Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1043 conjugacy classes of subgroups.
The subgroup lattice has 590 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #33175 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33174 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33171 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #33042 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #29842 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #29806 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×#1721 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×C8×M16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #9832 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #9830 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #9824 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #9824 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | #9822 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | #9820 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | #8820 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#492 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#486 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#486 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#469 | — | 2 | 2 | no | — |
| 23 | C2×#468 | — | 2 | 2 | no | — |
| 24 | C2×#466 | — | 2 | 2 | no | — |
| 25 | C2×#431 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×C4×M16 | C22 | 1 | 1 | yes | characteristic |
| 27 | C22×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2857 | C22 | 1 | 1 | yes | characteristic |
| 29 | #2856 | C22 | 1 | 1 | yes | characteristic |
| 30 | #2856 | C22 | 1 | 1 | yes | characteristic |
| 31 | #2856 | C22 | 1 | 1 | yes | characteristic |
| 32 | #2067 | — | 2 | 2 | no | — |
| 33 | #2054 | — | 2 | 2 | no | — |
| 34 | #2033 | — | 2 | 2 | no | — |
| 35 | #1999 | — | 2 | 2 | no | — |
| 36 | #1990 | C22 | 1 | 1 | yes | characteristic |
| 37 | #1990 | C22 | 1 | 1 | yes | characteristic |
| 38 | #1973 | — | 2 | 2 | no | — |
| 39 | #1972 | — | 2 | 2 | no | — |
| 40 | #1970 | — | 2 | 2 | no | — |
| 41 | #1957 | C22 | 1 | 1 | yes | characteristic |
| 42 | #1957 | C22 | 1 | 1 | yes | characteristic |
| 43 | #1957 | C22 | 1 | 1 | yes | characteristic |
| 44 | #1957 | C22 | 1 | 1 | yes | characteristic |
| 45 | #1957 | C22 | 1 | 1 | yes | characteristic |
| 46 | #1957 | C22 | 1 | 1 | yes | characteristic |
| 47 | #1957 | C22 | 1 | 1 | yes | characteristic |
| 48 | #1957 | C22 | 1 | 1 | yes | characteristic |
| 49 | #1957 | C22 | 1 | 1 | yes | characteristic |
| 50 | #1956 | C22 | 1 | 1 | yes | characteristic |
| 51 | #1934 | — | 2 | 2 | no | — |
| 52 | #1893 | — | 2 | 2 | no | — |
| 53 | #1845 | C22 | 1 | 1 | yes | characteristic |
| 54 | #1845 | C22 | 1 | 1 | yes | characteristic |
| 55 | #1845 | C22 | 1 | 1 | yes | characteristic |
| 56 | #1844 | C22 | 1 | 1 | yes | characteristic |
| 57 | #1834 | C22 | 1 | 1 | yes | characteristic |
| 58 | #1834 | C22 | 1 | 1 | yes | characteristic |
| 59 | #1721 | — | 2 | 4 | no | — |
| 60 | C8×M16 | — | 2 | 4 | no | — |
| 61 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 62 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 63 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 64 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 65 | C22×SD16 | — | 2 | 2 | no | — |
| 66 | C22×D8 | — | 2 | 2 | no | — |
| 67 | C22×D8 | — | 2 | 2 | no | — |
| 68 | C22×D8 | — | 2 | 2 | no | — |
| 69 | C22×M16 | D4 | 1 | 2 | yes | — |
| 70 | C23×C8 | D4 | 1 | 2 | yes | — |
| 71 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 72 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 73 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 74 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 75 | C2×C22⋊Q8 | — | 2 | 1 | no | — |
| 76 | C2×#121 | — | 2 | 1 | no | — |
| 77 | C2×#121 | — | 2 | 1 | no | — |
| 78 | C2×#121 | — | 2 | 1 | no | — |
| 79 | C2×#121 | — | 2 | 1 | no | — |
| 80 | C2×#121 | — | 2 | 1 | no | — |
| 81 | C2×#121 | — | 2 | 1 | no | — |
| 82 | C2×#121 | — | 2 | 1 | no | — |
| 83 | C2×#121 | — | 2 | 1 | no | — |
| 84 | C2×#121 | — | 2 | 1 | no | — |
| 85 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 86 | #493 | — | 2 | 2 | no | — |
| 87 | #493 | — | 2 | 2 | no | — |
| 88 | #493 | — | 2 | 2 | no | — |
| 89 | #493 | — | 2 | 2 | no | — |
| 90 | #493 | — | 2 | 2 | no | — |
| 91 | #493 | — | 2 | 2 | no | — |
| 92 | #492 | — | 2 | 2 | no | — |
| 93 | #492 | — | 2 | 2 | no | — |
| 94 | #486 | — | 2 | 2 | no | — |
| 95 | #486 | — | 2 | 2 | no | — |
| 96 | #486 | — | 2 | 2 | no | — |
| 97 | #486 | — | 2 | 2 | no | — |
| 98 | #469 | — | 4 | 8 | no | — |
| 99 | #468 | — | 4 | 8 | no | — |
| 100 | #466 | — | 4 | 8 | no | — |
| 101 | #431 | — | 2 | 2 | no | — |
| 102 | #431 | — | 2 | 2 | no | — |
| 103 | C2×#107 | — | 2 | 2 | no | — |
| 104 | C2×#107 | — | 2 | 2 | no | — |
| 105 | C2×#107 | — | 2 | 2 | no | — |
| 106 | C2×#106 | — | 2 | 2 | no | — |
| 107 | C2×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 108 | C2×#103 | — | 2 | 2 | no | — |
| 109 | C2×#102 | — | 2 | 2 | no | — |
| 110 | C2×#102 | — | 2 | 2 | no | — |
| 111 | C2×#102 | — | 2 | 2 | no | — |
| 112 | C2×#102 | — | 2 | 2 | no | — |
| 113 | C2×#102 | — | 2 | 2 | no | — |
| 114 | C2×#102 | — | 2 | 2 | no | — |
| 115 | C2×#102 | — | 2 | 2 | no | — |
| 116 | C2×#102 | — | 2 | 2 | no | — |
| 117 | C2×#102 | — | 2 | 2 | no | — |
| 118 | C2×#102 | — | 2 | 2 | no | — |
| 119 | C2×#102 | — | 2 | 2 | no | — |
| 120 | C2×C22⋊C8 | D4 | 1 | 2 | yes | — |
| 121 | C4×M16 | — | 2 | 2 | no | — |
| 122 | C4×M16 | — | 2 | 2 | no | — |
| 123 | C2×#97 | D4 | 1 | 2 | yes | — |
| 124 | C2×C4×C8 | — | 2 | 2 | no | — |
| 125 | C2×C4×C8 | — | 2 | 2 | no | — |
| 126 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 127 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 133 | #391 | — | 2 | 1 | no | — |
| 134 | #390 | — | 2 | 1 | no | — |
| 135 | #390 | — | 2 | 1 | no | — |
| 136 | #390 | — | 2 | 1 | no | — |
| 137 | #390 | — | 2 | 1 | no | — |
| 138 | #390 | — | 2 | 1 | no | — |
| 139 | #390 | — | 2 | 1 | no | — |
| 140 | #390 | — | 2 | 1 | no | — |
| 141 | #390 | — | 2 | 1 | no | — |
| 142 | #390 | — | 2 | 1 | no | — |
| 143 | #389 | — | 2 | 1 | no | — |
| 144 | #389 | — | 2 | 1 | no | — |
| 145 | #384 | C23 | 1 | 1 | yes | characteristic |
| 146 | #384 | C23 | 1 | 1 | yes | characteristic |
| 147 | #384 | C23 | 1 | 1 | yes | characteristic |
| 148 | #384 | C23 | 1 | 1 | yes | characteristic |
| 149 | #380 | — | 2 | 1 | no | — |
| 150 | #380 | — | 2 | 1 | no | — |
| 151 | #340 | — | 2 | 2 | no | — |
| 152 | #340 | — | 2 | 2 | no | — |
| 153 | #340 | — | 2 | 2 | no | — |
| 154 | #340 | — | 2 | 2 | no | — |
| 155 | #322 | — | 2 | 2 | no | — |
| 156 | #322 | D4 | 1 | 4 | yes | complemented |
| 157 | C82 | — | 2 | 2 | no | — |
| 158 | C82 | — | 2 | 2 | no | — |
| 159 | C22×Q8 | — | 2 | 1 | no | — |
| 160 | C22×D4 | — | 2 | 1 | no | — |
| 161 | C22×D4 | — | 2 | 1 | no | — |
| 162 | C22×D4 | — | 2 | 1 | no | — |
| 163 | C22×D4 | — | 2 | 1 | no | — |
| 164 | C22×D4 | — | 2 | 1 | no | — |
| 165 | C22×D4 | — | 2 | 1 | no | — |
| 166 | C22×D4 | — | 2 | 1 | no | — |
| 167 | C22×D4 | — | 2 | 1 | no | — |
| 168 | C22×D4 | — | 2 | 1 | no | — |
| 169 | C22×D4 | — | 4 | 1 | no | — |
| 170 | C22×D4 | — | 4 | 1 | no | — |
| 171 | C22×D4 | — | 4 | 1 | no | — |
| 172 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 173 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 174 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 175 | C2×SD16 | — | 4 | 4 | no | — |
| 176 | C2×SD16 | — | 4 | 4 | no | — |
| 177 | C2×SD16 | — | 4 | 4 | no | — |
| 178 | C2×D8 | — | 4 | 4 | no | — |
| 179 | C2×D8 | — | 4 | 4 | no | — |
| 180 | C2×D8 | — | 4 | 4 | no | — |
| 181 | C2×D8 | — | 4 | 4 | no | — |
| 182 | C2×D8 | — | 4 | 4 | no | — |
| 183 | C2×D8 | — | 4 | 4 | no | — |
| 184 | C2×D8 | — | 4 | 4 | no | — |
| 185 | C2×D8 | — | 8 | 2 | no | — |
| 186 | C2×D8 | — | 8 | 2 | no | — |
| 187 | C2×M16 | — | 2 | 4 | no | — |
| 188 | C2×M16 | — | 2 | 4 | no | — |
| 189 | C2×M16 | D8 | 1 | 8 | yes | complemented |
| 190 | C22×C8 | — | 2 | 4 | no | — |
| 191 | C22×C8 | — | 2 | 4 | no | — |
| 192 | C22×C8 | — | 4 | 2 | no | — |
| 193 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 194 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 195 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 196 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 197 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 198 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 199 | C4⋊Q8 | — | 2 | 1 | no | — |
| 200 | C4⋊Q8 | — | 2 | 1 | no | — |
| 201 | C4⋊Q8 | — | 2 | 1 | no | — |
| 202 | C4⋊Q8 | — | 2 | 1 | no | — |
| 203 | #127 | — | 2 | 1 | no | — |
| 204 | #127 | — | 2 | 1 | no | — |
| 205 | #127 | — | 2 | 1 | no | — |
| 206 | #127 | — | 2 | 1 | no | — |
| 207 | #127 | — | 2 | 1 | no | — |
| 208 | #127 | — | 2 | 1 | no | — |
| 209 | #127 | — | 2 | 1 | no | — |
| 210 | #127 | — | 2 | 1 | no | — |
| 211 | #127 | — | 2 | 1 | no | — |
| 212 | #127 | — | 2 | 1 | no | — |
| 213 | #127 | — | 2 | 1 | no | — |
| 214 | #127 | — | 2 | 1 | no | — |
| 215 | C22⋊Q8 | — | 4 | 4 | no | — |
| 216 | #121 | — | 4 | 4 | no | — |
| 217 | #121 | — | 4 | 4 | no | — |
| 218 | #121 | — | 4 | 4 | no | — |
| 219 | #121 | — | 4 | 4 | no | — |
| 220 | #121 | — | 4 | 4 | no | — |
| 221 | #121 | — | 4 | 4 | no | — |
| 222 | #121 | — | 8 | 2 | no | — |
| 223 | #121 | — | 8 | 2 | no | — |
| 224 | #121 | — | 8 | 2 | no | — |
| 225 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 226 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 227 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 228 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 229 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 230 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 231 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 232 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 233 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 234 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 235 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 236 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 237 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 238 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 239 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 240 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 241 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 242 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 243 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 244 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 245 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 246 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 247 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 248 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 249 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 250 | C2×C42 | — | 2 | 1 | no | — |
| 251 | C2×C42 | — | 2 | 1 | no | — |
| 252 | C2×C42 | — | 2 | 1 | no | — |
| 253 | C2×C42 | — | 2 | 1 | no | — |
| 254 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 255 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 256 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 257 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 258 | #107 | — | 4 | 4 | no | — |
| 259 | #107 | — | 4 | 4 | no | — |
| 260 | #107 | — | 8 | 2 | no | — |
| 261 | #106 | — | 4 | 4 | no | — |
| 262 | C4⋊C8 | — | 2 | 4 | no | — |
| 263 | #103 | — | 4 | 4 | no | — |
| 264 | #102 | — | 4 | 4 | no | — |
| 265 | #102 | — | 4 | 4 | no | — |
| 266 | #102 | — | 4 | 4 | no | — |
| 267 | #102 | — | 4 | 4 | no | — |
| 268 | #102 | — | 4 | 4 | no | — |
| 269 | #102 | — | 4 | 4 | no | — |
| 270 | #102 | — | 4 | 4 | no | — |
| 271 | #102 | — | 4 | 4 | no | — |
| 272 | #102 | — | 8 | 2 | no | — |
| 273 | #102 | — | 8 | 2 | no | — |
| 274 | #102 | — | 8 | 2 | no | — |
| 275 | C22⋊C8 | — | 2 | 4 | no | — |
| 276 | #97 | — | 2 | 4 | no | — |
| 277 | C4×C8 | — | 2 | 4 | no | — |
| 278 | C4×C8 | — | 2 | 1 | no | — |
| 279 | C4×C8 | — | 2 | 1 | no | — |
| 280 | C4×C8 | — | 2 | 1 | no | — |
| 281 | C4×C8 | — | 2 | 1 | no | — |
| 282 | C4×C8 | — | 2 | 1 | no | — |
| 283 | C4×C8 | — | 2 | 1 | no | — |
| 284 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 285 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 286 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 287 | C4×C8 | #41 | 1 | 2 | yes | — |
| 288 | C4×C8 | #41 | 1 | 2 | yes | — |
| 289 | C4×C8 | #41 | 1 | 2 | yes | — |
| 290 | #95 | — | 4 | 1 | no | — |
| 291 | #95 | — | 4 | 1 | no | — |
| 292 | C24 | — | 4 | 1 | no | — |
| 293 | C24 | — | 4 | 1 | no | — |
| 294 | C24 | — | 4 | 1 | no | — |
| 295 | C24 | #127 | 1 | 1 | yes | characteristic |
| 296 | C2×Q8 | — | 4 | 1 | no | — |
| 297 | C2×Q8 | — | 4 | 1 | no | — |
| 298 | C2×Q8 | — | 4 | 2 | no | — |
| 299 | C2×Q8 | — | 4 | 2 | no | — |
| 300 | C2×D4 | — | 4 | 1 | no | — |
| 301 | C2×D4 | — | 4 | 1 | no | — |
| 302 | C2×D4 | — | 4 | 1 | no | — |
| 303 | C2×D4 | — | 4 | 1 | no | — |
| 304 | C2×D4 | — | 4 | 1 | no | — |
| 305 | C2×D4 | — | 4 | 1 | no | — |
| 306 | C2×D4 | — | 4 | 1 | no | — |
| 307 | C2×D4 | — | 4 | 1 | no | — |
| 308 | C2×D4 | — | 4 | 1 | no | — |
| 309 | C2×D4 | — | 4 | 1 | no | — |
| 310 | C2×D4 | — | 4 | 1 | no | — |
| 311 | C2×D4 | — | 4 | 1 | no | — |
| 312 | C2×D4 | — | 4 | 1 | no | — |
| 313 | C2×D4 | — | 4 | 1 | no | — |
| 314 | C2×D4 | — | 4 | 1 | no | — |
| 315 | C2×D4 | — | 4 | 1 | no | — |
| 316 | C2×D4 | — | 4 | 1 | no | — |
| 317 | C2×D4 | — | 4 | 1 | no | — |
| 318 | C2×D4 | — | 4 | 2 | no | — |
| 319 | C2×D4 | — | 4 | 2 | no | — |
| 320 | C2×D4 | — | 4 | 2 | no | — |
| 321 | C2×D4 | — | 4 | 2 | no | — |
| 322 | C2×D4 | — | 4 | 2 | no | — |
| 323 | C2×D4 | — | 4 | 2 | no | — |
| 324 | C2×D4 | — | 4 | 2 | no | — |
| 325 | C2×D4 | — | 4 | 2 | no | — |
| 326 | C2×D4 | — | 4 | 2 | no | — |
| 327 | C2×D4 | — | 4 | 2 | no | — |
| 328 | C2×D4 | — | 4 | 2 | no | — |
| 329 | C2×D4 | — | 4 | 2 | no | — |
| 330 | C2×D4 | — | 4 | 2 | no | — |
| 331 | C2×D4 | — | 4 | 2 | no | — |
| 332 | C2×D4 | — | 4 | 2 | no | — |
| 333 | C2×D4 | — | 4 | 2 | no | — |
| 334 | C2×D4 | — | 4 | 2 | no | — |
| 335 | C2×D4 | — | 4 | 2 | no | — |
| 336 | C2×D4 | — | 8 | 2 | no | — |
| 337 | C2×D4 | — | 8 | 2 | no | — |
| 338 | C2×D4 | — | 8 | 2 | no | — |
| 339 | C2×D4 | — | 8 | 2 | no | — |
| 340 | C2×D4 | — | 8 | 2 | no | — |
| 341 | C2×D4 | — | 8 | 2 | no | — |
| 342 | C2×D4 | — | 8 | 2 | no | — |
| 343 | C2×D4 | — | 8 | 2 | no | — |
| 344 | C2×D4 | — | 8 | 2 | no | — |
| 345 | C22×C4 | — | 2 | 1 | no | — |
| 346 | C22×C4 | — | 2 | 1 | no | — |
| 347 | C22×C4 | — | 2 | 2 | no | — |
| 348 | C22×C4 | — | 2 | 1 | no | — |
| 349 | C22×C4 | — | 2 | 1 | no | — |
| 350 | C22×C4 | — | 2 | 1 | no | — |
| 351 | C22×C4 | — | 2 | 1 | no | — |
| 352 | C22×C4 | — | 2 | 2 | no | — |
| 353 | C22×C4 | — | 2 | 2 | no | — |
| 354 | C22×C4 | — | 2 | 2 | no | — |
| 355 | C22×C4 | — | 2 | 2 | no | — |
| 356 | C22×C4 | — | 4 | 1 | no | — |
| 357 | C22×C4 | — | 4 | 1 | no | — |
| 358 | C22×C4 | — | 4 | 1 | no | — |
| 359 | C22×C4 | — | 4 | 1 | no | — |
| 360 | C22×C4 | — | 4 | 1 | no | — |
| 361 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 362 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 363 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 364 | C22×C4 | C2×D8 | 1 | 4 | yes | — |
| 365 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 366 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 367 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 368 | SD16 | — | 8 | 8 | no | — |
| 369 | D8 | — | 8 | 8 | no | — |
| 370 | D8 | — | 8 | 8 | no | — |
| 371 | D8 | — | 8 | 8 | no | — |
| 372 | M16 | — | 2 | 16 | no | — |
| 373 | C2×C8 | — | 2 | 2 | no | — |
| 374 | C2×C8 | — | 2 | 2 | no | — |
| 375 | C2×C8 | — | 2 | 2 | no | — |
| 376 | C2×C8 | — | 2 | 2 | no | — |
| 377 | C2×C8 | — | 2 | 2 | no | — |
| 378 | C2×C8 | — | 2 | 2 | no | — |
| 379 | C2×C8 | — | 2 | 2 | no | — |
| 380 | C2×C8 | — | 2 | 2 | no | — |
| 381 | C2×C8 | — | 2 | 2 | no | — |
| 382 | C2×C8 | — | 2 | 2 | no | — |
| 383 | C2×C8 | — | 2 | 2 | no | — |
| 384 | C2×C8 | — | 4 | 8 | no | — |
| 385 | C2×C8 | #121 | 1 | 4 | yes | — |
| 386 | C2×C8 | #121 | 1 | 4 | yes | — |
| 387 | C2×C8 | #121 | 1 | 4 | yes | — |
| 388 | C2×C8 | C2×D8 | 1 | 4 | yes | — |
| 389 | C2×C8 | C2×D8 | 1 | 4 | yes | — |
| 390 | C2×C8 | #135 | 1 | 4 | yes | — |
| 391 | C2×C8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 392 | C4⋊C4 | — | 4 | 1 | no | — |
| 393 | C4⋊C4 | — | 4 | 1 | no | — |
| 394 | C4⋊C4 | — | 4 | 1 | no | — |
| 395 | C4⋊C4 | — | 4 | 1 | no | — |
| 396 | C4⋊C4 | — | 4 | 2 | no | — |
| 397 | C4⋊C4 | — | 4 | 2 | no | — |
| 398 | C4⋊C4 | — | 4 | 2 | no | — |
| 399 | C4⋊C4 | — | 4 | 2 | no | — |
| 400 | C4⋊C4 | — | 4 | 2 | no | — |
| 401 | C4⋊C4 | — | 4 | 2 | no | — |
| 402 | C4⋊C4 | — | 4 | 2 | no | — |
| 403 | C4⋊C4 | — | 4 | 2 | no | — |
| 404 | C4⋊C4 | — | 4 | 2 | no | — |
| 405 | C4⋊C4 | — | 4 | 2 | no | — |
| 406 | C4⋊C4 | — | 4 | 2 | no | — |
| 407 | C4⋊C4 | — | 4 | 2 | no | — |
| 408 | C4⋊C4 | — | 8 | 2 | no | — |
| 409 | C22⋊C4 | — | 8 | 2 | no | — |
| 410 | C22⋊C4 | — | 8 | 2 | no | — |
| 411 | C22⋊C4 | — | 8 | 2 | no | — |
| 412 | C22⋊C4 | — | 8 | 2 | no | — |
| 413 | C22⋊C4 | — | 8 | 2 | no | — |
| 414 | C22⋊C4 | — | 8 | 2 | no | — |
| 415 | C22⋊C4 | — | 8 | 2 | no | — |
| 416 | C22⋊C4 | — | 8 | 2 | no | — |
| 417 | C22⋊C4 | — | 8 | 2 | no | — |
| 418 | C22⋊C4 | — | 8 | 2 | no | — |
| 419 | C42 | — | 2 | 1 | no | — |
| 420 | C42 | — | 2 | 1 | no | — |
| 421 | C42 | — | 2 | 1 | no | — |
| 422 | C42 | — | 2 | 1 | no | — |
| 423 | C42 | — | 2 | 1 | no | — |
| 424 | C42 | — | 2 | 1 | no | — |
| 425 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 426 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 427 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 428 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 429 | C23 | — | 2 | 1 | no | — |
| 430 | C23 | — | 2 | 1 | no | — |
| 431 | C23 | — | 2 | 1 | no | — |
| 432 | C23 | — | 2 | 1 | no | — |
| 433 | C23 | — | 2 | 1 | no | — |
| 434 | C23 | — | 2 | 1 | no | — |
| 435 | C23 | — | 8 | 1 | no | — |
| 436 | C23 | — | 8 | 1 | no | — |
| 437 | C23 | — | 8 | 1 | no | — |
| 438 | C23 | — | 8 | 1 | no | — |
| 439 | C23 | — | 8 | 1 | no | — |
| 440 | C23 | — | 8 | 1 | no | — |
| 441 | C23 | — | 8 | 1 | no | — |
| 442 | C23 | — | 8 | 1 | no | — |
| 443 | C23 | — | 8 | 1 | no | — |
| 444 | C23 | — | 8 | 1 | no | — |
| 445 | C23 | — | 8 | 1 | no | — |
| 446 | C23 | — | 8 | 1 | no | — |
| 447 | C23 | — | 8 | 1 | no | — |
| 448 | C23 | — | 8 | 1 | no | — |
| 449 | C23 | — | 8 | 1 | no | — |
| 450 | C23 | — | 8 | 1 | no | — |
| 451 | C23 | — | 8 | 1 | no | — |
| 452 | C23 | — | 8 | 1 | no | — |
| 453 | C23 | — | 8 | 1 | no | — |
| 454 | C23 | — | 8 | 1 | no | — |
| 455 | C23 | — | 8 | 1 | no | — |
| 456 | C23 | #493 | 1 | 2 | yes | — |
| 457 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 458 | Q8 | — | 8 | 2 | no | — |
| 459 | Q8 | — | 8 | 2 | no | — |
| 460 | D4 | — | 8 | 2 | no | — |
| 461 | D4 | — | 8 | 2 | no | — |
| 462 | D4 | — | 8 | 2 | no | — |
| 463 | D4 | — | 8 | 2 | no | — |
| 464 | D4 | — | 8 | 2 | no | — |
| 465 | D4 | — | 8 | 2 | no | — |
| 466 | D4 | — | 8 | 2 | no | — |
| 467 | D4 | — | 8 | 2 | no | — |
| 468 | D4 | — | 8 | 2 | no | — |
| 469 | D4 | — | 8 | 2 | no | — |
| 470 | D4 | — | 8 | 2 | no | — |
| 471 | D4 | — | 8 | 2 | no | — |
| 472 | D4 | — | 8 | 2 | no | — |
| 473 | D4 | — | 8 | 2 | no | — |
| 474 | D4 | — | 8 | 2 | no | — |
| 475 | D4 | — | 8 | 2 | no | — |
| 476 | D4 | — | 8 | 2 | no | — |
| 477 | D4 | — | 8 | 2 | no | — |
| 478 | D4 | — | 16 | 4 | no | — |
| 479 | D4 | — | 16 | 4 | no | — |
| 480 | D4 | — | 16 | 4 | no | — |
| 481 | C2×C4 | — | 2 | 1 | no | — |
| 482 | C2×C4 | — | 2 | 1 | no | — |
| 483 | C2×C4 | — | 2 | 4 | no | — |
| 484 | C2×C4 | — | 2 | 4 | no | — |
| 485 | C2×C4 | — | 2 | 1 | no | — |
| 486 | C2×C4 | — | 2 | 1 | no | — |
| 487 | C2×C4 | — | 2 | 1 | no | — |
| 488 | C2×C4 | — | 2 | 1 | no | — |
| 489 | C2×C4 | — | 2 | 1 | no | — |
| 490 | C2×C4 | — | 2 | 1 | no | — |
| 491 | C2×C4 | — | 4 | 2 | no | — |
| 492 | C2×C4 | — | 4 | 2 | no | — |
| 493 | C2×C4 | — | 4 | 2 | no | — |
| 494 | C2×C4 | — | 4 | 2 | no | — |
| 495 | C2×C4 | — | 8 | 1 | no | — |
| 496 | C2×C4 | — | 8 | 1 | no | — |
| 497 | C2×C4 | — | 8 | 1 | no | — |
| 498 | C2×C4 | — | 8 | 1 | no | — |
| 499 | C2×C4 | — | 8 | 1 | no | — |
| 500 | C2×C4 | — | 8 | 1 | no | — |
| 501 | C2×C4 | — | 8 | 1 | no | — |
| 502 | C2×C4 | — | 8 | 1 | no | — |
| 503 | C2×C4 | — | 8 | 1 | no | — |
| 504 | C2×C4 | — | 8 | 1 | no | — |
| 505 | C2×C4 | — | 8 | 1 | no | — |
| 506 | C2×C4 | — | 8 | 1 | no | — |
| 507 | C2×C4 | — | 8 | 1 | no | — |
| 508 | C2×C4 | — | 8 | 1 | no | — |
| 509 | C2×C4 | — | 8 | 1 | no | — |
| 510 | C2×C4 | #493 | 1 | 2 | yes | — |
| 511 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 512 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 513 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 514 | C2×C4 | #532 | 1 | 1 | yes | characteristic |
| 515 | C2×C4 | #532 | 1 | 1 | yes | characteristic |
| 516 | C2×C4 | #532 | 1 | 1 | yes | characteristic |
| 517 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 518 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 519 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 520 | C2×C4 | #575 | 1 | 2 | yes | — |
| 521 | C2×C4 | #576 | 1 | 2 | yes | — |
| 522 | C2×C4 | #577 | 1 | 2 | yes | — |
| 523 | C8 | — | 2 | 4 | no | — |
| 524 | C8 | — | 2 | 4 | no | — |
| 525 | C8 | — | 2 | 4 | no | — |
| 526 | C8 | — | 2 | 4 | no | — |
| 527 | C8 | — | 4 | 2 | no | — |
| 528 | C8 | #466 | 1 | 8 | yes | complemented |
| 529 | C22 | — | 2 | 2 | no | — |
| 530 | C22 | — | 2 | 2 | no | — |
| 531 | C22 | — | 2 | 2 | no | — |
| 532 | C22 | — | 4 | 1 | no | — |
| 533 | C22 | — | 4 | 1 | no | — |
| 534 | C22 | — | 4 | 1 | no | — |
| 535 | C22 | — | 4 | 1 | no | — |
| 536 | C22 | — | 16 | 1 | no | — |
| 537 | C22 | — | 16 | 1 | no | — |
| 538 | C22 | — | 16 | 1 | no | — |
| 539 | C22 | — | 16 | 1 | no | — |
| 540 | C22 | — | 16 | 1 | no | — |
| 541 | C22 | — | 16 | 1 | no | — |
| 542 | C22 | — | 16 | 1 | no | — |
| 543 | C22 | — | 16 | 1 | no | — |
| 544 | C22 | — | 16 | 1 | no | — |
| 545 | C22 | — | 16 | 1 | no | — |
| 546 | C22 | — | 16 | 1 | no | — |
| 547 | C22 | — | 16 | 1 | no | — |
| 548 | C22 | — | 16 | 1 | no | — |
| 549 | C22 | — | 16 | 1 | no | — |
| 550 | C22 | — | 16 | 1 | no | — |
| 551 | C22 | — | 16 | 1 | no | — |
| 552 | C22 | — | 16 | 1 | no | — |
| 553 | C22 | — | 16 | 1 | no | — |
| 554 | C22 | — | 16 | 1 | no | — |
| 555 | C22 | — | 16 | 1 | no | — |
| 556 | C22 | — | 16 | 1 | no | — |
| 557 | C22 | #2856 | 1 | 1 | yes | characteristic |
| 558 | C22 | C2×#493 | 1 | 1 | yes | characteristic |
| 559 | C22 | #3147 | 1 | 1 | yes | characteristic |
| 560 | C22 | #3151 | 1 | 1 | yes | characteristic |
| 561 | C22 | #3153 | 1 | 1 | yes | characteristic |
| 562 | C22 | #3154 | 1 | 1 | yes | characteristic |
| 563 | C22 | #3155 | 1 | 1 | yes | characteristic |
| 564 | C4 | — | 2 | 2 | no | — |
| 565 | C4 | — | 4 | 1 | no | — |
| 566 | C4 | — | 4 | 1 | no | — |
| 567 | C4 | — | 16 | 1 | no | — |
| 568 | C4 | — | 16 | 1 | no | — |
| 569 | C4 | — | 16 | 1 | no | — |
| 570 | C4 | — | 16 | 1 | no | — |
| 571 | C4 | — | 16 | 1 | no | — |
| 572 | C4 | C2×#466 | 1 | 2 | yes | — |
| 573 | C4 | #3055 | 1 | 2 | yes | — |
| 574 | C4 | #3062 | 1 | 2 | yes | — |
| 575 | C4 | #3169 | 1 | 2 | yes | — |
| 576 | C4 | #3171 | 1 | 2 | yes | — |
| 577 | C4 | #3179 | 1 | 2 | yes | — |
| 578 | C2 | — | 4 | 2 | no | — |
| 579 | C2 | — | 32 | 1 | no | — |
| 580 | C2 | — | 32 | 1 | no | — |
| 581 | C2 | — | 32 | 1 | no | — |
| 582 | C2 | #19629 | 1 | 1 | yes | characteristic |
| 583 | C2 | #19631 | 1 | 1 | yes | characteristic |
| 584 | C2 | #19641 | 1 | 1 | yes | characteristic |
| 585 | C2 | #19644 | 1 | 1 | yes | characteristic |
| 586 | C2 | #33171 | 1 | 1 | yes | characteristic |
| 587 | C2 | #33185 | 1 | 1 | yes | characteristic |
| 588 | C2 | #33187 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33175 | C2 | semidirect |
| 2. #33174 | C2 | semidirect |
| 3. #33171 | C2 | semidirect |
| 4. #33042 | C2 | semidirect |
| 5. #29842 | C2 | semidirect |
| 6. #29806 | C2 | semidirect |
| 7. C2×#1721 | C2 | semidirect |
| 8. C2×C8×M16 | C2 | semidirect |
| 9. #9832 | C2 | semidirect |
| 10. #9830 | C2 | semidirect |
| 11. #9824 | C2 | semidirect |
| 12. #9824 | C2 | semidirect |
| 13. #9822 | C2 | semidirect |
| 14. #9820 | C2 | semidirect |
| 15. #8820 | C2 | semidirect |
| 156. #322 | D4 | semidirect |
| 189. C2×M16 | D8 | semidirect |
| 528. C8 | #466 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33175 | 2 |
| #33174 | 2 |
| #33171 | 2 |
| #33042 | 2 |
| #29842 | 2 |
| #29806 | 2 |
| C2×#1721 | 2 |
| C2×C8×M16 | 2 |
| #9832 | 2 |
| #9830 | 2 |
| #9824 | 2 |
| #9822 | 2 |
| #9820 | 2 |
| #8820 | 2 |