Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2018 conjugacy classes of subgroups.
The subgroup lattice has 544 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×#509 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#3407 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #60450 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#3058 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #33612 | C2 | 1 | 8 | yes | maximal, complemented |
| 6 | #33609 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #33607 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #33605 | C2 | 1 | 8 | yes | maximal, complemented |
| 9 | #33409 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #33380 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | C2×#2140 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C22×#135 | C22 | 1 | 1 | yes | characteristic |
| 13 | C23×D8 | C22 | 1 | 1 | yes | characteristic |
| 14 | C22×#131 | C22 | 1 | 1 | yes | characteristic |
| 15 | #3449 | — | 2 | 1 | no | — |
| 16 | C2×#535 | — | 2 | 1 | no | — |
| 17 | C2×#509 | — | 2 | 8 | no | — |
| 18 | C2×#509 | C22 | 1 | 4 | yes | — |
| 19 | C2×#509 | C22 | 1 | 4 | yes | complemented |
| 20 | C22×SD32 | C22 | 1 | 2 | yes | complemented |
| 21 | C22×D16 | C22 | 1 | 2 | yes | complemented |
| 22 | #3407 | — | 2 | 8 | no | — |
| 23 | C2×#504 | C22 | 1 | 4 | yes | complemented |
| 24 | C2×#504 | C22 | 1 | 4 | yes | complemented |
| 25 | C22×M32 | C22 | 1 | 1 | yes | characteristic |
| 26 | C22×M32 | C22 | 1 | 1 | yes | characteristic |
| 27 | C22×M32 | C22 | 1 | 4 | yes | complemented |
| 28 | #3061 | C22 | 1 | 4 | yes | complemented |
| 29 | #3060 | — | 2 | 2 | no | — |
| 30 | #3059 | C22 | 1 | 2 | yes | complemented |
| 31 | #3058 | — | 2 | 2 | no | — |
| 32 | #3058 | — | 2 | 2 | no | — |
| 33 | #3058 | C22 | 1 | 4 | yes | — |
| 34 | #3058 | C22 | 1 | 4 | yes | complemented |
| 35 | #3058 | C22 | 1 | 2 | yes | complemented |
| 36 | #3057 | — | 2 | 2 | no | — |
| 37 | #3054 | — | 2 | 2 | no | — |
| 38 | C2×#469 | — | 2 | 2 | no | — |
| 39 | #3050 | C22 | 1 | 4 | yes | complemented |
| 40 | C2×#466 | C22 | 1 | 4 | yes | complemented |
| 41 | C2×#427 | C22 | 1 | 1 | yes | characteristic |
| 42 | C2×#427 | C22 | 1 | 1 | yes | characteristic |
| 43 | #2896 | — | 2 | 1 | no | — |
| 44 | C2×#418 | — | 2 | 1 | no | — |
| 45 | #2220 | C22 | 1 | 16 | yes | complemented |
| 46 | #2218 | C22 | 1 | 16 | yes | complemented |
| 47 | #2217 | C22 | 1 | 16 | yes | complemented |
| 48 | #2215 | C22 | 1 | 16 | yes | complemented |
| 49 | #2159 | C22 | 1 | 4 | yes | complemented |
| 50 | #2159 | C22 | 1 | 4 | yes | complemented |
| 51 | #2158 | C22 | 1 | 4 | yes | complemented |
| 52 | #2141 | C22 | 1 | 2 | yes | complemented |
| 53 | #2140 | C22 | 1 | 4 | yes | complemented |
| 54 | #2140 | C22 | 1 | 4 | yes | complemented |
| 55 | #2140 | C22 | 1 | 2 | yes | complemented |
| 56 | #2139 | C22 | 1 | 4 | yes | complemented |
| 57 | #2138 | C22 | 1 | 4 | yes | complemented |
| 58 | C2×#357 | C22 | 1 | 4 | yes | complemented |
| 59 | C22×#41 | — | 2 | 1 | no | — |
| 60 | C22×#41 | D4 | 1 | 1 | yes | characteristic |
| 61 | C23×D4 | — | 2 | 1 | no | — |
| 62 | C2×#135 | — | 2 | 1 | no | — |
| 63 | C2×#135 | — | 2 | 1 | no | — |
| 64 | C2×#135 | — | 2 | 4 | no | — |
| 65 | C2×#135 | — | 2 | 4 | no | — |
| 66 | C2×#135 | C23 | 1 | 2 | yes | — |
| 67 | C2×#135 | C23 | 1 | 2 | yes | — |
| 68 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 69 | C22×SD16 | — | 2 | 1 | no | — |
| 70 | C22×D8 | — | 2 | 4 | no | — |
| 71 | C22×D8 | — | 2 | 1 | no | — |
| 72 | C22×D8 | — | 2 | 1 | no | — |
| 73 | C22×D8 | — | 2 | 1 | no | — |
| 74 | C22×D8 | — | 2 | 4 | no | — |
| 75 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 76 | C22×D8 | C23 | 1 | 2 | yes | — |
| 77 | C22×D8 | C23 | 1 | 2 | yes | — |
| 78 | C22×D8 | C23 | 1 | 2 | yes | — |
| 79 | C2×#131 | — | 2 | 4 | no | — |
| 80 | C2×#131 | — | 2 | 4 | no | — |
| 81 | C2×#131 | D4 | 1 | 2 | yes | — |
| 82 | C2×#131 | D4 | 1 | 2 | yes | — |
| 83 | C2×#131 | C23 | 1 | 2 | yes | — |
| 84 | C2×#131 | C23 | 1 | 2 | yes | — |
| 85 | C22×M16 | D4 | 1 | 1 | yes | characteristic |
| 86 | C22×M16 | D4 | 1 | 2 | yes | — |
| 87 | C23×C8 | C23 | 1 | 2 | yes | — |
| 88 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 89 | #537 | — | 2 | 4 | no | — |
| 90 | #536 | — | 2 | 2 | no | — |
| 91 | #536 | — | 4 | 1 | no | — |
| 92 | #535 | — | 2 | 4 | no | — |
| 93 | #535 | — | 2 | 4 | no | — |
| 94 | #535 | — | 2 | 2 | no | — |
| 95 | #535 | — | 4 | 2 | no | — |
| 96 | #535 | — | 4 | 2 | no | — |
| 97 | #535 | — | 4 | 1 | no | — |
| 98 | #534 | — | 4 | 2 | no | — |
| 99 | #532 | — | 4 | 2 | no | — |
| 100 | C2×#127 | — | 4 | 1 | no | — |
| 101 | C2×#124 | — | 4 | 1 | no | — |
| 102 | #525 | — | 2 | 4 | no | — |
| 103 | #525 | — | 4 | 2 | no | — |
| 104 | C2×#121 | — | 2 | 4 | no | — |
| 105 | C2×#120 | — | 4 | 2 | no | — |
| 106 | C2×#117 | — | 2 | 1 | no | — |
| 107 | C2×#117 | — | 2 | 1 | no | — |
| 108 | #509 | — | 2 | 16 | no | — |
| 109 | #509 | — | 2 | 16 | no | — |
| 110 | C2×SD32 | — | 2 | 8 | no | — |
| 111 | C2×SD32 | C23 | 1 | 8 | yes | — |
| 112 | C2×D16 | — | 2 | 8 | no | — |
| 113 | C2×D16 | C23 | 1 | 8 | yes | — |
| 114 | #504 | — | 2 | 16 | no | — |
| 115 | #504 | — | 2 | 16 | no | — |
| 116 | C2×M32 | — | 2 | 4 | no | — |
| 117 | C2×M32 | — | 2 | 16 | no | — |
| 118 | C2×M32 | D4 | 1 | 8 | yes | complemented |
| 119 | C2×M32 | C23 | 1 | 2 | yes | — |
| 120 | C2×M32 | C23 | 1 | 2 | yes | — |
| 121 | C2×M32 | C23 | 1 | 2 | yes | — |
| 122 | C2×M32 | C23 | 1 | 2 | yes | — |
| 123 | C2×M32 | C23 | 1 | 8 | yes | complemented |
| 124 | C22×C16 | C23 | 1 | 2 | yes | — |
| 125 | C22×C16 | C23 | 1 | 2 | yes | — |
| 126 | C22×C16 | C23 | 1 | 8 | yes | complemented |
| 127 | #470 | — | 2 | 4 | no | — |
| 128 | #469 | — | 2 | 8 | no | — |
| 129 | #469 | — | 2 | 8 | no | — |
| 130 | #469 | — | 2 | 4 | no | — |
| 131 | #468 | — | 2 | 8 | no | — |
| 132 | #467 | C23 | 1 | 8 | yes | complemented |
| 133 | #466 | — | 2 | 8 | no | — |
| 134 | #466 | C23 | 1 | 16 | yes | — |
| 135 | #466 | C23 | 1 | 8 | yes | complemented |
| 136 | #465 | — | 2 | 8 | no | — |
| 137 | #427 | — | 2 | 1 | no | — |
| 138 | #427 | — | 2 | 1 | no | — |
| 139 | #427 | — | 2 | 1 | no | — |
| 140 | #427 | — | 2 | 1 | no | — |
| 141 | #427 | C23 | 1 | 2 | yes | — |
| 142 | #427 | C23 | 1 | 2 | yes | — |
| 143 | #427 | C23 | 1 | 2 | yes | — |
| 144 | #427 | C23 | 1 | 2 | yes | — |
| 145 | C2×#107 | C23 | 1 | 2 | yes | — |
| 146 | C2×#107 | C23 | 1 | 2 | yes | — |
| 147 | C2×#106 | — | 2 | 1 | no | — |
| 148 | C2×#106 | — | 2 | 1 | no | — |
| 149 | #419 | — | 2 | 2 | no | — |
| 150 | #418 | — | 2 | 4 | no | — |
| 151 | #418 | — | 2 | 4 | no | — |
| 152 | #418 | — | 2 | 2 | no | — |
| 153 | #417 | — | 2 | 4 | no | — |
| 154 | #416 | — | 2 | 4 | no | — |
| 155 | C2×#102 | — | 2 | 4 | no | — |
| 156 | #367 | C23 | 1 | 8 | yes | complemented |
| 157 | #366 | C23 | 1 | 8 | yes | complemented |
| 158 | #358 | C23 | 1 | 8 | yes | complemented |
| 159 | #357 | C23 | 1 | 16 | yes | — |
| 160 | #357 | C23 | 1 | 8 | yes | complemented |
| 161 | C25 | — | 4 | 1 | no | — |
| 162 | C2×#41 | — | 2 | 4 | no | — |
| 163 | C2×#41 | — | 2 | 4 | no | — |
| 164 | C2×#41 | — | 2 | 2 | no | — |
| 165 | C2×#41 | — | 2 | 2 | no | — |
| 166 | C2×#41 | — | 4 | 1 | no | — |
| 167 | C2×#41 | — | 4 | 1 | no | — |
| 168 | C2×#41 | — | 4 | 4 | no | — |
| 169 | C2×#41 | — | 4 | 4 | no | — |
| 170 | C2×#41 | D8 | 1 | 2 | yes | — |
| 171 | C2×#41 | D8 | 1 | 2 | yes | — |
| 172 | C2×#41 | C2×D4 | 1 | 2 | yes | — |
| 173 | C2×#41 | C2×D4 | 1 | 2 | yes | — |
| 174 | C22×Q8 | — | 2 | 1 | no | — |
| 175 | C22×Q8 | D8 | 1 | 1 | yes | characteristic |
| 176 | C22×D4 | — | 2 | 2 | no | — |
| 177 | C22×D4 | — | 2 | 2 | no | — |
| 178 | C22×D4 | — | 2 | 2 | no | — |
| 179 | C22×D4 | — | 2 | 1 | no | — |
| 180 | C22×D4 | — | 4 | 1 | no | — |
| 181 | C22×D4 | — | 4 | 4 | no | — |
| 182 | C22×D4 | — | 4 | 4 | no | — |
| 183 | C22×D4 | — | 4 | 1 | no | — |
| 184 | C22×D4 | — | 4 | 1 | no | — |
| 185 | C22×D4 | — | 4 | 1 | no | — |
| 186 | C22×D4 | — | 4 | 4 | no | — |
| 187 | C22×D4 | D8 | 1 | 2 | yes | — |
| 188 | C22×D4 | D8 | 1 | 1 | yes | characteristic |
| 189 | C23×C4 | — | 4 | 1 | no | — |
| 190 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 191 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 192 | #135 | — | 2 | 4 | no | — |
| 193 | #135 | — | 2 | 4 | no | — |
| 194 | #135 | — | 2 | 4 | no | — |
| 195 | #135 | — | 2 | 4 | no | — |
| 196 | #135 | — | 4 | 4 | no | — |
| 197 | #135 | — | 4 | 4 | no | — |
| 198 | C2×Q16 | — | 2 | 2 | no | — |
| 199 | C2×Q16 | — | 2 | 1 | no | — |
| 200 | C2×Q16 | — | 2 | 2 | no | — |
| 201 | C2×Q16 | C24 | 1 | 2 | yes | — |
| 202 | C2×SD16 | — | 2 | 2 | no | — |
| 203 | C2×SD16 | — | 2 | 2 | no | — |
| 204 | C2×SD16 | — | 4 | 4 | no | — |
| 205 | C2×D8 | — | 2 | 4 | no | — |
| 206 | C2×D8 | — | 2 | 4 | no | — |
| 207 | C2×D8 | — | 2 | 4 | no | — |
| 208 | C2×D8 | — | 2 | 2 | no | — |
| 209 | C2×D8 | — | 2 | 2 | no | — |
| 210 | C2×D8 | — | 2 | 2 | no | — |
| 211 | C2×D8 | — | 2 | 2 | no | — |
| 212 | C2×D8 | — | 2 | 1 | no | — |
| 213 | C2×D8 | — | 2 | 4 | no | — |
| 214 | C2×D8 | — | 2 | 4 | no | — |
| 215 | C2×D8 | — | 2 | 4 | no | — |
| 216 | C2×D8 | — | 2 | 2 | no | — |
| 217 | C2×D8 | — | 4 | 4 | no | — |
| 218 | C2×D8 | — | 4 | 4 | no | — |
| 219 | C2×D8 | — | 4 | 4 | no | — |
| 220 | C2×D8 | C24 | 1 | 4 | yes | — |
| 221 | C2×D8 | C24 | 1 | 2 | yes | — |
| 222 | #131 | — | 2 | 8 | no | — |
| 223 | #131 | — | 2 | 8 | no | — |
| 224 | #131 | — | 2 | 4 | no | — |
| 225 | #131 | — | 2 | 4 | no | — |
| 226 | #131 | — | 2 | 4 | no | — |
| 227 | #131 | — | 2 | 4 | no | — |
| 228 | C2×M16 | — | 2 | 4 | no | — |
| 229 | C2×M16 | — | 2 | 8 | no | — |
| 230 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 231 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 232 | C2×M16 | C2×D4 | 1 | 4 | yes | — |
| 233 | C2×M16 | C2×D4 | 1 | 4 | yes | — |
| 234 | C22×C8 | — | 2 | 4 | no | — |
| 235 | C22×C8 | — | 2 | 4 | no | — |
| 236 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 237 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 238 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 239 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 240 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 241 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 242 | C22×C8 | #41 | 1 | 4 | yes | — |
| 243 | C22×C8 | C24 | 1 | 4 | yes | — |
| 244 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 245 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 246 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 247 | C4⋊Q8 | — | 4 | 2 | no | — |
| 248 | #127 | — | 4 | 4 | no | — |
| 249 | #127 | — | 4 | 4 | no | — |
| 250 | #127 | — | 4 | 2 | no | — |
| 251 | #124 | — | 4 | 4 | no | — |
| 252 | #124 | — | 4 | 4 | no | — |
| 253 | #124 | — | 4 | 2 | no | — |
| 254 | #124 | — | 4 | 2 | no | — |
| 255 | #123 | — | 4 | 4 | no | — |
| 256 | #123 | — | 4 | 4 | no | — |
| 257 | C22⋊Q8 | — | 2 | 8 | no | — |
| 258 | C22⋊Q8 | — | 4 | 4 | no | — |
| 259 | #121 | — | 2 | 16 | no | — |
| 260 | #121 | — | 2 | 8 | no | — |
| 261 | #121 | — | 4 | 4 | no | — |
| 262 | #121 | — | 4 | 8 | no | — |
| 263 | #121 | — | 4 | 8 | no | — |
| 264 | #120 | — | 4 | 4 | no | — |
| 265 | #120 | — | 4 | 8 | no | — |
| 266 | #120 | — | 4 | 8 | no | — |
| 267 | #120 | — | 4 | 4 | no | — |
| 268 | C4×D4 | — | 4 | 8 | no | — |
| 269 | #117 | — | 2 | 2 | no | — |
| 270 | #117 | — | 2 | 2 | no | — |
| 271 | #117 | — | 2 | 2 | no | — |
| 272 | #117 | — | 2 | 2 | no | — |
| 273 | #117 | — | 4 | 1 | no | — |
| 274 | #117 | — | 4 | 1 | no | — |
| 275 | #117 | — | 4 | 1 | no | — |
| 276 | #117 | — | 4 | 1 | no | — |
| 277 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 278 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 279 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 280 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 281 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 282 | C2×C42 | — | 4 | 1 | no | — |
| 283 | C2×C42 | — | 4 | 1 | no | — |
| 284 | SD32 | — | 2 | 16 | no | — |
| 285 | D16 | — | 2 | 16 | no | — |
| 286 | M32 | — | 2 | 4 | no | — |
| 287 | M32 | — | 2 | 4 | no | — |
| 288 | M32 | — | 2 | 16 | no | — |
| 289 | M32 | C2×D4 | 1 | 8 | yes | — |
| 290 | M32 | C2×D4 | 1 | 8 | yes | complemented |
| 291 | C2×C16 | — | 2 | 4 | no | — |
| 292 | C2×C16 | — | 2 | 16 | no | — |
| 293 | C2×C16 | C2×D4 | 1 | 8 | yes | complemented |
| 294 | C2×C16 | C24 | 1 | 4 | yes | — |
| 295 | C2×C16 | C24 | 1 | 4 | yes | — |
| 296 | #107 | — | 2 | 2 | no | — |
| 297 | #107 | — | 2 | 2 | no | — |
| 298 | #107 | C24 | 1 | 4 | yes | — |
| 299 | #107 | C24 | 1 | 4 | yes | — |
| 300 | #106 | — | 2 | 2 | no | — |
| 301 | #106 | — | 2 | 2 | no | — |
| 302 | #106 | — | 2 | 2 | no | — |
| 303 | #106 | — | 2 | 2 | no | — |
| 304 | #103 | — | 2 | 8 | no | — |
| 305 | #102 | — | 2 | 16 | no | — |
| 306 | #102 | — | 2 | 8 | no | — |
| 307 | C24 | — | 4 | 2 | no | — |
| 308 | C24 | — | 4 | 1 | no | — |
| 309 | C24 | — | 4 | 2 | no | — |
| 310 | C24 | — | 4 | 2 | no | — |
| 311 | C24 | — | 8 | 4 | no | — |
| 312 | C24 | — | 16 | 4 | no | — |
| 313 | C24 | C2×D8 | 1 | 2 | yes | — |
| 314 | C24 | C2×D8 | 1 | 1 | yes | characteristic |
| 315 | #41 | — | 2 | 4 | no | — |
| 316 | #41 | — | 2 | 4 | no | — |
| 317 | #41 | — | 2 | 4 | no | — |
| 318 | #41 | — | 2 | 4 | no | — |
| 319 | #41 | — | 2 | 8 | no | — |
| 320 | #41 | — | 2 | 8 | no | — |
| 321 | #41 | — | 4 | 4 | no | — |
| 322 | #41 | — | 4 | 4 | no | — |
| 323 | #41 | — | 4 | 4 | no | — |
| 324 | #41 | — | 4 | 4 | no | — |
| 325 | #41 | — | 8 | 4 | no | — |
| 326 | #41 | — | 8 | 4 | no | — |
| 327 | C2×Q8 | — | 2 | 4 | no | — |
| 328 | C2×Q8 | — | 2 | 2 | no | — |
| 329 | C2×Q8 | — | 4 | 1 | no | — |
| 330 | C2×Q8 | — | 4 | 2 | no | — |
| 331 | C2×Q8 | — | 4 | 2 | no | — |
| 332 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 333 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 334 | C2×D4 | — | 2 | 8 | no | — |
| 335 | C2×D4 | — | 2 | 4 | no | — |
| 336 | C2×D4 | — | 2 | 2 | no | — |
| 337 | C2×D4 | — | 2 | 4 | no | — |
| 338 | C2×D4 | — | 4 | 2 | no | — |
| 339 | C2×D4 | — | 4 | 2 | no | — |
| 340 | C2×D4 | — | 4 | 8 | no | — |
| 341 | C2×D4 | — | 4 | 16 | no | — |
| 342 | C2×D4 | — | 4 | 4 | no | — |
| 343 | C2×D4 | — | 4 | 4 | no | — |
| 344 | C2×D4 | — | 4 | 4 | no | — |
| 345 | C2×D4 | — | 4 | 2 | no | — |
| 346 | C2×D4 | — | 4 | 2 | no | — |
| 347 | C2×D4 | — | 4 | 2 | no | — |
| 348 | C2×D4 | — | 4 | 2 | no | — |
| 349 | C2×D4 | — | 4 | 1 | no | — |
| 350 | C2×D4 | — | 4 | 4 | no | — |
| 351 | C2×D4 | — | 4 | 4 | no | — |
| 352 | C2×D4 | — | 4 | 4 | no | — |
| 353 | C2×D4 | — | 4 | 2 | no | — |
| 354 | C2×D4 | — | 8 | 4 | no | — |
| 355 | C2×D4 | — | 8 | 4 | no | — |
| 356 | C2×D4 | — | 8 | 4 | no | — |
| 357 | C2×D4 | — | 8 | 4 | no | — |
| 358 | C2×D4 | — | 16 | 8 | no | — |
| 359 | C2×D4 | C2×D8 | 1 | 4 | yes | — |
| 360 | C2×D4 | C2×D8 | 1 | 4 | yes | — |
| 361 | C2×D4 | C2×D8 | 1 | 2 | yes | — |
| 362 | C2×D4 | C2×D8 | 1 | 2 | yes | — |
| 363 | C22×C4 | — | 2 | 4 | no | — |
| 364 | C22×C4 | — | 2 | 4 | no | — |
| 365 | C22×C4 | — | 4 | 2 | no | — |
| 366 | C22×C4 | — | 4 | 1 | no | — |
| 367 | C22×C4 | — | 4 | 2 | no | — |
| 368 | C22×C4 | — | 4 | 2 | no | — |
| 369 | C22×C4 | — | 4 | 2 | no | — |
| 370 | C22×C4 | — | 8 | 4 | no | — |
| 371 | C22×C4 | #121 | 1 | 4 | yes | — |
| 372 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 373 | C22×C4 | C2×D8 | 1 | 4 | yes | — |
| 374 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 375 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 376 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 377 | C22×C4 | #135 | 1 | 4 | yes | — |
| 378 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 379 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 380 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 381 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 382 | Q16 | — | 2 | 2 | no | — |
| 383 | Q16 | — | 2 | 2 | no | — |
| 384 | Q16 | — | 4 | 2 | no | — |
| 385 | SD16 | — | 4 | 4 | no | — |
| 386 | SD16 | — | 4 | 4 | no | — |
| 387 | D8 | — | 2 | 2 | no | — |
| 388 | D8 | — | 2 | 4 | no | — |
| 389 | D8 | — | 2 | 4 | no | — |
| 390 | D8 | — | 2 | 2 | no | — |
| 391 | D8 | — | 4 | 4 | no | — |
| 392 | D8 | — | 4 | 4 | no | — |
| 393 | D8 | — | 4 | 4 | no | — |
| 394 | D8 | — | 4 | 2 | no | — |
| 395 | M16 | — | 2 | 4 | no | — |
| 396 | M16 | — | 2 | 4 | no | — |
| 397 | M16 | — | 2 | 8 | no | — |
| 398 | M16 | — | 2 | 8 | no | — |
| 399 | C2×C8 | — | 2 | 4 | no | — |
| 400 | C2×C8 | — | 2 | 8 | no | — |
| 401 | C2×C8 | — | 2 | 4 | no | — |
| 402 | C2×C8 | — | 2 | 4 | no | — |
| 403 | C2×C8 | #121 | 1 | 4 | yes | — |
| 404 | C2×C8 | #121 | 1 | 4 | yes | — |
| 405 | C2×C8 | #121 | 1 | 4 | yes | — |
| 406 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 407 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 408 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 409 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 410 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 411 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 412 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 413 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 414 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 415 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 416 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 417 | C4⋊C4 | — | 2 | 4 | no | — |
| 418 | C4⋊C4 | — | 2 | 4 | no | — |
| 419 | C4⋊C4 | — | 4 | 2 | no | — |
| 420 | C4⋊C4 | — | 4 | 2 | no | — |
| 421 | C4⋊C4 | — | 4 | 8 | no | — |
| 422 | C22⋊C4 | — | 4 | 2 | no | — |
| 423 | C22⋊C4 | — | 4 | 2 | no | — |
| 424 | C22⋊C4 | — | 4 | 2 | no | — |
| 425 | C22⋊C4 | — | 4 | 2 | no | — |
| 426 | C22⋊C4 | — | 4 | 8 | no | — |
| 427 | C22⋊C4 | — | 4 | 16 | no | — |
| 428 | C22⋊C4 | — | 4 | 8 | no | — |
| 429 | C42 | — | 4 | 2 | no | — |
| 430 | C42 | — | 4 | 2 | no | — |
| 431 | C42 | — | 4 | 2 | no | — |
| 432 | C42 | — | 4 | 2 | no | — |
| 433 | C16 | — | 2 | 4 | no | — |
| 434 | C16 | C22×D4 | 1 | 8 | yes | — |
| 435 | C23 | — | 2 | 4 | no | — |
| 436 | C23 | — | 4 | 2 | no | — |
| 437 | C23 | — | 4 | 4 | no | — |
| 438 | C23 | — | 4 | 2 | no | — |
| 439 | C23 | — | 4 | 4 | no | — |
| 440 | C23 | — | 8 | 8 | no | — |
| 441 | C23 | — | 8 | 2 | no | — |
| 442 | C23 | — | 8 | 4 | no | — |
| 443 | C23 | — | 8 | 4 | no | — |
| 444 | C23 | — | 8 | 4 | no | — |
| 445 | C23 | — | 16 | 2 | no | — |
| 446 | C23 | — | 16 | 4 | no | — |
| 447 | C23 | — | 16 | 4 | no | — |
| 448 | C23 | — | 16 | 16 | no | — |
| 449 | C23 | #466 | 1 | 4 | yes | — |
| 450 | C23 | C22×D8 | 1 | 4 | yes | — |
| 451 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 452 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 453 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 454 | Q8 | — | 2 | 4 | no | — |
| 455 | Q8 | — | 2 | 4 | no | — |
| 456 | Q8 | — | 4 | 2 | no | — |
| 457 | Q8 | — | 4 | 2 | no | — |
| 458 | Q8 | — | 8 | 2 | no | — |
| 459 | D4 | — | 2 | 8 | no | — |
| 460 | D4 | — | 2 | 8 | no | — |
| 461 | D4 | — | 2 | 4 | no | — |
| 462 | D4 | — | 2 | 4 | no | — |
| 463 | D4 | — | 4 | 4 | no | — |
| 464 | D4 | — | 4 | 2 | no | — |
| 465 | D4 | — | 4 | 4 | no | — |
| 466 | D4 | — | 4 | 2 | no | — |
| 467 | D4 | — | 8 | 4 | no | — |
| 468 | D4 | — | 8 | 4 | no | — |
| 469 | D4 | — | 8 | 4 | no | — |
| 470 | D4 | — | 8 | 4 | no | — |
| 471 | D4 | — | 8 | 4 | no | — |
| 472 | D4 | — | 8 | 2 | no | — |
| 473 | D4 | — | 16 | 8 | no | — |
| 474 | D4 | — | 16 | 16 | no | — |
| 475 | C2×C4 | — | 2 | 4 | no | — |
| 476 | C2×C4 | — | 2 | 4 | no | — |
| 477 | C2×C4 | — | 2 | 8 | no | — |
| 478 | C2×C4 | — | 2 | 4 | no | — |
| 479 | C2×C4 | — | 4 | 4 | no | — |
| 480 | C2×C4 | — | 4 | 4 | no | — |
| 481 | C2×C4 | — | 4 | 2 | no | — |
| 482 | C2×C4 | — | 8 | 4 | no | — |
| 483 | C2×C4 | — | 8 | 2 | no | — |
| 484 | C2×C4 | — | 8 | 4 | no | — |
| 485 | C2×C4 | — | 8 | 4 | no | — |
| 486 | C2×C4 | — | 16 | 2 | no | — |
| 487 | C2×C4 | #466 | 1 | 4 | yes | — |
| 488 | C2×C4 | #466 | 1 | 4 | yes | — |
| 489 | C2×C4 | #466 | 1 | 4 | yes | — |
| 490 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 491 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 492 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 493 | C2×C4 | C22×D8 | 1 | 4 | yes | — |
| 494 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 495 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 496 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 497 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 498 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 499 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 500 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 501 | C8 | — | 2 | 4 | no | — |
| 502 | C8 | C2×#121 | 1 | 2 | yes | — |
| 503 | C8 | C2×#121 | 1 | 2 | yes | — |
| 504 | C8 | C2×#121 | 1 | 2 | yes | — |
| 505 | C8 | C23×D4 | 1 | 1 | yes | characteristic |
| 506 | C8 | C22×#41 | 1 | 1 | yes | characteristic |
| 507 | C22 | — | 2 | 2 | no | — |
| 508 | C22 | — | 2 | 2 | no | — |
| 509 | C22 | — | 2 | 2 | no | — |
| 510 | C22 | — | 2 | 4 | no | — |
| 511 | C22 | — | 2 | 8 | no | — |
| 512 | C22 | — | 4 | 4 | no | — |
| 513 | C22 | — | 8 | 4 | no | — |
| 514 | C22 | — | 8 | 2 | no | — |
| 515 | C22 | — | 8 | 4 | no | — |
| 516 | C22 | — | 8 | 8 | no | — |
| 517 | C22 | — | 8 | 4 | no | — |
| 518 | C22 | — | 16 | 8 | no | — |
| 519 | C22 | — | 16 | 4 | no | — |
| 520 | C22 | — | 16 | 8 | no | — |
| 521 | C22 | — | 16 | 8 | no | — |
| 522 | C22 | C2×#466 | 1 | 2 | yes | — |
| 523 | C22 | C2×#466 | 1 | 2 | yes | — |
| 524 | C22 | C2×#466 | 1 | 2 | yes | — |
| 525 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 526 | C4 | — | 2 | 4 | no | — |
| 527 | C4 | — | 8 | 2 | no | — |
| 528 | C4 | — | 8 | 4 | no | — |
| 529 | C4 | — | 8 | 4 | no | — |
| 530 | C4 | C2×#466 | 1 | 2 | yes | — |
| 531 | C4 | C2×#466 | 1 | 2 | yes | — |
| 532 | C4 | C2×#466 | 1 | 2 | yes | — |
| 533 | C4 | C22×#121 | 1 | 1 | yes | characteristic |
| 534 | C4 | C22×#135 | 1 | 1 | yes | characteristic |
| 535 | C2 | — | 2 | 2 | no | — |
| 536 | C2 | — | 2 | 2 | no | — |
| 537 | C2 | — | 2 | 2 | no | — |
| 538 | C2 | — | 4 | 4 | no | — |
| 539 | C2 | — | 16 | 2 | no | — |
| 540 | C2 | — | 16 | 4 | no | — |
| 541 | C2 | C22×#466 | 1 | 1 | yes | characteristic |
| 542 | C2 | #62636 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#509 | C2 | semidirect |
| 2. C2×#3407 | C2 | semidirect |
| 3. #60450 | C2 | semidirect |
| 4. C2×#3058 | C2 | semidirect |
| 5. #33612 | C2 | semidirect |
| 6. #33609 | C2 | semidirect |
| 7. #33607 | C2 | semidirect |
| 8. #33605 | C2 | semidirect |
| 9. #33409 | C2 | semidirect |
| 10. #33380 | C2 | semidirect |
| 11. C2×#2140 | C2 | semidirect |
| 19. C2×#509 | C22 | semidirect |
| 20. C22×SD32 | C22 | semidirect |
| 21. C22×D16 | C22 | semidirect |
| 23. C2×#504 | C22 | semidirect |
| 24. C2×#504 | C22 | semidirect |
| 27. C22×M32 | C22 | semidirect |
| 28. #3061 | C22 | semidirect |
| 30. #3059 | C22 | semidirect |
| 34. #3058 | C22 | semidirect |
| 35. #3058 | C22 | semidirect |
| 39. #3050 | C22 | semidirect |
| 40. C2×#466 | C22 | semidirect |
| 45. #2220 | C22 | semidirect |
| 46. #2218 | C22 | semidirect |
| 47. #2217 | C22 | semidirect |
| 48. #2215 | C22 | semidirect |
| 49. #2159 | C22 | semidirect |
| 50. #2159 | C22 | semidirect |
| 51. #2158 | C22 | semidirect |
| 52. #2141 | C22 | semidirect |
| 53. #2140 | C22 | semidirect |
| 54. #2140 | C22 | semidirect |
| 55. #2140 | C22 | semidirect |
| 56. #2139 | C22 | semidirect |
| 57. #2138 | C22 | semidirect |
| 58. C2×#357 | C22 | semidirect |
| 118. C2×M32 | D4 | semidirect |
| 123. C2×M32 | C23 | semidirect |
| 126. C22×C16 | C23 | semidirect |
| 132. #467 | C23 | semidirect |
| 135. #466 | C23 | semidirect |
| 156. #367 | C23 | semidirect |
| 157. #366 | C23 | semidirect |
| 158. #358 | C23 | semidirect |
| 160. #357 | C23 | semidirect |
| 290. M32 | C2×D4 | semidirect |
| 293. C2×C16 | C2×D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#509 | 2 |
| C2×#3407 | 2 |
| #60450 | 2 |
| C2×#3058 | 2 |
| #33612 | 2 |
| #33609 | 2 |
| #33607 | 2 |
| #33605 | 2 |
| #33409 | 2 |
| #33380 | 2 |
| C2×#2140 | 2 |