Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
756 conjugacy classes of subgroups.
The subgroup lattice has 516 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #61413 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33596 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33471 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #29270 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13417 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13414 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #13334 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #13332 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #13270 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #13216 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #13034 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #13027 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | #12954 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | #12951 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | #12826 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | #3510 | — | 2 | 1 | no | — |
| 17 | #3414 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#510 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#509 | C22 | 1 | 1 | yes | characteristic |
| 20 | #3401 | — | 2 | 4 | no | — |
| 21 | #3396 | — | 2 | 2 | no | — |
| 22 | #3393 | — | 2 | 2 | no | — |
| 23 | #3156 | C22 | 1 | 1 | yes | characteristic |
| 24 | #3155 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#495 | C22 | 1 | 1 | yes | characteristic |
| 26 | #3043 | — | 2 | 1 | no | — |
| 27 | #3042 | — | 2 | 1 | no | — |
| 28 | #3040 | — | 2 | 1 | no | — |
| 29 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 30 | #2957 | C22 | 1 | 1 | yes | characteristic |
| 31 | #2950 | C22 | 1 | 1 | yes | characteristic |
| 32 | #2948 | C22 | 1 | 1 | yes | characteristic |
| 33 | C4×#135 | C22 | 1 | 1 | yes | characteristic |
| 34 | #2214 | — | 2 | 2 | no | — |
| 35 | #2212 | — | 2 | 2 | no | — |
| 36 | AGL(1,ℤ/16) | — | 2 | 4 | no | — |
| 37 | #2155 | C22 | 1 | 1 | yes | characteristic |
| 38 | #2153 | C22 | 1 | 1 | yes | characteristic |
| 39 | #2148 | C22 | 1 | 1 | yes | characteristic |
| 40 | C2×#362 | C22 | 1 | 1 | yes | characteristic |
| 41 | C2×#362 | C22 | 1 | 1 | yes | characteristic |
| 42 | C2×#361 | C22 | 1 | 1 | yes | characteristic |
| 43 | C2×#361 | C22 | 1 | 1 | yes | characteristic |
| 44 | #2145 | C22 | 1 | 1 | yes | characteristic |
| 45 | #2145 | C22 | 1 | 1 | yes | characteristic |
| 46 | #2143 | C22 | 1 | 1 | yes | characteristic |
| 47 | #2141 | C22 | 1 | 1 | yes | characteristic |
| 48 | #2140 | C22 | 1 | 1 | yes | characteristic |
| 49 | #2110 | C22 | 1 | 1 | yes | characteristic |
| 50 | #2098 | — | 2 | 1 | no | — |
| 51 | #2031 | — | 2 | 1 | no | — |
| 52 | #1976 | C22 | 1 | 1 | yes | characteristic |
| 53 | #1974 | C22 | 1 | 1 | yes | characteristic |
| 54 | #1953 | C22 | 1 | 1 | yes | characteristic |
| 55 | #1945 | C22 | 1 | 1 | yes | characteristic |
| 56 | #1868 | — | 2 | 1 | no | — |
| 57 | #1861 | — | 2 | 1 | no | — |
| 58 | #1833 | C22 | 1 | 1 | yes | characteristic |
| 59 | #1389 | C22 | 1 | 1 | yes | characteristic |
| 60 | #1384 | C22 | 1 | 1 | yes | characteristic |
| 61 | #1378 | C22 | 1 | 1 | yes | characteristic |
| 62 | #1378 | C22 | 1 | 1 | yes | characteristic |
| 63 | #1376 | C22 | 1 | 1 | yes | characteristic |
| 64 | #1376 | C22 | 1 | 1 | yes | characteristic |
| 65 | C2×#137 | — | 2 | 1 | no | — |
| 66 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 67 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 68 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 69 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 70 | C22×SD16 | — | 2 | 1 | no | — |
| 71 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 72 | #568 | — | 2 | 1 | no | — |
| 73 | #559 | — | 2 | 4 | no | — |
| 74 | #556 | — | 2 | 4 | no | — |
| 75 | #551 | — | 4 | 4 | no | — |
| 76 | #542 | — | 4 | 2 | no | — |
| 77 | #536 | — | 2 | 1 | no | — |
| 78 | #535 | — | 2 | 1 | no | — |
| 79 | #533 | D4 | 1 | 1 | yes | characteristic |
| 80 | #529 | — | 4 | 2 | no | — |
| 81 | C2×#124 | — | 2 | 1 | no | — |
| 82 | #519 | — | 2 | 1 | no | — |
| 83 | #518 | — | 2 | 1 | no | — |
| 84 | C4×#41 | — | 2 | 1 | no | — |
| 85 | #510 | — | 2 | 2 | no | — |
| 86 | #510 | — | 2 | 1 | no | — |
| 87 | #510 | — | 2 | 1 | no | — |
| 88 | #510 | — | 4 | 1 | no | — |
| 89 | #509 | — | 2 | 2 | no | — |
| 90 | #509 | — | 2 | 1 | no | — |
| 91 | #509 | — | 2 | 1 | no | — |
| 92 | #509 | — | 2 | 2 | no | — |
| 93 | #508 | — | 4 | 1 | no | — |
| 94 | C2×Q32 | — | 2 | 1 | no | — |
| 95 | C2×SD32 | — | 2 | 1 | no | — |
| 96 | C2×SD32 | — | 2 | 1 | no | — |
| 97 | C2×D16 | — | 2 | 1 | no | — |
| 98 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 99 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 100 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 101 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 102 | #497 | — | 2 | 2 | no | — |
| 103 | #496 | — | 2 | 2 | no | — |
| 104 | #495 | — | 2 | 1 | no | — |
| 105 | #495 | — | 2 | 2 | no | — |
| 106 | #495 | — | 2 | 2 | no | — |
| 107 | #495 | D4 | 1 | 2 | yes | — |
| 108 | #494 | D4 | 1 | 2 | yes | — |
| 109 | #493 | — | 2 | 1 | no | — |
| 110 | #464 | — | 2 | 4 | no | — |
| 111 | #464 | — | 2 | 4 | no | — |
| 112 | #463 | — | 2 | 4 | no | — |
| 113 | #463 | — | 2 | 4 | no | — |
| 114 | #461 | — | 2 | 4 | no | — |
| 115 | #460 | — | 2 | 4 | no | — |
| 116 | #446 | — | 2 | 2 | no | — |
| 117 | C8×Q8 | — | 2 | 2 | no | — |
| 118 | #444 | — | 2 | 2 | no | — |
| 119 | #443 | — | 2 | 2 | no | — |
| 120 | #442 | D4 | 1 | 4 | yes | — |
| 121 | #441 | — | 2 | 2 | no | — |
| 122 | C4×Q16 | — | 2 | 1 | no | — |
| 123 | C4×Q16 | — | 2 | 2 | no | — |
| 124 | C4×SD16 | — | 2 | 2 | no | — |
| 125 | C4×D8 | — | 2 | 1 | no | — |
| 126 | C4×D8 | — | 2 | 2 | no | — |
| 127 | #430 | — | 2 | 1 | no | — |
| 128 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 129 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 130 | #427 | C23 | 1 | 1 | yes | characteristic |
| 131 | #427 | C23 | 1 | 1 | yes | characteristic |
| 132 | #424 | — | 2 | 1 | no | — |
| 133 | #424 | D4 | 1 | 1 | yes | characteristic |
| 134 | #424 | D4 | 1 | 1 | yes | characteristic |
| 135 | #423 | D4 | 1 | 1 | yes | characteristic |
| 136 | #421 | — | 2 | 1 | no | — |
| 137 | #421 | — | 2 | 1 | no | — |
| 138 | C2×#104 | — | 2 | 1 | no | — |
| 139 | #419 | — | 2 | 1 | no | — |
| 140 | #418 | — | 2 | 1 | no | — |
| 141 | #416 | — | 2 | 1 | no | — |
| 142 | #410 | — | 2 | 1 | no | — |
| 143 | #408 | — | 2 | 1 | no | — |
| 144 | #405 | C23 | 1 | 1 | yes | characteristic |
| 145 | #405 | C23 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 147 | #365 | — | 2 | 2 | no | — |
| 148 | #364 | — | 2 | 2 | no | — |
| 149 | #362 | — | 2 | 2 | no | — |
| 150 | #362 | — | 2 | 1 | no | — |
| 151 | #362 | — | 2 | 2 | no | — |
| 152 | #361 | — | 2 | 2 | no | — |
| 153 | #361 | — | 2 | 1 | no | — |
| 154 | #361 | — | 4 | 1 | no | — |
| 155 | #360 | — | 2 | 2 | no | — |
| 156 | #360 | — | 2 | 2 | no | — |
| 157 | #359 | — | 4 | 1 | no | — |
| 158 | #358 | — | 4 | 1 | no | — |
| 159 | #357 | — | 4 | 1 | no | — |
| 160 | #347 | — | 2 | 2 | no | — |
| 161 | #344 | — | 2 | 1 | no | — |
| 162 | #344 | — | 2 | 1 | no | — |
| 163 | #339 | — | 2 | 1 | no | — |
| 164 | H(3,ℤ/4) | — | 2 | 1 | no | — |
| 165 | C2×#41 | — | 2 | 1 | no | — |
| 166 | C2×#41 | — | 2 | 1 | no | — |
| 167 | C22×Q8 | — | 2 | 1 | no | — |
| 168 | C22×D4 | — | 2 | 1 | no | — |
| 169 | #137 | — | 8 | 1 | no | — |
| 170 | #137 | — | 8 | 1 | no | — |
| 171 | AGL(1,ℤ/8) | — | 8 | 1 | no | — |
| 172 | AGL(1,ℤ/8) | — | 8 | 1 | no | — |
| 173 | #135 | — | 2 | 1 | no | — |
| 174 | #135 | — | 2 | 1 | no | — |
| 175 | #135 | — | 2 | 1 | no | — |
| 176 | #135 | — | 2 | 1 | no | — |
| 177 | #135 | — | 2 | 1 | no | — |
| 178 | #135 | — | 2 | 1 | no | — |
| 179 | #135 | — | 2 | 1 | no | — |
| 180 | #135 | — | 2 | 1 | no | — |
| 181 | C2×Q16 | — | 2 | 2 | no | — |
| 182 | C2×Q16 | — | 2 | 1 | no | — |
| 183 | C2×Q16 | — | 2 | 1 | no | — |
| 184 | C2×Q16 | — | 2 | 1 | no | — |
| 185 | C2×Q16 | — | 2 | 1 | no | — |
| 186 | C2×Q16 | — | 2 | 1 | no | — |
| 187 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 188 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 189 | C2×Q16 | #41 | 1 | 1 | yes | characteristic |
| 190 | C2×SD16 | — | 2 | 4 | no | — |
| 191 | C2×SD16 | — | 2 | 2 | no | — |
| 192 | C2×SD16 | — | 2 | 2 | no | — |
| 193 | C2×SD16 | — | 2 | 1 | no | — |
| 194 | C2×SD16 | — | 2 | 1 | no | — |
| 195 | C2×SD16 | — | 8 | 1 | no | — |
| 196 | C2×SD16 | — | 8 | 1 | no | — |
| 197 | C2×D8 | — | 2 | 2 | no | — |
| 198 | C2×D8 | — | 2 | 1 | no | — |
| 199 | C2×D8 | — | 2 | 1 | no | — |
| 200 | C2×D8 | — | 2 | 1 | no | — |
| 201 | C2×D8 | — | 2 | 1 | no | — |
| 202 | C2×D8 | — | 2 | 1 | no | — |
| 203 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 204 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 205 | C2×D8 | #41 | 1 | 1 | yes | characteristic |
| 206 | #131 | — | 4 | 2 | no | — |
| 207 | #131 | — | 8 | 1 | no | — |
| 208 | C2×M16 | — | 2 | 1 | no | — |
| 209 | C2×M16 | — | 2 | 1 | no | — |
| 210 | C2×M16 | — | 8 | 1 | no | — |
| 211 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 212 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 213 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 214 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 215 | C4⋊Q8 | — | 2 | 2 | no | — |
| 216 | C4⋊Q8 | — | 2 | 1 | no | — |
| 217 | #127 | — | 2 | 2 | no | — |
| 218 | #126 | — | 4 | 4 | no | — |
| 219 | #126 | — | 4 | 4 | no | — |
| 220 | #125 | — | 2 | 1 | no | — |
| 221 | #124 | — | 2 | 4 | no | — |
| 222 | #124 | — | 2 | 2 | no | — |
| 223 | #124 | — | 2 | 2 | no | — |
| 224 | #124 | — | 4 | 4 | no | — |
| 225 | #124 | — | 4 | 1 | no | — |
| 226 | #124 | — | 4 | 1 | no | — |
| 227 | #123 | — | 4 | 2 | no | — |
| 228 | #123 | — | 4 | 4 | no | — |
| 229 | #123 | — | 4 | 4 | no | — |
| 230 | C22⋊Q8 | — | 2 | 2 | no | — |
| 231 | C22⋊Q8 | — | 4 | 2 | no | — |
| 232 | C22⋊Q8 | — | 4 | 4 | no | — |
| 233 | #121 | — | 4 | 2 | no | — |
| 234 | #120 | — | 4 | 2 | no | — |
| 235 | C4×Q8 | — | 2 | 2 | no | — |
| 236 | C4×Q8 | — | 2 | 2 | no | — |
| 237 | C4×Q8 | — | 2 | 4 | no | — |
| 238 | C4×Q8 | — | 4 | 2 | no | — |
| 239 | C4×D4 | — | 2 | 2 | no | — |
| 240 | C4×D4 | — | 2 | 4 | no | — |
| 241 | C4×D4 | — | 4 | 2 | no | — |
| 242 | C4×D4 | — | 4 | 4 | no | — |
| 243 | C4×D4 | — | 4 | 2 | no | — |
| 244 | #117 | — | 2 | 1 | no | — |
| 245 | #117 | — | 2 | 1 | no | — |
| 246 | #117 | — | 4 | 4 | no | — |
| 247 | #117 | — | 4 | 2 | no | — |
| 248 | #117 | — | 4 | 1 | no | — |
| 249 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 250 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 251 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 252 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 253 | C2×C42 | — | 4 | 1 | no | — |
| 254 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 255 | Q32 | — | 4 | 2 | no | — |
| 256 | Q32 | — | 4 | 1 | no | — |
| 257 | SD32 | — | 2 | 4 | no | — |
| 258 | SD32 | — | 4 | 2 | no | — |
| 259 | SD32 | — | 4 | 1 | no | — |
| 260 | SD32 | — | 4 | 1 | no | — |
| 261 | D16 | — | 2 | 4 | no | — |
| 262 | D16 | — | 4 | 1 | no | — |
| 263 | M32 | — | 2 | 1 | no | — |
| 264 | M32 | — | 2 | 1 | no | — |
| 265 | M32 | — | 2 | 1 | no | — |
| 266 | M32 | — | 2 | 1 | no | — |
| 267 | M32 | — | 2 | 1 | no | — |
| 268 | M32 | — | 2 | 1 | no | — |
| 269 | M32 | — | 2 | 1 | no | — |
| 270 | M32 | #41 | 1 | 2 | yes | — |
| 271 | C2×C16 | — | 2 | 1 | no | — |
| 272 | C2×C16 | — | 2 | 1 | no | — |
| 273 | C2×C16 | — | 2 | 1 | no | — |
| 274 | C2×C16 | — | 2 | 1 | no | — |
| 275 | #108 | — | 2 | 1 | no | — |
| 276 | #108 | — | 2 | 1 | no | — |
| 277 | #108 | — | 2 | 1 | no | — |
| 278 | #108 | — | 2 | 1 | no | — |
| 279 | #108 | — | 8 | 1 | no | — |
| 280 | #107 | — | 2 | 1 | no | — |
| 281 | #107 | — | 2 | 1 | no | — |
| 282 | #106 | — | 2 | 1 | no | — |
| 283 | #106 | C2×D4 | 1 | 2 | yes | — |
| 284 | C4⋊C8 | — | 2 | 2 | no | — |
| 285 | C4⋊C8 | — | 2 | 2 | no | — |
| 286 | C4⋊C8 | — | 2 | 2 | no | — |
| 287 | C4⋊C8 | — | 8 | 1 | no | — |
| 288 | #104 | — | 4 | 2 | no | — |
| 289 | #104 | — | 4 | 1 | no | — |
| 290 | #104 | — | 4 | 1 | no | — |
| 291 | #104 | — | 8 | 1 | no | — |
| 292 | #103 | — | 2 | 2 | no | — |
| 293 | #103 | — | 2 | 4 | no | — |
| 294 | #102 | — | 2 | 2 | no | — |
| 295 | #102 | — | 2 | 4 | no | — |
| 296 | #99 | — | 4 | 2 | no | — |
| 297 | C22⋊C8 | — | 8 | 1 | no | — |
| 298 | #97 | — | 2 | 1 | no | — |
| 299 | #97 | C2×D4 | 1 | 2 | yes | — |
| 300 | C4×C8 | — | 2 | 1 | no | — |
| 301 | C4×C8 | — | 2 | 1 | no | — |
| 302 | C4×C8 | — | 2 | 1 | no | — |
| 303 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 304 | C24 | — | 4 | 1 | no | — |
| 305 | #41 | — | 4 | 1 | no | — |
| 306 | #41 | — | 4 | 1 | no | — |
| 307 | #41 | — | 4 | 1 | no | — |
| 308 | #41 | — | 4 | 1 | no | — |
| 309 | #41 | — | 4 | 1 | no | — |
| 310 | #41 | — | 4 | 1 | no | — |
| 311 | #41 | — | 4 | 1 | no | — |
| 312 | #41 | — | 4 | 1 | no | — |
| 313 | C2×Q8 | — | 2 | 2 | no | — |
| 314 | C2×Q8 | — | 2 | 2 | no | — |
| 315 | C2×Q8 | — | 2 | 1 | no | — |
| 316 | C2×Q8 | — | 2 | 1 | no | — |
| 317 | C2×Q8 | — | 4 | 1 | no | — |
| 318 | C2×Q8 | — | 4 | 1 | no | — |
| 319 | C2×Q8 | — | 4 | 1 | no | — |
| 320 | C2×Q8 | — | 4 | 1 | no | — |
| 321 | C2×Q8 | — | 4 | 1 | no | — |
| 322 | C2×D4 | — | 2 | 2 | no | — |
| 323 | C2×D4 | — | 2 | 1 | no | — |
| 324 | C2×D4 | — | 2 | 1 | no | — |
| 325 | C2×D4 | — | 4 | 4 | no | — |
| 326 | C2×D4 | — | 4 | 1 | no | — |
| 327 | C2×D4 | — | 4 | 1 | no | — |
| 328 | C2×D4 | — | 4 | 1 | no | — |
| 329 | C2×D4 | — | 4 | 1 | no | — |
| 330 | C2×D4 | — | 4 | 1 | no | — |
| 331 | C2×D4 | — | 4 | 1 | no | — |
| 332 | C2×D4 | — | 4 | 1 | no | — |
| 333 | C22×C4 | — | 2 | 1 | no | — |
| 334 | C22×C4 | — | 4 | 2 | no | — |
| 335 | C22×C4 | — | 4 | 4 | no | — |
| 336 | C22×C4 | — | 4 | 1 | no | — |
| 337 | C22×C4 | — | 4 | 1 | no | — |
| 338 | C22×C4 | — | 4 | 1 | no | — |
| 339 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 340 | Q16 | — | 2 | 1 | no | — |
| 341 | Q16 | — | 2 | 1 | no | — |
| 342 | Q16 | — | 2 | 2 | no | — |
| 343 | Q16 | — | 2 | 1 | no | — |
| 344 | Q16 | — | 2 | 2 | no | — |
| 345 | Q16 | — | 2 | 1 | no | — |
| 346 | Q16 | — | 4 | 1 | no | — |
| 347 | Q16 | — | 4 | 1 | no | — |
| 348 | Q16 | — | 8 | 2 | no | — |
| 349 | SD16 | — | 4 | 1 | no | — |
| 350 | SD16 | — | 4 | 1 | no | — |
| 351 | SD16 | — | 4 | 1 | no | — |
| 352 | SD16 | — | 4 | 1 | no | — |
| 353 | SD16 | — | 8 | 2 | no | — |
| 354 | SD16 | — | 8 | 4 | no | — |
| 355 | SD16 | — | 8 | 2 | no | — |
| 356 | D8 | — | 2 | 1 | no | — |
| 357 | D8 | — | 2 | 2 | no | — |
| 358 | D8 | — | 2 | 2 | no | — |
| 359 | D8 | — | 2 | 1 | no | — |
| 360 | D8 | — | 2 | 1 | no | — |
| 361 | D8 | — | 2 | 1 | no | — |
| 362 | D8 | — | 4 | 1 | no | — |
| 363 | D8 | — | 4 | 1 | no | — |
| 364 | D8 | — | 8 | 2 | no | — |
| 365 | M16 | — | 2 | 1 | no | — |
| 366 | M16 | — | 2 | 1 | no | — |
| 367 | M16 | — | 2 | 1 | no | — |
| 368 | M16 | — | 2 | 1 | no | — |
| 369 | M16 | — | 4 | 1 | no | — |
| 370 | M16 | — | 4 | 1 | no | — |
| 371 | M16 | — | 4 | 1 | no | — |
| 372 | M16 | — | 4 | 1 | no | — |
| 373 | M16 | — | 8 | 2 | no | — |
| 374 | C2×C8 | — | 2 | 1 | no | — |
| 375 | C2×C8 | — | 2 | 1 | no | — |
| 376 | C2×C8 | — | 2 | 1 | no | — |
| 377 | C2×C8 | — | 2 | 1 | no | — |
| 378 | C2×C8 | — | 4 | 1 | no | — |
| 379 | C2×C8 | — | 4 | 1 | no | — |
| 380 | C2×C8 | — | 8 | 2 | no | — |
| 381 | C2×C8 | #120 | 1 | 2 | yes | — |
| 382 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 383 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 384 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 385 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 386 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 387 | C2×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 388 | C4⋊C4 | — | 2 | 2 | no | — |
| 389 | C4⋊C4 | — | 2 | 2 | no | — |
| 390 | C4⋊C4 | — | 2 | 2 | no | — |
| 391 | C4⋊C4 | — | 2 | 1 | no | — |
| 392 | C4⋊C4 | — | 2 | 1 | no | — |
| 393 | C4⋊C4 | — | 2 | 2 | no | — |
| 394 | C4⋊C4 | — | 4 | 2 | no | — |
| 395 | C4⋊C4 | — | 4 | 4 | no | — |
| 396 | C4⋊C4 | — | 4 | 4 | no | — |
| 397 | C4⋊C4 | — | 4 | 1 | no | — |
| 398 | C22⋊C4 | — | 2 | 1 | no | — |
| 399 | C22⋊C4 | — | 4 | 4 | no | — |
| 400 | C22⋊C4 | — | 4 | 4 | no | — |
| 401 | C22⋊C4 | — | 4 | 4 | no | — |
| 402 | C22⋊C4 | — | 4 | 2 | no | — |
| 403 | C22⋊C4 | — | 4 | 4 | no | — |
| 404 | C22⋊C4 | — | 4 | 4 | no | — |
| 405 | C22⋊C4 | — | 4 | 1 | no | — |
| 406 | C22⋊C4 | — | 4 | 1 | no | — |
| 407 | C22⋊C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 408 | C42 | — | 2 | 1 | no | — |
| 409 | C42 | — | 2 | 1 | no | — |
| 410 | C42 | — | 4 | 2 | no | — |
| 411 | C42 | — | 4 | 4 | no | — |
| 412 | C42 | — | 4 | 1 | no | — |
| 413 | C42 | — | 4 | 1 | no | — |
| 414 | C42 | #120 | 1 | 2 | yes | — |
| 415 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 416 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 417 | C16 | — | 2 | 2 | no | — |
| 418 | C16 | — | 4 | 1 | no | — |
| 419 | C16 | — | 4 | 1 | no | — |
| 420 | C16 | — | 4 | 1 | no | — |
| 421 | C23 | — | 4 | 2 | no | — |
| 422 | C23 | — | 4 | 1 | no | — |
| 423 | C23 | — | 4 | 1 | no | — |
| 424 | C23 | — | 8 | 1 | no | — |
| 425 | C23 | — | 8 | 1 | no | — |
| 426 | C23 | — | 16 | 2 | no | — |
| 427 | C23 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 428 | Q8 | — | 4 | 1 | no | — |
| 429 | Q8 | — | 4 | 1 | no | — |
| 430 | Q8 | — | 4 | 1 | no | — |
| 431 | Q8 | — | 4 | 2 | no | — |
| 432 | Q8 | — | 4 | 2 | no | — |
| 433 | Q8 | — | 4 | 1 | no | — |
| 434 | Q8 | — | 8 | 2 | no | — |
| 435 | Q8 | — | 8 | 1 | no | — |
| 436 | Q8 | — | 8 | 1 | no | — |
| 437 | D4 | — | 4 | 1 | no | — |
| 438 | D4 | — | 4 | 2 | no | — |
| 439 | D4 | — | 4 | 1 | no | — |
| 440 | D4 | — | 4 | 1 | no | — |
| 441 | D4 | — | 4 | 1 | no | — |
| 442 | D4 | — | 4 | 2 | no | — |
| 443 | D4 | — | 8 | 1 | no | — |
| 444 | D4 | — | 8 | 1 | no | — |
| 445 | D4 | — | 8 | 1 | no | — |
| 446 | D4 | — | 8 | 1 | no | — |
| 447 | D4 | — | 8 | 1 | no | — |
| 448 | D4 | — | 8 | 1 | no | — |
| 449 | D4 | — | 16 | 4 | no | — |
| 450 | C2×C4 | — | 2 | 1 | no | — |
| 451 | C2×C4 | — | 2 | 2 | no | — |
| 452 | C2×C4 | — | 2 | 1 | no | — |
| 453 | C2×C4 | — | 2 | 2 | no | — |
| 454 | C2×C4 | — | 4 | 1 | no | — |
| 455 | C2×C4 | — | 4 | 2 | no | — |
| 456 | C2×C4 | — | 4 | 2 | no | — |
| 457 | C2×C4 | — | 4 | 1 | no | — |
| 458 | C2×C4 | — | 8 | 1 | no | — |
| 459 | C2×C4 | — | 8 | 1 | no | — |
| 460 | C2×C4 | — | 8 | 1 | no | — |
| 461 | C2×C4 | — | 8 | 1 | no | — |
| 462 | C2×C4 | — | 8 | 1 | no | — |
| 463 | C2×C4 | — | 8 | 1 | no | — |
| 464 | C2×C4 | — | 8 | 1 | no | — |
| 465 | C2×C4 | — | 8 | 1 | no | — |
| 466 | C2×C4 | — | 16 | 2 | no | — |
| 467 | C2×C4 | — | 16 | 4 | no | — |
| 468 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 469 | C2×C4 | #548 | 1 | 1 | yes | characteristic |
| 470 | C2×C4 | #550 | 1 | 1 | yes | characteristic |
| 471 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 472 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 473 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 474 | C8 | — | 2 | 2 | no | — |
| 475 | C8 | — | 4 | 1 | no | — |
| 476 | C8 | — | 4 | 1 | no | — |
| 477 | C8 | — | 8 | 1 | no | — |
| 478 | C8 | — | 8 | 1 | no | — |
| 479 | C8 | #525 | 1 | 1 | yes | characteristic |
| 480 | C8 | #537 | 1 | 1 | yes | characteristic |
| 481 | C8 | #546 | 1 | 1 | yes | characteristic |
| 482 | C8 | #547 | 1 | 1 | yes | characteristic |
| 483 | C22 | — | 4 | 1 | no | — |
| 484 | C22 | — | 8 | 2 | no | — |
| 485 | C22 | — | 8 | 4 | no | — |
| 486 | C22 | — | 8 | 1 | no | — |
| 487 | C22 | — | 8 | 1 | no | — |
| 488 | C22 | — | 8 | 2 | no | — |
| 489 | C22 | — | 16 | 1 | no | — |
| 490 | C22 | — | 32 | 1 | no | — |
| 491 | C22 | — | 32 | 1 | no | — |
| 492 | C22 | #3003 | 1 | 1 | yes | characteristic |
| 493 | C22 | #3366 | 1 | 1 | yes | characteristic |
| 494 | C22 | #3399 | 1 | 1 | yes | characteristic |
| 495 | C4 | — | 4 | 1 | no | — |
| 496 | C4 | — | 4 | 1 | no | — |
| 497 | C4 | — | 4 | 2 | no | — |
| 498 | C4 | — | 4 | 2 | no | — |
| 499 | C4 | — | 8 | 2 | no | — |
| 500 | C4 | — | 8 | 1 | no | — |
| 501 | C4 | — | 8 | 1 | no | — |
| 502 | C4 | — | 8 | 2 | no | — |
| 503 | C4 | — | 16 | 1 | no | — |
| 504 | C4 | #2407 | 1 | 1 | yes | characteristic |
| 505 | C4 | #3008 | 1 | 1 | yes | characteristic |
| 506 | C4 | #3366 | 1 | 1 | yes | characteristic |
| 507 | C4 | #3399 | 1 | 1 | yes | characteristic |
| 508 | C2 | — | 2 | 1 | no | — |
| 509 | C2 | — | 2 | 1 | no | — |
| 510 | C2 | — | 2 | 1 | no | — |
| 511 | C2 | — | 16 | 1 | no | — |
| 512 | C2 | — | 16 | 2 | no | — |
| 513 | C2 | — | 16 | 1 | no | — |
| 514 | C2 | #24196 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24196 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13334 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #61413 | C2 | semidirect |
| 2. #33596 | C2 | semidirect |
| 3. #33471 | C2 | semidirect |
| 4. #29270 | C2 | semidirect |
| 5. #13417 | C2 | semidirect |
| 6. #13414 | C2 | semidirect |
| 8. #13332 | C2 | semidirect |
| 9. #13270 | C2 | semidirect |
| 10. #13216 | C2 | semidirect |
| 11. #13034 | C2 | semidirect |
| 12. #13027 | C2 | semidirect |
| 13. #12954 | C2 | semidirect |
| 14. #12951 | C2 | semidirect |
| 15. #12826 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #61413 | 2 |
| #33596 | 2 |
| #33471 | 2 |
| #29270 | 2 |
| #13417 | 2 |
| #13414 | 2 |
| #13334 | 2 |
| #13332 | 2 |
| #13270 | 2 |
| #13216 | 2 |
| #13034 | 2 |
| #13027 | 2 |
| #12954 | 2 |
| #12951 | 2 |
| #12826 | 2 |