Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2127 conjugacy classes of subgroups.
The subgroup lattice has 456 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #46328 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #36179 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33078 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #32800 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #32039 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #30723 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #30510 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #24454 | C2 | 1 | 6 | yes | maximal, complemented |
| 9 | #24054 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | #23864 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | #23357 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C22×AGL(1,ℤ/8) | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#560 | — | 2 | 1 | no | — |
| 14 | C2×#547 | C22 | 1 | 3 | yes | — |
| 15 | C2×#547 | C22 | 1 | 3 | yes | complemented |
| 16 | C2×#484 | C22 | 1 | 3 | yes | complemented |
| 17 | C2×#468 | C22 | 1 | 6 | yes | — |
| 18 | C2×#449 | C22 | 1 | 6 | yes | — |
| 19 | C2×#450 | C22 | 1 | 3 | yes | — |
| 20 | C2×#417 | C22 | 1 | 3 | yes | complemented |
| 21 | #2868 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | #2860 | C22 | 1 | 2 | yes | — |
| 23 | #2856 | — | 2 | 1 | no | — |
| 24 | #2823 | — | 2 | 1 | no | — |
| 25 | #2699 | — | 2 | 3 | no | — |
| 26 | #2697 | — | 2 | 3 | no | — |
| 27 | #2542 | C22 | 1 | 3 | yes | — |
| 28 | #2542 | C22 | 1 | 3 | yes | complemented |
| 29 | #2501 | — | 2 | 3 | no | — |
| 30 | #2501 | — | 2 | 3 | no | — |
| 31 | #2486 | C22 | 1 | 3 | yes | — |
| 32 | #2416 | C22 | 1 | 6 | yes | — |
| 33 | #2408 | C22 | 1 | 3 | yes | — |
| 34 | C4×#127 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×#384 | — | 2 | 1 | no | — |
| 36 | C42×D4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 37 | #2043 | C22 | 1 | 6 | yes | complemented |
| 38 | #2043 | C22 | 1 | 6 | yes | complemented |
| 39 | #2036 | C22 | 1 | 6 | yes | — |
| 40 | #2036 | C22 | 1 | 6 | yes | complemented |
| 41 | #2013 | C22 | 1 | 6 | yes | — |
| 42 | #2013 | C22 | 1 | 6 | yes | — |
| 43 | #2000 | C22 | 1 | 6 | yes | — |
| 44 | #2000 | C22 | 1 | 6 | yes | — |
| 45 | #1925 | C22 | 1 | 6 | yes | — |
| 46 | #1918 | C22 | 1 | 6 | yes | complemented |
| 47 | #1894 | C22 | 1 | 3 | yes | — |
| 48 | #1894 | C22 | 1 | 3 | yes | complemented |
| 49 | #1879 | C22 | 1 | 6 | yes | — |
| 50 | #1872 | C22 | 1 | 6 | yes | complemented |
| 51 | #1870 | C22 | 1 | 3 | yes | — |
| 52 | #1870 | C22 | 1 | 3 | yes | — |
| 53 | #1870 | C22 | 1 | 6 | yes | complemented |
| 54 | #1862 | C22 | 1 | 6 | yes | complemented |
| 55 | #1839 | C22 | 1 | 2 | yes | complemented |
| 56 | #1788 | C22 | 1 | 3 | yes | — |
| 57 | #1788 | C22 | 1 | 3 | yes | complemented |
| 58 | C22×#41 | C23 | 1 | 3 | yes | — |
| 59 | C22×#41 | C23 | 1 | 1 | yes | characteristic |
| 60 | C23×D4 | C23 | 1 | 1 | yes | characteristic |
| 61 | C2×AGL(1,ℤ/8) | — | 4 | 6 | no | — |
| 62 | C22×SD16 | C23 | 1 | 6 | yes | — |
| 63 | C22×SD16 | C23 | 1 | 2 | yes | — |
| 64 | C22×D8 | C23 | 1 | 2 | yes | — |
| 65 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 66 | C22×M16 | C23 | 1 | 3 | yes | — |
| 67 | #560 | — | 4 | 8 | no | — |
| 68 | #547 | — | 4 | 6 | no | — |
| 69 | #547 | — | 4 | 6 | no | — |
| 70 | #547 | — | 4 | 6 | no | — |
| 71 | #547 | — | 4 | 6 | no | — |
| 72 | C2×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 73 | C2×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 74 | C2×#127 | D4 | 1 | 1 | yes | characteristic |
| 75 | C2×#126 | — | 2 | 2 | no | — |
| 76 | C2×#123 | — | 2 | 3 | no | — |
| 77 | C2×#123 | — | 2 | 3 | no | — |
| 78 | C2×#123 | — | 2 | 3 | no | — |
| 79 | C2×#123 | — | 2 | 6 | no | — |
| 80 | C2×C22⋊Q8 | C23 | 1 | 3 | yes | — |
| 81 | C2×C22⋊Q8 | C23 | 1 | 6 | yes | — |
| 82 | C2×C22⋊Q8 | C23 | 1 | 3 | yes | — |
| 83 | C2×#121 | — | 2 | 3 | no | — |
| 84 | C2×#121 | — | 2 | 3 | no | — |
| 85 | C2×#121 | — | 2 | 6 | no | — |
| 86 | C2×#121 | C23 | 1 | 6 | yes | — |
| 87 | C2×#120 | — | 2 | 3 | no | — |
| 88 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 89 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 90 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 91 | C22×C4⋊C4 | — | 2 | 3 | no | — |
| 92 | C22×C4⋊C4 | — | 2 | 3 | no | — |
| 93 | C22×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 94 | C22×C42 | — | 2 | 1 | no | — |
| 95 | #484 | — | 4 | 6 | no | — |
| 96 | #468 | — | 4 | 12 | no | — |
| 97 | #450 | — | 4 | 6 | no | — |
| 98 | #449 | — | 4 | 12 | no | — |
| 99 | C2×#107 | C23 | 1 | 6 | yes | — |
| 100 | #417 | — | 4 | 6 | no | — |
| 101 | C2×#103 | C23 | 1 | 6 | yes | — |
| 102 | C2×#103 | C23 | 1 | 6 | yes | — |
| 103 | C2×#103 | C23 | 1 | 6 | yes | — |
| 104 | C2×#103 | C23 | 1 | 6 | yes | — |
| 105 | C2×#102 | C23 | 1 | 6 | yes | — |
| 106 | C2×#102 | C23 | 1 | 6 | yes | — |
| 107 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 108 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 109 | C2×C22⋊C8 | C23 | 1 | 6 | yes | — |
| 110 | #399 | — | 2 | 3 | no | — |
| 111 | #399 | — | 2 | 3 | no | — |
| 112 | #399 | — | 2 | 6 | no | — |
| 113 | #398 | — | 2 | 3 | no | — |
| 114 | #397 | — | 2 | 3 | no | — |
| 115 | #396 | — | 2 | 6 | no | — |
| 116 | #394 | — | 2 | 3 | no | — |
| 117 | #394 | — | 2 | 6 | no | — |
| 118 | #392 | — | 2 | 3 | no | — |
| 119 | #390 | — | 2 | 3 | no | — |
| 120 | #390 | — | 2 | 3 | no | — |
| 121 | #390 | D4 | 1 | 3 | yes | — |
| 122 | #390 | D4 | 1 | 3 | yes | — |
| 123 | #390 | C23 | 1 | 6 | yes | — |
| 124 | #389 | C23 | 1 | 6 | yes | — |
| 125 | #388 | — | 2 | 6 | no | — |
| 126 | #386 | — | 2 | 6 | no | — |
| 127 | #384 | — | 2 | 3 | no | — |
| 128 | #384 | — | 2 | 3 | no | — |
| 129 | #384 | — | 2 | 1 | no | — |
| 130 | #384 | — | 2 | 3 | no | — |
| 131 | #384 | D4 | 1 | 1 | yes | characteristic |
| 132 | #381 | — | 2 | 3 | no | — |
| 133 | #380 | C23 | 1 | 3 | yes | — |
| 134 | #380 | C23 | 1 | 3 | yes | — |
| 135 | #379 | — | 2 | 3 | no | — |
| 136 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 137 | C4×C22⋊C4 | — | 2 | 3 | no | — |
| 138 | C43 | C23 | 1 | 1 | yes | characteristic |
| 139 | #340 | C23 | 1 | 6 | yes | — |
| 140 | #340 | C23 | 1 | 6 | yes | — |
| 141 | #336 | C23 | 1 | 6 | yes | — |
| 142 | #336 | C23 | 1 | 6 | yes | — |
| 143 | C25 | — | 2 | 1 | no | — |
| 144 | C2×#41 | — | 2 | 6 | no | — |
| 145 | C2×#41 | — | 2 | 6 | no | — |
| 146 | C2×#41 | — | 2 | 12 | no | — |
| 147 | C2×#41 | — | 4 | 3 | no | — |
| 148 | C2×#41 | — | 4 | 3 | no | — |
| 149 | C2×#41 | — | 4 | 6 | no | — |
| 150 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 151 | C22×Q8 | C24 | 1 | 3 | yes | — |
| 152 | C22×D4 | — | 2 | 1 | no | — |
| 153 | C22×D4 | — | 2 | 3 | no | — |
| 154 | C22×D4 | — | 2 | 6 | no | — |
| 155 | C22×D4 | — | 2 | 1 | no | — |
| 156 | C22×D4 | — | 4 | 3 | no | — |
| 157 | C22×D4 | — | 4 | 3 | no | — |
| 158 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 159 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 160 | C22×D4 | C24 | 1 | 2 | yes | — |
| 161 | C22×D4 | C24 | 1 | 1 | yes | characteristic |
| 162 | C23×C4 | — | 2 | 3 | no | — |
| 163 | C23×C4 | — | 2 | 3 | no | — |
| 164 | C23×C4 | — | 2 | 1 | no | — |
| 165 | C23×C4 | — | 2 | 3 | no | — |
| 166 | C23×C4 | C24 | 1 | 3 | yes | — |
| 167 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 168 | AGL(1,ℤ/8) | — | 4 | 16 | no | — |
| 169 | C2×SD16 | — | 4 | 6 | no | — |
| 170 | C2×SD16 | — | 4 | 6 | no | — |
| 171 | C2×SD16 | — | 4 | 6 | no | — |
| 172 | C2×SD16 | — | 4 | 6 | no | — |
| 173 | C2×D8 | — | 2 | 12 | no | — |
| 174 | C2×M16 | — | 4 | 3 | no | — |
| 175 | C2×M16 | — | 4 | 3 | no | — |
| 176 | C2×M16 | — | 4 | 6 | no | — |
| 177 | C22×C8 | C24 | 1 | 2 | yes | — |
| 178 | C22×C8 | C24 | 1 | 6 | yes | — |
| 179 | C4⋊Q8 | — | 4 | 3 | no | — |
| 180 | C4⋊Q8 | — | 4 | 3 | no | — |
| 181 | C4⋊Q8 | — | 4 | 3 | no | — |
| 182 | C4⋊Q8 | — | 4 | 3 | no | — |
| 183 | #127 | — | 2 | 4 | no | — |
| 184 | #126 | — | 4 | 8 | no | — |
| 185 | #123 | — | 4 | 12 | no | — |
| 186 | #123 | — | 4 | 12 | no | — |
| 187 | #123 | — | 4 | 24 | no | — |
| 188 | #123 | — | 4 | 12 | no | — |
| 189 | C22⋊Q8 | — | 2 | 6 | no | — |
| 190 | C22⋊Q8 | — | 2 | 12 | no | — |
| 191 | C22⋊Q8 | — | 2 | 6 | no | — |
| 192 | C22⋊Q8 | — | 4 | 6 | no | — |
| 193 | C22⋊Q8 | — | 4 | 3 | no | — |
| 194 | C22⋊Q8 | — | 4 | 3 | no | — |
| 195 | #121 | — | 4 | 12 | no | — |
| 196 | #121 | — | 4 | 12 | no | — |
| 197 | #121 | — | 4 | 24 | no | — |
| 198 | #121 | — | 4 | 6 | no | — |
| 199 | #121 | — | 4 | 6 | no | — |
| 200 | #120 | — | 4 | 12 | no | — |
| 201 | C4×D4 | — | 2 | 6 | no | — |
| 202 | C4×D4 | — | 2 | 6 | no | — |
| 203 | C4×D4 | — | 4 | 3 | no | — |
| 204 | C4×D4 | — | 4 | 3 | no | — |
| 205 | C4×D4 | — | 4 | 3 | no | — |
| 206 | C4×D4 | — | 4 | 3 | no | — |
| 207 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 208 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 209 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 210 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 211 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 212 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 213 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 214 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 215 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 216 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 217 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 218 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 219 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 220 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 221 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 222 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 223 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 224 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 225 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 226 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 227 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 228 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 229 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 230 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 231 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 232 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 233 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 234 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 235 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 236 | C2×C42 | — | 2 | 3 | no | — |
| 237 | C2×C42 | — | 4 | 4 | no | — |
| 238 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 239 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 240 | C2×C42 | C24 | 1 | 3 | yes | — |
| 241 | #107 | — | 2 | 12 | no | — |
| 242 | #103 | — | 4 | 6 | no | — |
| 243 | #103 | — | 4 | 6 | no | — |
| 244 | #103 | — | 4 | 6 | no | — |
| 245 | #103 | — | 4 | 6 | no | — |
| 246 | #102 | — | 2 | 12 | no | — |
| 247 | #102 | — | 2 | 12 | no | — |
| 248 | C22⋊C8 | — | 4 | 6 | no | — |
| 249 | C22⋊C8 | — | 4 | 3 | no | — |
| 250 | C22⋊C8 | — | 4 | 3 | no | — |
| 251 | #95 | — | 2 | 3 | no | — |
| 252 | #95 | — | 2 | 3 | no | — |
| 253 | #95 | — | 2 | 3 | no | — |
| 254 | #95 | — | 2 | 6 | no | — |
| 255 | C24 | — | 2 | 1 | no | — |
| 256 | C24 | — | 2 | 2 | no | — |
| 257 | C24 | — | 4 | 2 | no | — |
| 258 | C24 | — | 8 | 3 | no | — |
| 259 | C24 | — | 8 | 3 | no | — |
| 260 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 261 | C24 | C22×D4 | 1 | 3 | yes | — |
| 262 | #41 | — | 4 | 4 | no | — |
| 263 | #41 | — | 4 | 12 | no | — |
| 264 | #41 | — | 8 | 6 | no | — |
| 265 | #41 | — | 8 | 6 | no | — |
| 266 | #41 | — | 8 | 12 | no | — |
| 267 | C2×Q8 | — | 2 | 3 | no | — |
| 268 | C2×Q8 | — | 2 | 3 | no | — |
| 269 | C2×Q8 | — | 2 | 6 | no | — |
| 270 | C2×Q8 | — | 2 | 6 | no | — |
| 271 | C2×Q8 | — | 2 | 3 | no | — |
| 272 | C2×Q8 | — | 2 | 3 | no | — |
| 273 | C2×D4 | — | 2 | 6 | no | — |
| 274 | C2×D4 | — | 2 | 6 | no | — |
| 275 | C2×D4 | — | 2 | 3 | no | — |
| 276 | C2×D4 | — | 2 | 6 | no | — |
| 277 | C2×D4 | — | 2 | 6 | no | — |
| 278 | C2×D4 | — | 4 | 3 | no | — |
| 279 | C2×D4 | — | 4 | 3 | no | — |
| 280 | C2×D4 | — | 4 | 3 | no | — |
| 281 | C2×D4 | — | 4 | 3 | no | — |
| 282 | C2×D4 | — | 4 | 4 | no | — |
| 283 | C2×D4 | — | 4 | 6 | no | — |
| 284 | C2×D4 | — | 4 | 12 | no | — |
| 285 | C2×D4 | — | 4 | 6 | no | — |
| 286 | C2×D4 | — | 4 | 12 | no | — |
| 287 | C2×D4 | — | 8 | 3 | no | — |
| 288 | C2×D4 | — | 8 | 6 | no | — |
| 289 | C2×D4 | — | 8 | 6 | no | — |
| 290 | C2×D4 | — | 8 | 6 | no | — |
| 291 | C2×D4 | H(5,𝔽2) | 1 | 6 | yes | — |
| 292 | C22×C4 | — | 2 | 12 | no | — |
| 293 | C22×C4 | — | 2 | 3 | no | — |
| 294 | C22×C4 | — | 2 | 3 | no | — |
| 295 | C22×C4 | — | 2 | 6 | no | — |
| 296 | C22×C4 | — | 2 | 3 | no | — |
| 297 | C22×C4 | — | 2 | 3 | no | — |
| 298 | C22×C4 | — | 2 | 6 | no | — |
| 299 | C22×C4 | — | 2 | 3 | no | — |
| 300 | C22×C4 | — | 2 | 6 | no | — |
| 301 | C22×C4 | — | 2 | 1 | no | — |
| 302 | C22×C4 | — | 2 | 3 | no | — |
| 303 | C22×C4 | — | 4 | 6 | no | — |
| 304 | C22×C4 | — | 4 | 12 | no | — |
| 305 | C22×C4 | — | 4 | 6 | no | — |
| 306 | C22×C4 | — | 4 | 6 | no | — |
| 307 | C22×C4 | — | 4 | 6 | no | — |
| 308 | C22×C4 | — | 4 | 6 | no | — |
| 309 | C22×C4 | — | 8 | 3 | no | — |
| 310 | C22×C4 | — | 8 | 3 | no | — |
| 311 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 312 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 313 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 314 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 315 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 316 | SD16 | — | 4 | 8 | no | — |
| 317 | SD16 | — | 8 | 12 | no | — |
| 318 | D8 | — | 4 | 8 | no | — |
| 319 | M16 | — | 4 | 4 | no | — |
| 320 | M16 | — | 8 | 6 | no | — |
| 321 | C2×C8 | — | 2 | 6 | no | — |
| 322 | C2×C8 | — | 2 | 6 | no | — |
| 323 | C2×C8 | — | 2 | 6 | no | — |
| 324 | C2×C8 | — | 2 | 6 | no | — |
| 325 | C4⋊C4 | — | 2 | 3 | no | — |
| 326 | C4⋊C4 | — | 2 | 6 | no | — |
| 327 | C4⋊C4 | — | 2 | 6 | no | — |
| 328 | C4⋊C4 | — | 4 | 3 | no | — |
| 329 | C4⋊C4 | — | 4 | 3 | no | — |
| 330 | C4⋊C4 | — | 4 | 12 | no | — |
| 331 | C4⋊C4 | — | 4 | 6 | no | — |
| 332 | C4⋊C4 | — | 4 | 6 | no | — |
| 333 | C4⋊C4 | — | 4 | 6 | no | — |
| 334 | C4⋊C4 | — | 4 | 6 | no | — |
| 335 | C4⋊C4 | — | 4 | 6 | no | — |
| 336 | C4⋊C4 | — | 4 | 6 | no | — |
| 337 | C4⋊C4 | — | 4 | 12 | no | — |
| 338 | C4⋊C4 | — | 8 | 3 | no | — |
| 339 | C4⋊C4 | H(5,𝔽2) | 1 | 6 | yes | — |
| 340 | C22⋊C4 | — | 2 | 12 | no | — |
| 341 | C22⋊C4 | — | 4 | 6 | no | — |
| 342 | C22⋊C4 | — | 4 | 3 | no | — |
| 343 | C22⋊C4 | — | 4 | 3 | no | — |
| 344 | C22⋊C4 | — | 4 | 6 | no | — |
| 345 | C22⋊C4 | — | 4 | 12 | no | — |
| 346 | C22⋊C4 | — | 4 | 6 | no | — |
| 347 | C22⋊C4 | — | 4 | 6 | no | — |
| 348 | C22⋊C4 | — | 4 | 6 | no | — |
| 349 | C22⋊C4 | — | 4 | 12 | no | — |
| 350 | C22⋊C4 | — | 8 | 6 | no | — |
| 351 | C22⋊C4 | — | 8 | 6 | no | — |
| 352 | C42 | — | 2 | 3 | no | — |
| 353 | C42 | — | 2 | 3 | no | — |
| 354 | C42 | — | 4 | 3 | no | — |
| 355 | C42 | — | 4 | 6 | no | — |
| 356 | C42 | AGL(1,ℤ/8) | 1 | 4 | yes | complemented |
| 357 | C23 | — | 2 | 6 | no | — |
| 358 | C23 | — | 2 | 3 | no | — |
| 359 | C23 | — | 2 | 6 | no | — |
| 360 | C23 | — | 2 | 3 | no | — |
| 361 | C23 | — | 2 | 3 | no | — |
| 362 | C23 | — | 2 | 6 | no | — |
| 363 | C23 | — | 2 | 3 | no | — |
| 364 | C23 | — | 2 | 1 | no | — |
| 365 | C23 | — | 2 | 3 | no | — |
| 366 | C23 | — | 4 | 3 | no | — |
| 367 | C23 | — | 4 | 1 | no | — |
| 368 | C23 | — | 4 | 6 | no | — |
| 369 | C23 | — | 4 | 6 | no | — |
| 370 | C23 | — | 4 | 2 | no | — |
| 371 | C23 | — | 8 | 4 | no | — |
| 372 | C23 | — | 16 | 6 | no | — |
| 373 | C23 | — | 16 | 6 | no | — |
| 374 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 375 | Q8 | — | 4 | 1 | no | — |
| 376 | Q8 | — | 4 | 3 | no | — |
| 377 | Q8 | — | 4 | 6 | no | — |
| 378 | Q8 | — | 4 | 3 | no | — |
| 379 | Q8 | — | 4 | 3 | no | — |
| 380 | D4 | — | 2 | 6 | no | — |
| 381 | D4 | — | 4 | 2 | no | — |
| 382 | D4 | — | 4 | 6 | no | — |
| 383 | D4 | — | 8 | 6 | no | — |
| 384 | D4 | — | 8 | 6 | no | — |
| 385 | D4 | — | 8 | 4 | no | — |
| 386 | D4 | — | 8 | 12 | no | — |
| 387 | D4 | — | 8 | 4 | no | — |
| 388 | D4 | — | 16 | 12 | no | — |
| 389 | D4 | #560 | 1 | 4 | yes | complemented |
| 390 | C2×C4 | — | 2 | 3 | no | — |
| 391 | C2×C4 | — | 2 | 3 | no | — |
| 392 | C2×C4 | — | 2 | 6 | no | — |
| 393 | C2×C4 | — | 2 | 3 | no | — |
| 394 | C2×C4 | — | 2 | 6 | no | — |
| 395 | C2×C4 | — | 2 | 6 | no | — |
| 396 | C2×C4 | — | 2 | 3 | no | — |
| 397 | C2×C4 | — | 2 | 3 | no | — |
| 398 | C2×C4 | — | 4 | 24 | no | — |
| 399 | C2×C4 | — | 4 | 3 | no | — |
| 400 | C2×C4 | — | 4 | 3 | no | — |
| 401 | C2×C4 | — | 4 | 1 | no | — |
| 402 | C2×C4 | — | 4 | 3 | no | — |
| 403 | C2×C4 | — | 4 | 3 | no | — |
| 404 | C2×C4 | — | 4 | 3 | no | — |
| 405 | C2×C4 | — | 4 | 6 | no | — |
| 406 | C2×C4 | — | 4 | 6 | no | — |
| 407 | C2×C4 | — | 4 | 3 | no | — |
| 408 | C2×C4 | — | 4 | 3 | no | — |
| 409 | C2×C4 | — | 4 | 6 | no | — |
| 410 | C2×C4 | — | 4 | 6 | no | — |
| 411 | C2×C4 | — | 4 | 6 | no | — |
| 412 | C2×C4 | — | 8 | 12 | no | — |
| 413 | C2×C4 | — | 8 | 4 | no | — |
| 414 | C2×C4 | — | 16 | 6 | no | — |
| 415 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 416 | C2×C4 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 417 | C2×C4 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 418 | C8 | — | 4 | 2 | no | — |
| 419 | C8 | — | 4 | 6 | no | — |
| 420 | C22 | — | 2 | 4 | no | — |
| 421 | C22 | — | 4 | 6 | no | — |
| 422 | C22 | — | 4 | 6 | no | — |
| 423 | C22 | — | 4 | 3 | no | — |
| 424 | C22 | — | 4 | 6 | no | — |
| 425 | C22 | — | 4 | 3 | no | — |
| 426 | C22 | — | 4 | 6 | no | — |
| 427 | C22 | — | 4 | 3 | no | — |
| 428 | C22 | — | 4 | 3 | no | — |
| 429 | C22 | — | 4 | 3 | no | — |
| 430 | C22 | — | 4 | 1 | no | — |
| 431 | C22 | — | 8 | 6 | no | — |
| 432 | C22 | — | 8 | 6 | no | — |
| 433 | C22 | — | 8 | 2 | no | — |
| 434 | C22 | — | 16 | 8 | no | — |
| 435 | C22 | #3484 | 1 | 3 | yes | — |
| 436 | C22 | C22×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 437 | C22 | #3586 | 1 | 3 | yes | — |
| 438 | C4 | — | 2 | 6 | no | — |
| 439 | C4 | — | 4 | 6 | no | — |
| 440 | C4 | — | 4 | 3 | no | — |
| 441 | C4 | — | 4 | 3 | no | — |
| 442 | C4 | — | 4 | 1 | no | — |
| 443 | C4 | — | 8 | 6 | no | — |
| 444 | C4 | — | 8 | 1 | no | — |
| 445 | C4 | — | 8 | 3 | no | — |
| 446 | C4 | C2×#560 | 1 | 1 | yes | characteristic |
| 447 | C4 | #3543 | 1 | 3 | yes | — |
| 448 | C2 | — | 4 | 4 | no | — |
| 449 | C2 | — | 8 | 3 | no | — |
| 450 | C2 | — | 8 | 1 | no | — |
| 451 | C2 | — | 16 | 2 | no | — |
| 452 | C2 | #46324 | 1 | 1 | yes | characteristic |
| 453 | C2 | #61585 | 1 | 3 | yes | — |
| 454 | C2 | #61676 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #46328 | C2 | semidirect |
| 2. #36179 | C2 | semidirect |
| 3. #33078 | C2 | semidirect |
| 4. #32800 | C2 | semidirect |
| 5. #32039 | C2 | semidirect |
| 6. #30723 | C2 | semidirect |
| 7. #30510 | C2 | semidirect |
| 8. #24454 | C2 | semidirect |
| 9. #24054 | C2 | semidirect |
| 10. #23864 | C2 | semidirect |
| 11. #23357 | C2 | semidirect |
| 15. C2×#547 | C22 | semidirect |
| 16. C2×#484 | C22 | semidirect |
| 20. C2×#417 | C22 | semidirect |
| 21. #2868 | C22 | semidirect |
| 28. #2542 | C22 | semidirect |
| 36. C42×D4 | C22 | semidirect |
| 37. #2043 | C22 | semidirect |
| 38. #2043 | C22 | semidirect |
| 40. #2036 | C22 | semidirect |
| 46. #1918 | C22 | semidirect |
| 48. #1894 | C22 | semidirect |
| 50. #1872 | C22 | semidirect |
| 53. #1870 | C22 | semidirect |
| 54. #1862 | C22 | semidirect |
| 55. #1839 | C22 | semidirect |
| 57. #1788 | C22 | semidirect |
| 356. C42 | AGL(1,ℤ/8) | semidirect |
| 389. D4 | #560 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #46328 | 2 |
| #36179 | 2 |
| #33078 | 2 |
| #32800 | 2 |
| #32039 | 2 |
| #30723 | 2 |
| #30510 | 2 |
| #24454 | 2 |
| #24054 | 2 |
| #23864 | 2 |
| #23357 | 2 |