Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
851 conjugacy classes of subgroups.
The subgroup lattice has 487 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #61408 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #28241 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19189 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #18668 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #17976 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10459 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10255 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #9862 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #9658 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #8683 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #8357 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #3507 | C22 | 1 | 2 | yes | — |
| 13 | #3402 | — | 2 | 2 | no | — |
| 14 | #3400 | — | 2 | 2 | no | — |
| 15 | #3396 | — | 2 | 1 | no | — |
| 16 | #3395 | — | 2 | 1 | no | — |
| 17 | #3393 | — | 2 | 2 | no | — |
| 18 | #3245 | C22 | 1 | 2 | yes | — |
| 19 | #3167 | C22 | 1 | 1 | yes | characteristic |
| 20 | #3156 | — | 2 | 1 | no | — |
| 21 | #3147 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#473 | — | 2 | 1 | no | — |
| 23 | #3046 | — | 2 | 2 | no | — |
| 24 | #3037 | C22 | 1 | 2 | yes | — |
| 25 | #3037 | C22 | 1 | 2 | yes | — |
| 26 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 27 | #2951 | — | 2 | 1 | no | — |
| 28 | C2×C4×Q16 | C22 | 1 | 1 | yes | characteristic |
| 29 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 30 | #2052 | — | 2 | 2 | no | — |
| 31 | #1984 | C22 | 1 | 2 | yes | — |
| 32 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1816 | — | 2 | 1 | no | — |
| 34 | #1801 | C22 | 1 | 2 | yes | — |
| 35 | #1689 | — | 2 | 1 | no | — |
| 36 | #1687 | — | 2 | 1 | no | — |
| 37 | #1665 | — | 2 | 2 | no | — |
| 38 | #1652 | — | 2 | 2 | no | — |
| 39 | #1644 | — | 2 | 2 | no | — |
| 40 | #1628 | — | 2 | 2 | no | — |
| 41 | #1622 | — | 2 | 2 | no | — |
| 42 | #1583 | — | 2 | 2 | no | — |
| 43 | #1582 | — | 2 | 2 | no | — |
| 44 | #1566 | C22 | 1 | 1 | yes | characteristic |
| 45 | C2×#334 | C22 | 1 | 1 | yes | characteristic |
| 46 | #1550 | C22 | 1 | 2 | yes | — |
| 47 | #1547 | C22 | 1 | 2 | yes | — |
| 48 | #1493 | C22 | 1 | 1 | yes | characteristic |
| 49 | #1492 | C22 | 1 | 1 | yes | characteristic |
| 50 | #1492 | C22 | 1 | 2 | yes | — |
| 51 | #1492 | C22 | 1 | 2 | yes | — |
| 52 | C2×Q8⋊C8 | C22 | 1 | 2 | yes | — |
| 53 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 54 | #1292 | C22 | 1 | 2 | yes | — |
| 55 | C2×#137 | — | 2 | 2 | no | — |
| 56 | C2×#137 | — | 2 | 2 | no | — |
| 57 | C2×#135 | — | 2 | 1 | no | — |
| 58 | C2×#135 | — | 2 | 1 | no | — |
| 59 | C22×Q16 | D4 | 1 | 1 | yes | characteristic |
| 60 | C2×#131 | — | 2 | 1 | no | — |
| 61 | #559 | — | 2 | 2 | no | — |
| 62 | #557 | — | 2 | 2 | no | — |
| 63 | #556 | — | 2 | 4 | no | — |
| 64 | #556 | — | 2 | 2 | no | — |
| 65 | #554 | — | 2 | 2 | no | — |
| 66 | #552 | — | 4 | 2 | no | — |
| 67 | #539 | — | 2 | 4 | no | — |
| 68 | #536 | — | 2 | 2 | no | — |
| 69 | #533 | C23 | 1 | 1 | yes | characteristic |
| 70 | #532 | C23 | 1 | 2 | yes | — |
| 71 | #532 | C23 | 1 | 1 | yes | characteristic |
| 72 | #529 | — | 4 | 2 | no | — |
| 73 | #520 | — | 2 | 2 | no | — |
| 74 | C2×C4×Q8 | C23 | 1 | 2 | yes | — |
| 75 | #498 | D4 | 1 | 1 | yes | characteristic |
| 76 | #497 | — | 2 | 4 | no | — |
| 77 | #495 | — | 2 | 2 | no | — |
| 78 | #495 | — | 2 | 1 | no | — |
| 79 | #495 | — | 2 | 1 | no | — |
| 80 | #494 | — | 2 | 2 | no | — |
| 81 | #494 | D4 | 1 | 1 | yes | characteristic |
| 82 | #493 | D4 | 1 | 1 | yes | characteristic |
| 83 | #492 | D4 | 1 | 2 | yes | — |
| 84 | #492 | D4 | 1 | 1 | yes | characteristic |
| 85 | #487 | D4 | 1 | 1 | yes | characteristic |
| 86 | #486 | D4 | 1 | 1 | yes | characteristic |
| 87 | #474 | — | 2 | 2 | no | — |
| 88 | #474 | — | 2 | 2 | no | — |
| 89 | #473 | — | 4 | 4 | no | — |
| 90 | #466 | — | 4 | 2 | no | — |
| 91 | #465 | — | 2 | 2 | no | — |
| 92 | #464 | — | 2 | 4 | no | — |
| 93 | #464 | — | 2 | 4 | no | — |
| 94 | #464 | — | 4 | 2 | no | — |
| 95 | #463 | — | 2 | 4 | no | — |
| 96 | #462 | — | 2 | 4 | no | — |
| 97 | #462 | — | 2 | 2 | no | — |
| 98 | #462 | — | 2 | 2 | no | — |
| 99 | #461 | — | 2 | 4 | no | — |
| 100 | #460 | — | 2 | 2 | no | — |
| 101 | #460 | — | 2 | 2 | no | — |
| 102 | #459 | — | 2 | 2 | no | — |
| 103 | #459 | — | 2 | 2 | no | — |
| 104 | #446 | — | 2 | 1 | no | — |
| 105 | #446 | D4 | 1 | 2 | yes | — |
| 106 | C8×Q8 | — | 2 | 1 | no | — |
| 107 | C8×Q8 | D4 | 1 | 2 | yes | — |
| 108 | #440 | — | 2 | 4 | no | — |
| 109 | C4×Q16 | — | 2 | 2 | no | — |
| 110 | C4×Q16 | — | 2 | 1 | no | — |
| 111 | C4×Q16 | — | 2 | 1 | no | — |
| 112 | C4×Q16 | — | 2 | 1 | no | — |
| 113 | C4×Q16 | — | 2 | 1 | no | — |
| 114 | C4×SD16 | — | 2 | 2 | no | — |
| 115 | C4×D8 | — | 2 | 2 | no | — |
| 116 | #436 | — | 2 | 1 | no | — |
| 117 | #436 | — | 2 | 1 | no | — |
| 118 | C8×D4 | — | 2 | 1 | no | — |
| 119 | C8×D4 | — | 2 | 1 | no | — |
| 120 | C2×#108 | — | 2 | 1 | no | — |
| 121 | C2×#108 | — | 2 | 1 | no | — |
| 122 | C2×#107 | D4 | 1 | 1 | yes | characteristic |
| 123 | #424 | — | 2 | 1 | no | — |
| 124 | #423 | C23 | 1 | 1 | yes | characteristic |
| 125 | #423 | C23 | 1 | 2 | yes | — |
| 126 | #423 | C23 | 1 | 1 | yes | characteristic |
| 127 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 128 | #417 | — | 2 | 2 | no | — |
| 129 | #417 | — | 2 | 2 | no | — |
| 130 | C2×#103 | D4 | 1 | 2 | yes | — |
| 131 | C2×#101 | — | 2 | 2 | no | — |
| 132 | #408 | — | 2 | 1 | no | — |
| 133 | #405 | — | 2 | 1 | no | — |
| 134 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 136 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 137 | #343 | — | 2 | 2 | no | — |
| 138 | #336 | — | 4 | 2 | no | — |
| 139 | #334 | — | 2 | 2 | no | — |
| 140 | #334 | — | 2 | 1 | no | — |
| 141 | #334 | — | 2 | 1 | no | — |
| 142 | #333 | — | 2 | 2 | no | — |
| 143 | #332 | — | 2 | 2 | no | — |
| 144 | #332 | — | 2 | 2 | no | — |
| 145 | #331 | — | 2 | 2 | no | — |
| 146 | Q8⋊C8 | — | 2 | 2 | no | — |
| 147 | Q8⋊C8 | — | 2 | 2 | no | — |
| 148 | Q8⋊C8 | — | 2 | 2 | no | — |
| 149 | D4⋊C8 | — | 2 | 2 | no | — |
| 150 | D4⋊C8 | — | 2 | 2 | no | — |
| 151 | D4⋊C8 | — | 2 | 2 | no | — |
| 152 | D4⋊C8 | — | 4 | 2 | no | — |
| 153 | #322 | — | 2 | 1 | no | — |
| 154 | #322 | — | 2 | 1 | no | — |
| 155 | C2×#41 | — | 2 | 2 | no | — |
| 156 | C2×#41 | — | 2 | 1 | no | — |
| 157 | C22×Q8 | C2×D4 | 1 | 2 | yes | — |
| 158 | #137 | — | 4 | 4 | no | — |
| 159 | #137 | — | 8 | 2 | no | — |
| 160 | #137 | — | 8 | 2 | no | — |
| 161 | #137 | — | 8 | 2 | no | — |
| 162 | #135 | — | 4 | 2 | no | — |
| 163 | #135 | — | 4 | 2 | no | — |
| 164 | #135 | — | 8 | 1 | no | — |
| 165 | #135 | — | 8 | 1 | no | — |
| 166 | C2×Q16 | — | 2 | 4 | no | — |
| 167 | C2×Q16 | — | 2 | 1 | no | — |
| 168 | C2×Q16 | — | 2 | 1 | no | — |
| 169 | C2×Q16 | — | 2 | 1 | no | — |
| 170 | C2×Q16 | — | 2 | 1 | no | — |
| 171 | C2×Q16 | — | 2 | 2 | no | — |
| 172 | C2×Q16 | — | 2 | 2 | no | — |
| 173 | C2×Q16 | — | 4 | 2 | no | — |
| 174 | C2×SD16 | — | 2 | 4 | no | — |
| 175 | C2×SD16 | — | 2 | 2 | no | — |
| 176 | C2×SD16 | — | 2 | 2 | no | — |
| 177 | C2×SD16 | — | 2 | 2 | no | — |
| 178 | C2×SD16 | — | 4 | 2 | no | — |
| 179 | C2×D8 | — | 2 | 1 | no | — |
| 180 | C2×D8 | — | 2 | 1 | no | — |
| 181 | C2×D8 | — | 4 | 2 | no | — |
| 182 | #131 | — | 4 | 2 | no | — |
| 183 | #131 | — | 4 | 2 | no | — |
| 184 | C2×M16 | — | 2 | 1 | no | — |
| 185 | C2×M16 | — | 2 | 1 | no | — |
| 186 | C2×M16 | — | 2 | 2 | no | — |
| 187 | C2×M16 | — | 4 | 1 | no | — |
| 188 | C22×C8 | — | 2 | 1 | no | — |
| 189 | C22×C8 | — | 2 | 1 | no | — |
| 190 | C22×C8 | — | 2 | 2 | no | — |
| 191 | C22×C8 | — | 4 | 1 | no | — |
| 192 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 193 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 194 | C4⋊Q8 | — | 2 | 4 | no | — |
| 195 | C4⋊Q8 | C2×D4 | 1 | 2 | yes | — |
| 196 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 197 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 198 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 199 | #127 | C2×D4 | 1 | 2 | yes | — |
| 200 | #127 | C2×D4 | 1 | 1 | yes | characteristic |
| 201 | #126 | — | 4 | 4 | no | — |
| 202 | #126 | — | 4 | 4 | no | — |
| 203 | #125 | — | 2 | 1 | no | — |
| 204 | #125 | — | 4 | 2 | no | — |
| 205 | #125 | — | 4 | 4 | no | — |
| 206 | #124 | — | 2 | 4 | no | — |
| 207 | #124 | — | 2 | 2 | no | — |
| 208 | #124 | — | 2 | 1 | no | — |
| 209 | #124 | — | 4 | 2 | no | — |
| 210 | #123 | — | 4 | 4 | no | — |
| 211 | #123 | — | 4 | 4 | no | — |
| 212 | C22⋊Q8 | — | 2 | 1 | no | — |
| 213 | C22⋊Q8 | — | 4 | 2 | no | — |
| 214 | C22⋊Q8 | — | 4 | 2 | no | — |
| 215 | C22⋊Q8 | — | 4 | 4 | no | — |
| 216 | C22⋊Q8 | #41 | 1 | 2 | yes | — |
| 217 | #121 | — | 2 | 2 | no | — |
| 218 | #121 | — | 2 | 1 | no | — |
| 219 | #121 | — | 2 | 2 | no | — |
| 220 | #121 | — | 2 | 1 | no | — |
| 221 | C4×Q8 | — | 2 | 1 | no | — |
| 222 | C4×Q8 | — | 2 | 4 | no | — |
| 223 | C4×Q8 | — | 2 | 2 | no | — |
| 224 | C4×Q8 | — | 2 | 2 | no | — |
| 225 | C4×Q8 | — | 2 | 2 | no | — |
| 226 | C4×Q8 | — | 2 | 2 | no | — |
| 227 | C4×Q8 | — | 4 | 2 | no | — |
| 228 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 229 | C4×D4 | — | 2 | 4 | no | — |
| 230 | C4×D4 | — | 2 | 2 | no | — |
| 231 | C4×D4 | — | 2 | 1 | no | — |
| 232 | C4×D4 | — | 2 | 1 | no | — |
| 233 | C4×D4 | — | 2 | 2 | no | — |
| 234 | C4×D4 | — | 4 | 4 | no | — |
| 235 | C4×D4 | — | 4 | 2 | no | — |
| 236 | #117 | — | 2 | 1 | no | — |
| 237 | #117 | — | 4 | 2 | no | — |
| 238 | #117 | — | 4 | 4 | no | — |
| 239 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 240 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 241 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 242 | C2×C42 | — | 2 | 2 | no | — |
| 243 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 244 | #108 | — | 4 | 2 | no | — |
| 245 | #108 | — | 8 | 1 | no | — |
| 246 | #107 | — | 2 | 1 | no | — |
| 247 | #107 | — | 2 | 1 | no | — |
| 248 | #107 | — | 4 | 2 | no | — |
| 249 | #106 | — | 2 | 2 | no | — |
| 250 | C4⋊C8 | — | 2 | 2 | no | — |
| 251 | C4⋊C8 | — | 2 | 2 | no | — |
| 252 | C4⋊C8 | — | 2 | 1 | no | — |
| 253 | C4⋊C8 | — | 2 | 2 | no | — |
| 254 | C4⋊C8 | — | 2 | 2 | no | — |
| 255 | C4⋊C8 | — | 2 | 1 | no | — |
| 256 | C4⋊C8 | — | 2 | 1 | no | — |
| 257 | C4⋊C8 | — | 2 | 2 | no | — |
| 258 | C4⋊C8 | — | 2 | 2 | no | — |
| 259 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 260 | #103 | — | 2 | 4 | no | — |
| 261 | #103 | — | 2 | 2 | no | — |
| 262 | #103 | — | 2 | 2 | no | — |
| 263 | #103 | — | 2 | 2 | no | — |
| 264 | #103 | — | 4 | 2 | no | — |
| 265 | #102 | — | 2 | 4 | no | — |
| 266 | #102 | — | 2 | 2 | no | — |
| 267 | #102 | — | 4 | 2 | no | — |
| 268 | #102 | — | 4 | 2 | no | — |
| 269 | #102 | — | 4 | 2 | no | — |
| 270 | #101 | — | 4 | 4 | no | — |
| 271 | C22⋊C8 | — | 4 | 1 | no | — |
| 272 | C22⋊C8 | — | 4 | 1 | no | — |
| 273 | #97 | — | 2 | 2 | no | — |
| 274 | C4×C8 | — | 2 | 2 | no | — |
| 275 | C4×C8 | — | 2 | 1 | no | — |
| 276 | C4×C8 | — | 2 | 1 | no | — |
| 277 | C4×C8 | — | 2 | 1 | no | — |
| 278 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 279 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 280 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 281 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 282 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 283 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 284 | #41 | — | 4 | 1 | no | — |
| 285 | #41 | — | 4 | 2 | no | — |
| 286 | #41 | — | 4 | 1 | no | — |
| 287 | #41 | — | 4 | 2 | no | — |
| 288 | #41 | — | 4 | 2 | no | — |
| 289 | #41 | — | 4 | 1 | no | — |
| 290 | #41 | — | 4 | 2 | no | — |
| 291 | #41 | — | 4 | 1 | no | — |
| 292 | C2×Q8 | — | 2 | 1 | no | — |
| 293 | C2×Q8 | — | 2 | 2 | no | — |
| 294 | C2×Q8 | — | 2 | 2 | no | — |
| 295 | C2×Q8 | — | 2 | 1 | no | — |
| 296 | C2×Q8 | — | 2 | 2 | no | — |
| 297 | C2×Q8 | — | 2 | 2 | no | — |
| 298 | C2×Q8 | — | 2 | 2 | no | — |
| 299 | C2×Q8 | — | 2 | 2 | no | — |
| 300 | C2×Q8 | — | 2 | 2 | no | — |
| 301 | C2×Q8 | #121 | 1 | 2 | yes | — |
| 302 | C2×D4 | — | 2 | 2 | no | — |
| 303 | C2×D4 | — | 2 | 1 | no | — |
| 304 | C2×D4 | — | 2 | 1 | no | — |
| 305 | C2×D4 | — | 2 | 1 | no | — |
| 306 | C2×D4 | — | 2 | 2 | no | — |
| 307 | C2×D4 | — | 2 | 2 | no | — |
| 308 | C2×D4 | — | 4 | 2 | no | — |
| 309 | C2×D4 | — | 4 | 1 | no | — |
| 310 | C22×C4 | — | 2 | 2 | no | — |
| 311 | C22×C4 | — | 2 | 2 | no | — |
| 312 | C22×C4 | — | 4 | 4 | no | — |
| 313 | C22×C4 | — | 4 | 1 | no | — |
| 314 | C22×C4 | — | 4 | 2 | no | — |
| 315 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 316 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 317 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 318 | Q16 | — | 4 | 2 | no | — |
| 319 | Q16 | — | 4 | 2 | no | — |
| 320 | Q16 | — | 4 | 2 | no | — |
| 321 | Q16 | — | 8 | 1 | no | — |
| 322 | Q16 | — | 8 | 4 | no | — |
| 323 | Q16 | — | 8 | 4 | no | — |
| 324 | SD16 | — | 4 | 4 | no | — |
| 325 | SD16 | — | 8 | 2 | no | — |
| 326 | SD16 | — | 8 | 2 | no | — |
| 327 | SD16 | — | 8 | 4 | no | — |
| 328 | SD16 | — | 8 | 4 | no | — |
| 329 | D8 | — | 4 | 2 | no | — |
| 330 | D8 | — | 8 | 1 | no | — |
| 331 | D8 | — | 16 | 2 | no | — |
| 332 | M16 | — | 4 | 1 | no | — |
| 333 | M16 | — | 4 | 1 | no | — |
| 334 | M16 | — | 4 | 1 | no | — |
| 335 | M16 | — | 4 | 1 | no | — |
| 336 | M16 | — | 4 | 2 | no | — |
| 337 | M16 | — | 4 | 2 | no | — |
| 338 | M16 | — | 8 | 2 | no | — |
| 339 | C2×C8 | — | 2 | 1 | no | — |
| 340 | C2×C8 | — | 2 | 1 | no | — |
| 341 | C2×C8 | — | 2 | 2 | no | — |
| 342 | C2×C8 | — | 2 | 1 | no | — |
| 343 | C2×C8 | — | 2 | 1 | no | — |
| 344 | C2×C8 | — | 4 | 2 | no | — |
| 345 | C2×C8 | — | 4 | 1 | no | — |
| 346 | C2×C8 | — | 4 | 1 | no | — |
| 347 | C2×C8 | — | 4 | 2 | no | — |
| 348 | C2×C8 | — | 4 | 2 | no | — |
| 349 | C2×C8 | — | 8 | 1 | no | — |
| 350 | C2×C8 | — | 8 | 2 | no | — |
| 351 | C2×C8 | — | 8 | 2 | no | — |
| 352 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 353 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 354 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 355 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 356 | C2×C8 | #135 | 1 | 2 | yes | — |
| 357 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 358 | C4⋊C4 | — | 2 | 2 | no | — |
| 359 | C4⋊C4 | — | 2 | 1 | no | — |
| 360 | C4⋊C4 | — | 2 | 1 | no | — |
| 361 | C4⋊C4 | — | 2 | 1 | no | — |
| 362 | C4⋊C4 | — | 2 | 2 | no | — |
| 363 | C4⋊C4 | — | 2 | 2 | no | — |
| 364 | C4⋊C4 | — | 2 | 1 | no | — |
| 365 | C4⋊C4 | — | 2 | 1 | no | — |
| 366 | C4⋊C4 | — | 2 | 2 | no | — |
| 367 | C4⋊C4 | — | 2 | 2 | no | — |
| 368 | C4⋊C4 | — | 2 | 2 | no | — |
| 369 | C4⋊C4 | — | 4 | 4 | no | — |
| 370 | C4⋊C4 | — | 4 | 4 | no | — |
| 371 | C4⋊C4 | — | 4 | 4 | no | — |
| 372 | C4⋊C4 | — | 4 | 2 | no | — |
| 373 | C4⋊C4 | — | 4 | 2 | no | — |
| 374 | C4⋊C4 | #121 | 1 | 2 | yes | — |
| 375 | C22⋊C4 | — | 2 | 2 | no | — |
| 376 | C22⋊C4 | — | 4 | 4 | no | — |
| 377 | C22⋊C4 | — | 4 | 4 | no | — |
| 378 | C22⋊C4 | — | 4 | 4 | no | — |
| 379 | C22⋊C4 | — | 4 | 2 | no | — |
| 380 | C22⋊C4 | — | 4 | 2 | no | — |
| 381 | C22⋊C4 | — | 4 | 1 | no | — |
| 382 | C22⋊C4 | — | 4 | 1 | no | — |
| 383 | C42 | — | 2 | 2 | no | — |
| 384 | C42 | — | 4 | 4 | no | — |
| 385 | C42 | — | 4 | 2 | no | — |
| 386 | C42 | — | 4 | 2 | no | — |
| 387 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 388 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 389 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 390 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 391 | C23 | — | 4 | 2 | no | — |
| 392 | C23 | — | 4 | 1 | no | — |
| 393 | C23 | #546 | 1 | 1 | yes | characteristic |
| 394 | Q8 | — | 4 | 2 | no | — |
| 395 | Q8 | — | 4 | 2 | no | — |
| 396 | Q8 | — | 4 | 2 | no | — |
| 397 | Q8 | — | 4 | 2 | no | — |
| 398 | Q8 | — | 4 | 2 | no | — |
| 399 | Q8 | — | 4 | 2 | no | — |
| 400 | Q8 | — | 4 | 2 | no | — |
| 401 | Q8 | — | 4 | 1 | no | — |
| 402 | Q8 | — | 4 | 2 | no | — |
| 403 | Q8 | — | 4 | 1 | no | — |
| 404 | D4 | — | 4 | 2 | no | — |
| 405 | D4 | — | 4 | 2 | no | — |
| 406 | D4 | — | 4 | 2 | no | — |
| 407 | D4 | — | 4 | 1 | no | — |
| 408 | D4 | — | 4 | 1 | no | — |
| 409 | D4 | — | 4 | 2 | no | — |
| 410 | D4 | — | 4 | 2 | no | — |
| 411 | D4 | — | 4 | 2 | no | — |
| 412 | D4 | — | 8 | 2 | no | — |
| 413 | D4 | — | 8 | 2 | no | — |
| 414 | D4 | — | 8 | 1 | no | — |
| 415 | D4 | — | 8 | 2 | no | — |
| 416 | D4 | — | 8 | 1 | no | — |
| 417 | C2×C4 | — | 2 | 2 | no | — |
| 418 | C2×C4 | — | 2 | 2 | no | — |
| 419 | C2×C4 | — | 2 | 2 | no | — |
| 420 | C2×C4 | — | 4 | 1 | no | — |
| 421 | C2×C4 | — | 4 | 2 | no | — |
| 422 | C2×C4 | — | 4 | 2 | no | — |
| 423 | C2×C4 | — | 4 | 2 | no | — |
| 424 | C2×C4 | — | 4 | 2 | no | — |
| 425 | C2×C4 | — | 4 | 1 | no | — |
| 426 | C2×C4 | — | 4 | 2 | no | — |
| 427 | C2×C4 | — | 8 | 1 | no | — |
| 428 | C2×C4 | — | 8 | 1 | no | — |
| 429 | C2×C4 | — | 8 | 2 | no | — |
| 430 | C2×C4 | — | 8 | 2 | no | — |
| 431 | C2×C4 | — | 8 | 2 | no | — |
| 432 | C2×C4 | — | 16 | 4 | no | — |
| 433 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 434 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 435 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 436 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 437 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 438 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 439 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 440 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 441 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 442 | C2×C4 | #577 | 1 | 1 | yes | characteristic |
| 443 | C8 | — | 2 | 1 | no | — |
| 444 | C8 | — | 2 | 1 | no | — |
| 445 | C8 | — | 4 | 1 | no | — |
| 446 | C8 | — | 4 | 1 | no | — |
| 447 | C8 | — | 4 | 2 | no | — |
| 448 | C8 | — | 8 | 1 | no | — |
| 449 | C8 | — | 8 | 1 | no | — |
| 450 | C8 | — | 8 | 2 | no | — |
| 451 | C8 | — | 8 | 2 | no | — |
| 452 | C8 | #449 | 1 | 2 | yes | — |
| 453 | C8 | #449 | 1 | 2 | yes | — |
| 454 | C22 | — | 2 | 1 | no | — |
| 455 | C22 | — | 2 | 1 | no | — |
| 456 | C22 | — | 8 | 1 | no | — |
| 457 | C22 | — | 8 | 2 | no | — |
| 458 | C22 | — | 8 | 2 | no | — |
| 459 | C22 | — | 8 | 1 | no | — |
| 460 | C22 | — | 8 | 1 | no | — |
| 461 | C22 | — | 8 | 2 | no | — |
| 462 | C22 | #2408 | 1 | 1 | yes | characteristic |
| 463 | C22 | #3305 | 1 | 1 | yes | characteristic |
| 464 | C22 | #3332 | 1 | 1 | yes | characteristic |
| 465 | C4 | — | 2 | 1 | no | — |
| 466 | C4 | — | 2 | 1 | no | — |
| 467 | C4 | — | 2 | 1 | no | — |
| 468 | C4 | — | 4 | 2 | no | — |
| 469 | C4 | — | 4 | 2 | no | — |
| 470 | C4 | — | 8 | 2 | no | — |
| 471 | C4 | — | 8 | 2 | no | — |
| 472 | C4 | — | 8 | 1 | no | — |
| 473 | C4 | — | 8 | 2 | no | — |
| 474 | C4 | C2×#449 | 1 | 1 | yes | characteristic |
| 475 | C4 | #3012 | 1 | 1 | yes | characteristic |
| 476 | C4 | #3274 | 1 | 1 | yes | characteristic |
| 477 | C4 | #3284 | 1 | 1 | yes | characteristic |
| 478 | C4 | #3289 | 1 | 2 | yes | — |
| 479 | C2 | — | 2 | 1 | no | — |
| 480 | C2 | — | 2 | 1 | no | — |
| 481 | C2 | — | 16 | 2 | no | — |
| 482 | C2 | — | 16 | 1 | no | — |
| 483 | C2 | #24238 | 1 | 1 | yes | characteristic |
| 484 | C2 | #24288 | 1 | 1 | yes | characteristic |
| 485 | C2 | #24293 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #19189 | 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. #61408 | C2 | semidirect |
| 2. #28241 | C2 | semidirect |
| 4. #18668 | C2 | semidirect |
| 5. #17976 | C2 | semidirect |
| 6. #10459 | C2 | semidirect |
| 7. #10255 | C2 | semidirect |
| 8. #9862 | C2 | semidirect |
| 9. #9658 | C2 | semidirect |
| 10. #8683 | C2 | semidirect |
| 11. #8357 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #61408 | 2 |
| #28241 | 2 |
| #19189 | 2 |
| #18668 | 2 |
| #17976 | 2 |
| #10459 | 2 |
| #10255 | 2 |
| #9862 | 2 |
| #9658 | 2 |
| #8683 | 2 |
| #8357 | 2 |