Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
983 conjugacy classes of subgroups.
The subgroup lattice has 532 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #62471 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #28675 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #24233 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #19261 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C8×#137 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10503 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10243 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #9799 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #9719 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #8667 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #8421 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #3499 | C22 | 1 | 2 | yes | — |
| 13 | Q8×D8 | — | 2 | 1 | no | — |
| 14 | Q8×D8 | — | 2 | 1 | no | — |
| 15 | #3334 | — | 2 | 1 | no | — |
| 16 | #3334 | — | 2 | 1 | no | — |
| 17 | #3334 | — | 2 | 2 | no | — |
| 18 | #3334 | — | 2 | 2 | no | — |
| 19 | #3226 | — | 2 | 2 | no | — |
| 20 | #3226 | C22 | 1 | 2 | yes | — |
| 21 | #3169 | — | 2 | 1 | no | — |
| 22 | #3169 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#500 | — | 2 | 1 | no | — |
| 24 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 25 | #3037 | C22 | 1 | 2 | yes | — |
| 26 | C2×#456 | — | 2 | 2 | no | — |
| 27 | C8×#41 | C22 | 1 | 1 | yes | characteristic |
| 28 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 29 | C4×#137 | C22 | 1 | 1 | yes | characteristic |
| 30 | #2546 | C22 | 1 | 2 | yes | — |
| 31 | #2086 | — | 2 | 2 | no | — |
| 32 | #2038 | — | 2 | 2 | no | — |
| 33 | #2005 | — | 2 | 2 | no | — |
| 34 | #1988 | C22 | 1 | 2 | yes | — |
| 35 | #1980 | C22 | 1 | 2 | yes | — |
| 36 | #1966 | C22 | 1 | 2 | yes | — |
| 37 | #1942 | — | 2 | 2 | no | — |
| 38 | #1806 | C22 | 1 | 2 | yes | — |
| 39 | #1793 | — | 2 | 2 | no | — |
| 40 | C4×#107 | C22 | 1 | 1 | yes | characteristic |
| 41 | #1682 | — | 2 | 2 | no | — |
| 42 | #1674 | — | 2 | 1 | no | — |
| 43 | #1668 | — | 2 | 1 | no | — |
| 44 | #1579 | — | 2 | 1 | no | — |
| 45 | SD16⋊C8 | — | 2 | 1 | no | — |
| 46 | C8×Q16 | — | 2 | 1 | no | — |
| 47 | C8×SD16 | — | 2 | 1 | no | — |
| 48 | #1567 | C22 | 1 | 1 | yes | characteristic |
| 49 | #1565 | C22 | 1 | 1 | yes | characteristic |
| 50 | #1547 | C22 | 1 | 2 | yes | — |
| 51 | #1546 | C22 | 1 | 2 | yes | — |
| 52 | #1502 | C22 | 1 | 2 | yes | — |
| 53 | #1479 | C22 | 1 | 1 | yes | characteristic |
| 54 | #1477 | C22 | 1 | 1 | yes | characteristic |
| 55 | C8×M16 | C22 | 1 | 1 | yes | characteristic |
| 56 | #1303 | C22 | 1 | 2 | yes | — |
| 57 | C2×2−1+4 | — | 2 | 2 | no | — |
| 58 | C2×#137 | — | 2 | 2 | no | — |
| 59 | C2×#137 | D4 | 1 | 1 | yes | characteristic |
| 60 | D4×Q8 | — | 2 | 2 | no | — |
| 61 | D4×Q8 | — | 2 | 2 | no | — |
| 62 | #547 | — | 2 | 2 | no | — |
| 63 | #547 | — | 2 | 4 | no | — |
| 64 | #547 | — | 2 | 2 | no | — |
| 65 | #547 | — | 4 | 2 | no | — |
| 66 | #536 | — | 2 | 2 | no | — |
| 67 | #536 | — | 4 | 2 | no | — |
| 68 | #532 | — | 2 | 2 | no | — |
| 69 | #532 | C23 | 1 | 2 | yes | — |
| 70 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 71 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 72 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 73 | C4×#41 | C23 | 1 | 1 | yes | characteristic |
| 74 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 75 | #500 | — | 2 | 2 | no | — |
| 76 | #500 | — | 2 | 2 | no | — |
| 77 | #500 | — | 2 | 2 | no | — |
| 78 | #500 | — | 2 | 1 | no | — |
| 79 | #500 | — | 2 | 1 | no | — |
| 80 | #500 | — | 2 | 1 | no | — |
| 81 | #500 | — | 2 | 1 | no | — |
| 82 | #500 | D4 | 1 | 1 | yes | characteristic |
| 83 | #493 | D4 | 1 | 1 | yes | characteristic |
| 84 | #486 | — | 2 | 2 | no | — |
| 85 | #486 | D4 | 1 | 2 | yes | — |
| 86 | #480 | — | 2 | 4 | no | — |
| 87 | #480 | — | 2 | 4 | no | — |
| 88 | #480 | — | 2 | 4 | no | — |
| 89 | #474 | — | 2 | 4 | no | — |
| 90 | #474 | — | 2 | 4 | no | — |
| 91 | #474 | — | 2 | 2 | no | — |
| 92 | #474 | — | 2 | 2 | no | — |
| 93 | #466 | — | 2 | 2 | no | — |
| 94 | #466 | — | 2 | 2 | no | — |
| 95 | #466 | — | 2 | 1 | no | — |
| 96 | #466 | — | 2 | 1 | no | — |
| 97 | #464 | — | 4 | 2 | no | — |
| 98 | #462 | — | 2 | 2 | no | — |
| 99 | #460 | — | 2 | 2 | no | — |
| 100 | #456 | — | 4 | 8 | no | — |
| 101 | C8×Q8 | — | 2 | 1 | no | — |
| 102 | C8×Q8 | — | 2 | 1 | no | — |
| 103 | C8×Q8 | — | 2 | 1 | no | — |
| 104 | C8×Q8 | — | 2 | 1 | no | — |
| 105 | C8×Q8 | D4 | 1 | 1 | yes | characteristic |
| 106 | C8×Q8 | D4 | 1 | 1 | yes | characteristic |
| 107 | #441 | — | 2 | 1 | no | — |
| 108 | #440 | — | 2 | 1 | no | — |
| 109 | C4×Q16 | — | 2 | 1 | no | — |
| 110 | C4×SD16 | — | 2 | 1 | no | — |
| 111 | C4×D8 | — | 2 | 2 | no | — |
| 112 | C4×D8 | — | 2 | 2 | no | — |
| 113 | C4×D8 | — | 2 | 1 | no | — |
| 114 | C4×D8 | — | 2 | 1 | no | — |
| 115 | C8×D4 | — | 2 | 1 | no | — |
| 116 | C8×D4 | — | 2 | 1 | no | — |
| 117 | C8×D4 | D4 | 1 | 1 | yes | characteristic |
| 118 | C8×D4 | D4 | 1 | 1 | yes | characteristic |
| 119 | #431 | — | 2 | 1 | no | — |
| 120 | #431 | C23 | 1 | 1 | yes | characteristic |
| 121 | #431 | C23 | 1 | 1 | yes | characteristic |
| 122 | #428 | D4 | 1 | 1 | yes | characteristic |
| 123 | C2×#107 | — | 2 | 2 | no | — |
| 124 | C2×#107 | — | 2 | 2 | no | — |
| 125 | C2×#107 | — | 2 | 1 | no | — |
| 126 | C2×#107 | D4 | 1 | 2 | yes | — |
| 127 | C2×#107 | D4 | 1 | 2 | yes | — |
| 128 | #423 | C23 | 1 | 2 | yes | — |
| 129 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 130 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 131 | C2×#104 | — | 2 | 2 | no | — |
| 132 | C2×#104 | — | 2 | 2 | no | — |
| 133 | C2×#104 | — | 2 | 2 | no | — |
| 134 | #419 | D4 | 1 | 1 | yes | characteristic |
| 135 | #417 | — | 2 | 2 | no | — |
| 136 | #417 | D4 | 1 | 1 | yes | characteristic |
| 137 | C2×#101 | — | 2 | 2 | no | — |
| 138 | C2×#101 | — | 2 | 2 | no | — |
| 139 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 140 | C2×C4×C8 | — | 2 | 1 | no | — |
| 141 | C2×C4×C8 | — | 2 | 1 | no | — |
| 142 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 143 | #398 | — | 4 | 2 | no | — |
| 144 | #391 | — | 2 | 2 | no | — |
| 145 | #389 | — | 2 | 4 | no | — |
| 146 | C4×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 147 | #340 | — | 2 | 1 | no | — |
| 148 | #340 | — | 2 | 4 | no | — |
| 149 | #339 | — | 2 | 2 | no | — |
| 150 | #335 | — | 2 | 1 | no | — |
| 151 | #335 | — | 2 | 1 | no | — |
| 152 | #335 | — | 2 | 1 | no | — |
| 153 | #334 | — | 2 | 1 | no | — |
| 154 | #332 | — | 2 | 2 | no | — |
| 155 | #332 | — | 4 | 2 | no | — |
| 156 | #331 | — | 2 | 2 | no | — |
| 157 | #327 | — | 4 | 2 | no | — |
| 158 | Q8⋊C8 | — | 2 | 1 | no | — |
| 159 | Q8⋊C8 | — | 2 | 1 | no | — |
| 160 | Q8⋊C8 | — | 2 | 1 | no | — |
| 161 | D4⋊C8 | — | 2 | 1 | no | — |
| 162 | #322 | — | 2 | 1 | no | — |
| 163 | C82 | — | 2 | 1 | no | — |
| 164 | 2−1+4 | — | 4 | 4 | no | — |
| 165 | 2−1+4 | — | 4 | 4 | no | — |
| 166 | 2−1+4 | — | 8 | 2 | no | — |
| 167 | 2−1+4 | — | 8 | 2 | no | — |
| 168 | C2×#41 | — | 2 | 2 | no | — |
| 169 | C2×#41 | — | 4 | 2 | no | — |
| 170 | C2×#41 | — | 4 | 4 | no | — |
| 171 | C2×#41 | — | 4 | 2 | no | — |
| 172 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 173 | C22×Q8 | — | 2 | 2 | no | — |
| 174 | C22×Q8 | — | 4 | 2 | no | — |
| 175 | C22×Q8 | — | 4 | 2 | no | — |
| 176 | C22×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 177 | #137 | — | 2 | 2 | no | — |
| 178 | #137 | — | 4 | 4 | no | — |
| 179 | #137 | — | 8 | 2 | no | — |
| 180 | #137 | D8 | 1 | 4 | yes | complemented |
| 181 | C2×Q16 | — | 2 | 1 | no | — |
| 182 | C2×Q16 | — | 4 | 2 | no | — |
| 183 | C2×SD16 | — | 2 | 1 | no | — |
| 184 | C2×SD16 | — | 4 | 2 | no | — |
| 185 | C2×D8 | — | 2 | 1 | no | — |
| 186 | C2×D8 | — | 2 | 1 | no | — |
| 187 | C2×M16 | — | 2 | 2 | no | — |
| 188 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 189 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 190 | C22×C8 | — | 2 | 1 | no | — |
| 191 | C22×C8 | — | 2 | 1 | no | — |
| 192 | C22×C8 | — | 2 | 1 | no | — |
| 193 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 194 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 195 | C4⋊Q8 | — | 2 | 4 | no | — |
| 196 | C4⋊Q8 | — | 2 | 2 | no | — |
| 197 | C4⋊Q8 | — | 2 | 2 | no | — |
| 198 | C4⋊Q8 | — | 2 | 2 | no | — |
| 199 | C4⋊Q8 | — | 2 | 2 | no | — |
| 200 | C4⋊Q8 | — | 4 | 2 | no | — |
| 201 | C4⋊Q8 | — | 4 | 2 | no | — |
| 202 | C4⋊Q8 | — | 4 | 4 | no | — |
| 203 | C4⋊Q8 | — | 8 | 2 | no | — |
| 204 | C4⋊Q8 | C2×D4 | 1 | 2 | yes | — |
| 205 | #127 | C2×D4 | 1 | 2 | yes | — |
| 206 | #124 | — | 2 | 2 | no | — |
| 207 | #124 | — | 4 | 2 | no | — |
| 208 | #123 | — | 4 | 4 | no | — |
| 209 | #123 | — | 4 | 4 | no | — |
| 210 | #123 | — | 4 | 4 | no | — |
| 211 | C22⋊Q8 | — | 4 | 4 | no | — |
| 212 | C22⋊Q8 | — | 4 | 4 | no | — |
| 213 | C22⋊Q8 | — | 4 | 2 | no | — |
| 214 | C22⋊Q8 | — | 4 | 2 | no | — |
| 215 | #121 | — | 2 | 4 | no | — |
| 216 | #121 | — | 2 | 2 | no | — |
| 217 | #121 | — | 2 | 2 | no | — |
| 218 | #121 | — | 4 | 2 | no | — |
| 219 | C4×Q8 | — | 2 | 1 | no | — |
| 220 | C4×Q8 | — | 2 | 1 | no | — |
| 221 | C4×Q8 | — | 2 | 1 | no | — |
| 222 | C4×Q8 | — | 2 | 1 | no | — |
| 223 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 224 | C4×Q8 | #41 | 1 | 1 | yes | characteristic |
| 225 | C4×D4 | — | 2 | 1 | no | — |
| 226 | C4×D4 | — | 2 | 1 | no | — |
| 227 | C4×D4 | — | 2 | 4 | no | — |
| 228 | C4×D4 | — | 2 | 2 | no | — |
| 229 | C4×D4 | — | 2 | 2 | no | — |
| 230 | C4×D4 | — | 4 | 2 | no | — |
| 231 | C4×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 232 | C4×D4 | #41 | 1 | 1 | yes | characteristic |
| 233 | #117 | — | 2 | 1 | no | — |
| 234 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 235 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 236 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 237 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 238 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 239 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 240 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 241 | C2×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 242 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 243 | C2×C42 | — | 2 | 1 | no | — |
| 244 | C2×C42 | — | 2 | 1 | no | — |
| 245 | C2×C42 | — | 2 | 2 | no | — |
| 246 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 247 | #107 | — | 2 | 4 | no | — |
| 248 | #107 | — | 2 | 4 | no | — |
| 249 | #107 | — | 2 | 4 | no | — |
| 250 | #107 | — | 2 | 1 | no | — |
| 251 | #107 | — | 2 | 1 | no | — |
| 252 | #107 | — | 2 | 2 | no | — |
| 253 | #107 | — | 2 | 2 | no | — |
| 254 | #107 | — | 2 | 2 | no | — |
| 255 | #107 | — | 2 | 2 | no | — |
| 256 | #107 | — | 2 | 1 | no | — |
| 257 | #106 | — | 2 | 1 | no | — |
| 258 | C4⋊C8 | — | 2 | 1 | no | — |
| 259 | C4⋊C8 | — | 2 | 1 | no | — |
| 260 | C4⋊C8 | — | 2 | 1 | no | — |
| 261 | C4⋊C8 | — | 2 | 1 | no | — |
| 262 | C4⋊C8 | — | 2 | 2 | no | — |
| 263 | C4⋊C8 | — | 2 | 1 | no | — |
| 264 | C4⋊C8 | — | 2 | 2 | no | — |
| 265 | C4⋊C8 | — | 4 | 1 | no | — |
| 266 | C4⋊C8 | — | 4 | 1 | no | — |
| 267 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 268 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 269 | #104 | — | 4 | 4 | no | — |
| 270 | #104 | — | 4 | 4 | no | — |
| 271 | #104 | — | 8 | 2 | no | — |
| 272 | #103 | — | 2 | 1 | no | — |
| 273 | #103 | — | 2 | 1 | no | — |
| 274 | #103 | — | 2 | 1 | no | — |
| 275 | #103 | — | 4 | 2 | no | — |
| 276 | #102 | — | 2 | 4 | no | — |
| 277 | #102 | — | 2 | 4 | no | — |
| 278 | #102 | — | 2 | 1 | no | — |
| 279 | #102 | — | 2 | 2 | no | — |
| 280 | #102 | — | 2 | 2 | no | — |
| 281 | #102 | — | 4 | 2 | no | — |
| 282 | #101 | — | 4 | 4 | no | — |
| 283 | #101 | — | 8 | 2 | no | — |
| 284 | C22⋊C8 | — | 2 | 1 | no | — |
| 285 | C22⋊C8 | — | 2 | 1 | no | — |
| 286 | C22⋊C8 | — | 2 | 1 | no | — |
| 287 | C22⋊C8 | — | 2 | 1 | no | — |
| 288 | #97 | — | 2 | 1 | no | — |
| 289 | C4×C8 | — | 2 | 1 | no | — |
| 290 | C4×C8 | — | 2 | 1 | no | — |
| 291 | C4×C8 | — | 2 | 1 | no | — |
| 292 | C4×C8 | — | 2 | 1 | no | — |
| 293 | C4×C8 | — | 2 | 1 | no | — |
| 294 | C4×C8 | — | 4 | 1 | no | — |
| 295 | C4×C8 | — | 4 | 1 | no | — |
| 296 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 297 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 298 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 299 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 300 | #95 | — | 2 | 2 | no | — |
| 301 | #95 | — | 4 | 4 | no | — |
| 302 | #41 | — | 2 | 1 | no | — |
| 303 | #41 | — | 2 | 1 | no | — |
| 304 | #41 | — | 4 | 2 | no | — |
| 305 | #41 | — | 4 | 2 | no | — |
| 306 | #41 | — | 4 | 2 | no | — |
| 307 | #41 | — | 4 | 2 | no | — |
| 308 | #41 | — | 8 | 4 | no | — |
| 309 | #41 | — | 8 | 4 | no | — |
| 310 | #41 | — | 8 | 8 | no | — |
| 311 | #41 | — | 8 | 8 | no | — |
| 312 | #41 | — | 8 | 4 | no | — |
| 313 | #41 | — | 8 | 4 | no | — |
| 314 | #41 | C2×D8 | 1 | 2 | yes | — |
| 315 | #41 | #135 | 1 | 2 | yes | — |
| 316 | C2×Q8 | — | 2 | 1 | no | — |
| 317 | C2×Q8 | — | 2 | 1 | no | — |
| 318 | C2×Q8 | — | 2 | 1 | no | — |
| 319 | C2×Q8 | — | 2 | 2 | no | — |
| 320 | C2×Q8 | — | 2 | 1 | no | — |
| 321 | C2×Q8 | — | 2 | 1 | no | — |
| 322 | C2×Q8 | — | 4 | 4 | no | — |
| 323 | C2×Q8 | — | 4 | 2 | no | — |
| 324 | C2×Q8 | — | 4 | 2 | no | — |
| 325 | C2×Q8 | — | 4 | 2 | no | — |
| 326 | C2×Q8 | — | 4 | 2 | no | — |
| 327 | C2×Q8 | — | 4 | 2 | no | — |
| 328 | C2×Q8 | — | 4 | 2 | no | — |
| 329 | C2×Q8 | — | 8 | 4 | no | — |
| 330 | C2×Q8 | — | 8 | 4 | no | — |
| 331 | C2×Q8 | — | 8 | 4 | no | — |
| 332 | C2×Q8 | — | 8 | 4 | no | — |
| 333 | C2×Q8 | #121 | 1 | 1 | yes | characteristic |
| 334 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 335 | C2×D4 | — | 2 | 1 | no | — |
| 336 | C2×D4 | — | 2 | 2 | no | — |
| 337 | C2×D4 | — | 2 | 2 | no | — |
| 338 | C2×D4 | — | 2 | 2 | no | — |
| 339 | C2×D4 | — | 4 | 4 | no | — |
| 340 | C2×D4 | — | 4 | 2 | no | — |
| 341 | C2×D4 | #121 | 1 | 1 | yes | characteristic |
| 342 | C22×C4 | — | 2 | 1 | no | — |
| 343 | C22×C4 | — | 2 | 1 | no | — |
| 344 | C22×C4 | — | 2 | 1 | no | — |
| 345 | C22×C4 | — | 2 | 1 | no | — |
| 346 | C22×C4 | — | 2 | 1 | no | — |
| 347 | C22×C4 | — | 2 | 4 | no | — |
| 348 | C22×C4 | — | 4 | 4 | no | — |
| 349 | C22×C4 | — | 4 | 4 | no | — |
| 350 | C22×C4 | — | 4 | 2 | no | — |
| 351 | C22×C4 | — | 4 | 2 | no | — |
| 352 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 353 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 354 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 355 | Q16 | — | 2 | 4 | no | — |
| 356 | Q16 | — | 8 | 4 | no | — |
| 357 | SD16 | — | 2 | 4 | no | — |
| 358 | SD16 | — | 8 | 4 | no | — |
| 359 | D8 | — | 4 | 2 | no | — |
| 360 | D8 | — | 4 | 2 | no | — |
| 361 | M16 | — | 2 | 1 | no | — |
| 362 | M16 | — | 2 | 1 | no | — |
| 363 | M16 | — | 2 | 1 | no | — |
| 364 | M16 | — | 4 | 2 | no | — |
| 365 | M16 | — | 4 | 2 | no | — |
| 366 | M16 | C2×D8 | 1 | 2 | yes | — |
| 367 | C2×C8 | — | 2 | 1 | no | — |
| 368 | C2×C8 | — | 2 | 2 | no | — |
| 369 | C2×C8 | — | 2 | 1 | no | — |
| 370 | C2×C8 | — | 2 | 1 | no | — |
| 371 | C2×C8 | — | 2 | 1 | no | — |
| 372 | C2×C8 | — | 4 | 1 | no | — |
| 373 | C2×C8 | — | 4 | 2 | no | — |
| 374 | C2×C8 | — | 4 | 2 | no | — |
| 375 | C2×C8 | — | 4 | 2 | no | — |
| 376 | C2×C8 | — | 4 | 1 | no | — |
| 377 | C2×C8 | — | 4 | 1 | no | — |
| 378 | C2×C8 | — | 4 | 2 | no | — |
| 379 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 380 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 381 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 382 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 383 | C4⋊C4 | — | 2 | 1 | no | — |
| 384 | C4⋊C4 | — | 2 | 1 | no | — |
| 385 | C4⋊C4 | — | 2 | 4 | no | — |
| 386 | C4⋊C4 | — | 2 | 2 | no | — |
| 387 | C4⋊C4 | — | 2 | 4 | no | — |
| 388 | C4⋊C4 | — | 2 | 1 | no | — |
| 389 | C4⋊C4 | — | 2 | 1 | no | — |
| 390 | C4⋊C4 | — | 2 | 2 | no | — |
| 391 | C4⋊C4 | — | 2 | 4 | no | — |
| 392 | C4⋊C4 | — | 4 | 2 | no | — |
| 393 | C4⋊C4 | — | 4 | 4 | no | — |
| 394 | C4⋊C4 | — | 4 | 4 | no | — |
| 395 | C4⋊C4 | — | 4 | 1 | no | — |
| 396 | C4⋊C4 | — | 4 | 2 | no | — |
| 397 | C4⋊C4 | — | 4 | 2 | no | — |
| 398 | C4⋊C4 | — | 4 | 1 | no | — |
| 399 | C4⋊C4 | — | 8 | 8 | no | — |
| 400 | C4⋊C4 | #121 | 1 | 1 | yes | characteristic |
| 401 | C4⋊C4 | #121 | 1 | 1 | yes | characteristic |
| 402 | C22⋊C4 | — | 2 | 1 | no | — |
| 403 | C22⋊C4 | — | 2 | 1 | no | — |
| 404 | C22⋊C4 | — | 2 | 1 | no | — |
| 405 | C22⋊C4 | — | 4 | 4 | no | — |
| 406 | C22⋊C4 | — | 4 | 4 | no | — |
| 407 | C22⋊C4 | — | 4 | 2 | no | — |
| 408 | C22⋊C4 | — | 4 | 2 | no | — |
| 409 | C42 | — | 2 | 1 | no | — |
| 410 | C42 | — | 2 | 1 | no | — |
| 411 | C42 | — | 2 | 1 | no | — |
| 412 | C42 | — | 4 | 1 | no | — |
| 413 | C42 | — | 4 | 1 | no | — |
| 414 | C42 | — | 4 | 2 | no | — |
| 415 | C42 | — | 4 | 2 | no | — |
| 416 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 417 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 418 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 419 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 420 | C23 | — | 2 | 1 | no | — |
| 421 | C23 | — | 4 | 2 | no | — |
| 422 | C23 | D42 | 1 | 1 | yes | characteristic |
| 423 | Q8 | — | 2 | 1 | no | — |
| 424 | Q8 | — | 2 | 2 | no | — |
| 425 | Q8 | — | 2 | 2 | no | — |
| 426 | Q8 | — | 4 | 1 | no | — |
| 427 | Q8 | — | 4 | 1 | no | — |
| 428 | Q8 | — | 4 | 2 | no | — |
| 429 | Q8 | — | 4 | 2 | no | — |
| 430 | Q8 | — | 8 | 8 | no | — |
| 431 | Q8 | — | 8 | 2 | no | — |
| 432 | Q8 | — | 8 | 2 | no | — |
| 433 | Q8 | — | 8 | 2 | no | — |
| 434 | Q8 | — | 8 | 2 | no | — |
| 435 | Q8 | #466 | 1 | 2 | yes | — |
| 436 | D4 | — | 2 | 2 | no | — |
| 437 | D4 | — | 2 | 2 | no | — |
| 438 | D4 | — | 2 | 1 | no | — |
| 439 | D4 | — | 4 | 2 | no | — |
| 440 | D4 | — | 4 | 2 | no | — |
| 441 | D4 | — | 4 | 2 | no | — |
| 442 | D4 | — | 4 | 2 | no | — |
| 443 | D4 | — | 8 | 2 | no | — |
| 444 | D4 | — | 8 | 8 | no | — |
| 445 | D4 | — | 8 | 2 | no | — |
| 446 | D4 | — | 8 | 2 | no | — |
| 447 | D4 | #466 | 1 | 2 | yes | — |
| 448 | C2×C4 | — | 2 | 1 | no | — |
| 449 | C2×C4 | — | 2 | 1 | no | — |
| 450 | C2×C4 | — | 2 | 1 | no | — |
| 451 | C2×C4 | — | 2 | 2 | no | — |
| 452 | C2×C4 | — | 2 | 2 | no | — |
| 453 | C2×C4 | — | 2 | 1 | no | — |
| 454 | C2×C4 | — | 2 | 1 | no | — |
| 455 | C2×C4 | — | 2 | 2 | no | — |
| 456 | C2×C4 | — | 4 | 1 | no | — |
| 457 | C2×C4 | — | 4 | 2 | no | — |
| 458 | C2×C4 | — | 4 | 1 | no | — |
| 459 | C2×C4 | — | 4 | 1 | no | — |
| 460 | C2×C4 | — | 4 | 1 | no | — |
| 461 | C2×C4 | — | 4 | 1 | no | — |
| 462 | C2×C4 | — | 4 | 4 | no | — |
| 463 | C2×C4 | — | 4 | 2 | no | — |
| 464 | C2×C4 | — | 4 | 1 | no | — |
| 465 | C2×C4 | — | 4 | 4 | no | — |
| 466 | C2×C4 | — | 4 | 2 | no | — |
| 467 | C2×C4 | — | 4 | 4 | no | — |
| 468 | C2×C4 | — | 4 | 4 | no | — |
| 469 | C2×C4 | — | 8 | 8 | no | — |
| 470 | C2×C4 | — | 8 | 2 | no | — |
| 471 | C2×C4 | — | 8 | 8 | no | — |
| 472 | C2×C4 | — | 8 | 2 | no | — |
| 473 | C2×C4 | — | 8 | 2 | no | — |
| 474 | C2×C4 | — | 8 | 2 | no | — |
| 475 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 476 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 477 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 478 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 479 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 480 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 481 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 482 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 483 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 484 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 485 | C8 | — | 2 | 1 | no | — |
| 486 | C8 | — | 2 | 1 | no | — |
| 487 | C8 | — | 2 | 1 | no | — |
| 488 | C8 | — | 2 | 2 | no | — |
| 489 | C8 | — | 4 | 1 | no | — |
| 490 | C8 | — | 4 | 1 | no | — |
| 491 | C8 | — | 4 | 1 | no | — |
| 492 | C8 | — | 8 | 1 | no | — |
| 493 | C8 | — | 8 | 1 | no | — |
| 494 | C8 | — | 8 | 2 | no | — |
| 495 | C8 | #456 | 1 | 2 | yes | complemented |
| 496 | C22 | — | 2 | 1 | no | — |
| 497 | C22 | — | 2 | 1 | no | — |
| 498 | C22 | — | 2 | 2 | no | — |
| 499 | C22 | — | 4 | 1 | no | — |
| 500 | C22 | — | 4 | 1 | no | — |
| 501 | C22 | — | 8 | 2 | no | — |
| 502 | C22 | — | 8 | 2 | no | — |
| 503 | C22 | — | 8 | 2 | no | — |
| 504 | C22 | #2408 | 1 | 1 | yes | characteristic |
| 505 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 506 | C22 | #3280 | 1 | 1 | yes | characteristic |
| 507 | C4 | — | 2 | 1 | no | — |
| 508 | C4 | — | 2 | 2 | no | — |
| 509 | C4 | — | 2 | 1 | no | — |
| 510 | C4 | — | 2 | 1 | no | — |
| 511 | C4 | — | 4 | 1 | no | — |
| 512 | C4 | — | 4 | 2 | no | — |
| 513 | C4 | — | 4 | 1 | no | — |
| 514 | C4 | — | 8 | 4 | no | — |
| 515 | C4 | — | 8 | 4 | no | — |
| 516 | C4 | — | 8 | 2 | no | — |
| 517 | C4 | — | 8 | 1 | no | — |
| 518 | C4 | C2×#456 | 1 | 1 | yes | characteristic |
| 519 | C4 | #3020 | 1 | 1 | yes | characteristic |
| 520 | C4 | C2×#466 | 1 | 1 | yes | characteristic |
| 521 | C4 | #3058 | 1 | 1 | yes | characteristic |
| 522 | C4 | D4×D8 | 1 | 1 | yes | characteristic |
| 523 | C4 | #3280 | 1 | 1 | yes | characteristic |
| 524 | C2 | — | 2 | 1 | no | — |
| 525 | C2 | — | 2 | 1 | no | — |
| 526 | C2 | — | 4 | 2 | no | — |
| 527 | C2 | — | 16 | 2 | no | — |
| 528 | C2 | #24233 | 1 | 1 | yes | characteristic |
| 529 | C2 | #24267 | 1 | 1 | yes | characteristic |
| 530 | C2 | #24296 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62471 | 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 |
|---|---|---|
| 2. #28675 | C2 | semidirect |
| 3. #24233 | C2 | semidirect |
| 4. #19261 | C2 | semidirect |
| 5. C8×#137 | C2 | semidirect |
| 6. #10503 | C2 | semidirect |
| 7. #10243 | C2 | semidirect |
| 8. #9799 | C2 | semidirect |
| 9. #9719 | C2 | semidirect |
| 10. #8667 | C2 | semidirect |
| 11. #8421 | C2 | semidirect |
| 180. #137 | D8 | semidirect |
| 495. C8 | #456 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62471 | 2 |
| #28675 | 2 |
| #24233 | 2 |
| #19261 | 2 |
| C8×#137 | 2 |
| #10503 | 2 |
| #10243 | 2 |
| #9799 | 2 |
| #9719 | 2 |
| #8667 | 2 |
| #8421 | 2 |