Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2223 conjugacy classes of subgroups.
The subgroup lattice has 459 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #52993 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #35921 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #33092 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #32803 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #32038 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #30713 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #30506 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #24432 | C2 | 1 | 6 | yes | maximal, complemented |
| 9 | #24038 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | #23862 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | #23358 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C22×#137 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#546 | C22 | 1 | 3 | yes | — |
| 14 | C2×D42 | C22 | 1 | 3 | yes | — |
| 15 | C2×#483 | C22 | 1 | 3 | yes | complemented |
| 16 | C2×#465 | C22 | 1 | 6 | yes | complemented |
| 17 | C2×#448 | C22 | 1 | 3 | yes | complemented |
| 18 | C2×#450 | C22 | 1 | 6 | yes | — |
| 19 | C22×#102 | C22 | 1 | 3 | yes | — |
| 20 | #2841 | — | 2 | 3 | no | — |
| 21 | #2834 | — | 2 | 1 | no | — |
| 22 | #2760 | — | 2 | 1 | no | — |
| 23 | #2693 | — | 2 | 3 | no | — |
| 24 | #2657 | C22 | 1 | 1 | yes | characteristic |
| 25 | #2656 | — | 2 | 1 | no | — |
| 26 | #2645 | C22 | 1 | 2 | yes | complemented |
| 27 | #2541 | C22 | 1 | 3 | yes | — |
| 28 | #2540 | C22 | 1 | 3 | yes | — |
| 29 | #2539 | — | 2 | 3 | no | — |
| 30 | #2486 | C22 | 1 | 3 | yes | complemented |
| 31 | #2433 | — | 2 | 3 | no | — |
| 32 | #2417 | C22 | 1 | 6 | yes | complemented |
| 33 | #2409 | C22 | 1 | 3 | yes | complemented |
| 34 | #2397 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×#392 | — | 2 | 1 | no | — |
| 36 | #2275 | C22 | 1 | 1 | yes | characteristic |
| 37 | #2042 | C22 | 1 | 6 | yes | complemented |
| 38 | #2041 | C22 | 1 | 6 | yes | complemented |
| 39 | #2034 | C22 | 1 | 6 | yes | complemented |
| 40 | #2034 | C22 | 1 | 6 | yes | complemented |
| 41 | #2015 | C22 | 1 | 6 | yes | complemented |
| 42 | #2014 | C22 | 1 | 6 | yes | complemented |
| 43 | #2001 | C22 | 1 | 6 | yes | complemented |
| 44 | #2000 | C22 | 1 | 6 | yes | complemented |
| 45 | #1927 | C22 | 1 | 6 | yes | complemented |
| 46 | #1919 | C22 | 1 | 6 | yes | complemented |
| 47 | #1895 | C22 | 1 | 3 | yes | complemented |
| 48 | #1892 | C22 | 1 | 3 | yes | complemented |
| 49 | #1877 | C22 | 1 | 6 | yes | — |
| 50 | #1874 | C22 | 1 | 6 | yes | complemented |
| 51 | #1870 | C22 | 1 | 3 | yes | complemented |
| 52 | #1869 | C22 | 1 | 3 | yes | complemented |
| 53 | #1869 | C22 | 1 | 6 | yes | complemented |
| 54 | #1864 | C22 | 1 | 6 | yes | — |
| 55 | #1840 | C22 | 1 | 2 | yes | complemented |
| 56 | #1788 | C22 | 1 | 3 | yes | — |
| 57 | #1786 | C22 | 1 | 3 | yes | — |
| 58 | C22×#41 | C23 | 1 | 1 | yes | characteristic |
| 59 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 60 | C23×D4 | — | 2 | 3 | no | — |
| 61 | C23×D4 | C23 | 1 | 3 | yes | — |
| 62 | C2×#137 | — | 4 | 6 | no | — |
| 63 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 64 | C22×SD16 | C23 | 1 | 2 | yes | — |
| 65 | C22×SD16 | C23 | 1 | 6 | yes | — |
| 66 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 67 | C23×C8 | C23 | 1 | 3 | yes | — |
| 68 | #546 | — | 4 | 6 | no | — |
| 69 | #546 | — | 4 | 6 | no | — |
| 70 | D42 | — | 4 | 6 | no | — |
| 71 | D42 | — | 4 | 6 | no | — |
| 72 | C2×#127 | — | 2 | 3 | no | — |
| 73 | C2×#127 | C23 | 1 | 3 | yes | — |
| 74 | C2×#124 | C23 | 1 | 3 | yes | — |
| 75 | C2×#123 | — | 2 | 3 | no | — |
| 76 | C2×#123 | — | 2 | 3 | no | — |
| 77 | C2×C22⋊Q8 | C23 | 1 | 3 | yes | — |
| 78 | C2×C22⋊Q8 | C23 | 1 | 6 | yes | — |
| 79 | C2×#121 | — | 2 | 3 | no | — |
| 80 | C2×#121 | — | 2 | 3 | no | — |
| 81 | C2×#121 | C23 | 1 | 6 | yes | — |
| 82 | C2×#121 | C23 | 1 | 3 | yes | — |
| 83 | C2×#120 | — | 2 | 3 | no | — |
| 84 | C2×#120 | — | 2 | 6 | no | — |
| 85 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 86 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 87 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 88 | C22×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 89 | C22×C22⋊C4 | — | 2 | 3 | no | — |
| 90 | #483 | — | 4 | 6 | no | — |
| 91 | #465 | — | 4 | 12 | no | — |
| 92 | #450 | — | 4 | 12 | no | — |
| 93 | #448 | — | 4 | 6 | no | — |
| 94 | C2×#106 | C23 | 1 | 6 | yes | — |
| 95 | C2×#103 | C23 | 1 | 6 | yes | — |
| 96 | C2×#103 | C23 | 1 | 6 | yes | — |
| 97 | C2×#102 | — | 4 | 6 | no | — |
| 98 | C2×#102 | C23 | 1 | 6 | yes | — |
| 99 | C2×#102 | C23 | 1 | 6 | yes | — |
| 100 | C2×#102 | C23 | 1 | 6 | yes | — |
| 101 | C2×#102 | C23 | 1 | 6 | yes | — |
| 102 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 103 | C2×C22⋊C8 | C23 | 1 | 6 | yes | — |
| 104 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 105 | #401 | — | 2 | 2 | no | — |
| 106 | #400 | — | 2 | 6 | no | — |
| 107 | #399 | — | 2 | 3 | no | — |
| 108 | #399 | D4 | 1 | 1 | yes | characteristic |
| 109 | #399 | D4 | 1 | 3 | yes | — |
| 110 | #398 | — | 2 | 6 | no | — |
| 111 | #398 | — | 2 | 3 | no | — |
| 112 | #397 | — | 2 | 3 | no | — |
| 113 | #396 | — | 2 | 6 | no | — |
| 114 | #395 | — | 2 | 3 | no | — |
| 115 | #393 | — | 2 | 3 | no | — |
| 116 | #392 | — | 2 | 3 | no | — |
| 117 | #392 | — | 2 | 3 | no | — |
| 118 | #392 | — | 2 | 3 | no | — |
| 119 | #392 | — | 2 | 1 | no | — |
| 120 | #392 | — | 2 | 3 | no | — |
| 121 | #392 | D4 | 1 | 1 | yes | characteristic |
| 122 | #392 | D4 | 1 | 3 | yes | — |
| 123 | #391 | C23 | 1 | 6 | yes | — |
| 124 | #390 | — | 2 | 3 | no | — |
| 125 | #390 | — | 2 | 6 | no | — |
| 126 | #389 | C23 | 1 | 6 | yes | — |
| 127 | #386 | — | 2 | 3 | no | — |
| 128 | #386 | — | 2 | 6 | no | — |
| 129 | #385 | — | 2 | 3 | no | — |
| 130 | #381 | — | 2 | 3 | no | — |
| 131 | #380 | — | 2 | 6 | no | — |
| 132 | #380 | C23 | 1 | 3 | yes | — |
| 133 | #380 | C23 | 1 | 3 | yes | — |
| 134 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 135 | C4×C22⋊C4 | — | 2 | 3 | no | — |
| 136 | #376 | C23 | 1 | 1 | yes | characteristic |
| 137 | C2×#95 | — | 2 | 1 | no | — |
| 138 | #340 | C23 | 1 | 6 | yes | — |
| 139 | #340 | C23 | 1 | 6 | yes | — |
| 140 | #336 | C23 | 1 | 6 | yes | — |
| 141 | #336 | C23 | 1 | 6 | yes | — |
| 142 | C25 | — | 2 | 3 | no | — |
| 143 | C2×#41 | — | 2 | 6 | no | — |
| 144 | C2×#41 | — | 4 | 3 | no | — |
| 145 | C22×Q8 | — | 2 | 1 | no | — |
| 146 | C22×Q8 | — | 4 | 3 | no | — |
| 147 | C22×Q8 | — | 4 | 3 | no | — |
| 148 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 149 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 150 | C22×D4 | — | 2 | 12 | no | — |
| 151 | C22×D4 | — | 2 | 6 | no | — |
| 152 | C22×D4 | — | 2 | 1 | no | — |
| 153 | C22×D4 | — | 2 | 6 | no | — |
| 154 | C22×D4 | — | 2 | 3 | no | — |
| 155 | C22×D4 | — | 2 | 3 | no | — |
| 156 | C22×D4 | — | 2 | 3 | no | — |
| 157 | C22×D4 | — | 2 | 3 | no | — |
| 158 | C22×D4 | — | 4 | 6 | no | — |
| 159 | C22×D4 | — | 4 | 3 | no | — |
| 160 | C22×D4 | — | 4 | 12 | no | — |
| 161 | C22×D4 | — | 8 | 12 | no | — |
| 162 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 163 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 164 | C22×D4 | C24 | 1 | 3 | yes | — |
| 165 | C22×D4 | C24 | 1 | 1 | yes | characteristic |
| 166 | C23×C4 | — | 2 | 3 | no | — |
| 167 | C23×C4 | — | 2 | 3 | no | — |
| 168 | C23×C4 | — | 2 | 1 | no | — |
| 169 | C23×C4 | — | 2 | 1 | no | — |
| 170 | C23×C4 | C24 | 1 | 3 | yes | — |
| 171 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 172 | #137 | — | 4 | 16 | no | — |
| 173 | C2×Q16 | — | 4 | 6 | no | — |
| 174 | C2×SD16 | — | 2 | 12 | no | — |
| 175 | C2×SD16 | — | 4 | 6 | no | — |
| 176 | C2×SD16 | — | 4 | 6 | no | — |
| 177 | C2×SD16 | — | 4 | 6 | no | — |
| 178 | C2×M16 | — | 4 | 3 | no | — |
| 179 | C22×C8 | — | 4 | 3 | no | — |
| 180 | C22×C8 | — | 4 | 6 | no | — |
| 181 | C22×C8 | C24 | 1 | 6 | yes | — |
| 182 | C22×C8 | C24 | 1 | 2 | yes | — |
| 183 | #127 | — | 4 | 3 | no | — |
| 184 | #127 | — | 4 | 3 | no | — |
| 185 | #127 | — | 8 | 6 | no | — |
| 186 | #124 | — | 4 | 3 | no | — |
| 187 | #124 | — | 4 | 3 | no | — |
| 188 | #123 | — | 4 | 12 | no | — |
| 189 | #123 | — | 8 | 6 | no | — |
| 190 | C22⋊Q8 | — | 2 | 6 | no | — |
| 191 | C22⋊Q8 | — | 4 | 3 | no | — |
| 192 | C22⋊Q8 | — | 4 | 6 | no | — |
| 193 | C22⋊Q8 | — | 4 | 6 | no | — |
| 194 | #121 | — | 2 | 12 | no | — |
| 195 | #121 | — | 2 | 6 | no | — |
| 196 | #121 | — | 4 | 6 | no | — |
| 197 | #121 | — | 4 | 3 | no | — |
| 198 | #121 | — | 4 | 12 | no | — |
| 199 | #121 | — | 4 | 12 | no | — |
| 200 | #120 | — | 4 | 24 | no | — |
| 201 | #120 | — | 4 | 12 | no | — |
| 202 | C4×D4 | — | 2 | 6 | no | — |
| 203 | C4×D4 | — | 2 | 6 | no | — |
| 204 | C4×D4 | — | 4 | 3 | no | — |
| 205 | C4×D4 | — | 4 | 3 | no | — |
| 206 | C4×D4 | — | 4 | 3 | no | — |
| 207 | C4×D4 | — | 4 | 3 | no | — |
| 208 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 209 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 210 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 211 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 212 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 213 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 214 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 215 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 216 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 217 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 218 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 219 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 220 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 221 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 222 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 223 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 224 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 225 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 226 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 227 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 228 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 229 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 230 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 231 | C2×C42 | — | 2 | 3 | no | — |
| 232 | C2×C42 | C24 | 1 | 3 | yes | — |
| 233 | #106 | — | 2 | 12 | no | — |
| 234 | #103 | — | 2 | 12 | no | — |
| 235 | #103 | — | 4 | 6 | no | — |
| 236 | #102 | — | 2 | 12 | no | — |
| 237 | #102 | — | 4 | 6 | no | — |
| 238 | #102 | — | 4 | 6 | no | — |
| 239 | #102 | — | 4 | 6 | no | — |
| 240 | C22⋊C8 | — | 4 | 3 | no | — |
| 241 | C22⋊C8 | — | 4 | 6 | no | — |
| 242 | C22⋊C8 | — | 4 | 3 | no | — |
| 243 | #95 | — | 2 | 3 | no | — |
| 244 | #95 | — | 2 | 3 | no | — |
| 245 | #95 | — | 2 | 6 | no | — |
| 246 | #95 | — | 2 | 3 | no | — |
| 247 | #95 | — | 2 | 6 | no | — |
| 248 | #95 | — | 2 | 6 | no | — |
| 249 | #95 | C2×D4 | 1 | 1 | yes | characteristic |
| 250 | #95 | C2×D4 | 1 | 3 | yes | — |
| 251 | C24 | — | 2 | 1 | no | — |
| 252 | C24 | — | 2 | 3 | no | — |
| 253 | C24 | — | 4 | 6 | no | — |
| 254 | C24 | — | 4 | 12 | no | — |
| 255 | C24 | — | 4 | 6 | no | — |
| 256 | C24 | — | 8 | 6 | no | — |
| 257 | C24 | — | 8 | 3 | no | — |
| 258 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 259 | C24 | C22×D4 | 1 | 3 | yes | — |
| 260 | #41 | — | 4 | 4 | no | — |
| 261 | #41 | — | 8 | 6 | no | — |
| 262 | C2×Q8 | — | 2 | 6 | no | — |
| 263 | C2×Q8 | — | 2 | 3 | no | — |
| 264 | C2×Q8 | — | 2 | 3 | no | — |
| 265 | C2×Q8 | — | 2 | 6 | no | — |
| 266 | C2×Q8 | — | 4 | 4 | no | — |
| 267 | C2×Q8 | — | 8 | 3 | no | — |
| 268 | C2×Q8 | — | 8 | 6 | no | — |
| 269 | C2×D4 | — | 2 | 12 | no | — |
| 270 | C2×D4 | — | 2 | 6 | no | — |
| 271 | C2×D4 | — | 2 | 3 | no | — |
| 272 | C2×D4 | — | 2 | 3 | no | — |
| 273 | C2×D4 | — | 2 | 6 | no | — |
| 274 | C2×D4 | — | 2 | 6 | no | — |
| 275 | C2×D4 | — | 2 | 3 | no | — |
| 276 | C2×D4 | — | 4 | 3 | no | — |
| 277 | C2×D4 | — | 4 | 3 | no | — |
| 278 | C2×D4 | — | 4 | 3 | no | — |
| 279 | C2×D4 | — | 4 | 12 | no | — |
| 280 | C2×D4 | — | 4 | 12 | no | — |
| 281 | C2×D4 | — | 4 | 6 | no | — |
| 282 | C2×D4 | — | 4 | 6 | no | — |
| 283 | C2×D4 | — | 4 | 6 | no | — |
| 284 | C2×D4 | — | 4 | 6 | no | — |
| 285 | C2×D4 | — | 4 | 12 | no | — |
| 286 | C2×D4 | — | 4 | 6 | no | — |
| 287 | C2×D4 | — | 4 | 12 | no | — |
| 288 | C2×D4 | — | 8 | 6 | no | — |
| 289 | C2×D4 | — | 8 | 3 | no | — |
| 290 | C2×D4 | — | 8 | 3 | no | — |
| 291 | C2×D4 | — | 8 | 6 | no | — |
| 292 | C2×D4 | — | 8 | 12 | no | — |
| 293 | C2×D4 | — | 8 | 6 | no | — |
| 294 | C2×D4 | — | 8 | 6 | no | — |
| 295 | C2×D4 | — | 8 | 48 | no | — |
| 296 | C2×D4 | H(5,𝔽2) | 1 | 6 | yes | — |
| 297 | C22×C4 | — | 2 | 3 | no | — |
| 298 | C22×C4 | — | 2 | 12 | no | — |
| 299 | C22×C4 | — | 2 | 3 | no | — |
| 300 | C22×C4 | — | 2 | 3 | no | — |
| 301 | C22×C4 | — | 2 | 6 | no | — |
| 302 | C22×C4 | — | 2 | 3 | no | — |
| 303 | C22×C4 | — | 2 | 6 | no | — |
| 304 | C22×C4 | — | 2 | 3 | no | — |
| 305 | C22×C4 | — | 2 | 6 | no | — |
| 306 | C22×C4 | — | 2 | 2 | no | — |
| 307 | C22×C4 | — | 2 | 1 | no | — |
| 308 | C22×C4 | — | 4 | 6 | no | — |
| 309 | C22×C4 | — | 4 | 6 | no | — |
| 310 | C22×C4 | — | 8 | 6 | no | — |
| 311 | C22×C4 | — | 8 | 3 | no | — |
| 312 | C22×C4 | — | 8 | 3 | no | — |
| 313 | C22×C4 | — | 8 | 3 | no | — |
| 314 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 315 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 316 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 317 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 318 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 319 | Q16 | — | 4 | 8 | no | — |
| 320 | SD16 | — | 4 | 8 | no | — |
| 321 | SD16 | — | 8 | 12 | no | — |
| 322 | M16 | — | 4 | 4 | no | — |
| 323 | C2×C8 | — | 2 | 6 | no | — |
| 324 | C2×C8 | — | 2 | 6 | no | — |
| 325 | C2×C8 | — | 2 | 6 | no | — |
| 326 | C2×C8 | — | 2 | 6 | no | — |
| 327 | C2×C8 | — | 8 | 6 | no | — |
| 328 | C4⋊C4 | — | 2 | 3 | no | — |
| 329 | C4⋊C4 | — | 2 | 6 | no | — |
| 330 | C4⋊C4 | — | 2 | 6 | no | — |
| 331 | C4⋊C4 | — | 4 | 3 | no | — |
| 332 | C4⋊C4 | — | 4 | 3 | no | — |
| 333 | C4⋊C4 | — | 4 | 6 | no | — |
| 334 | C4⋊C4 | — | 8 | 3 | no | — |
| 335 | C4⋊C4 | — | 8 | 6 | no | — |
| 336 | C4⋊C4 | — | 8 | 6 | no | — |
| 337 | C4⋊C4 | H(5,𝔽2) | 1 | 6 | yes | — |
| 338 | C22⋊C4 | — | 2 | 12 | no | — |
| 339 | C22⋊C4 | — | 4 | 3 | no | — |
| 340 | C22⋊C4 | — | 4 | 6 | no | — |
| 341 | C22⋊C4 | — | 4 | 3 | no | — |
| 342 | C22⋊C4 | — | 4 | 6 | no | — |
| 343 | C22⋊C4 | — | 4 | 12 | no | — |
| 344 | C22⋊C4 | — | 4 | 6 | no | — |
| 345 | C22⋊C4 | — | 4 | 6 | no | — |
| 346 | C22⋊C4 | — | 4 | 6 | no | — |
| 347 | C22⋊C4 | — | 4 | 6 | no | — |
| 348 | C22⋊C4 | — | 8 | 6 | no | — |
| 349 | C22⋊C4 | — | 8 | 3 | no | — |
| 350 | C42 | — | 2 | 3 | no | — |
| 351 | C42 | — | 2 | 3 | no | — |
| 352 | C42 | — | 8 | 3 | no | — |
| 353 | C23 | — | 2 | 3 | no | — |
| 354 | C23 | — | 2 | 6 | no | — |
| 355 | C23 | — | 2 | 3 | no | — |
| 356 | C23 | — | 2 | 6 | no | — |
| 357 | C23 | — | 2 | 6 | no | — |
| 358 | C23 | — | 2 | 1 | no | — |
| 359 | C23 | — | 2 | 3 | no | — |
| 360 | C23 | — | 2 | 3 | no | — |
| 361 | C23 | — | 4 | 1 | no | — |
| 362 | C23 | — | 4 | 3 | no | — |
| 363 | C23 | — | 4 | 3 | no | — |
| 364 | C23 | — | 4 | 3 | no | — |
| 365 | C23 | — | 4 | 6 | no | — |
| 366 | C23 | — | 4 | 6 | no | — |
| 367 | C23 | — | 4 | 3 | no | — |
| 368 | C23 | — | 8 | 12 | no | — |
| 369 | C23 | — | 8 | 24 | no | — |
| 370 | C23 | — | 8 | 12 | no | — |
| 371 | C23 | — | 16 | 12 | no | — |
| 372 | C23 | — | 16 | 6 | no | — |
| 373 | C23 | #577 | 1 | 6 | yes | — |
| 374 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 375 | Q8 | — | 4 | 2 | no | — |
| 376 | Q8 | — | 4 | 1 | no | — |
| 377 | Q8 | — | 4 | 6 | no | — |
| 378 | Q8 | — | 4 | 3 | no | — |
| 379 | Q8 | — | 8 | 4 | no | — |
| 380 | D4 | — | 2 | 6 | no | — |
| 381 | D4 | — | 4 | 12 | no | — |
| 382 | D4 | — | 4 | 3 | no | — |
| 383 | D4 | — | 4 | 6 | no | — |
| 384 | D4 | — | 4 | 3 | no | — |
| 385 | D4 | — | 8 | 6 | no | — |
| 386 | D4 | — | 8 | 12 | no | — |
| 387 | D4 | — | 8 | 24 | no | — |
| 388 | D4 | — | 8 | 12 | no | — |
| 389 | D4 | — | 8 | 4 | no | — |
| 390 | D4 | — | 16 | 6 | no | — |
| 391 | D4 | — | 16 | 6 | no | — |
| 392 | D4 | #560 | 1 | 4 | yes | — |
| 393 | C2×C4 | — | 2 | 3 | no | — |
| 394 | C2×C4 | — | 2 | 3 | no | — |
| 395 | C2×C4 | — | 2 | 3 | no | — |
| 396 | C2×C4 | — | 2 | 6 | no | — |
| 397 | C2×C4 | — | 2 | 6 | no | — |
| 398 | C2×C4 | — | 2 | 3 | no | — |
| 399 | C2×C4 | — | 2 | 3 | no | — |
| 400 | C2×C4 | — | 2 | 3 | no | — |
| 401 | C2×C4 | — | 2 | 6 | no | — |
| 402 | C2×C4 | — | 4 | 6 | no | — |
| 403 | C2×C4 | — | 4 | 1 | no | — |
| 404 | C2×C4 | — | 4 | 3 | no | — |
| 405 | C2×C4 | — | 4 | 3 | no | — |
| 406 | C2×C4 | — | 4 | 6 | no | — |
| 407 | C2×C4 | — | 4 | 3 | no | — |
| 408 | C2×C4 | — | 4 | 6 | no | — |
| 409 | C2×C4 | — | 4 | 3 | no | — |
| 410 | C2×C4 | — | 4 | 6 | no | — |
| 411 | C2×C4 | — | 4 | 6 | no | — |
| 412 | C2×C4 | — | 4 | 6 | no | — |
| 413 | C2×C4 | — | 8 | 12 | no | — |
| 414 | C2×C4 | — | 8 | 4 | no | — |
| 415 | C2×C4 | — | 8 | 4 | no | — |
| 416 | C2×C4 | — | 16 | 6 | no | — |
| 417 | C2×C4 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 418 | C2×C4 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 419 | C8 | — | 4 | 2 | no | — |
| 420 | C8 | — | 4 | 6 | no | — |
| 421 | C22 | — | 2 | 6 | no | — |
| 422 | C22 | — | 2 | 12 | no | — |
| 423 | C22 | — | 2 | 4 | no | — |
| 424 | C22 | — | 4 | 6 | no | — |
| 425 | C22 | — | 4 | 6 | no | — |
| 426 | C22 | — | 4 | 6 | no | — |
| 427 | C22 | — | 4 | 3 | no | — |
| 428 | C22 | — | 4 | 3 | no | — |
| 429 | C22 | — | 4 | 3 | no | — |
| 430 | C22 | — | 4 | 3 | no | — |
| 431 | C22 | — | 4 | 1 | no | — |
| 432 | C22 | — | 8 | 6 | no | — |
| 433 | C22 | — | 8 | 3 | no | — |
| 434 | C22 | — | 8 | 6 | no | — |
| 435 | C22 | — | 8 | 3 | no | — |
| 436 | C22 | — | 8 | 3 | no | — |
| 437 | C22 | — | 16 | 24 | no | — |
| 438 | C22 | #3484 | 1 | 3 | yes | — |
| 439 | C22 | C2×#577 | 1 | 3 | yes | — |
| 440 | C22 | #3585 | 1 | 1 | yes | characteristic |
| 441 | C4 | — | 4 | 3 | no | — |
| 442 | C4 | — | 4 | 3 | no | — |
| 443 | C4 | — | 4 | 6 | no | — |
| 444 | C4 | — | 4 | 1 | no | — |
| 445 | C4 | — | 4 | 3 | no | — |
| 446 | C4 | — | 8 | 6 | no | — |
| 447 | C4 | — | 8 | 1 | no | — |
| 448 | C4 | — | 8 | 2 | no | — |
| 449 | C4 | C2×#560 | 1 | 1 | yes | characteristic |
| 450 | C4 | #3543 | 1 | 3 | yes | — |
| 451 | C2 | — | 4 | 6 | no | — |
| 452 | C2 | — | 4 | 4 | no | — |
| 453 | C2 | — | 8 | 1 | no | — |
| 454 | C2 | — | 16 | 3 | no | — |
| 455 | C2 | #46324 | 1 | 1 | yes | characteristic |
| 456 | C2 | #61623 | 1 | 3 | yes | — |
| 457 | C2 | #61631 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35921 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #52993 | C2 | semidirect |
| 3. #33092 | C2 | semidirect |
| 4. #32803 | C2 | semidirect |
| 5. #32038 | C2 | semidirect |
| 6. #30713 | C2 | semidirect |
| 7. #30506 | C2 | semidirect |
| 8. #24432 | C2 | semidirect |
| 9. #24038 | C2 | semidirect |
| 10. #23862 | C2 | semidirect |
| 11. #23358 | C2 | semidirect |
| 15. C2×#483 | C22 | semidirect |
| 16. C2×#465 | C22 | semidirect |
| 17. C2×#448 | C22 | semidirect |
| 26. #2645 | C22 | semidirect |
| 30. #2486 | C22 | semidirect |
| 32. #2417 | C22 | semidirect |
| 33. #2409 | C22 | semidirect |
| 37. #2042 | C22 | semidirect |
| 38. #2041 | C22 | semidirect |
| 39. #2034 | C22 | semidirect |
| 40. #2034 | C22 | semidirect |
| 41. #2015 | C22 | semidirect |
| 42. #2014 | C22 | semidirect |
| 43. #2001 | C22 | semidirect |
| 44. #2000 | C22 | semidirect |
| 45. #1927 | C22 | semidirect |
| 46. #1919 | C22 | semidirect |
| 47. #1895 | C22 | semidirect |
| 48. #1892 | C22 | semidirect |
| 50. #1874 | C22 | semidirect |
| 51. #1870 | C22 | semidirect |
| 52. #1869 | C22 | semidirect |
| 53. #1869 | C22 | semidirect |
| 55. #1840 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #52993 | 2 |
| #35921 | 2 |
| #33092 | 2 |
| #32803 | 2 |
| #32038 | 2 |
| #30713 | 2 |
| #30506 | 2 |
| #24432 | 2 |
| #24038 | 2 |
| #23862 | 2 |
| #23358 | 2 |