Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
886 conjugacy classes of subgroups.
The subgroup lattice has 499 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #33161 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #28437 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22734 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #21505 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#1668 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10591 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10536 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #9706 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #9604 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #9222 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #8588 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#493 | C22 | 1 | 2 | yes | — |
| 13 | C2×#460 | — | 2 | 1 | no | — |
| 14 | C2×#460 | — | 2 | 1 | no | — |
| 15 | C2×#460 | C22 | 1 | 2 | yes | — |
| 16 | C2×#460 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#459 | — | 2 | 2 | no | — |
| 18 | C2×C8×Q8 | C22 | 1 | 2 | yes | — |
| 19 | C2×#440 | — | 2 | 1 | no | — |
| 20 | #2867 | C22 | 1 | 1 | yes | characteristic |
| 21 | C4×#127 | C22 | 1 | 1 | yes | characteristic |
| 22 | C42×Q8 | C22 | 1 | 1 | yes | characteristic |
| 23 | #2011 | — | 2 | 1 | no | — |
| 24 | #2011 | — | 2 | 1 | no | — |
| 25 | #1982 | — | 2 | 1 | no | — |
| 26 | #1982 | — | 2 | 1 | no | — |
| 27 | #1982 | C22 | 1 | 1 | yes | characteristic |
| 28 | #1982 | C22 | 1 | 2 | yes | — |
| 29 | #1957 | C22 | 1 | 2 | yes | — |
| 30 | #1947 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1939 | — | 2 | 2 | no | — |
| 32 | #1939 | C22 | 1 | 2 | yes | — |
| 33 | #1925 | — | 2 | 1 | no | — |
| 34 | #1880 | — | 2 | 1 | no | — |
| 35 | #1842 | C22 | 1 | 1 | yes | characteristic |
| 36 | #1831 | — | 2 | 1 | no | — |
| 37 | #1831 | C22 | 1 | 1 | yes | characteristic |
| 38 | #1803 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1767 | C22 | 1 | 1 | yes | characteristic |
| 40 | C4×C4⋊C8 | — | 2 | 1 | no | — |
| 41 | C4×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 42 | #1762 | — | 2 | 1 | no | — |
| 43 | #1668 | — | 2 | 8 | no | — |
| 44 | C2×#335 | C22 | 1 | 2 | yes | — |
| 45 | C2×#331 | C22 | 1 | 2 | yes | — |
| 46 | C2×#331 | C22 | 1 | 2 | yes | — |
| 47 | C2×#331 | C22 | 1 | 2 | yes | — |
| 48 | #1296 | C22 | 1 | 2 | yes | — |
| 49 | #1277 | C22 | 1 | 2 | yes | — |
| 50 | C22×SD16 | — | 2 | 1 | no | — |
| 51 | C22×SD16 | — | 2 | 2 | no | — |
| 52 | C22×D8 | — | 2 | 2 | no | — |
| 53 | C22×D8 | — | 2 | 2 | no | — |
| 54 | C2×#127 | — | 2 | 1 | no | — |
| 55 | C2×#127 | — | 2 | 1 | no | — |
| 56 | C2×#127 | — | 2 | 1 | no | — |
| 57 | C2×#127 | — | 2 | 1 | no | — |
| 58 | C2×#127 | — | 2 | 1 | no | — |
| 59 | C2×#127 | — | 2 | 1 | no | — |
| 60 | C2×#127 | C23 | 1 | 2 | yes | — |
| 61 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 62 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 63 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 64 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×C4×D4 | — | 2 | 1 | no | — |
| 66 | C2×C4×D4 | — | 2 | 1 | no | — |
| 67 | C2×C4×D4 | — | 2 | 1 | no | — |
| 68 | #493 | — | 2 | 4 | no | — |
| 69 | #493 | — | 4 | 2 | no | — |
| 70 | #460 | — | 2 | 2 | no | — |
| 71 | #460 | — | 2 | 4 | no | — |
| 72 | #460 | — | 2 | 4 | no | — |
| 73 | #460 | — | 4 | 4 | no | — |
| 74 | #460 | — | 4 | 4 | no | — |
| 75 | #460 | D4 | 1 | 4 | yes | complemented |
| 76 | #459 | — | 4 | 8 | no | — |
| 77 | C8×Q8 | — | 2 | 4 | no | — |
| 78 | C8×Q8 | — | 4 | 2 | no | — |
| 79 | #440 | — | 4 | 4 | no | — |
| 80 | C2×#107 | — | 2 | 1 | no | — |
| 81 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 82 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 83 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 84 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 85 | C2×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 86 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 87 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 88 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 89 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 90 | C2×#103 | — | 2 | 1 | no | — |
| 91 | C2×#102 | — | 2 | 1 | no | — |
| 92 | C2×#102 | — | 2 | 2 | no | — |
| 93 | C2×#102 | — | 2 | 2 | no | — |
| 94 | C2×#102 | — | 2 | 1 | no | — |
| 95 | C2×#102 | — | 2 | 1 | no | — |
| 96 | C2×#102 | — | 2 | 2 | no | — |
| 97 | C2×#102 | — | 2 | 2 | no | — |
| 98 | C2×#102 | — | 2 | 2 | no | — |
| 99 | C2×#97 | — | 2 | 1 | no | — |
| 100 | C2×C4×C8 | — | 2 | 2 | no | — |
| 101 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 102 | #390 | — | 2 | 1 | no | — |
| 103 | #390 | — | 2 | 1 | no | — |
| 104 | #390 | — | 2 | 1 | no | — |
| 105 | #384 | C23 | 1 | 1 | yes | characteristic |
| 106 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 107 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 108 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 109 | C4×C4⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 110 | C4×C4⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 111 | C4×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 112 | C43 | — | 2 | 1 | no | — |
| 113 | C43 | C23 | 1 | 1 | yes | characteristic |
| 114 | #340 | — | 2 | 1 | no | — |
| 115 | #340 | — | 2 | 1 | no | — |
| 116 | #336 | — | 2 | 2 | no | — |
| 117 | #336 | — | 2 | 1 | no | — |
| 118 | #336 | — | 2 | 1 | no | — |
| 119 | #336 | — | 2 | 1 | no | — |
| 120 | #336 | D4 | 1 | 1 | yes | characteristic |
| 121 | #336 | D4 | 1 | 1 | yes | characteristic |
| 122 | #335 | — | 2 | 4 | no | — |
| 123 | #331 | — | 2 | 4 | no | — |
| 124 | #331 | — | 2 | 4 | no | — |
| 125 | #331 | — | 4 | 2 | no | — |
| 126 | C22×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 127 | C22×D4 | — | 2 | 1 | no | — |
| 128 | C22×D4 | — | 2 | 1 | no | — |
| 129 | C22×D4 | — | 2 | 2 | no | — |
| 130 | C22×D4 | — | 2 | 2 | no | — |
| 131 | C22×D4 | — | 2 | 1 | no | — |
| 132 | C22×D4 | — | 2 | 2 | no | — |
| 133 | C22×D4 | — | 4 | 1 | no | — |
| 134 | C22×D4 | — | 4 | 1 | no | — |
| 135 | C22×D4 | — | 4 | 1 | no | — |
| 136 | C22×D4 | — | 4 | 1 | no | — |
| 137 | C23×C4 | — | 4 | 1 | no | — |
| 138 | C2×SD16 | — | 2 | 4 | no | — |
| 139 | C2×SD16 | — | 4 | 2 | no | — |
| 140 | C2×SD16 | — | 4 | 2 | no | — |
| 141 | C2×SD16 | — | 4 | 4 | no | — |
| 142 | C2×SD16 | — | 4 | 4 | no | — |
| 143 | C2×SD16 | — | 4 | 4 | no | — |
| 144 | C2×D8 | — | 4 | 4 | no | — |
| 145 | C2×D8 | — | 4 | 4 | no | — |
| 146 | C2×D8 | — | 4 | 4 | no | — |
| 147 | C2×D8 | — | 4 | 4 | no | — |
| 148 | C2×D8 | — | 8 | 2 | no | — |
| 149 | C2×D8 | — | 8 | 2 | no | — |
| 150 | C22×C8 | — | 2 | 1 | no | — |
| 151 | C22×C8 | — | 2 | 2 | no | — |
| 152 | C22×C8 | — | 2 | 2 | no | — |
| 153 | C22×C8 | — | 2 | 1 | no | — |
| 154 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 155 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 156 | #127 | — | 2 | 2 | no | — |
| 157 | #127 | — | 2 | 1 | no | — |
| 158 | #127 | — | 2 | 2 | no | — |
| 159 | #127 | — | 2 | 1 | no | — |
| 160 | #127 | — | 2 | 2 | no | — |
| 161 | #127 | — | 2 | 1 | no | — |
| 162 | #127 | — | 2 | 2 | no | — |
| 163 | #127 | — | 4 | 2 | no | — |
| 164 | #127 | — | 4 | 4 | no | — |
| 165 | #127 | — | 4 | 2 | no | — |
| 166 | #127 | — | 4 | 2 | no | — |
| 167 | #127 | — | 4 | 2 | no | — |
| 168 | #127 | — | 8 | 2 | no | — |
| 169 | #127 | — | 8 | 2 | no | — |
| 170 | #127 | — | 8 | 2 | no | — |
| 171 | #127 | C2×D4 | 1 | 2 | yes | — |
| 172 | C4×Q8 | — | 2 | 2 | no | — |
| 173 | C4×Q8 | — | 2 | 1 | no | — |
| 174 | C4×Q8 | — | 2 | 1 | no | — |
| 175 | C4×Q8 | — | 2 | 2 | no | — |
| 176 | C4×Q8 | — | 4 | 1 | no | — |
| 177 | C4×Q8 | D8 | 1 | 4 | yes | complemented |
| 178 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 179 | C4×Q8 | #41 | 1 | 2 | yes | — |
| 180 | C4×D4 | — | 4 | 4 | no | — |
| 181 | C4×D4 | — | 4 | 2 | no | — |
| 182 | C4×D4 | — | 4 | 2 | no | — |
| 183 | C4×D4 | — | 8 | 2 | no | — |
| 184 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 185 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 186 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 187 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 188 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 189 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 190 | C2×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 191 | C2×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 192 | C2×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 193 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 194 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 195 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 196 | C2×C42 | — | 2 | 1 | no | — |
| 197 | C2×C42 | — | 2 | 1 | no | — |
| 198 | C2×C42 | — | 2 | 1 | no | — |
| 199 | C2×C42 | — | 2 | 1 | no | — |
| 200 | C2×C42 | — | 2 | 1 | no | — |
| 201 | C2×C42 | — | 2 | 1 | no | — |
| 202 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 203 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 204 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 205 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 206 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 207 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 208 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 209 | #107 | — | 4 | 2 | no | — |
| 210 | C4⋊C8 | — | 2 | 4 | no | — |
| 211 | C4⋊C8 | — | 2 | 2 | no | — |
| 212 | C4⋊C8 | — | 2 | 2 | no | — |
| 213 | C4⋊C8 | — | 2 | 1 | no | — |
| 214 | C4⋊C8 | — | 2 | 2 | no | — |
| 215 | C4⋊C8 | — | 2 | 1 | no | — |
| 216 | C4⋊C8 | — | 2 | 2 | no | — |
| 217 | C4⋊C8 | — | 2 | 1 | no | — |
| 218 | C4⋊C8 | — | 4 | 4 | no | — |
| 219 | C4⋊C8 | — | 4 | 2 | no | — |
| 220 | C4⋊C8 | — | 4 | 4 | no | — |
| 221 | C4⋊C8 | — | 4 | 2 | no | — |
| 222 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 223 | #103 | — | 4 | 2 | no | — |
| 224 | #102 | — | 2 | 4 | no | — |
| 225 | #102 | — | 4 | 2 | no | — |
| 226 | #102 | — | 4 | 4 | no | — |
| 227 | #102 | — | 4 | 4 | no | — |
| 228 | #102 | — | 4 | 4 | no | — |
| 229 | #102 | — | 4 | 4 | no | — |
| 230 | #102 | — | 4 | 2 | no | — |
| 231 | #102 | — | 4 | 4 | no | — |
| 232 | #97 | — | 4 | 2 | no | — |
| 233 | C4×C8 | — | 2 | 2 | no | — |
| 234 | C4×C8 | — | 2 | 2 | no | — |
| 235 | C4×C8 | — | 4 | 4 | no | — |
| 236 | #95 | — | 2 | 1 | no | — |
| 237 | #95 | — | 2 | 1 | no | — |
| 238 | #95 | — | 2 | 1 | no | — |
| 239 | C24 | — | 4 | 1 | no | — |
| 240 | C24 | — | 4 | 2 | no | — |
| 241 | C2×Q8 | — | 2 | 1 | no | — |
| 242 | C2×Q8 | — | 2 | 1 | no | — |
| 243 | C2×Q8 | — | 2 | 1 | no | — |
| 244 | C2×Q8 | #121 | 1 | 2 | yes | — |
| 245 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 246 | C2×Q8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 247 | C2×D4 | — | 2 | 2 | no | — |
| 248 | C2×D4 | — | 2 | 2 | no | — |
| 249 | C2×D4 | — | 2 | 2 | no | — |
| 250 | C2×D4 | — | 4 | 2 | no | — |
| 251 | C2×D4 | — | 4 | 1 | no | — |
| 252 | C2×D4 | — | 4 | 2 | no | — |
| 253 | C2×D4 | — | 4 | 2 | no | — |
| 254 | C2×D4 | — | 4 | 2 | no | — |
| 255 | C2×D4 | — | 4 | 1 | no | — |
| 256 | C2×D4 | — | 4 | 2 | no | — |
| 257 | C2×D4 | — | 4 | 1 | no | — |
| 258 | C2×D4 | — | 4 | 2 | no | — |
| 259 | C2×D4 | — | 4 | 1 | no | — |
| 260 | C2×D4 | — | 4 | 2 | no | — |
| 261 | C2×D4 | — | 4 | 1 | no | — |
| 262 | C2×D4 | — | 4 | 2 | no | — |
| 263 | C2×D4 | — | 4 | 2 | no | — |
| 264 | C2×D4 | — | 4 | 1 | no | — |
| 265 | C2×D4 | — | 4 | 2 | no | — |
| 266 | C2×D4 | — | 4 | 2 | no | — |
| 267 | C2×D4 | — | 4 | 2 | no | — |
| 268 | C2×D4 | — | 4 | 1 | no | — |
| 269 | C2×D4 | — | 4 | 2 | no | — |
| 270 | C2×D4 | — | 4 | 2 | no | — |
| 271 | C2×D4 | — | 4 | 2 | no | — |
| 272 | C2×D4 | — | 4 | 2 | no | — |
| 273 | C2×D4 | — | 8 | 1 | no | — |
| 274 | C2×D4 | — | 8 | 1 | no | — |
| 275 | C2×D4 | — | 8 | 2 | no | — |
| 276 | C2×D4 | — | 8 | 2 | no | — |
| 277 | C2×D4 | — | 8 | 2 | no | — |
| 278 | C2×D4 | — | 8 | 2 | no | — |
| 279 | C2×D4 | — | 8 | 2 | no | — |
| 280 | C2×D4 | — | 8 | 2 | no | — |
| 281 | C2×D4 | — | 8 | 2 | no | — |
| 282 | C2×D4 | — | 8 | 2 | no | — |
| 283 | C2×D4 | — | 8 | 2 | no | — |
| 284 | C2×D4 | — | 8 | 2 | no | — |
| 285 | C2×D4 | — | 8 | 2 | no | — |
| 286 | C2×D4 | — | 8 | 2 | no | — |
| 287 | C2×D4 | — | 8 | 2 | no | — |
| 288 | C2×D4 | — | 8 | 2 | no | — |
| 289 | C22×C4 | — | 2 | 1 | no | — |
| 290 | C22×C4 | — | 2 | 1 | no | — |
| 291 | C22×C4 | — | 2 | 1 | no | — |
| 292 | C22×C4 | — | 2 | 1 | no | — |
| 293 | C22×C4 | — | 4 | 1 | no | — |
| 294 | C22×C4 | — | 8 | 2 | no | — |
| 295 | C22×C4 | — | 8 | 2 | no | — |
| 296 | C22×C4 | — | 8 | 2 | no | — |
| 297 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 298 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 299 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 300 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 301 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 302 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 303 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 304 | SD16 | — | 4 | 8 | no | — |
| 305 | SD16 | — | 8 | 8 | no | — |
| 306 | D8 | — | 8 | 8 | no | — |
| 307 | D8 | — | 8 | 8 | no | — |
| 308 | C2×C8 | — | 2 | 2 | no | — |
| 309 | C2×C8 | — | 2 | 2 | no | — |
| 310 | C2×C8 | — | 2 | 2 | no | — |
| 311 | C2×C8 | — | 2 | 2 | no | — |
| 312 | C2×C8 | — | 2 | 2 | no | — |
| 313 | C2×C8 | — | 2 | 2 | no | — |
| 314 | C2×C8 | — | 4 | 1 | no | — |
| 315 | C2×C8 | — | 4 | 1 | no | — |
| 316 | C2×C8 | — | 4 | 2 | no | — |
| 317 | C2×C8 | — | 4 | 1 | no | — |
| 318 | C2×C8 | — | 4 | 2 | no | — |
| 319 | C2×C8 | — | 4 | 1 | no | — |
| 320 | C2×C8 | — | 4 | 2 | no | — |
| 321 | C2×C8 | — | 4 | 2 | no | — |
| 322 | C2×C8 | — | 4 | 2 | no | — |
| 323 | C2×C8 | — | 4 | 1 | no | — |
| 324 | C2×C8 | — | 4 | 2 | no | — |
| 325 | C2×C8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 326 | C4⋊C4 | — | 2 | 1 | no | — |
| 327 | C4⋊C4 | — | 2 | 2 | no | — |
| 328 | C4⋊C4 | — | 2 | 2 | no | — |
| 329 | C4⋊C4 | — | 2 | 1 | no | — |
| 330 | C4⋊C4 | — | 2 | 4 | no | — |
| 331 | C4⋊C4 | — | 2 | 1 | no | — |
| 332 | C4⋊C4 | — | 2 | 1 | no | — |
| 333 | C4⋊C4 | — | 4 | 2 | no | — |
| 334 | C4⋊C4 | — | 4 | 1 | no | — |
| 335 | C4⋊C4 | — | 4 | 1 | no | — |
| 336 | C4⋊C4 | — | 4 | 2 | no | — |
| 337 | C4⋊C4 | — | 8 | 1 | no | — |
| 338 | C4⋊C4 | #121 | 1 | 2 | yes | — |
| 339 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 340 | C22⋊C4 | — | 8 | 2 | no | — |
| 341 | C22⋊C4 | — | 8 | 2 | no | — |
| 342 | C22⋊C4 | — | 8 | 2 | no | — |
| 343 | C42 | — | 2 | 2 | no | — |
| 344 | C42 | — | 2 | 2 | no | — |
| 345 | C42 | — | 2 | 2 | no | — |
| 346 | C42 | — | 2 | 1 | no | — |
| 347 | C42 | — | 2 | 1 | no | — |
| 348 | C42 | — | 2 | 2 | no | — |
| 349 | C42 | — | 2 | 4 | no | — |
| 350 | C42 | — | 2 | 2 | no | — |
| 351 | C42 | — | 4 | 1 | no | — |
| 352 | C42 | — | 4 | 2 | no | — |
| 353 | C42 | — | 4 | 2 | no | — |
| 354 | C42 | — | 4 | 2 | no | — |
| 355 | C42 | — | 8 | 1 | no | — |
| 356 | C42 | C2×D8 | 1 | 2 | yes | — |
| 357 | C42 | #135 | 1 | 2 | yes | — |
| 358 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 359 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 360 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 361 | C42 | 2−1+4 | 1 | 1 | yes | characteristic |
| 362 | C23 | — | 4 | 2 | no | — |
| 363 | C23 | — | 8 | 1 | no | — |
| 364 | C23 | — | 8 | 1 | no | — |
| 365 | C23 | — | 8 | 1 | no | — |
| 366 | C23 | — | 8 | 1 | no | — |
| 367 | C23 | — | 8 | 1 | no | — |
| 368 | C23 | — | 8 | 1 | no | — |
| 369 | C23 | — | 8 | 2 | no | — |
| 370 | C23 | — | 8 | 2 | no | — |
| 371 | C23 | — | 8 | 2 | no | — |
| 372 | C23 | — | 8 | 2 | no | — |
| 373 | C23 | — | 8 | 2 | no | — |
| 374 | C23 | — | 8 | 2 | no | — |
| 375 | C23 | — | 8 | 2 | no | — |
| 376 | C23 | C2×#121 | 1 | 1 | yes | characteristic |
| 377 | Q8 | — | 2 | 2 | no | — |
| 378 | Q8 | — | 2 | 2 | no | — |
| 379 | Q8 | — | 2 | 2 | no | — |
| 380 | Q8 | #459 | 1 | 4 | yes | complemented |
| 381 | D4 | — | 4 | 4 | no | — |
| 382 | D4 | — | 4 | 2 | no | — |
| 383 | D4 | — | 4 | 2 | no | — |
| 384 | D4 | — | 8 | 2 | no | — |
| 385 | D4 | — | 8 | 2 | no | — |
| 386 | D4 | — | 8 | 2 | no | — |
| 387 | D4 | — | 8 | 2 | no | — |
| 388 | D4 | — | 8 | 2 | no | — |
| 389 | D4 | — | 8 | 1 | no | — |
| 390 | D4 | — | 8 | 2 | no | — |
| 391 | D4 | — | 8 | 1 | no | — |
| 392 | D4 | — | 8 | 2 | no | — |
| 393 | D4 | — | 8 | 2 | no | — |
| 394 | D4 | — | 8 | 2 | no | — |
| 395 | D4 | — | 8 | 2 | no | — |
| 396 | D4 | — | 8 | 2 | no | — |
| 397 | D4 | — | 16 | 2 | no | — |
| 398 | D4 | — | 16 | 2 | no | — |
| 399 | D4 | — | 16 | 4 | no | — |
| 400 | D4 | — | 16 | 4 | no | — |
| 401 | D4 | — | 16 | 4 | no | — |
| 402 | D4 | — | 16 | 4 | no | — |
| 403 | C2×C4 | — | 2 | 2 | no | — |
| 404 | C2×C4 | — | 2 | 1 | no | — |
| 405 | C2×C4 | — | 2 | 1 | no | — |
| 406 | C2×C4 | — | 2 | 1 | no | — |
| 407 | C2×C4 | — | 2 | 2 | no | — |
| 408 | C2×C4 | — | 2 | 1 | no | — |
| 409 | C2×C4 | — | 2 | 2 | no | — |
| 410 | C2×C4 | — | 2 | 1 | no | — |
| 411 | C2×C4 | — | 2 | 1 | no | — |
| 412 | C2×C4 | — | 2 | 1 | no | — |
| 413 | C2×C4 | — | 2 | 1 | no | — |
| 414 | C2×C4 | — | 2 | 2 | no | — |
| 415 | C2×C4 | — | 2 | 1 | no | — |
| 416 | C2×C4 | — | 2 | 2 | no | — |
| 417 | C2×C4 | — | 2 | 1 | no | — |
| 418 | C2×C4 | — | 2 | 1 | no | — |
| 419 | C2×C4 | — | 4 | 1 | no | — |
| 420 | C2×C4 | — | 4 | 1 | no | — |
| 421 | C2×C4 | — | 4 | 1 | no | — |
| 422 | C2×C4 | — | 4 | 1 | no | — |
| 423 | C2×C4 | — | 4 | 1 | no | — |
| 424 | C2×C4 | — | 4 | 1 | no | — |
| 425 | C2×C4 | — | 4 | 1 | no | — |
| 426 | C2×C4 | — | 8 | 1 | no | — |
| 427 | C2×C4 | — | 8 | 1 | no | — |
| 428 | C2×C4 | — | 8 | 1 | no | — |
| 429 | C2×C4 | — | 16 | 4 | no | — |
| 430 | C2×C4 | #466 | 1 | 2 | yes | — |
| 431 | C2×C4 | #466 | 1 | 2 | yes | — |
| 432 | C2×C4 | #472 | 1 | 2 | yes | — |
| 433 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 434 | C2×C4 | #537 | 1 | 1 | yes | characteristic |
| 435 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 436 | C2×C4 | #547 | 1 | 1 | yes | characteristic |
| 437 | C2×C4 | #548 | 1 | 1 | yes | characteristic |
| 438 | C2×C4 | #553 | 1 | 1 | yes | characteristic |
| 439 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 440 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 441 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 442 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 443 | C2×C4 | #575 | 1 | 1 | yes | characteristic |
| 444 | C2×C4 | #577 | 1 | 2 | yes | — |
| 445 | C8 | — | 2 | 4 | no | — |
| 446 | C8 | — | 4 | 2 | no | — |
| 447 | C8 | — | 4 | 2 | no | — |
| 448 | C8 | — | 8 | 1 | no | — |
| 449 | C8 | — | 8 | 2 | no | — |
| 450 | C8 | — | 8 | 2 | no | — |
| 451 | C22 | — | 8 | 2 | no | — |
| 452 | C22 | — | 8 | 2 | no | — |
| 453 | C22 | — | 8 | 2 | no | — |
| 454 | C22 | — | 16 | 1 | no | — |
| 455 | C22 | — | 16 | 1 | no | — |
| 456 | C22 | — | 16 | 1 | no | — |
| 457 | C22 | — | 16 | 1 | no | — |
| 458 | C22 | — | 16 | 2 | no | — |
| 459 | C22 | — | 16 | 2 | no | — |
| 460 | C22 | — | 16 | 2 | no | — |
| 461 | C22 | — | 16 | 2 | no | — |
| 462 | C22 | — | 16 | 2 | no | — |
| 463 | C22 | — | 16 | 2 | no | — |
| 464 | C22 | — | 16 | 2 | no | — |
| 465 | C22 | #2416 | 1 | 1 | yes | characteristic |
| 466 | C22 | #3038 | 1 | 1 | yes | characteristic |
| 467 | C22 | #3043 | 1 | 1 | yes | characteristic |
| 468 | C22 | C2×#466 | 1 | 1 | yes | characteristic |
| 469 | C22 | #3055 | 1 | 1 | yes | characteristic |
| 470 | C22 | C2×#472 | 1 | 1 | yes | characteristic |
| 471 | C22 | #3068 | 1 | 1 | yes | characteristic |
| 472 | C4 | — | 2 | 2 | no | — |
| 473 | C4 | — | 2 | 2 | no | — |
| 474 | C4 | — | 2 | 4 | no | — |
| 475 | C4 | — | 2 | 1 | no | — |
| 476 | C4 | — | 2 | 1 | no | — |
| 477 | C4 | — | 2 | 2 | no | — |
| 478 | C4 | — | 4 | 2 | no | — |
| 479 | C4 | — | 4 | 1 | no | — |
| 480 | C4 | — | 4 | 2 | no | — |
| 481 | C4 | — | 8 | 1 | no | — |
| 482 | C4 | — | 16 | 1 | no | — |
| 483 | C4 | C2×#459 | 1 | 1 | yes | characteristic |
| 484 | C4 | #3035 | 1 | 1 | yes | characteristic |
| 485 | C4 | #3245 | 1 | 1 | yes | characteristic |
| 486 | C4 | #3247 | 1 | 1 | yes | characteristic |
| 487 | C4 | #3265 | 1 | 2 | yes | — |
| 488 | C4 | #3368 | 1 | 2 | yes | — |
| 489 | C2 | — | 16 | 2 | no | — |
| 490 | C2 | — | 32 | 2 | no | — |
| 491 | C2 | #19267 | 1 | 1 | yes | characteristic |
| 492 | C2 | #19320 | 1 | 1 | yes | characteristic |
| 493 | C2 | #19398 | 1 | 1 | yes | characteristic |
| 494 | C2 | #19470 | 1 | 1 | yes | characteristic |
| 495 | C2 | #24742 | 1 | 1 | yes | characteristic |
| 496 | C2 | #24750 | 1 | 1 | yes | characteristic |
| 497 | C2 | #24770 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #28437 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33161 | C2 | semidirect |
| 3. #22734 | C2 | semidirect |
| 4. #21505 | C2 | semidirect |
| 5. C2×#1668 | C2 | semidirect |
| 6. #10591 | C2 | semidirect |
| 7. #10536 | C2 | semidirect |
| 8. #9706 | C2 | semidirect |
| 9. #9604 | C2 | semidirect |
| 10. #9222 | C2 | semidirect |
| 11. #8588 | C2 | semidirect |
| 75. #460 | D4 | semidirect |
| 177. C4×Q8 | D8 | semidirect |
| 380. Q8 | #459 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33161 | 2 |
| #28437 | 2 |
| #22734 | 2 |
| #21505 | 2 |
| C2×#1668 | 2 |
| #10591 | 2 |
| #10536 | 2 |
| #9706 | 2 |
| #9604 | 2 |
| #9222 | 2 |
| #8588 | 2 |