Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
409 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 77 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 | #513022 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #483996 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #150977 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #150147 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #140775 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #94407 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #94120 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #33748 | C22 | 1 | 1 | yes | characteristic |
| 9 | #33463 | C22 | 1 | 1 | yes | characteristic |
| 10 | #17970 | — | 2 | 1 | no | — |
| 11 | #13717 | C4 | 1 | 2 | yes | — |
| 12 | #13710 | — | 2 | 1 | no | — |
| 13 | #13692 | — | 2 | 1 | no | — |
| 14 | #13667 | C4 | 1 | 2 | yes | — |
| 15 | #13658 | — | 2 | 2 | no | — |
| 16 | #13642 | C22 | 1 | 1 | yes | characteristic |
| 17 | #13642 | C22 | 1 | 1 | yes | characteristic |
| 18 | #13196 | — | 2 | 1 | no | — |
| 19 | #12078 | — | 2 | 2 | no | — |
| 20 | #12078 | — | 2 | 2 | no | — |
| 21 | #12021 | C22 | 1 | 1 | yes | characteristic |
| 22 | #12011 | C22 | 1 | 1 | yes | characteristic |
| 23 | #11678 | — | 2 | 1 | no | — |
| 24 | #11602 | — | 2 | 1 | no | — |
| 25 | #11405 | C22 | 1 | 1 | yes | characteristic |
| 26 | #7453 | — | 2 | 1 | no | — |
| 27 | #7448 | — | 2 | 1 | no | — |
| 28 | #7389 | — | 2 | 2 | no | — |
| 29 | #3414 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | #3169 | — | 2 | 1 | no | — |
| 31 | #2981 | — | 4 | 1 | no | — |
| 32 | C2×#443 | — | 2 | 1 | no | — |
| 33 | #2263 | — | 2 | 2 | no | — |
| 34 | #2254 | D4 | 1 | 1 | yes | characteristic |
| 35 | #2246 | D4 | 1 | 1 | yes | characteristic |
| 36 | #2240 | — | 2 | 1 | no | — |
| 37 | #2218 | — | 2 | 1 | no | — |
| 38 | #2218 | C2×C4 | 1 | 2 | yes | — |
| 39 | #2179 | — | 2 | 1 | no | — |
| 40 | #2179 | — | 2 | 1 | no | — |
| 41 | #2178 | — | 2 | 1 | no | — |
| 42 | #2178 | — | 2 | 1 | no | — |
| 43 | #2177 | — | 2 | 1 | no | — |
| 44 | #2177 | D4 | 1 | 2 | yes | — |
| 45 | #2161 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | C2×#364 | — | 2 | 1 | no | — |
| 47 | #2143 | — | 2 | 1 | no | — |
| 48 | C2×#350 | — | 4 | 1 | no | — |
| 49 | #2113 | — | 4 | 1 | no | — |
| 50 | C4×M32 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×#346 | D4 | 1 | 1 | yes | characteristic |
| 52 | #1594 | — | 2 | 2 | no | — |
| 53 | #1589 | — | 4 | 1 | no | — |
| 54 | #1580 | — | 2 | 2 | no | — |
| 55 | C8×D8 | — | 4 | 1 | no | — |
| 56 | #1568 | — | 2 | 1 | no | — |
| 57 | #1496 | — | 4 | 1 | no | — |
| 58 | #1450 | D4 | 1 | 1 | yes | characteristic |
| 59 | #1426 | — | 2 | 2 | no | — |
| 60 | #1422 | — | 2 | 1 | no | — |
| 61 | #1422 | — | 2 | 1 | no | — |
| 62 | #1419 | — | 2 | 2 | no | — |
| 63 | #1418 | — | 2 | 2 | no | — |
| 64 | #1400 | — | 2 | 1 | no | — |
| 65 | #1400 | C2×C4 | 1 | 2 | yes | — |
| 66 | #1398 | — | 2 | 1 | no | — |
| 67 | #1398 | — | 2 | 1 | no | — |
| 68 | #1373 | — | 2 | 1 | no | — |
| 69 | #1373 | — | 2 | 1 | no | — |
| 70 | #1368 | — | 2 | 1 | no | — |
| 71 | #1368 | D4 | 1 | 2 | yes | — |
| 72 | #1335 | — | 4 | 1 | no | — |
| 73 | #1335 | — | 4 | 1 | no | — |
| 74 | #1333 | — | 2 | 4 | no | — |
| 75 | #1313 | — | 2 | 1 | no | — |
| 76 | #1313 | — | 2 | 1 | no | — |
| 77 | C2×#135 | — | 2 | 1 | no | — |
| 78 | C2×#131 | — | 4 | 1 | no | — |
| 79 | #532 | — | 4 | 1 | no | — |
| 80 | #510 | — | 4 | 1 | no | — |
| 81 | #509 | — | 4 | 1 | no | — |
| 82 | #508 | — | 2 | 1 | no | — |
| 83 | #508 | — | 2 | 1 | no | — |
| 84 | C2×D16 | — | 2 | 1 | no | — |
| 85 | C2×D16 | C22⋊C4 | 1 | 2 | yes | — |
| 86 | C2×M32 | — | 4 | 1 | no | — |
| 87 | C2×M32 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C2×M32 | #41 | 1 | 1 | yes | characteristic |
| 89 | C22×C16 | — | 2 | 1 | no | — |
| 90 | #500 | — | 2 | 1 | no | — |
| 91 | #493 | — | 2 | 1 | no | — |
| 92 | #486 | — | 4 | 1 | no | — |
| 93 | #466 | — | 2 | 2 | no | — |
| 94 | #466 | — | 4 | 1 | no | — |
| 95 | #443 | — | 4 | 1 | no | — |
| 96 | #443 | — | 4 | 1 | no | — |
| 97 | #443 | — | 4 | 1 | no | — |
| 98 | #443 | — | 4 | 1 | no | — |
| 99 | C4×D8 | — | 2 | 2 | no | — |
| 100 | C4×D8 | — | 4 | 1 | no | — |
| 101 | #436 | — | 4 | 2 | no | — |
| 102 | #435 | — | 8 | 2 | no | — |
| 103 | C8×D4 | — | 4 | 2 | no | — |
| 104 | #432 | — | 4 | 1 | no | — |
| 105 | #431 | — | 2 | 1 | no | — |
| 106 | C2×#108 | — | 2 | 1 | no | — |
| 107 | C2×#104 | — | 4 | 1 | no | — |
| 108 | #408 | — | 8 | 2 | no | — |
| 109 | C2×C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | SD64 | — | 4 | 2 | no | — |
| 111 | D32 | — | 4 | 2 | no | — |
| 112 | M64 | — | 2 | 2 | no | — |
| 113 | M64 | — | 4 | 1 | no | — |
| 114 | #368 | — | 2 | 1 | no | — |
| 115 | #368 | — | 2 | 1 | no | — |
| 116 | #367 | — | 2 | 1 | no | — |
| 117 | #367 | C22⋊C4 | 1 | 2 | yes | — |
| 118 | #364 | — | 4 | 1 | no | — |
| 119 | #364 | — | 4 | 1 | no | — |
| 120 | #359 | — | 4 | 1 | no | — |
| 121 | #359 | — | 4 | 1 | no | — |
| 122 | #357 | — | 2 | 2 | no | — |
| 123 | #357 | — | 2 | 2 | no | — |
| 124 | #350 | — | 8 | 2 | no | — |
| 125 | C22⋊C16 | — | 8 | 2 | no | — |
| 126 | #346 | — | 2 | 1 | no | — |
| 127 | #346 | — | 2 | 1 | no | — |
| 128 | #346 | C2×D4 | 1 | 1 | yes | characteristic |
| 129 | #346 | C2×D4 | 1 | 1 | yes | characteristic |
| 130 | C4×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 131 | C4×C16 | #41 | 1 | 1 | yes | characteristic |
| 132 | #335 | — | 4 | 1 | no | — |
| 133 | #334 | — | 2 | 2 | no | — |
| 134 | D4⋊C8 | — | 4 | 2 | no | — |
| 135 | D4⋊C8 | — | 4 | 2 | no | — |
| 136 | #322 | — | 2 | 1 | no | — |
| 137 | C82 | D8 | 1 | 1 | yes | characteristic |
| 138 | C82 | D8 | 1 | 1 | yes | characteristic |
| 139 | C2×#41 | — | 4 | 1 | no | — |
| 140 | C2×#41 | — | 8 | 1 | no | — |
| 141 | #135 | — | 4 | 1 | no | — |
| 142 | #135 | — | 4 | 1 | no | — |
| 143 | #135 | — | 4 | 1 | no | — |
| 144 | #135 | — | 4 | 1 | no | — |
| 145 | C2×Q16 | — | 2 | 1 | no | — |
| 146 | C2×SD16 | — | 4 | 1 | no | — |
| 147 | C2×D8 | — | 2 | 1 | no | — |
| 148 | C2×D8 | — | 2 | 2 | no | — |
| 149 | C2×D8 | — | 4 | 1 | no | — |
| 150 | #131 | — | 8 | 1 | no | — |
| 151 | #131 | — | 8 | 1 | no | — |
| 152 | #131 | — | 8 | 1 | no | — |
| 153 | #131 | — | 8 | 1 | no | — |
| 154 | C2×M16 | — | 4 | 1 | no | — |
| 155 | C2×M16 | — | 8 | 1 | no | — |
| 156 | C2×M16 | — | 8 | 2 | no | — |
| 157 | C22×C8 | — | 8 | 1 | no | — |
| 158 | C22×C8 | — | 8 | 2 | no | — |
| 159 | C22×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 160 | C22×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 161 | C4⋊Q8 | — | 4 | 1 | no | — |
| 162 | #127 | — | 4 | 1 | no | — |
| 163 | #124 | — | 8 | 1 | no | — |
| 164 | #121 | — | 4 | 2 | no | — |
| 165 | #121 | — | 8 | 1 | no | — |
| 166 | C4×D4 | — | 4 | 2 | no | — |
| 167 | C4×D4 | — | 8 | 1 | no | — |
| 168 | C2×C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 169 | Q32 | — | 2 | 1 | no | — |
| 170 | Q32 | — | 2 | 1 | no | — |
| 171 | SD32 | — | 4 | 1 | no | — |
| 172 | SD32 | — | 4 | 1 | no | — |
| 173 | D16 | — | 2 | 1 | no | — |
| 174 | D16 | — | 2 | 1 | no | — |
| 175 | D16 | — | 4 | 2 | no | — |
| 176 | D16 | — | 8 | 1 | no | — |
| 177 | M32 | — | 4 | 1 | no | — |
| 178 | M32 | — | 4 | 1 | no | — |
| 179 | M32 | — | 8 | 1 | no | — |
| 180 | M32 | — | 8 | 1 | no | — |
| 181 | C2×C16 | — | 2 | 2 | no | — |
| 182 | C2×C16 | — | 4 | 1 | no | — |
| 183 | C2×C16 | — | 4 | 1 | no | — |
| 184 | C2×C16 | — | 8 | 1 | no | — |
| 185 | C2×C16 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 186 | C2×C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 187 | C2×C16 | #121 | 1 | 1 | yes | characteristic |
| 188 | C2×C16 | #123 | 1 | 1 | yes | characteristic |
| 189 | #108 | — | 4 | 1 | no | — |
| 190 | #108 | — | 4 | 1 | no | — |
| 191 | #107 | — | 2 | 2 | no | — |
| 192 | #107 | — | 4 | 1 | no | — |
| 193 | C4⋊C8 | — | 4 | 1 | no | — |
| 194 | C4⋊C8 | — | 4 | 2 | no | — |
| 195 | #104 | — | 8 | 1 | no | — |
| 196 | #104 | — | 8 | 1 | no | — |
| 197 | #102 | — | 4 | 2 | no | — |
| 198 | #102 | — | 4 | 2 | no | — |
| 199 | C22⋊C8 | — | 4 | 1 | no | — |
| 200 | C22⋊C8 | — | 8 | 2 | no | — |
| 201 | C22⋊C8 | — | 8 | 2 | no | — |
| 202 | C22⋊C8 | — | 8 | 1 | no | — |
| 203 | #97 | — | 8 | 1 | no | — |
| 204 | C4×C8 | — | 2 | 1 | no | — |
| 205 | C4×C8 | — | 2 | 1 | no | — |
| 206 | C4×C8 | #120 | 1 | 1 | yes | characteristic |
| 207 | C4×C8 | #121 | 1 | 1 | yes | characteristic |
| 208 | C4×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 209 | C4×C8 | #135 | 1 | 1 | yes | characteristic |
| 210 | C32 | — | 4 | 2 | no | — |
| 211 | C32 | — | 8 | 1 | no | — |
| 212 | #41 | — | 8 | 1 | no | — |
| 213 | #41 | — | 8 | 1 | no | — |
| 214 | #41 | — | 8 | 1 | no | — |
| 215 | #41 | — | 8 | 1 | no | — |
| 216 | #41 | — | 32 | 2 | no | — |
| 217 | C2×Q8 | — | 4 | 1 | no | — |
| 218 | C2×Q8 | — | 8 | 1 | no | — |
| 219 | C2×D4 | — | 4 | 1 | no | — |
| 220 | C2×D4 | — | 4 | 2 | no | — |
| 221 | C2×D4 | — | 8 | 1 | no | — |
| 222 | C2×D4 | — | 8 | 1 | no | — |
| 223 | C2×D4 | — | 8 | 1 | no | — |
| 224 | C2×D4 | — | 8 | 2 | no | — |
| 225 | C22×C4 | — | 2 | 1 | no | — |
| 226 | C22×C4 | — | 8 | 2 | no | — |
| 227 | C22×C4 | — | 8 | 1 | no | — |
| 228 | C22×C4 | #442 | 1 | 1 | yes | characteristic |
| 229 | Q16 | — | 4 | 1 | no | — |
| 230 | Q16 | — | 4 | 1 | no | — |
| 231 | SD16 | — | 8 | 1 | no | — |
| 232 | SD16 | — | 8 | 1 | no | — |
| 233 | D8 | — | 4 | 1 | no | — |
| 234 | D8 | — | 4 | 2 | no | — |
| 235 | D8 | — | 4 | 1 | no | — |
| 236 | D8 | — | 4 | 2 | no | — |
| 237 | D8 | — | 8 | 1 | no | — |
| 238 | D8 | — | 8 | 1 | no | — |
| 239 | M16 | — | 8 | 1 | no | — |
| 240 | M16 | — | 8 | 1 | no | — |
| 241 | M16 | — | 16 | 1 | no | — |
| 242 | M16 | — | 16 | 1 | no | — |
| 243 | M16 | — | 32 | 2 | no | — |
| 244 | C2×C8 | — | 2 | 1 | no | — |
| 245 | C2×C8 | — | 2 | 1 | no | — |
| 246 | C2×C8 | — | 2 | 1 | no | — |
| 247 | C2×C8 | — | 2 | 1 | no | — |
| 248 | C2×C8 | — | 4 | 1 | no | — |
| 249 | C2×C8 | — | 4 | 1 | no | — |
| 250 | C2×C8 | — | 8 | 2 | no | — |
| 251 | C2×C8 | — | 8 | 1 | no | — |
| 252 | C2×C8 | — | 16 | 1 | no | — |
| 253 | C2×C8 | — | 16 | 1 | no | — |
| 254 | C2×C8 | — | 32 | 2 | no | — |
| 255 | C2×C8 | #386 | 1 | 1 | yes | characteristic |
| 256 | C2×C8 | C4×D8 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | #449 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | #459 | 1 | 1 | yes | characteristic |
| 259 | C4⋊C4 | — | 4 | 2 | no | — |
| 260 | C4⋊C4 | — | 8 | 1 | no | — |
| 261 | C22⋊C4 | — | 8 | 2 | no | — |
| 262 | C22⋊C4 | — | 8 | 2 | no | — |
| 263 | C42 | — | 2 | 1 | no | — |
| 264 | C42 | #447 | 1 | 1 | yes | characteristic |
| 265 | C42 | #463 | 1 | 1 | yes | characteristic |
| 266 | C16 | — | 2 | 1 | no | — |
| 267 | C16 | — | 2 | 1 | no | — |
| 268 | C16 | — | 2 | 1 | no | — |
| 269 | C16 | — | 2 | 1 | no | — |
| 270 | C16 | — | 4 | 2 | no | — |
| 271 | C16 | — | 16 | 1 | no | — |
| 272 | C23 | — | 8 | 1 | no | — |
| 273 | C23 | — | 8 | 2 | no | — |
| 274 | C23 | AGL(1,ℤ/16) | 1 | 1 | yes | characteristic |
| 275 | Q8 | — | 8 | 1 | no | — |
| 276 | Q8 | — | 8 | 1 | no | — |
| 277 | Q8 | — | 32 | 1 | no | — |
| 278 | D4 | — | 8 | 1 | no | — |
| 279 | D4 | — | 8 | 1 | no | — |
| 280 | D4 | — | 8 | 2 | no | — |
| 281 | D4 | — | 8 | 2 | no | — |
| 282 | D4 | — | 16 | 1 | no | — |
| 283 | D4 | — | 16 | 1 | no | — |
| 284 | D4 | — | 32 | 1 | no | — |
| 285 | D4 | — | 32 | 1 | no | — |
| 286 | D4 | — | 32 | 2 | no | — |
| 287 | C2×C4 | — | 2 | 1 | no | — |
| 288 | C2×C4 | — | 2 | 1 | no | — |
| 289 | C2×C4 | — | 2 | 1 | no | — |
| 290 | C2×C4 | — | 2 | 1 | no | — |
| 291 | C2×C4 | — | 8 | 1 | no | — |
| 292 | C2×C4 | — | 8 | 1 | no | — |
| 293 | C2×C4 | — | 8 | 2 | no | — |
| 294 | C2×C4 | — | 16 | 1 | no | — |
| 295 | C2×C4 | — | 16 | 2 | no | — |
| 296 | C2×C4 | — | 16 | 2 | no | — |
| 297 | C2×C4 | — | 16 | 1 | no | — |
| 298 | C2×C4 | #1879 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | AGL(1,ℤ/16) | 1 | 1 | yes | characteristic |
| 300 | C8 | — | 2 | 1 | no | — |
| 301 | C8 | — | 2 | 1 | no | — |
| 302 | C8 | — | 8 | 1 | no | — |
| 303 | C8 | — | 8 | 1 | no | — |
| 304 | C8 | — | 16 | 2 | no | — |
| 305 | C8 | — | 16 | 1 | no | — |
| 306 | C8 | #1881 | 1 | 1 | yes | characteristic |
| 307 | C8 | #1891 | 1 | 1 | yes | characteristic |
| 308 | C8 | #2212 | 1 | 1 | yes | characteristic |
| 309 | C8 | #2213 | 1 | 1 | yes | characteristic |
| 310 | C22 | — | 2 | 1 | no | — |
| 311 | C22 | — | 4 | 1 | no | — |
| 312 | C22 | — | 16 | 2 | no | — |
| 313 | C22 | — | 16 | 2 | no | — |
| 314 | C22 | — | 16 | 2 | no | — |
| 315 | C22 | — | 16 | 1 | no | — |
| 316 | C22 | — | 16 | 2 | no | — |
| 317 | C22 | #13037 | 1 | 1 | yes | characteristic |
| 318 | C4 | — | 2 | 1 | no | — |
| 319 | C4 | — | 4 | 1 | no | — |
| 320 | C4 | — | 4 | 1 | no | — |
| 321 | C4 | — | 16 | 2 | no | — |
| 322 | C4 | — | 16 | 1 | no | — |
| 323 | C4 | #9049 | 1 | 1 | yes | characteristic |
| 324 | C4 | #13029 | 1 | 1 | yes | characteristic |
| 325 | C2 | — | 2 | 1 | no | — |
| 326 | C2 | — | 4 | 1 | no | — |
| 327 | C2 | — | 32 | 2 | no | — |
| 328 | C2 | — | 32 | 1 | no | — |
| 329 | C2 | #137479 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #137479 |
| commutator | C2×M32 | C22×C4 |
| Frattini, cosocle | C4×M32 | C23 |
| prime radical | #513022 | C2 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ #11405 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #483996 | C2 | semidirect |
| 3. #150977 | C2 | semidirect |
| 4. #150147 | C2 | semidirect |
| 5. #140775 | C2 | semidirect |
| 6. #94407 | C2 | semidirect |
| 7. #94120 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #513022 | 2 |
| #483996 | 2 |
| #150977 | 2 |
| #150147 | 2 |
| #140775 | 2 |
| #94407 | 2 |
| #94120 | 2 |