Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
749 conjugacy classes of subgroups.
The subgroup lattice has 635 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #25405 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22536 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19171 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #18770 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#1642 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #17997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10607 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #10456 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #10445 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #10168 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #9796 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #9550 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | #9273 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | #9252 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | #8907 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C2×#550 | C22 | 1 | 1 | yes | characteristic |
| 17 | #3205 | C22 | 1 | 1 | yes | characteristic |
| 18 | #3152 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#486 | C22 | 1 | 1 | yes | characteristic |
| 20 | #3103 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#475 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#471 | — | 2 | 1 | no | — |
| 23 | #3036 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×#461 | C22 | 1 | 1 | yes | characteristic |
| 25 | Q8×M16 | C22 | 1 | 1 | yes | characteristic |
| 26 | Q8×M16 | C22 | 1 | 1 | yes | characteristic |
| 27 | #2935 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2052 | — | 2 | 1 | no | — |
| 29 | #2019 | — | 2 | 1 | no | — |
| 30 | #1966 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1965 | — | 2 | 1 | no | — |
| 32 | #1965 | — | 2 | 1 | no | — |
| 33 | #1965 | C22 | 1 | 1 | yes | characteristic |
| 34 | #1965 | C22 | 1 | 1 | yes | characteristic |
| 35 | #1950 | C22 | 1 | 1 | yes | characteristic |
| 36 | #1908 | — | 2 | 1 | no | — |
| 37 | #1800 | — | 2 | 1 | no | — |
| 38 | #1800 | C22 | 1 | 1 | yes | characteristic |
| 39 | C4×#100 | — | 2 | 1 | no | — |
| 40 | C4×#100 | C22 | 1 | 1 | yes | characteristic |
| 41 | #1690 | — | 2 | 1 | no | — |
| 42 | #1686 | — | 2 | 1 | no | — |
| 43 | #1670 | — | 2 | 1 | no | — |
| 44 | #1666 | — | 2 | 1 | no | — |
| 45 | #1642 | — | 2 | 1 | no | — |
| 46 | #1642 | — | 2 | 1 | no | — |
| 47 | #1642 | — | 2 | 1 | no | — |
| 48 | #1642 | — | 2 | 1 | no | — |
| 49 | #1642 | — | 2 | 1 | no | — |
| 50 | #1642 | — | 2 | 1 | no | — |
| 51 | #1621 | — | 2 | 1 | no | — |
| 52 | #1621 | — | 2 | 1 | no | — |
| 53 | #1590 | — | 2 | 1 | no | — |
| 54 | #1583 | — | 2 | 1 | no | — |
| 55 | #1582 | — | 2 | 1 | no | — |
| 56 | C8×SD16 | — | 2 | 1 | no | — |
| 57 | #1570 | C22 | 1 | 1 | yes | characteristic |
| 58 | #1567 | C22 | 1 | 1 | yes | characteristic |
| 59 | #1565 | C22 | 1 | 1 | yes | characteristic |
| 60 | #1547 | C22 | 1 | 1 | yes | characteristic |
| 61 | #1546 | C22 | 1 | 1 | yes | characteristic |
| 62 | #1546 | C22 | 1 | 1 | yes | characteristic |
| 63 | #1545 | C22 | 1 | 1 | yes | characteristic |
| 64 | C2×#331 | C22 | 1 | 1 | yes | characteristic |
| 65 | #1492 | C22 | 1 | 1 | yes | characteristic |
| 66 | #1487 | C22 | 1 | 1 | yes | characteristic |
| 67 | #1478 | C22 | 1 | 1 | yes | characteristic |
| 68 | C2×Q8⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 69 | C2×D4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 70 | C8×M16 | C22 | 1 | 1 | yes | characteristic |
| 71 | #1301 | C22 | 1 | 1 | yes | characteristic |
| 72 | #1292 | C22 | 1 | 1 | yes | characteristic |
| 73 | #1283 | C22 | 1 | 1 | yes | characteristic |
| 74 | #1278 | C22 | 1 | 1 | yes | characteristic |
| 75 | C22×#41 | — | 2 | 1 | no | — |
| 76 | C2×#137 | — | 2 | 1 | no | — |
| 77 | C2×#137 | — | 2 | 1 | no | — |
| 78 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 79 | C22×SD16 | — | 2 | 1 | no | — |
| 80 | C2×#131 | — | 2 | 1 | no | — |
| 81 | #550 | — | 2 | 1 | no | — |
| 82 | #550 | — | 2 | 1 | no | — |
| 83 | #550 | — | 2 | 1 | no | — |
| 84 | #550 | — | 2 | 1 | no | — |
| 85 | #550 | — | 4 | 1 | no | — |
| 86 | #550 | — | 4 | 1 | no | — |
| 87 | #532 | C23 | 1 | 1 | yes | characteristic |
| 88 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 89 | C2×#127 | — | 2 | 1 | no | — |
| 90 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 91 | C2×#121 | — | 2 | 2 | no | — |
| 92 | C2×#121 | — | 2 | 1 | no | — |
| 93 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 94 | C2×C4×D4 | — | 2 | 1 | no | — |
| 95 | C2×C4×D4 | C23 | 1 | 1 | yes | characteristic |
| 96 | #497 | — | 4 | 1 | no | — |
| 97 | #494 | — | 2 | 1 | no | — |
| 98 | #493 | — | 2 | 1 | no | — |
| 99 | #492 | — | 2 | 1 | no | — |
| 100 | #486 | — | 2 | 1 | no | — |
| 101 | #486 | — | 2 | 1 | no | — |
| 102 | #486 | — | 2 | 1 | no | — |
| 103 | #486 | — | 4 | 1 | no | — |
| 104 | #481 | — | 2 | 1 | no | — |
| 105 | #475 | — | 2 | 1 | no | — |
| 106 | #475 | — | 2 | 1 | no | — |
| 107 | #475 | — | 2 | 1 | no | — |
| 108 | #475 | — | 2 | 1 | no | — |
| 109 | #475 | D4 | 1 | 1 | yes | characteristic |
| 110 | #475 | D4 | 1 | 1 | yes | characteristic |
| 111 | #471 | — | 4 | 4 | no | — |
| 112 | #463 | — | 4 | 1 | no | — |
| 113 | #461 | — | 2 | 1 | no | — |
| 114 | #461 | — | 2 | 1 | no | — |
| 115 | #461 | — | 2 | 1 | no | — |
| 116 | #461 | — | 4 | 1 | no | — |
| 117 | #460 | D4 | 1 | 1 | yes | characteristic |
| 118 | #460 | D4 | 1 | 1 | yes | characteristic |
| 119 | #459 | — | 2 | 1 | no | — |
| 120 | #459 | — | 4 | 1 | no | — |
| 121 | #448 | — | 2 | 1 | no | — |
| 122 | #447 | — | 2 | 2 | no | — |
| 123 | #446 | — | 2 | 1 | no | — |
| 124 | #446 | — | 2 | 1 | no | — |
| 125 | #446 | — | 4 | 1 | no | — |
| 126 | #446 | — | 4 | 1 | no | — |
| 127 | C8×Q8 | — | 2 | 1 | no | — |
| 128 | C8×Q8 | — | 2 | 1 | no | — |
| 129 | C4×SD16 | — | 2 | 1 | no | — |
| 130 | C4×SD16 | — | 2 | 1 | no | — |
| 131 | #436 | — | 2 | 1 | no | — |
| 132 | #435 | — | 4 | 1 | no | — |
| 133 | C8×D4 | — | 2 | 1 | no | — |
| 134 | #431 | C23 | 1 | 1 | yes | characteristic |
| 135 | #431 | C23 | 1 | 1 | yes | characteristic |
| 136 | C2×#108 | — | 2 | 1 | no | — |
| 137 | C2×#106 | — | 2 | 1 | no | — |
| 138 | #423 | — | 2 | 1 | no | — |
| 139 | #423 | — | 2 | 1 | no | — |
| 140 | #423 | C23 | 1 | 1 | yes | characteristic |
| 141 | #423 | C23 | 1 | 1 | yes | characteristic |
| 142 | #423 | C23 | 1 | 1 | yes | characteristic |
| 143 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 144 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 145 | #417 | — | 2 | 1 | no | — |
| 146 | C2×#103 | — | 2 | 1 | no | — |
| 147 | C2×#103 | — | 2 | 1 | no | — |
| 148 | C2×#102 | — | 2 | 2 | no | — |
| 149 | C2×#101 | — | 2 | 1 | no | — |
| 150 | C2×#101 | — | 2 | 1 | no | — |
| 151 | C2×#100 | — | 2 | 1 | no | — |
| 152 | C2×#100 | — | 2 | 1 | no | — |
| 153 | C2×#100 | — | 2 | 1 | no | — |
| 154 | C2×#100 | — | 2 | 1 | no | — |
| 155 | C2×#100 | — | 2 | 1 | no | — |
| 156 | C2×#100 | D4 | 1 | 1 | yes | characteristic |
| 157 | C2×#100 | D4 | 1 | 1 | yes | characteristic |
| 158 | #408 | — | 2 | 1 | no | — |
| 159 | C4×M16 | — | 2 | 1 | no | — |
| 160 | C4×M16 | — | 2 | 1 | no | — |
| 161 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 162 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 163 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 164 | #343 | — | 2 | 1 | no | — |
| 165 | #343 | D4 | 1 | 1 | yes | characteristic |
| 166 | #343 | D4 | 1 | 1 | yes | characteristic |
| 167 | #341 | — | 2 | 1 | no | — |
| 168 | #335 | — | 2 | 1 | no | — |
| 169 | #335 | — | 4 | 1 | no | — |
| 170 | #334 | — | 2 | 1 | no | — |
| 171 | #334 | — | 2 | 1 | no | — |
| 172 | #334 | — | 2 | 1 | no | — |
| 173 | #332 | — | 2 | 1 | no | — |
| 174 | #332 | — | 2 | 1 | no | — |
| 175 | #332 | — | 2 | 1 | no | — |
| 176 | #332 | — | 4 | 1 | no | — |
| 177 | #331 | — | 2 | 1 | no | — |
| 178 | #331 | — | 2 | 1 | no | — |
| 179 | #331 | — | 2 | 1 | no | — |
| 180 | #331 | — | 2 | 1 | no | — |
| 181 | #331 | — | 2 | 1 | no | — |
| 182 | Q8⋊C8 | — | 2 | 1 | no | — |
| 183 | Q8⋊C8 | — | 2 | 1 | no | — |
| 184 | Q8⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 185 | Q8⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 186 | Q8⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 187 | Q8⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 188 | D4⋊C8 | — | 2 | 1 | no | — |
| 189 | D4⋊C8 | — | 2 | 1 | no | — |
| 190 | D4⋊C8 | — | 2 | 1 | no | — |
| 191 | D4⋊C8 | — | 2 | 1 | no | — |
| 192 | D4⋊C8 | — | 2 | 1 | no | — |
| 193 | D4⋊C8 | — | 2 | 1 | no | — |
| 194 | #322 | — | 2 | 1 | no | — |
| 195 | C82 | — | 2 | 1 | no | — |
| 196 | C2×#41 | — | 2 | 1 | no | — |
| 197 | C2×#41 | — | 4 | 2 | no | — |
| 198 | C2×#41 | — | 4 | 1 | no | — |
| 199 | C2×#41 | — | 4 | 1 | no | — |
| 200 | C2×#41 | — | 8 | 2 | no | — |
| 201 | C2×#41 | — | 8 | 2 | no | — |
| 202 | C22×Q8 | — | 2 | 1 | no | — |
| 203 | C22×Q8 | #41 | 1 | 1 | yes | characteristic |
| 204 | C22×D4 | — | 2 | 2 | no | — |
| 205 | C22×D4 | — | 2 | 2 | no | — |
| 206 | C22×D4 | — | 2 | 1 | no | — |
| 207 | C22×D4 | — | 2 | 1 | no | — |
| 208 | C22×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 209 | C22×D4 | #41 | 1 | 2 | yes | — |
| 210 | C23×C4 | — | 2 | 2 | no | — |
| 211 | C23×C4 | — | 2 | 1 | no | — |
| 212 | #137 | — | 4 | 2 | no | — |
| 213 | #137 | — | 4 | 2 | no | — |
| 214 | #137 | — | 8 | 1 | no | — |
| 215 | #137 | — | 8 | 1 | no | — |
| 216 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 217 | AGL(1,ℤ/8) | — | 8 | 1 | no | — |
| 218 | C2×Q16 | — | 4 | 1 | no | — |
| 219 | C2×Q16 | — | 4 | 1 | no | — |
| 220 | C2×SD16 | — | 2 | 1 | no | — |
| 221 | C2×SD16 | — | 2 | 1 | no | — |
| 222 | C2×SD16 | — | 4 | 1 | no | — |
| 223 | C2×SD16 | — | 4 | 1 | no | — |
| 224 | C2×SD16 | — | 4 | 1 | no | — |
| 225 | C2×SD16 | — | 4 | 1 | no | — |
| 226 | C2×SD16 | — | 4 | 1 | no | — |
| 227 | C2×SD16 | — | 4 | 2 | no | — |
| 228 | C2×SD16 | — | 8 | 1 | no | — |
| 229 | C2×D8 | — | 4 | 1 | no | — |
| 230 | C2×D8 | — | 4 | 2 | no | — |
| 231 | #131 | — | 4 | 2 | no | — |
| 232 | #131 | — | 8 | 1 | no | — |
| 233 | C2×M16 | — | 2 | 1 | no | — |
| 234 | C2×M16 | — | 2 | 1 | no | — |
| 235 | C2×M16 | — | 2 | 1 | no | — |
| 236 | C2×M16 | — | 4 | 1 | no | — |
| 237 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 238 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 239 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 240 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 241 | C22×C8 | — | 2 | 1 | no | — |
| 242 | C22×C8 | — | 2 | 1 | no | — |
| 243 | C22×C8 | — | 2 | 1 | no | — |
| 244 | C22×C8 | — | 4 | 1 | no | — |
| 245 | C4⋊Q8 | — | 2 | 1 | no | — |
| 246 | C4⋊Q8 | — | 2 | 1 | no | — |
| 247 | C4⋊Q8 | — | 2 | 1 | no | — |
| 248 | C4⋊Q8 | — | 2 | 1 | no | — |
| 249 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 250 | #127 | — | 2 | 1 | no | — |
| 251 | #127 | — | 2 | 1 | no | — |
| 252 | #127 | — | 2 | 1 | no | — |
| 253 | #127 | — | 2 | 1 | no | — |
| 254 | #127 | — | 4 | 1 | no | — |
| 255 | #127 | — | 4 | 1 | no | — |
| 256 | #127 | — | 4 | 1 | no | — |
| 257 | #127 | — | 4 | 1 | no | — |
| 258 | #127 | C2×D4 | 1 | 1 | yes | characteristic |
| 259 | #124 | — | 2 | 1 | no | — |
| 260 | #121 | — | 2 | 1 | no | — |
| 261 | #121 | — | 2 | 1 | no | — |
| 262 | #121 | — | 4 | 2 | no | — |
| 263 | #121 | — | 4 | 2 | no | — |
| 264 | #121 | — | 4 | 2 | no | — |
| 265 | #121 | — | 4 | 2 | no | — |
| 266 | #121 | — | 4 | 2 | no | — |
| 267 | #121 | — | 4 | 1 | no | — |
| 268 | #121 | — | 4 | 1 | no | — |
| 269 | C4×Q8 | — | 2 | 1 | no | — |
| 270 | C4×Q8 | — | 2 | 1 | no | — |
| 271 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 272 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 273 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 274 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 275 | C4×D4 | — | 2 | 1 | no | — |
| 276 | C4×D4 | — | 2 | 1 | no | — |
| 277 | C4×D4 | — | 2 | 1 | no | — |
| 278 | C4×D4 | — | 2 | 1 | no | — |
| 279 | C4×D4 | — | 2 | 1 | no | — |
| 280 | C4×D4 | — | 2 | 1 | no | — |
| 281 | C4×D4 | — | 4 | 1 | no | — |
| 282 | C4×D4 | — | 4 | 1 | no | — |
| 283 | C4×D4 | — | 4 | 1 | no | — |
| 284 | C4×D4 | — | 4 | 1 | no | — |
| 285 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 286 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 287 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 288 | C2×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 289 | C2×C4⋊C4 | #41 | 1 | 1 | yes | characteristic |
| 290 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 291 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 292 | C2×C42 | — | 2 | 1 | no | — |
| 293 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 294 | #108 | — | 4 | 2 | no | — |
| 295 | #106 | — | 2 | 1 | no | — |
| 296 | #106 | — | 2 | 1 | no | — |
| 297 | #106 | — | 4 | 1 | no | — |
| 298 | #106 | — | 4 | 1 | no | — |
| 299 | C4⋊C8 | — | 2 | 1 | no | — |
| 300 | C4⋊C8 | — | 2 | 1 | no | — |
| 301 | C4⋊C8 | — | 2 | 1 | no | — |
| 302 | C4⋊C8 | — | 2 | 1 | no | — |
| 303 | C4⋊C8 | — | 2 | 1 | no | — |
| 304 | C4⋊C8 | — | 2 | 1 | no | — |
| 305 | C4⋊C8 | — | 2 | 1 | no | — |
| 306 | C4⋊C8 | — | 2 | 1 | no | — |
| 307 | C4⋊C8 | — | 2 | 1 | no | — |
| 308 | C4⋊C8 | — | 2 | 1 | no | — |
| 309 | C4⋊C8 | — | 4 | 1 | no | — |
| 310 | C4⋊C8 | — | 4 | 1 | no | — |
| 311 | C4⋊C8 | — | 4 | 1 | no | — |
| 312 | C4⋊C8 | — | 4 | 1 | no | — |
| 313 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 314 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 315 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 316 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 317 | #103 | — | 2 | 1 | no | — |
| 318 | #103 | — | 2 | 1 | no | — |
| 319 | #103 | — | 4 | 1 | no | — |
| 320 | #103 | — | 4 | 1 | no | — |
| 321 | #103 | — | 4 | 1 | no | — |
| 322 | #103 | — | 4 | 1 | no | — |
| 323 | #103 | — | 4 | 1 | no | — |
| 324 | #102 | — | 2 | 1 | no | — |
| 325 | #102 | — | 2 | 1 | no | — |
| 326 | #102 | — | 4 | 1 | no | — |
| 327 | #102 | — | 4 | 2 | no | — |
| 328 | #102 | — | 4 | 2 | no | — |
| 329 | #102 | — | 4 | 2 | no | — |
| 330 | #101 | — | 4 | 2 | no | — |
| 331 | #101 | — | 8 | 1 | no | — |
| 332 | #100 | — | 2 | 2 | no | — |
| 333 | #100 | — | 2 | 2 | no | — |
| 334 | #100 | — | 4 | 2 | no | — |
| 335 | #100 | — | 4 | 2 | no | — |
| 336 | #100 | — | 8 | 1 | no | — |
| 337 | #100 | — | 8 | 1 | no | — |
| 338 | #100 | — | 8 | 1 | no | — |
| 339 | C22⋊C8 | — | 2 | 1 | no | — |
| 340 | C22⋊C8 | — | 2 | 1 | no | — |
| 341 | C22⋊C8 | — | 4 | 1 | no | — |
| 342 | C22⋊C8 | — | 4 | 1 | no | — |
| 343 | #97 | — | 2 | 1 | no | — |
| 344 | #97 | — | 2 | 1 | no | — |
| 345 | #97 | — | 4 | 1 | no | — |
| 346 | #97 | — | 4 | 1 | no | — |
| 347 | C4×C8 | — | 2 | 1 | no | — |
| 348 | C4×C8 | — | 2 | 1 | no | — |
| 349 | C4×C8 | — | 2 | 1 | no | — |
| 350 | C4×C8 | — | 2 | 1 | no | — |
| 351 | C4×C8 | — | 2 | 1 | no | — |
| 352 | C4×C8 | — | 4 | 1 | no | — |
| 353 | C4×C8 | — | 4 | 1 | no | — |
| 354 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 355 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 356 | C24 | — | 2 | 1 | no | — |
| 357 | C24 | — | 2 | 2 | no | — |
| 358 | #41 | — | 4 | 1 | no | — |
| 359 | #41 | — | 4 | 1 | no | — |
| 360 | #41 | — | 4 | 1 | no | — |
| 361 | #41 | — | 4 | 1 | no | — |
| 362 | #41 | — | 8 | 2 | no | — |
| 363 | #41 | — | 8 | 2 | no | — |
| 364 | #41 | — | 8 | 2 | no | — |
| 365 | #41 | — | 8 | 8 | no | — |
| 366 | #41 | — | 8 | 2 | no | — |
| 367 | C2×Q8 | — | 2 | 1 | no | — |
| 368 | C2×Q8 | — | 2 | 2 | no | — |
| 369 | C2×Q8 | — | 2 | 1 | no | — |
| 370 | C2×Q8 | — | 4 | 1 | no | — |
| 371 | C2×Q8 | — | 4 | 1 | no | — |
| 372 | C2×Q8 | — | 4 | 1 | no | — |
| 373 | C2×Q8 | — | 4 | 1 | no | — |
| 374 | C2×Q8 | — | 4 | 1 | no | — |
| 375 | C2×Q8 | — | 4 | 1 | no | — |
| 376 | C2×Q8 | — | 4 | 1 | no | — |
| 377 | C2×Q8 | #121 | 1 | 1 | yes | characteristic |
| 378 | C2×Q8 | #121 | 1 | 1 | yes | characteristic |
| 379 | C2×D4 | — | 2 | 1 | no | — |
| 380 | C2×D4 | — | 2 | 1 | no | — |
| 381 | C2×D4 | — | 2 | 1 | no | — |
| 382 | C2×D4 | — | 2 | 1 | no | — |
| 383 | C2×D4 | — | 2 | 2 | no | — |
| 384 | C2×D4 | — | 2 | 2 | no | — |
| 385 | C2×D4 | — | 2 | 1 | no | — |
| 386 | C2×D4 | — | 2 | 1 | no | — |
| 387 | C2×D4 | — | 2 | 1 | no | — |
| 388 | C2×D4 | — | 2 | 1 | no | — |
| 389 | C2×D4 | — | 2 | 1 | no | — |
| 390 | C2×D4 | — | 4 | 2 | no | — |
| 391 | C2×D4 | — | 4 | 2 | no | — |
| 392 | C2×D4 | — | 4 | 2 | no | — |
| 393 | C2×D4 | — | 4 | 2 | no | — |
| 394 | C2×D4 | — | 4 | 1 | no | — |
| 395 | C2×D4 | — | 4 | 1 | no | — |
| 396 | C2×D4 | — | 4 | 1 | no | — |
| 397 | C2×D4 | — | 4 | 1 | no | — |
| 398 | C2×D4 | — | 4 | 1 | no | — |
| 399 | C2×D4 | — | 4 | 1 | no | — |
| 400 | C2×D4 | — | 4 | 1 | no | — |
| 401 | C2×D4 | — | 4 | 1 | no | — |
| 402 | C2×D4 | — | 4 | 1 | no | — |
| 403 | C2×D4 | — | 8 | 1 | no | — |
| 404 | C2×D4 | — | 8 | 1 | no | — |
| 405 | C2×D4 | — | 8 | 4 | no | — |
| 406 | C2×D4 | — | 8 | 2 | no | — |
| 407 | C2×D4 | — | 16 | 2 | no | — |
| 408 | C22×C4 | — | 2 | 1 | no | — |
| 409 | C22×C4 | — | 2 | 1 | no | — |
| 410 | C22×C4 | — | 2 | 1 | no | — |
| 411 | C22×C4 | — | 4 | 2 | no | — |
| 412 | C22×C4 | — | 4 | 2 | no | — |
| 413 | C22×C4 | — | 4 | 1 | no | — |
| 414 | C22×C4 | — | 4 | 1 | no | — |
| 415 | C22×C4 | — | 4 | 1 | no | — |
| 416 | C22×C4 | — | 4 | 1 | no | — |
| 417 | C22×C4 | — | 8 | 1 | no | — |
| 418 | C22×C4 | — | 8 | 1 | no | — |
| 419 | C22×C4 | — | 8 | 4 | no | — |
| 420 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 421 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 422 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 423 | Q16 | — | 8 | 2 | no | — |
| 424 | Q16 | — | 8 | 2 | no | — |
| 425 | SD16 | — | 4 | 2 | no | — |
| 426 | SD16 | — | 8 | 2 | no | — |
| 427 | SD16 | — | 8 | 1 | no | — |
| 428 | SD16 | — | 8 | 2 | no | — |
| 429 | SD16 | — | 8 | 2 | no | — |
| 430 | SD16 | — | 8 | 2 | no | — |
| 431 | SD16 | — | 8 | 2 | no | — |
| 432 | D8 | — | 8 | 2 | no | — |
| 433 | D8 | — | 16 | 2 | no | — |
| 434 | M16 | — | 2 | 1 | no | — |
| 435 | M16 | — | 2 | 1 | no | — |
| 436 | M16 | — | 2 | 1 | no | — |
| 437 | M16 | — | 2 | 1 | no | — |
| 438 | M16 | — | 2 | 1 | no | — |
| 439 | M16 | — | 2 | 1 | no | — |
| 440 | M16 | — | 4 | 1 | no | — |
| 441 | M16 | — | 4 | 1 | no | — |
| 442 | M16 | — | 4 | 1 | no | — |
| 443 | M16 | — | 4 | 1 | no | — |
| 444 | M16 | — | 4 | 1 | no | — |
| 445 | M16 | — | 4 | 1 | no | — |
| 446 | M16 | — | 8 | 2 | no | — |
| 447 | M16 | #135 | 1 | 2 | yes | — |
| 448 | M16 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 449 | C2×C8 | — | 2 | 1 | no | — |
| 450 | C2×C8 | — | 2 | 1 | no | — |
| 451 | C2×C8 | — | 2 | 1 | no | — |
| 452 | C2×C8 | — | 2 | 1 | no | — |
| 453 | C2×C8 | — | 2 | 1 | no | — |
| 454 | C2×C8 | — | 2 | 1 | no | — |
| 455 | C2×C8 | — | 2 | 1 | no | — |
| 456 | C2×C8 | — | 2 | 1 | no | — |
| 457 | C2×C8 | — | 2 | 1 | no | — |
| 458 | C2×C8 | — | 2 | 1 | no | — |
| 459 | C2×C8 | — | 4 | 1 | no | — |
| 460 | C2×C8 | — | 4 | 1 | no | — |
| 461 | C2×C8 | — | 4 | 1 | no | — |
| 462 | C2×C8 | — | 4 | 1 | no | — |
| 463 | C2×C8 | — | 4 | 1 | no | — |
| 464 | C2×C8 | — | 4 | 1 | no | — |
| 465 | C2×C8 | — | 4 | 1 | no | — |
| 466 | C2×C8 | — | 8 | 2 | no | — |
| 467 | C2×C8 | — | 8 | 1 | no | — |
| 468 | C2×C8 | — | 8 | 1 | no | — |
| 469 | C4⋊C4 | — | 2 | 1 | no | — |
| 470 | C4⋊C4 | — | 2 | 1 | no | — |
| 471 | C4⋊C4 | — | 2 | 1 | no | — |
| 472 | C4⋊C4 | — | 2 | 1 | no | — |
| 473 | C4⋊C4 | — | 2 | 1 | no | — |
| 474 | C4⋊C4 | — | 2 | 1 | no | — |
| 475 | C4⋊C4 | — | 2 | 1 | no | — |
| 476 | C4⋊C4 | — | 2 | 1 | no | — |
| 477 | C4⋊C4 | — | 2 | 1 | no | — |
| 478 | C4⋊C4 | — | 4 | 1 | no | — |
| 479 | C4⋊C4 | — | 4 | 1 | no | — |
| 480 | C4⋊C4 | — | 4 | 1 | no | — |
| 481 | C4⋊C4 | — | 4 | 1 | no | — |
| 482 | C4⋊C4 | #121 | 1 | 1 | yes | characteristic |
| 483 | C4⋊C4 | #121 | 1 | 1 | yes | characteristic |
| 484 | C22⋊C4 | — | 4 | 2 | no | — |
| 485 | C22⋊C4 | — | 4 | 2 | no | — |
| 486 | C22⋊C4 | — | 4 | 1 | no | — |
| 487 | C22⋊C4 | — | 4 | 1 | no | — |
| 488 | C22⋊C4 | — | 4 | 1 | no | — |
| 489 | C22⋊C4 | — | 4 | 1 | no | — |
| 490 | C42 | — | 2 | 1 | no | — |
| 491 | C42 | — | 2 | 1 | no | — |
| 492 | C42 | — | 2 | 1 | no | — |
| 493 | C42 | — | 2 | 1 | no | — |
| 494 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 495 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 496 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 497 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 498 | C23 | — | 4 | 1 | no | — |
| 499 | C23 | — | 4 | 2 | no | — |
| 500 | C23 | — | 4 | 1 | no | — |
| 501 | C23 | — | 4 | 1 | no | — |
| 502 | C23 | — | 4 | 2 | no | — |
| 503 | C23 | — | 4 | 1 | no | — |
| 504 | C23 | — | 4 | 2 | no | — |
| 505 | C23 | — | 8 | 1 | no | — |
| 506 | C23 | — | 8 | 2 | no | — |
| 507 | C23 | — | 8 | 2 | no | — |
| 508 | C23 | — | 8 | 1 | no | — |
| 509 | C23 | #525 | 1 | 1 | yes | characteristic |
| 510 | Q8 | — | 2 | 1 | no | — |
| 511 | Q8 | — | 2 | 1 | no | — |
| 512 | Q8 | — | 2 | 1 | no | — |
| 513 | Q8 | — | 4 | 2 | no | — |
| 514 | Q8 | — | 4 | 1 | no | — |
| 515 | Q8 | — | 4 | 1 | no | — |
| 516 | Q8 | — | 8 | 1 | no | — |
| 517 | Q8 | — | 8 | 1 | no | — |
| 518 | Q8 | — | 8 | 1 | no | — |
| 519 | Q8 | — | 8 | 1 | no | — |
| 520 | Q8 | #471 | 1 | 2 | yes | complemented |
| 521 | D4 | — | 4 | 1 | no | — |
| 522 | D4 | — | 4 | 1 | no | — |
| 523 | D4 | — | 4 | 1 | no | — |
| 524 | D4 | — | 4 | 1 | no | — |
| 525 | D4 | — | 4 | 1 | no | — |
| 526 | D4 | — | 4 | 1 | no | — |
| 527 | D4 | — | 4 | 1 | no | — |
| 528 | D4 | — | 4 | 1 | no | — |
| 529 | D4 | — | 8 | 1 | no | — |
| 530 | D4 | — | 8 | 2 | no | — |
| 531 | D4 | — | 8 | 2 | no | — |
| 532 | D4 | — | 8 | 2 | no | — |
| 533 | D4 | — | 8 | 4 | no | — |
| 534 | D4 | — | 8 | 4 | no | — |
| 535 | D4 | — | 8 | 1 | no | — |
| 536 | D4 | — | 8 | 1 | no | — |
| 537 | D4 | — | 8 | 1 | no | — |
| 538 | D4 | — | 8 | 1 | no | — |
| 539 | D4 | — | 8 | 1 | no | — |
| 540 | D4 | — | 8 | 1 | no | — |
| 541 | D4 | — | 8 | 1 | no | — |
| 542 | D4 | — | 8 | 1 | no | — |
| 543 | D4 | — | 16 | 2 | no | — |
| 544 | D4 | — | 16 | 2 | no | — |
| 545 | C2×C4 | — | 2 | 1 | no | — |
| 546 | C2×C4 | — | 2 | 1 | no | — |
| 547 | C2×C4 | — | 2 | 1 | no | — |
| 548 | C2×C4 | — | 2 | 1 | no | — |
| 549 | C2×C4 | — | 2 | 1 | no | — |
| 550 | C2×C4 | — | 2 | 1 | no | — |
| 551 | C2×C4 | — | 4 | 1 | no | — |
| 552 | C2×C4 | — | 4 | 2 | no | — |
| 553 | C2×C4 | — | 4 | 1 | no | — |
| 554 | C2×C4 | — | 4 | 1 | no | — |
| 555 | C2×C4 | — | 8 | 2 | no | — |
| 556 | C2×C4 | — | 8 | 4 | no | — |
| 557 | C2×C4 | — | 8 | 1 | no | — |
| 558 | C2×C4 | — | 8 | 4 | no | — |
| 559 | C2×C4 | — | 8 | 1 | no | — |
| 560 | C2×C4 | — | 8 | 1 | no | — |
| 561 | C2×C4 | — | 8 | 1 | no | — |
| 562 | C2×C4 | — | 8 | 1 | no | — |
| 563 | C2×C4 | — | 8 | 1 | no | — |
| 564 | C2×C4 | — | 8 | 1 | no | — |
| 565 | C2×C4 | — | 8 | 1 | no | — |
| 566 | C2×C4 | — | 8 | 1 | no | — |
| 567 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 568 | C2×C4 | #538 | 1 | 1 | yes | characteristic |
| 569 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 570 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 571 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 572 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 573 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 574 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 575 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 576 | C2×C4 | #577 | 1 | 1 | yes | characteristic |
| 577 | C8 | — | 2 | 2 | no | — |
| 578 | C8 | — | 2 | 2 | no | — |
| 579 | C8 | — | 4 | 1 | no | — |
| 580 | C8 | — | 4 | 1 | no | — |
| 581 | C8 | — | 4 | 1 | no | — |
| 582 | C8 | — | 4 | 1 | no | — |
| 583 | C8 | — | 4 | 1 | no | — |
| 584 | C8 | — | 4 | 1 | no | — |
| 585 | C8 | — | 4 | 1 | no | — |
| 586 | C8 | — | 4 | 1 | no | — |
| 587 | C8 | — | 8 | 1 | no | — |
| 588 | C8 | — | 8 | 1 | no | — |
| 589 | C8 | — | 8 | 1 | no | — |
| 590 | C8 | — | 8 | 1 | no | — |
| 591 | C8 | — | 8 | 1 | no | — |
| 592 | C22 | — | 2 | 1 | no | — |
| 593 | C22 | — | 2 | 1 | no | — |
| 594 | C22 | — | 8 | 1 | no | — |
| 595 | C22 | — | 8 | 2 | no | — |
| 596 | C22 | — | 8 | 1 | no | — |
| 597 | C22 | — | 8 | 1 | no | — |
| 598 | C22 | — | 8 | 1 | no | — |
| 599 | C22 | — | 8 | 2 | no | — |
| 600 | C22 | — | 8 | 2 | no | — |
| 601 | C22 | — | 8 | 1 | no | — |
| 602 | C22 | — | 8 | 1 | no | — |
| 603 | C22 | — | 16 | 1 | no | — |
| 604 | C22 | — | 16 | 2 | no | — |
| 605 | C22 | — | 16 | 1 | no | — |
| 606 | C22 | — | 16 | 2 | no | — |
| 607 | C22 | #2424 | 1 | 1 | yes | characteristic |
| 608 | C22 | #3106 | 1 | 1 | yes | characteristic |
| 609 | C22 | #3119 | 1 | 1 | yes | characteristic |
| 610 | C4 | — | 2 | 1 | no | — |
| 611 | C4 | — | 2 | 2 | no | — |
| 612 | C4 | — | 2 | 1 | no | — |
| 613 | C4 | — | 2 | 1 | no | — |
| 614 | C4 | — | 4 | 1 | no | — |
| 615 | C4 | — | 4 | 1 | no | — |
| 616 | C4 | — | 4 | 1 | no | — |
| 617 | C4 | — | 8 | 1 | no | — |
| 618 | C4 | — | 8 | 1 | no | — |
| 619 | C4 | — | 16 | 1 | no | — |
| 620 | C4 | C2×#471 | 1 | 1 | yes | characteristic |
| 621 | C4 | #3068 | 1 | 1 | yes | characteristic |
| 622 | C4 | #3265 | 1 | 1 | yes | characteristic |
| 623 | C4 | #3274 | 1 | 1 | yes | characteristic |
| 624 | C4 | #3282 | 1 | 1 | yes | characteristic |
| 625 | C4 | #3287 | 1 | 1 | yes | characteristic |
| 626 | C2 | — | 2 | 1 | no | — |
| 627 | C2 | — | 2 | 1 | no | — |
| 628 | C2 | — | 16 | 1 | no | — |
| 629 | C2 | — | 16 | 1 | no | — |
| 630 | C2 | — | 16 | 2 | no | — |
| 631 | C2 | #25144 | 1 | 1 | yes | characteristic |
| 632 | C2 | #25159 | 1 | 1 | yes | characteristic |
| 633 | C2 | #25199 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #18770 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #25405 | C2 | semidirect |
| 2. #22536 | C2 | semidirect |
| 3. #19171 | C2 | semidirect |
| 5. C2×#1642 | C2 | semidirect |
| 6. #17997 | C2 | semidirect |
| 7. #10607 | C2 | semidirect |
| 8. #10456 | C2 | semidirect |
| 9. #10445 | C2 | semidirect |
| 10. #10168 | C2 | semidirect |
| 11. #9796 | C2 | semidirect |
| 12. #9550 | C2 | semidirect |
| 13. #9273 | C2 | semidirect |
| 14. #9252 | C2 | semidirect |
| 15. #8907 | C2 | semidirect |
| 520. Q8 | #471 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #25405 | 2 |
| #22536 | 2 |
| #19171 | 2 |
| #18770 | 2 |
| C2×#1642 | 2 |
| #17997 | 2 |
| #10607 | 2 |
| #10456 | 2 |
| #10445 | 2 |
| #10168 | 2 |
| #9796 | 2 |
| #9550 | 2 |
| #9273 | 2 |
| #9252 | 2 |
| #8907 | 2 |