no | — |
| 206 | M16 | — | 4 | 2 | no | — |
| 207 | M16 | — | 8 | 2 | no | — |
| 208 | C2×C8 | — | 2 | 1 | no | — |
| 209 | C2×C8 | — | 2 | 1 | no | — |
| 210 | C2×C8 | — | 2 | 1 | no | — |
| 211 | C2×C8 | — | 2 | 2 | no | — |
| 212 | C2×C8 | — | 4 | 2 | no | — |
| 213 | C2×C8 | — | 4 | 1 | no | — |
| 214 | C2×C8 | — | 4 | 1 | no | — |
| 215 | C2×C8 | — | 4 | 1 | no | — |
| 216 | C2×C8 | — | 4 | 1 | no | — |
| 217 | C2×C8 | — | 4 | 2 | no | — |
| 218 | C2×C8 | — | 8 | 2 | no | — |
| 219 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 220 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 221 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 222 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 223 | C4⋊C4 | — | 4 | 1 | no | — |
| 224 | C4⋊C4 | — | 4 | 2 | no | — |
| 225 | C22⋊C4 | — | 4 | 1 | no | — |
| 226 | C22⋊C4 | — | 4 | 1 | no | — |
| 227 | C22⋊C4 | — | 4 | 2 | no | — |
| 228 | C42 | — | 2 | 1 | no | — |
| 229 | C42 | — | 4 | 2 | no | — |
| 230 | C42 | C2×SD16 | 1 | 1 | yes | characteristic |
| 231 | C42 | C2×SD16 | 1 | 1 | yes | characteristic |
| 232 | C16 | — | 4 | 2 | no | — |
| 233 | C16 | — | 4 | 2 | no | — |
| 234 | C23 | — | 2 | 1 | no | — |
| 235 | C23 | — | 2 | 1 | no | — |
| 236 | C23 | — | 8 | 2 | no | — |
| 237 | C23 | — | 8 | 2 | no | — |
| 238 | C23 | — | 8 | 2 | no | — |
| 239 | C23 | — | 16 | 4 | no | — |
| 240 | C23 | #575 | 1 | 1 | yes | characteristic |
| 241 | Q8 | — | 2 | 2 | no | — |
| 242 | Q8 | — | 2 | 2 | no | — |
| 243 | D4 | — | 2 | 2 | no | — |
| 244 | D4 | — | 2 | 2 | no | — |
| 245 | D4 | — | 4 | 2 | no | — |
| 246 | D4 | — | 4 | 2 | no | — |
| 247 | D4 | — | 8 | 2 | no | — |
| 248 | D4 | — | 8 | 2 | no | — |
| 249 | D4 | — | 8 | 2 | no | — |
| 250 | D4 | — | 8 | 2 | no | — |
| 251 | D4 | — | 8 | 2 | no | — |
| 252 | D4 | — | 16 | 4 | no | — |
| 253 | D4 | — | 16 | 4 | no | — |
| 254 | C2×C4 | — | 2 | 1 | no | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 4 | 2 | no | — |
| 259 | C2×C4 | — | 4 | 1 | no | — |
| 260 | C2×C4 | — | 4 | 1 | no | — |
| 261 | C2×C4 | — | 4 | 1 | no | — |
| 262 | C2×C4 | — | 4 | 1 | no | — |
| 263 | C2×C4 | — | 8 | 2 | no | — |
| 264 | C2×C4 | #534 | 1 | 1 | yes | characteristic |
| 265 | C2×C4 | #546 | 1 | 2 | yes | — |
| 266 | C2×C4 | C22×SD16 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | #576 | 1 | 2 | yes | — |
| 268 | C8 | — | 4 | 1 | no | — |
| 269 | C8 | — | 4 | 1 | no | — |
| 270 | C8 | — | 4 | 1 | no | — |
| 271 | C8 | — | 4 | 4 | no | — |
| 272 | C8 | — | 8 | 2 | no | — |
| 273 | C8 | #540 | 1 | 2 | yes | — |
| 274 | C8 | #559 | 1 | 1 | yes | characteristic |
| 275 | C8 | #560 | 1 | 1 | yes | characteristic |
| 276 | C22 | — | 4 | 1 | no | — |
| 277 | C22 | — | 4 | 1 | no | — |
| 278 | C22 | — | 4 | 1 | no | — |
| 279 | C22 | — | 8 | 1 | no | — |
| 280 | C22 | — | 8 | 1 | no | — |
| 281 | C22 | — | 16 | 4 | no | — |
| 282 | C22 | — | 16 | 2 | no | — |
| 283 | C22 | — | 32 | 2 | no | — |
| 284 | C22 | — | 32 | 2 | no | — |
| 285 | C22 | #3187 | 1 | 1 | yes | characteristic |
| 286 | C22 | #3309 | 1 | 2 | yes | — |
| 287 | C4 | — | 2 | 2 | no | — |
| 288 | C4 | — | 4 | 1 | no | — |
| 289 | C4 | — | 4 | 1 | no | — |
| 290 | C4 | — | 4 | 1 | no | — |
| 291 | C4 | — | 16 | 2 | no | — |
| 292 | C4 | #2709 | 1 | 1 | yes | characteristic |
| 293 | C4 | #3199 | 1 | 1 | yes | characteristic |
| 294 | C4 | #3295 | 1 | 2 | yes | — |
| 295 | C2 | — | 2 | 1 | no | — |
| 296 | C2 | — | 2 | 2 | no | — |
| 297 | C2 | — | 8 | 1 | no | — |
| 298 | C2 | — | 8 | 1 | no | — |
| 299 | C2 | — | 32 | 2 | no | — |
| 300 | C2 | #32028 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32028 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #28497 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #27693 | C2 | semidirect |
| 3. #13413 | C2 | semidirect |
| 4. #13404 | C2 | semidirect |
| 5. #13402 | C2 | semidirect |
| 6. #13117 | C2 | semidirect |
| 7. #13068 | C2 | semidirect |
| 8. #13007 | C2 | semidirect |
| 9. #12982 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28497 | 2 |
| #27693 | 2 |
| #13413 | 2 |
| #13404 | 2 |
| #13402 | 2 |
| #13117 | 2 |
| #13068 | 2 |
| #13007 | 2 |
| #12982 | 2 |