Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
744 conjugacy classes of subgroups.
The subgroup lattice has 627 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#2222 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33612 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#2193 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #33525 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #28503 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #24290 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #13393 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13388 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #13332 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #13329 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #13321 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #13308 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | #13135 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | #13134 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | #12980 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C2×#578 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#510 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#509 | C22 | 1 | 1 | yes | characteristic |
| 19 | C22×Q32 | C22 | 1 | 1 | yes | characteristic |
| 20 | C22×SD32 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#504 | C22 | 1 | 1 | yes | characteristic |
| 22 | #3154 | — | 2 | 1 | no | — |
| 23 | C2×#495 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×#473 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#473 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#472 | C22 | 1 | 1 | yes | characteristic |
| 27 | #3061 | C22 | 1 | 1 | yes | characteristic |
| 28 | #3059 | C22 | 1 | 1 | yes | characteristic |
| 29 | #3058 | C22 | 1 | 1 | yes | characteristic |
| 30 | #3013 | — | 2 | 1 | no | — |
| 31 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 32 | #2222 | — | 2 | 4 | no | — |
| 33 | #2220 | — | 2 | 1 | no | — |
| 34 | #2218 | — | 2 | 1 | no | — |
| 35 | #2217 | — | 2 | 1 | no | — |
| 36 | #2215 | — | 2 | 1 | no | — |
| 37 | #2193 | — | 2 | 4 | no | — |
| 38 | #2189 | — | 2 | 1 | no | — |
| 39 | #2189 | — | 2 | 1 | no | — |
| 40 | #2188 | — | 2 | 1 | no | — |
| 41 | #2188 | — | 2 | 1 | no | — |
| 42 | C2×#368 | C22 | 1 | 1 | yes | characteristic |
| 43 | #2159 | C22 | 1 | 1 | yes | characteristic |
| 44 | C2×#362 | C22 | 1 | 1 | yes | characteristic |
| 45 | C2×#362 | C22 | 1 | 1 | yes | characteristic |
| 46 | C2×#361 | C22 | 1 | 1 | yes | characteristic |
| 47 | C2×#359 | C22 | 1 | 1 | yes | characteristic |
| 48 | #2141 | C22 | 1 | 1 | yes | characteristic |
| 49 | #2138 | C22 | 1 | 1 | yes | characteristic |
| 50 | C2×#358 | C22 | 1 | 1 | yes | characteristic |
| 51 | C2×#357 | C22 | 1 | 1 | yes | characteristic |
| 52 | C2×#350 | C22 | 1 | 1 | yes | characteristic |
| 53 | C2×#350 | C22 | 1 | 1 | yes | characteristic |
| 54 | #2113 | C22 | 1 | 1 | yes | characteristic |
| 55 | #2052 | — | 2 | 1 | no | — |
| 56 | #2007 | — | 2 | 1 | no | — |
| 57 | #2004 | — | 2 | 1 | no | — |
| 58 | #2003 | — | 2 | 1 | no | — |
| 59 | #1977 | — | 2 | 1 | no | — |
| 60 | #1974 | C22 | 1 | 1 | yes | characteristic |
| 61 | #1969 | C22 | 1 | 1 | yes | characteristic |
| 62 | #1816 | C22 | 1 | 1 | yes | characteristic |
| 63 | #1793 | — | 2 | 1 | no | — |
| 64 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 65 | #1387 | C22 | 1 | 1 | yes | characteristic |
| 66 | #1382 | C22 | 1 | 1 | yes | characteristic |
| 67 | #1376 | C22 | 1 | 1 | yes | characteristic |
| 68 | #1374 | C22 | 1 | 1 | yes | characteristic |
| 69 | C2×2−1+4 | — | 2 | 1 | no | — |
| 70 | #578 | — | 2 | 2 | no | — |
| 71 | #578 | — | 4 | 1 | no | — |
| 72 | #578 | — | 4 | 1 | no | — |
| 73 | #578 | — | 4 | 1 | no | — |
| 74 | C2×#137 | — | 2 | 1 | no | — |
| 75 | C2×#137 | — | 2 | 1 | no | — |
| 76 | C2×#137 | — | 2 | 1 | no | — |
| 77 | C2×#137 | — | 4 | 1 | no | — |
| 78 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 79 | C2×#135 | — | 2 | 1 | no | — |
| 80 | C2×#135 | D4 | 1 | 1 | yes | characteristic |
| 81 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 82 | C22×Q16 | — | 2 | 1 | no | — |
| 83 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 84 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 85 | C22×SD16 | — | 2 | 1 | no | — |
| 86 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 87 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 88 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 89 | #537 | — | 2 | 1 | no | — |
| 90 | #536 | — | 2 | 1 | no | — |
| 91 | #535 | — | 2 | 1 | no | — |
| 92 | C2×#124 | — | 2 | 1 | no | — |
| 93 | #510 | — | 4 | 1 | no | — |
| 94 | #510 | — | 4 | 1 | no | — |
| 95 | #509 | — | 2 | 2 | no | — |
| 96 | #509 | — | 4 | 1 | no | — |
| 97 | C2×Q32 | — | 2 | 2 | no | — |
| 98 | C2×Q32 | — | 2 | 1 | no | — |
| 99 | C2×Q32 | — | 2 | 1 | no | — |
| 100 | C2×Q32 | — | 2 | 1 | no | — |
| 101 | C2×Q32 | — | 4 | 1 | no | — |
| 102 | C2×SD32 | — | 2 | 2 | no | — |
| 103 | C2×SD32 | — | 2 | 1 | no | — |
| 104 | C2×SD32 | — | 2 | 1 | no | — |
| 105 | C2×SD32 | — | 4 | 1 | no | — |
| 106 | C2×SD32 | — | 4 | 1 | no | — |
| 107 | C2×D16 | — | 2 | 1 | no | — |
| 108 | #504 | — | 2 | 2 | no | — |
| 109 | #504 | — | 4 | 1 | no | — |
| 110 | C2×M32 | — | 2 | 1 | no | — |
| 111 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 112 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 113 | C22×C16 | — | 2 | 1 | no | — |
| 114 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 115 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 116 | #497 | — | 4 | 1 | no | — |
| 117 | #496 | — | 4 | 1 | no | — |
| 118 | #495 | — | 4 | 1 | no | — |
| 119 | #495 | — | 4 | 1 | no | — |
| 120 | #495 | — | 4 | 1 | no | — |
| 121 | #492 | — | 4 | 1 | no | — |
| 122 | #473 | — | 2 | 2 | no | — |
| 123 | #473 | — | 2 | 2 | no | — |
| 124 | #473 | — | 4 | 1 | no | — |
| 125 | #473 | — | 4 | 1 | no | — |
| 126 | #472 | — | 2 | 2 | no | — |
| 127 | #472 | — | 4 | 1 | no | — |
| 128 | #470 | — | 4 | 1 | no | — |
| 129 | #469 | — | 4 | 1 | no | — |
| 130 | #468 | — | 4 | 1 | no | — |
| 131 | #467 | — | 2 | 1 | no | — |
| 132 | #467 | — | 2 | 1 | no | — |
| 133 | #467 | — | 2 | 1 | no | — |
| 134 | #466 | — | 2 | 1 | no | — |
| 135 | #466 | — | 2 | 1 | no | — |
| 136 | #466 | — | 2 | 1 | no | — |
| 137 | #452 | — | 4 | 1 | no | — |
| 138 | #451 | — | 4 | 1 | no | — |
| 139 | #449 | — | 4 | 1 | no | — |
| 140 | #448 | — | 4 | 1 | no | — |
| 141 | #443 | — | 4 | 1 | no | — |
| 142 | #443 | — | 4 | 1 | no | — |
| 143 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 144 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 145 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 146 | #427 | C23 | 1 | 1 | yes | characteristic |
| 147 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 148 | C2×#104 | — | 2 | 1 | no | — |
| 149 | C2×#104 | D4 | 1 | 1 | yes | characteristic |
| 150 | C2×#104 | D4 | 1 | 1 | yes | characteristic |
| 151 | #419 | — | 2 | 1 | no | — |
| 152 | #418 | — | 2 | 1 | no | — |
| 153 | #417 | — | 2 | 1 | no | — |
| 154 | #417 | — | 2 | 1 | no | — |
| 155 | C2×#103 | — | 2 | 1 | no | — |
| 156 | C2×#101 | — | 2 | 1 | no | — |
| 157 | C2×#101 | — | 2 | 1 | no | — |
| 158 | C2×#100 | — | 2 | 1 | no | — |
| 159 | #408 | — | 2 | 1 | no | — |
| 160 | #405 | — | 2 | 1 | no | — |
| 161 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 162 | #391 | — | 2 | 1 | no | — |
| 163 | #368 | — | 2 | 2 | no | — |
| 164 | #367 | — | 2 | 1 | no | — |
| 165 | #366 | — | 2 | 1 | no | — |
| 166 | #362 | — | 2 | 2 | no | — |
| 167 | #362 | D4 | 1 | 4 | yes | complemented |
| 168 | #361 | — | 4 | 1 | no | — |
| 169 | #359 | — | 2 | 2 | no | — |
| 170 | #358 | — | 2 | 1 | no | — |
| 171 | #358 | — | 2 | 1 | no | — |
| 172 | #358 | — | 4 | 1 | no | — |
| 173 | #358 | D4 | 1 | 1 | yes | characteristic |
| 174 | #358 | D4 | 1 | 1 | yes | characteristic |
| 175 | #357 | — | 2 | 1 | no | — |
| 176 | #357 | — | 2 | 1 | no | — |
| 177 | #357 | D4 | 1 | 1 | yes | characteristic |
| 178 | #357 | D4 | 1 | 1 | yes | characteristic |
| 179 | #350 | — | 2 | 2 | no | — |
| 180 | #350 | — | 4 | 1 | no | — |
| 181 | C22⋊C16 | — | 2 | 1 | no | — |
| 182 | C22⋊C16 | — | 2 | 1 | no | — |
| 183 | #343 | — | 2 | 1 | no | — |
| 184 | #339 | — | 2 | 1 | no | — |
| 185 | 2−1+4 | — | 4 | 2 | no | — |
| 186 | 2−1+4 | — | 4 | 2 | no | — |
| 187 | 2−1+4 | — | 4 | 2 | no | — |
| 188 | 2−1+4 | — | 8 | 1 | no | — |
| 189 | C2×#41 | — | 2 | 1 | no | — |
| 190 | C2×#41 | — | 4 | 1 | no | — |
| 191 | C2×#41 | — | 4 | 1 | no | — |
| 192 | C2×#41 | — | 8 | 1 | no | — |
| 193 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 194 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 195 | C22×Q8 | — | 2 | 1 | no | — |
| 196 | C22×Q8 | — | 2 | 1 | no | — |
| 197 | C22×Q8 | — | 4 | 1 | no | — |
| 198 | C22×Q8 | — | 4 | 1 | no | — |
| 199 | C22×D4 | — | 2 | 1 | no | — |
| 200 | #137 | — | 4 | 2 | no | — |
| 201 | #137 | — | 4 | 2 | no | — |
| 202 | #137 | — | 4 | 2 | no | — |
| 203 | #137 | — | 4 | 4 | no | — |
| 204 | #137 | — | 4 | 4 | no | — |
| 205 | #137 | — | 4 | 2 | no | — |
| 206 | #137 | — | 8 | 1 | no | — |
| 207 | #137 | — | 8 | 1 | no | — |
| 208 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 209 | AGL(1,ℤ/8) | — | 8 | 1 | no | — |
| 210 | #135 | — | 2 | 1 | no | — |
| 211 | #135 | — | 2 | 1 | no | — |
| 212 | #135 | — | 2 | 1 | no | — |
| 213 | #135 | — | 2 | 1 | no | — |
| 214 | #135 | — | 4 | 1 | no | — |
| 215 | #135 | — | 4 | 1 | no | — |
| 216 | #135 | — | 4 | 2 | no | — |
| 217 | #135 | — | 4 | 2 | no | — |
| 218 | C2×Q16 | — | 2 | 1 | no | — |
| 219 | C2×Q16 | — | 2 | 1 | no | — |
| 220 | C2×Q16 | — | 2 | 1 | no | — |
| 221 | C2×Q16 | — | 2 | 1 | no | — |
| 222 | C2×Q16 | — | 2 | 1 | no | — |
| 223 | C2×Q16 | — | 2 | 1 | no | — |
| 224 | C2×Q16 | — | 2 | 1 | no | — |
| 225 | C2×Q16 | — | 2 | 1 | no | — |
| 226 | C2×Q16 | — | 2 | 1 | no | — |
| 227 | C2×Q16 | — | 2 | 1 | no | — |
| 228 | C2×Q16 | — | 2 | 1 | no | — |
| 229 | C2×Q16 | — | 4 | 1 | no | — |
| 230 | C2×Q16 | — | 4 | 2 | no | — |
| 231 | C2×Q16 | — | 4 | 2 | no | — |
| 232 | C2×Q16 | — | 4 | 1 | no | — |
| 233 | C2×Q16 | — | 4 | 1 | no | — |
| 234 | C2×Q16 | D8 | 1 | 1 | yes | characteristic |
| 235 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 236 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 237 | C2×SD16 | — | 2 | 1 | no | — |
| 238 | C2×SD16 | — | 4 | 1 | no | — |
| 239 | C2×SD16 | — | 4 | 2 | no | — |
| 240 | C2×SD16 | — | 4 | 2 | no | — |
| 241 | C2×SD16 | — | 4 | 1 | no | — |
| 242 | C2×SD16 | — | 4 | 1 | no | — |
| 243 | C2×SD16 | — | 4 | 1 | no | — |
| 244 | C2×SD16 | — | 4 | 1 | no | — |
| 245 | C2×SD16 | — | 4 | 1 | no | — |
| 246 | C2×SD16 | D8 | 1 | 1 | yes | characteristic |
| 247 | C2×SD16 | D8 | 1 | 1 | yes | characteristic |
| 248 | C2×D8 | — | 2 | 1 | no | — |
| 249 | C2×D8 | — | 2 | 1 | no | — |
| 250 | C2×D8 | — | 2 | 1 | no | — |
| 251 | C2×D8 | — | 2 | 1 | no | — |
| 252 | C2×D8 | — | 2 | 1 | no | — |
| 253 | C2×D8 | — | 2 | 1 | no | — |
| 254 | C2×D8 | — | 4 | 1 | no | — |
| 255 | C2×D8 | D8 | 1 | 1 | yes | characteristic |
| 256 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 257 | #131 | — | 2 | 1 | no | — |
| 258 | #131 | — | 2 | 1 | no | — |
| 259 | #131 | — | 2 | 1 | no | — |
| 260 | #131 | — | 2 | 1 | no | — |
| 261 | #131 | — | 2 | 1 | no | — |
| 262 | #131 | — | 2 | 1 | no | — |
| 263 | #131 | — | 2 | 1 | no | — |
| 264 | #131 | — | 2 | 1 | no | — |
| 265 | C2×M16 | — | 2 | 1 | no | — |
| 266 | C2×M16 | — | 2 | 1 | no | — |
| 267 | C2×M16 | — | 2 | 1 | no | — |
| 268 | C2×M16 | — | 2 | 1 | no | — |
| 269 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 270 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 271 | C22×C8 | — | 2 | 1 | no | — |
| 272 | C22×C8 | — | 2 | 1 | no | — |
| 273 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 274 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 275 | C4⋊Q8 | — | 4 | 1 | no | — |
| 276 | #127 | — | 4 | 1 | no | — |
| 277 | #124 | — | 4 | 2 | no | — |
| 278 | #124 | — | 4 | 1 | no | — |
| 279 | #124 | — | 4 | 1 | no | — |
| 280 | #124 | — | 8 | 1 | no | — |
| 281 | #123 | — | 8 | 1 | no | — |
| 282 | C22⋊Q8 | — | 4 | 1 | no | — |
| 283 | C22⋊Q8 | — | 4 | 1 | no | — |
| 284 | C22⋊Q8 | — | 4 | 1 | no | — |
| 285 | C22⋊Q8 | — | 4 | 1 | no | — |
| 286 | #121 | — | 4 | 1 | no | — |
| 287 | #121 | — | 4 | 1 | no | — |
| 288 | #121 | — | 4 | 1 | no | — |
| 289 | #121 | — | 4 | 1 | no | — |
| 290 | #121 | — | 8 | 1 | no | — |
| 291 | #120 | — | 8 | 1 | no | — |
| 292 | #117 | — | 2 | 1 | no | — |
| 293 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 294 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 295 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 296 | Q32 | — | 4 | 2 | no | — |
| 297 | Q32 | — | 4 | 2 | no | — |
| 298 | Q32 | — | 4 | 2 | no | — |
| 299 | SD32 | — | 4 | 2 | no | — |
| 300 | SD32 | — | 4 | 2 | no | — |
| 301 | SD32 | — | 4 | 4 | no | — |
| 302 | D16 | — | 4 | 2 | no | — |
| 303 | M32 | — | 2 | 1 | no | — |
| 304 | M32 | — | 2 | 1 | no | — |
| 305 | M32 | — | 2 | 1 | no | — |
| 306 | M32 | — | 4 | 2 | no | — |
| 307 | M32 | C2×D4 | 1 | 2 | yes | — |
| 308 | C2×C16 | — | 2 | 1 | no | — |
| 309 | C2×C16 | — | 2 | 1 | no | — |
| 310 | C2×C16 | — | 2 | 1 | no | — |
| 311 | C2×C16 | — | 2 | 1 | no | — |
| 312 | C2×C16 | — | 2 | 1 | no | — |
| 313 | C2×C16 | — | 2 | 1 | no | — |
| 314 | C2×C16 | — | 2 | 1 | no | — |
| 315 | C2×C16 | — | 4 | 2 | no | — |
| 316 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 317 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 318 | #108 | — | 2 | 1 | no | — |
| 319 | #108 | — | 2 | 1 | no | — |
| 320 | #108 | — | 2 | 1 | no | — |
| 321 | #108 | — | 4 | 1 | no | — |
| 322 | #108 | C2×D4 | 1 | 2 | yes | — |
| 323 | #107 | — | 2 | 1 | no | — |
| 324 | #107 | — | 2 | 1 | no | — |
| 325 | #107 | C2×D4 | 1 | 1 | yes | characteristic |
| 326 | #107 | C2×D4 | 1 | 1 | yes | characteristic |
| 327 | #106 | — | 2 | 1 | no | — |
| 328 | #104 | — | 4 | 1 | no | — |
| 329 | #104 | — | 4 | 1 | no | — |
| 330 | #104 | — | 8 | 1 | no | — |
| 331 | #103 | — | 4 | 1 | no | — |
| 332 | #103 | — | 4 | 1 | no | — |
| 333 | #103 | — | 4 | 1 | no | — |
| 334 | #103 | — | 4 | 1 | no | — |
| 335 | #103 | — | 4 | 1 | no | — |
| 336 | #103 | — | 4 | 1 | no | — |
| 337 | #102 | — | 4 | 1 | no | — |
| 338 | #102 | — | 4 | 1 | no | — |
| 339 | #102 | — | 4 | 1 | no | — |
| 340 | #102 | — | 4 | 1 | no | — |
| 341 | #101 | — | 4 | 2 | no | — |
| 342 | #101 | — | 4 | 2 | no | — |
| 343 | #100 | — | 4 | 2 | no | — |
| 344 | C22⋊C8 | — | 4 | 1 | no | — |
| 345 | C22⋊C8 | — | 4 | 1 | no | — |
| 346 | #97 | — | 4 | 1 | no | — |
| 347 | C4×C8 | — | 2 | 1 | no | — |
| 348 | C4×C8 | — | 2 | 1 | no | — |
| 349 | C4×C8 | — | 4 | 1 | no | — |
| 350 | #95 | — | 8 | 1 | no | — |
| 351 | C24 | — | 4 | 1 | no | — |
| 352 | #41 | — | 2 | 1 | no | — |
| 353 | #41 | — | 2 | 1 | no | — |
| 354 | #41 | — | 2 | 1 | no | — |
| 355 | #41 | — | 2 | 1 | no | — |
| 356 | #41 | — | 2 | 1 | no | — |
| 357 | #41 | — | 2 | 1 | no | — |
| 358 | #41 | — | 2 | 1 | no | — |
| 359 | #41 | — | 2 | 1 | no | — |
| 360 | #41 | — | 4 | 4 | no | — |
| 361 | #41 | — | 4 | 1 | no | — |
| 362 | #41 | — | 4 | 1 | no | — |
| 363 | #41 | — | 4 | 1 | no | — |
| 364 | #41 | — | 4 | 1 | no | — |
| 365 | #41 | — | 8 | 2 | no | — |
| 366 | #41 | — | 8 | 4 | no | — |
| 367 | #41 | — | 8 | 2 | no | — |
| 368 | #41 | — | 8 | 2 | no | — |
| 369 | #41 | — | 8 | 4 | no | — |
| 370 | C2×Q8 | — | 2 | 2 | no | — |
| 371 | C2×Q8 | — | 2 | 1 | no | — |
| 372 | C2×Q8 | — | 4 | 1 | no | — |
| 373 | C2×Q8 | — | 4 | 4 | no | — |
| 374 | C2×Q8 | — | 4 | 1 | no | — |
| 375 | C2×Q8 | — | 4 | 1 | no | — |
| 376 | C2×Q8 | — | 4 | 1 | no | — |
| 377 | C2×Q8 | — | 4 | 1 | no | — |
| 378 | C2×Q8 | — | 4 | 1 | no | — |
| 379 | C2×Q8 | — | 4 | 1 | no | — |
| 380 | C2×Q8 | — | 4 | 1 | no | — |
| 381 | C2×Q8 | — | 4 | 1 | no | — |
| 382 | C2×Q8 | — | 4 | 1 | no | — |
| 383 | C2×Q8 | — | 4 | 1 | no | — |
| 384 | C2×Q8 | — | 8 | 1 | no | — |
| 385 | C2×Q8 | — | 8 | 2 | no | — |
| 386 | C2×Q8 | — | 8 | 2 | no | — |
| 387 | C2×Q8 | — | 8 | 2 | no | — |
| 388 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 389 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 390 | C2×D4 | — | 2 | 1 | no | — |
| 391 | C2×D4 | — | 2 | 1 | no | — |
| 392 | C2×D4 | — | 2 | 1 | no | — |
| 393 | C2×D4 | — | 4 | 1 | 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 | — | 8 | 1 | no | — |
| 401 | C2×D4 | — | 8 | 1 | no | — |
| 402 | C2×D4 | — | 8 | 1 | no | — |
| 403 | C2×D4 | — | 8 | 1 | no | — |
| 404 | C2×D4 | C2×D8 | 1 | 1 | yes | characteristic |
| 405 | C2×D4 | C2×D8 | 1 | 1 | yes | characteristic |
| 406 | C22×C4 | — | 2 | 1 | no | — |
| 407 | C22×C4 | — | 2 | 1 | no | — |
| 408 | C22×C4 | — | 2 | 1 | no | — |
| 409 | C22×C4 | — | 4 | 1 | no | — |
| 410 | C22×C4 | — | 4 | 1 | no | — |
| 411 | C22×C4 | — | 4 | 1 | no | — |
| 412 | C22×C4 | — | 4 | 1 | no | — |
| 413 | C22×C4 | — | 8 | 1 | no | — |
| 414 | C22×C4 | — | 8 | 1 | no | — |
| 415 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 416 | Q16 | — | 2 | 2 | no | — |
| 417 | Q16 | — | 2 | 1 | no | — |
| 418 | Q16 | — | 2 | 1 | no | — |
| 419 | Q16 | — | 2 | 2 | no | — |
| 420 | Q16 | — | 4 | 1 | no | — |
| 421 | Q16 | — | 4 | 4 | no | — |
| 422 | Q16 | — | 4 | 1 | no | — |
| 423 | Q16 | — | 4 | 1 | no | — |
| 424 | Q16 | — | 4 | 1 | no | — |
| 425 | Q16 | — | 4 | 1 | no | — |
| 426 | Q16 | — | 4 | 1 | no | — |
| 427 | Q16 | — | 4 | 1 | no | — |
| 428 | Q16 | — | 8 | 2 | no | — |
| 429 | Q16 | — | 8 | 2 | no | — |
| 430 | SD16 | — | 4 | 1 | no | — |
| 431 | SD16 | — | 4 | 1 | no | — |
| 432 | SD16 | — | 4 | 4 | no | — |
| 433 | SD16 | — | 4 | 1 | no | — |
| 434 | SD16 | — | 4 | 1 | no | — |
| 435 | SD16 | — | 8 | 2 | no | — |
| 436 | SD16 | — | 8 | 2 | no | — |
| 437 | SD16 | — | 8 | 2 | no | — |
| 438 | SD16 | — | 8 | 2 | no | — |
| 439 | SD16 | — | 8 | 2 | no | — |
| 440 | D8 | — | 2 | 1 | no | — |
| 441 | D8 | — | 2 | 1 | no | — |
| 442 | D8 | — | 4 | 1 | no | — |
| 443 | D8 | — | 4 | 1 | no | — |
| 444 | D8 | — | 4 | 1 | no | — |
| 445 | D8 | — | 4 | 1 | no | — |
| 446 | D8 | — | 4 | 1 | no | — |
| 447 | D8 | — | 8 | 2 | no | — |
| 448 | M16 | — | 2 | 1 | no | — |
| 449 | M16 | — | 2 | 1 | no | — |
| 450 | M16 | — | 2 | 1 | no | — |
| 451 | M16 | — | 2 | 2 | no | — |
| 452 | M16 | — | 2 | 1 | no | — |
| 453 | M16 | — | 4 | 1 | no | — |
| 454 | M16 | — | 4 | 1 | no | — |
| 455 | M16 | — | 4 | 1 | no | — |
| 456 | M16 | — | 4 | 1 | no | — |
| 457 | M16 | — | 4 | 1 | no | — |
| 458 | M16 | — | 4 | 1 | no | — |
| 459 | M16 | — | 4 | 1 | no | — |
| 460 | C2×C8 | — | 2 | 1 | no | — |
| 461 | C2×C8 | — | 2 | 1 | no | — |
| 462 | C2×C8 | — | 2 | 1 | no | — |
| 463 | C2×C8 | — | 2 | 1 | no | — |
| 464 | C2×C8 | — | 4 | 1 | no | — |
| 465 | C2×C8 | — | 4 | 1 | no | — |
| 466 | C2×C8 | — | 4 | 1 | no | — |
| 467 | C2×C8 | — | 4 | 1 | no | — |
| 468 | C2×C8 | — | 4 | 1 | no | — |
| 469 | C2×C8 | — | 4 | 1 | no | — |
| 470 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 471 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 472 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 473 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 474 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 475 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 476 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 477 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 478 | C4⋊C4 | — | 2 | 1 | no | — |
| 479 | C4⋊C4 | — | 2 | 1 | no | — |
| 480 | C4⋊C4 | — | 4 | 1 | no | — |
| 481 | C4⋊C4 | — | 4 | 1 | no | — |
| 482 | C4⋊C4 | — | 8 | 1 | no | — |
| 483 | C4⋊C4 | — | 8 | 1 | no | — |
| 484 | C4⋊C4 | — | 8 | 1 | no | — |
| 485 | C22⋊C4 | — | 4 | 1 | no | — |
| 486 | C22⋊C4 | — | 8 | 1 | no | — |
| 487 | C22⋊C4 | — | 8 | 1 | no | — |
| 488 | C22⋊C4 | — | 8 | 1 | no | — |
| 489 | C22⋊C4 | — | 8 | 1 | no | — |
| 490 | C22⋊C4 | — | 8 | 1 | no | — |
| 491 | C22⋊C4 | — | 8 | 1 | no | — |
| 492 | C22⋊C4 | — | 8 | 4 | no | — |
| 493 | C42 | — | 2 | 1 | no | — |
| 494 | C42 | — | 2 | 1 | no | — |
| 495 | C42 | — | 4 | 1 | no | — |
| 496 | C16 | — | 2 | 1 | no | — |
| 497 | C16 | — | 2 | 1 | no | — |
| 498 | C16 | — | 2 | 2 | no | — |
| 499 | C16 | — | 4 | 1 | no | — |
| 500 | C16 | — | 4 | 1 | no | — |
| 501 | C23 | — | 2 | 1 | no | — |
| 502 | C23 | — | 2 | 1 | no | — |
| 503 | C23 | — | 4 | 1 | no | — |
| 504 | C23 | — | 8 | 1 | no | — |
| 505 | C23 | — | 8 | 1 | no | — |
| 506 | C23 | — | 8 | 1 | no | — |
| 507 | C23 | — | 16 | 1 | no | — |
| 508 | C23 | — | 16 | 1 | no | — |
| 509 | C23 | #576 | 1 | 1 | yes | characteristic |
| 510 | Q8 | — | 2 | 1 | no | — |
| 511 | Q8 | — | 2 | 1 | no | — |
| 512 | Q8 | — | 2 | 1 | no | — |
| 513 | Q8 | — | 2 | 1 | no | — |
| 514 | Q8 | — | 4 | 2 | no | — |
| 515 | Q8 | — | 4 | 1 | no | — |
| 516 | Q8 | — | 4 | 1 | no | — |
| 517 | Q8 | — | 4 | 2 | no | — |
| 518 | Q8 | — | 8 | 4 | no | — |
| 519 | Q8 | — | 8 | 1 | no | — |
| 520 | Q8 | — | 8 | 1 | no | — |
| 521 | Q8 | — | 8 | 1 | no | — |
| 522 | Q8 | — | 8 | 1 | no | — |
| 523 | Q8 | — | 8 | 1 | no | — |
| 524 | Q8 | — | 8 | 1 | no | — |
| 525 | D4 | — | 2 | 1 | no | — |
| 526 | D4 | — | 2 | 1 | no | — |
| 527 | D4 | — | 2 | 1 | no | — |
| 528 | D4 | — | 2 | 1 | no | — |
| 529 | D4 | — | 4 | 1 | no | — |
| 530 | D4 | — | 4 | 1 | no | — |
| 531 | D4 | — | 4 | 1 | no | — |
| 532 | D4 | — | 4 | 1 | no | — |
| 533 | D4 | — | 4 | 1 | no | — |
| 534 | D4 | — | 4 | 1 | no | — |
| 535 | D4 | — | 8 | 4 | 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 | — | 32 | 1 | no | — |
| 543 | D4 | — | 32 | 1 | no | — |
| 544 | D4 | — | 32 | 1 | no | — |
| 545 | C2×C4 | — | 2 | 1 | no | — |
| 546 | C2×C4 | — | 2 | 1 | no | — |
| 547 | C2×C4 | — | 4 | 1 | no | — |
| 548 | C2×C4 | — | 4 | 1 | no | — |
| 549 | C2×C4 | — | 4 | 1 | no | — |
| 550 | C2×C4 | — | 4 | 1 | no | — |
| 551 | C2×C4 | — | 4 | 1 | no | — |
| 552 | C2×C4 | — | 4 | 1 | no | — |
| 553 | C2×C4 | — | 4 | 1 | no | — |
| 554 | C2×C4 | — | 4 | 2 | no | — |
| 555 | C2×C4 | — | 4 | 1 | no | — |
| 556 | C2×C4 | — | 4 | 1 | no | — |
| 557 | C2×C4 | — | 4 | 1 | no | — |
| 558 | C2×C4 | — | 8 | 4 | no | — |
| 559 | C2×C4 | — | 8 | 4 | 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 | — | 8 | 1 | no | — |
| 568 | C2×C4 | — | 16 | 1 | no | — |
| 569 | C2×C4 | #534 | 1 | 1 | yes | characteristic |
| 570 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 571 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 572 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 573 | C2×C4 | #575 | 1 | 1 | yes | characteristic |
| 574 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 575 | C8 | — | 2 | 1 | no | — |
| 576 | C8 | — | 2 | 1 | no | — |
| 577 | C8 | — | 4 | 1 | no | — |
| 578 | C8 | — | 4 | 1 | no | — |
| 579 | C8 | — | 4 | 2 | no | — |
| 580 | C8 | — | 8 | 1 | no | — |
| 581 | C8 | #535 | 1 | 1 | yes | characteristic |
| 582 | C8 | #535 | 1 | 1 | yes | characteristic |
| 583 | C8 | D42 | 1 | 1 | yes | characteristic |
| 584 | C8 | #560 | 1 | 1 | yes | characteristic |
| 585 | C22 | — | 2 | 1 | no | — |
| 586 | C22 | — | 2 | 1 | no | — |
| 587 | C22 | — | 4 | 1 | no | — |
| 588 | C22 | — | 4 | 1 | no | — |
| 589 | C22 | — | 4 | 1 | no | — |
| 590 | C22 | — | 4 | 1 | no | — |
| 591 | C22 | — | 4 | 1 | no | — |
| 592 | C22 | — | 4 | 1 | no | — |
| 593 | C22 | — | 8 | 1 | no | — |
| 594 | C22 | — | 8 | 1 | no | — |
| 595 | C22 | — | 8 | 1 | no | — |
| 596 | C22 | — | 16 | 2 | no | — |
| 597 | C22 | — | 16 | 1 | no | — |
| 598 | C22 | — | 16 | 1 | no | — |
| 599 | C22 | — | 16 | 1 | no | — |
| 600 | C22 | — | 32 | 1 | no | — |
| 601 | C22 | #3199 | 1 | 1 | yes | characteristic |
| 602 | C22 | #3268 | 1 | 1 | yes | characteristic |
| 603 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 604 | C4 | — | 4 | 1 | no | — |
| 605 | C4 | — | 4 | 1 | no | — |
| 606 | C4 | — | 8 | 1 | no | — |
| 607 | C4 | — | 8 | 2 | no | — |
| 608 | C4 | — | 8 | 1 | no | — |
| 609 | C4 | — | 8 | 1 | no | — |
| 610 | C4 | — | 8 | 1 | no | — |
| 611 | C4 | — | 16 | 1 | no | — |
| 612 | C4 | — | 16 | 1 | no | — |
| 613 | C4 | #2669 | 1 | 1 | yes | characteristic |
| 614 | C4 | #3186 | 1 | 1 | yes | characteristic |
| 615 | C4 | #3268 | 1 | 1 | yes | characteristic |
| 616 | C4 | D4×D8 | 1 | 1 | yes | characteristic |
| 617 | C2 | — | 2 | 1 | no | — |
| 618 | C2 | — | 2 | 1 | no | — |
| 619 | C2 | — | 8 | 1 | no | — |
| 620 | C2 | — | 8 | 1 | no | — |
| 621 | C2 | — | 16 | 1 | no | — |
| 622 | C2 | — | 32 | 1 | no | — |
| 623 | C2 | #30499 | 1 | 1 | yes | characteristic |
| 624 | C2 | #33855 | 1 | 1 | yes | characteristic |
| 625 | C2 | #33859 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2222 | C2 | semidirect |
| 2. #33612 | C2 | semidirect |
| 3. C2×#2193 | C2 | semidirect |
| 4. #33525 | C2 | semidirect |
| 5. #28503 | C2 | semidirect |
| 6. #24290 | C2 | semidirect |
| 7. #13393 | C2 | semidirect |
| 8. #13388 | C2 | semidirect |
| 9. #13332 | C2 | semidirect |
| 10. #13329 | C2 | semidirect |
| 11. #13321 | C2 | semidirect |
| 12. #13308 | C2 | semidirect |
| 13. #13135 | C2 | semidirect |
| 14. #13134 | C2 | semidirect |
| 15. #12980 | C2 | semidirect |
| 167. #362 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2222 | 2 |
| #33612 | 2 |
| C2×#2193 | 2 |
| #33525 | 2 |
| #28503 | 2 |
| #24290 | 2 |
| #13393 | 2 |
| #13388 | 2 |
| #13332 | 2 |
| #13329 | 2 |
| #13321 | 2 |
| #13308 | 2 |
| #13135 | 2 |
| #13134 | 2 |
| #12980 | 2 |