Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
13038 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 | C22×#560 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C23×#127 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#2709 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×#2673 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#2669 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #46324 | C2 | 1 | 32 | yes | maximal, direct product factor |
| 7 | C2×#2818 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C2×#2852 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C2×#2405 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | C2×#2410 | C2 | 1 | 6 | yes | maximal, complemented |
| 11 | C2×#2403 | C2 | 1 | 3 | yes | maximal, complemented |
| 12 | C2×#2380 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | C24×D4 | C22 | 1 | 3 | yes | — |
| 14 | C24×D4 | C22 | 1 | 1 | yes | characteristic |
| 15 | C24×D4 | C22 | 1 | 3 | yes | complemented |
| 16 | C2×#560 | — | 2 | 16 | no | — |
| 17 | C2×#560 | C22 | 1 | 16 | yes | complemented |
| 18 | C22×#127 | — | 2 | 16 | no | — |
| 19 | C22×#127 | C22 | 1 | 3 | yes | — |
| 20 | C22×#127 | C22 | 1 | 1 | yes | characteristic |
| 21 | C22×#127 | C22 | 1 | 16 | yes | complemented |
| 22 | C22×#127 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | C22×#127 | C22 | 1 | 3 | yes | complemented |
| 24 | C22×#126 | C22 | 1 | 2 | yes | complemented |
| 25 | C22×#123 | C22 | 1 | 3 | yes | complemented |
| 26 | C22×#123 | C22 | 1 | 3 | yes | complemented |
| 27 | C22×#121 | C22 | 1 | 6 | yes | complemented |
| 28 | C22×#121 | C22 | 1 | 6 | yes | complemented |
| 29 | C22×#120 | C22 | 1 | 6 | yes | — |
| 30 | C22×#120 | C22 | 1 | 3 | yes | complemented |
| 31 | C22×#120 | C22 | 1 | 3 | yes | complemented |
| 32 | C23×C22⋊C4 | C22 | 1 | 3 | yes | complemented |
| 33 | C23×C42 | C22 | 1 | 1 | yes | characteristic, complemented |
| 34 | #2852 | C22 | 1 | 32 | yes | complemented |
| 35 | #2818 | C22 | 1 | 32 | yes | complemented |
| 36 | #2709 | C22 | 1 | 48 | yes | complemented |
| 37 | #2673 | C22 | 1 | 96 | yes | complemented |
| 38 | #2669 | C22 | 1 | 48 | yes | complemented |
| 39 | #2410 | C22 | 1 | 96 | yes | complemented |
| 40 | #2405 | C22 | 1 | 48 | yes | complemented |
| 41 | #2403 | C22 | 1 | 48 | yes | complemented |
| 42 | C2×#394 | C22 | 1 | 6 | yes | complemented |
| 43 | C2×#394 | C22 | 1 | 6 | yes | complemented |
| 44 | C2×#394 | C22 | 1 | 6 | yes | complemented |
| 45 | C2×#394 | C22 | 1 | 6 | yes | complemented |
| 46 | C2×#392 | C22 | 1 | 6 | yes | complemented |
| 47 | C2×#392 | C22 | 1 | 6 | yes | complemented |
| 48 | C2×#392 | C22 | 1 | 6 | yes | complemented |
| 49 | C2×#392 | C22 | 1 | 6 | yes | complemented |
| 50 | #2380 | C22 | 1 | 16 | yes | complemented |
| 51 | C2×#388 | C22 | 1 | 6 | yes | complemented |
| 52 | C2×#387 | C22 | 1 | 6 | yes | complemented |
| 53 | C2×#386 | C22 | 1 | 6 | yes | — |
| 54 | C2×#386 | C22 | 1 | 3 | yes | complemented |
| 55 | C2×#386 | C22 | 1 | 6 | yes | complemented |
| 56 | C2×#386 | C22 | 1 | 3 | yes | complemented |
| 57 | C2×#386 | C22 | 1 | 3 | yes | complemented |
| 58 | C2×#386 | C22 | 1 | 6 | yes | complemented |
| 59 | C2×#386 | C22 | 1 | 3 | yes | complemented |
| 60 | C2×#385 | C22 | 1 | 6 | yes | complemented |
| 61 | C2×#383 | C22 | 1 | 2 | yes | complemented |
| 62 | C2×#379 | C22 | 1 | 3 | yes | — |
| 63 | C2×#379 | C22 | 1 | 3 | yes | complemented |
| 64 | C26 | C23 | 1 | 1 | yes | characteristic |
| 65 | C26 | C23 | 1 | 3 | yes | — |
| 66 | C26 | C23 | 1 | 1 | yes | characteristic |
| 67 | C23×D4 | — | 2 | 24 | no | — |
| 68 | C23×D4 | — | 2 | 24 | no | — |
| 69 | C23×D4 | — | 4 | 12 | no | — |
| 70 | C23×D4 | — | 4 | 12 | no | — |
| 71 | C23×D4 | — | 4 | 12 | no | — |
| 72 | C23×D4 | — | 4 | 12 | no | — |
| 73 | C23×D4 | — | 4 | 12 | no | — |
| 74 | C23×D4 | C23 | 1 | 2 | yes | — |
| 75 | C23×D4 | C23 | 1 | 3 | yes | — |
| 76 | C23×D4 | C23 | 1 | 3 | yes | — |
| 77 | C23×D4 | C23 | 1 | 6 | yes | — |
| 78 | C23×D4 | C23 | 1 | 3 | yes | — |
| 79 | C23×D4 | C23 | 1 | 24 | yes | — |
| 80 | C23×D4 | C23 | 1 | 3 | yes | — |
| 81 | C23×D4 | C23 | 1 | 6 | yes | — |
| 82 | C23×D4 | C23 | 1 | 2 | yes | — |
| 83 | C23×D4 | C23 | 1 | 8 | yes | — |
| 84 | C23×D4 | C23 | 1 | 6 | yes | — |
| 85 | C23×D4 | C23 | 1 | 3 | yes | — |
| 86 | C23×D4 | C23 | 1 | 3 | yes | — |
| 87 | C23×D4 | C23 | 1 | 24 | yes | complemented |
| 88 | C24×C4 | C23 | 1 | 3 | yes | — |
| 89 | C24×C4 | C23 | 1 | 3 | yes | — |
| 90 | C24×C4 | C23 | 1 | 1 | yes | characteristic |
| 91 | C24×C4 | C23 | 1 | 3 | yes | — |
| 92 | #560 | — | 2 | 128 | no | — |
| 93 | C2×#127 | — | 2 | 24 | no | — |
| 94 | C2×#127 | — | 2 | 24 | no | — |
| 95 | C2×#127 | — | 2 | 128 | no | — |
| 96 | C2×#127 | — | 2 | 8 | no | — |
| 97 | C2×#127 | D4 | 1 | 16 | yes | complemented |
| 98 | C2×#127 | C23 | 1 | 24 | yes | — |
| 99 | C2×#127 | C23 | 1 | 8 | yes | — |
| 100 | C2×#127 | C23 | 1 | 8 | yes | complemented |
| 101 | C2×#127 | C23 | 1 | 24 | yes | complemented |
| 102 | C2×#126 | — | 2 | 16 | no | — |
| 103 | C2×#126 | C23 | 1 | 16 | yes | complemented |
| 104 | C2×#123 | — | 2 | 24 | no | — |
| 105 | C2×#123 | — | 4 | 12 | no | — |
| 106 | C2×#123 | C23 | 1 | 24 | yes | complemented |
| 107 | C2×#123 | C23 | 1 | 24 | yes | complemented |
| 108 | C2×#121 | — | 2 | 48 | no | — |
| 109 | C2×#121 | — | 4 | 24 | no | — |
| 110 | C2×#121 | C23 | 1 | 48 | yes | complemented |
| 111 | C2×#121 | C23 | 1 | 48 | yes | complemented |
| 112 | C2×#120 | — | 2 | 24 | no | — |
| 113 | C2×#120 | — | 4 | 12 | no | — |
| 114 | C2×#120 | — | 4 | 24 | no | — |
| 115 | C2×#120 | C23 | 1 | 48 | yes | — |
| 116 | C2×#120 | C23 | 1 | 24 | yes | complemented |
| 117 | C2×#120 | C23 | 1 | 24 | yes | complemented |
| 118 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 119 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 120 | C22×C22⋊C4 | — | 4 | 12 | no | — |
| 121 | C22×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 122 | C22×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 123 | C22×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 124 | C22×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 125 | C22×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 126 | C22×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 127 | C22×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 128 | C22×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 129 | C22×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 130 | C22×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 131 | C22×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 132 | C22×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 133 | C22×C22⋊C4 | C23 | 1 | 24 | yes | complemented |
| 134 | C22×C42 | — | 2 | 8 | no | — |
| 135 | C22×C42 | C23 | 1 | 3 | yes | — |
| 136 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 137 | C22×C42 | C23 | 1 | 8 | yes | complemented |
| 138 | #394 | C23 | 1 | 48 | yes | complemented |
| 139 | #394 | C23 | 1 | 48 | yes | complemented |
| 140 | #394 | C23 | 1 | 48 | yes | complemented |
| 141 | #394 | C23 | 1 | 48 | yes | complemented |
| 142 | #392 | C23 | 1 | 48 | yes | complemented |
| 143 | #392 | C23 | 1 | 48 | yes | complemented |
| 144 | #392 | C23 | 1 | 48 | yes | complemented |
| 145 | #392 | C23 | 1 | 48 | yes | complemented |
| 146 | #388 | C23 | 1 | 48 | yes | complemented |
| 147 | #387 | C23 | 1 | 48 | yes | complemented |
| 148 | #386 | C23 | 1 | 48 | yes | — |
| 149 | #386 | C23 | 1 | 24 | yes | complemented |
| 150 | #386 | C23 | 1 | 24 | yes | complemented |
| 151 | #386 | C23 | 1 | 48 | yes | complemented |
| 152 | #386 | C23 | 1 | 24 | yes | complemented |
| 153 | #386 | C23 | 1 | 24 | yes | complemented |
| 154 | #386 | C23 | 1 | 48 | yes | complemented |
| 155 | #385 | C23 | 1 | 48 | yes | complemented |
| 156 | #383 | C23 | 1 | 16 | yes | complemented |
| 157 | #379 | C23 | 1 | 24 | yes | — |
| 158 | #379 | C23 | 1 | 24 | yes | complemented |
| 159 | C2×#95 | C23 | 1 | 6 | yes | — |
| 160 | C2×#95 | C23 | 1 | 6 | yes | — |
| 161 | C2×#95 | C23 | 1 | 6 | yes | — |
| 162 | C2×#95 | C23 | 1 | 6 | yes | — |
| 163 | C25 | — | 2 | 4 | no | — |
| 164 | C25 | — | 2 | 12 | no | — |
| 165 | C25 | — | 2 | 12 | no | — |
| 166 | C25 | — | 2 | 24 | no | — |
| 167 | C25 | — | 2 | 12 | no | — |
| 168 | C25 | — | 2 | 4 | no | — |
| 169 | C25 | — | 4 | 6 | no | — |
| 170 | C25 | — | 4 | 6 | no | — |
| 171 | C25 | — | 4 | 12 | no | — |
| 172 | C25 | — | 4 | 6 | no | — |
| 173 | C25 | — | 4 | 6 | no | — |
| 174 | C25 | C24 | 1 | 2 | yes | — |
| 175 | C25 | C24 | 1 | 1 | yes | characteristic |
| 176 | C25 | C24 | 1 | 3 | yes | — |
| 177 | C25 | C24 | 1 | 1 | yes | characteristic |
| 178 | C25 | C24 | 1 | 3 | yes | — |
| 179 | C25 | C24 | 1 | 12 | yes | — |
| 180 | C25 | C24 | 1 | 4 | yes | — |
| 181 | C25 | C24 | 1 | 1 | yes | characteristic |
| 182 | C25 | C24 | 1 | 4 | yes | — |
| 183 | C22×D4 | — | 2 | 12 | no | — |
| 184 | C22×D4 | — | 2 | 24 | no | — |
| 185 | C22×D4 | — | 2 | 12 | no | — |
| 186 | C22×D4 | — | 2 | 24 | no | — |
| 187 | C22×D4 | — | 2 | 24 | no | — |
| 188 | C22×D4 | — | 2 | 96 | no | — |
| 189 | C22×D4 | — | 2 | 96 | no | — |
| 190 | C22×D4 | — | 2 | 12 | no | — |
| 191 | C22×D4 | — | 2 | 12 | no | — |
| 192 | C22×D4 | — | 2 | 12 | no | — |
| 193 | C22×D4 | — | 2 | 24 | no | — |
| 194 | C22×D4 | — | 2 | 12 | no | — |
| 195 | C22×D4 | — | 4 | 48 | no | — |
| 196 | C22×D4 | — | 4 | 48 | no | — |
| 197 | C22×D4 | — | 4 | 6 | no | — |
| 198 | C22×D4 | — | 4 | 6 | no | — |
| 199 | C22×D4 | — | 4 | 6 | no | — |
| 200 | C22×D4 | — | 4 | 6 | no | — |
| 201 | C22×D4 | — | 4 | 12 | no | — |
| 202 | C22×D4 | — | 4 | 6 | no | — |
| 203 | C22×D4 | — | 4 | 6 | no | — |
| 204 | C22×D4 | — | 4 | 6 | no | — |
| 205 | C22×D4 | — | 4 | 6 | no | — |
| 206 | C22×D4 | — | 4 | 6 | no | — |
| 207 | C22×D4 | — | 4 | 6 | no | — |
| 208 | C22×D4 | — | 4 | 48 | no | — |
| 209 | C22×D4 | — | 4 | 12 | no | — |
| 210 | C22×D4 | — | 4 | 48 | no | — |
| 211 | C22×D4 | — | 4 | 192 | no | — |
| 212 | C22×D4 | — | 4 | 192 | no | — |
| 213 | C22×D4 | — | 4 | 12 | no | — |
| 214 | C22×D4 | — | 4 | 12 | no | — |
| 215 | C22×D4 | — | 4 | 12 | no | — |
| 216 | C22×D4 | — | 4 | 12 | no | — |
| 217 | C22×D4 | — | 4 | 12 | no | — |
| 218 | C22×D4 | — | 4 | 48 | no | — |
| 219 | C22×D4 | — | 4 | 64 | no | — |
| 220 | C22×D4 | C2×D4 | 1 | 24 | yes | — |
| 221 | C22×D4 | C2×D4 | 1 | 24 | yes | complemented |
| 222 | C22×D4 | C24 | 1 | 8 | yes | — |
| 223 | C22×D4 | C24 | 1 | 12 | yes | — |
| 224 | C22×D4 | C24 | 1 | 24 | yes | — |
| 225 | C22×D4 | C24 | 1 | 12 | yes | — |
| 226 | C22×D4 | C24 | 1 | 24 | yes | — |
| 227 | C22×D4 | C24 | 1 | 12 | yes | — |
| 228 | C22×D4 | C24 | 1 | 12 | yes | — |
| 229 | C22×D4 | C24 | 1 | 24 | yes | — |
| 230 | C22×D4 | C24 | 1 | 8 | yes | — |
| 231 | C22×D4 | C24 | 1 | 12 | yes | — |
| 232 | C22×D4 | C24 | 1 | 12 | yes | — |
| 233 | C23×C4 | — | 2 | 12 | no | — |
| 234 | C23×C4 | — | 2 | 12 | no | — |
| 235 | C23×C4 | — | 2 | 12 | no | — |
| 236 | C23×C4 | — | 2 | 12 | no | — |
| 237 | C23×C4 | — | 4 | 6 | no | — |
| 238 | C23×C4 | — | 4 | 6 | no | — |
| 239 | C23×C4 | — | 4 | 6 | no | — |
| 240 | C23×C4 | — | 4 | 12 | no | — |
| 241 | C23×C4 | — | 4 | 6 | no | — |
| 242 | C23×C4 | C24 | 1 | 12 | yes | — |
| 243 | C23×C4 | C24 | 1 | 6 | yes | — |
| 244 | C23×C4 | C24 | 1 | 6 | yes | — |
| 245 | C23×C4 | C24 | 1 | 2 | yes | — |
| 246 | C23×C4 | C24 | 1 | 3 | yes | — |
| 247 | C23×C4 | C24 | 1 | 3 | yes | — |
| 248 | C23×C4 | C24 | 1 | 12 | yes | — |
| 249 | C23×C4 | C24 | 1 | 12 | yes | — |
| 250 | C23×C4 | C24 | 1 | 4 | yes | — |
| 251 | #127 | — | 2 | 96 | no | — |
| 252 | #127 | — | 2 | 96 | no | — |
| 253 | #127 | — | 2 | 32 | no | — |
| 254 | #127 | C2×D4 | 1 | 64 | yes | complemented |
| 255 | #126 | — | 2 | 64 | no | — |
| 256 | #123 | — | 2 | 96 | no | — |
| 257 | #123 | — | 4 | 48 | no | — |
| 258 | #121 | — | 2 | 192 | no | — |
| 259 | #121 | — | 4 | 96 | no | — |
| 260 | #120 | — | 2 | 96 | no | — |
| 261 | #120 | — | 4 | 48 | no | — |
| 262 | #120 | — | 4 | 96 | no | — |
| 263 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 264 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 265 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 266 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 267 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 268 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 269 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 270 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 271 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 272 | C2×C22⋊C4 | — | 4 | 48 | no | — |
| 273 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 274 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 275 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 276 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 277 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 278 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 279 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 280 | C2×C22⋊C4 | C24 | 1 | 24 | yes | — |
| 281 | C2×C22⋊C4 | C24 | 1 | 24 | yes | — |
| 282 | C2×C22⋊C4 | C24 | 1 | 24 | yes | — |
| 283 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 284 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 285 | C2×C22⋊C4 | C24 | 1 | 24 | yes | — |
| 286 | C2×C22⋊C4 | C24 | 1 | 24 | yes | — |
| 287 | C2×C22⋊C4 | C24 | 1 | 24 | yes | — |
| 288 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 289 | C2×C22⋊C4 | C24 | 1 | 24 | yes | — |
| 290 | C2×C22⋊C4 | C24 | 1 | 24 | yes | — |
| 291 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 292 | C2×C42 | — | 2 | 12 | no | — |
| 293 | C2×C42 | — | 2 | 32 | no | — |
| 294 | C2×C42 | C2×D4 | 1 | 8 | yes | complemented |
| 295 | C2×C42 | C24 | 1 | 12 | yes | — |
| 296 | C2×C42 | C24 | 1 | 4 | yes | — |
| 297 | #95 | C24 | 1 | 24 | yes | — |
| 298 | #95 | C24 | 1 | 24 | yes | — |
| 299 | #95 | C24 | 1 | 24 | yes | — |
| 300 | #95 | C24 | 1 | 24 | yes | — |
| 301 | C24 | — | 2 | 2 | no | — |
| 302 | C24 | — | 2 | 6 | no | — |
| 303 | C24 | — | 2 | 6 | no | — |
| 304 | C24 | — | 2 | 6 | no | — |
| 305 | C24 | — | 2 | 12 | no | — |
| 306 | C24 | — | 2 | 6 | no | — |
| 307 | C24 | — | 2 | 12 | no | — |
| 308 | C24 | — | 2 | 6 | no | — |
| 309 | C24 | — | 2 | 4 | no | — |
| 310 | C24 | — | 2 | 8 | no | — |
| 311 | C24 | — | 2 | 24 | no | — |
| 312 | C24 | — | 2 | 2 | no | — |
| 313 | C24 | — | 2 | 6 | no | — |
| 314 | C24 | — | 2 | 12 | no | — |
| 315 | C24 | — | 2 | 6 | no | — |
| 316 | C24 | — | 2 | 6 | no | — |
| 317 | C24 | — | 2 | 12 | no | — |
| 318 | C24 | — | 2 | 12 | no | — |
| 319 | C24 | — | 2 | 6 | no | — |
| 320 | C24 | — | 2 | 12 | no | — |
| 321 | C24 | — | 2 | 6 | no | — |
| 322 | C24 | — | 2 | 24 | no | — |
| 323 | C24 | — | 2 | 48 | no | — |
| 324 | C24 | — | 2 | 24 | no | — |
| 325 | C24 | — | 2 | 8 | no | — |
| 326 | C24 | — | 4 | 12 | no | — |
| 327 | C24 | — | 4 | 12 | no | — |
| 328 | C24 | — | 4 | 24 | no | — |
| 329 | C24 | — | 4 | 12 | no | — |
| 330 | C24 | — | 4 | 12 | no | — |
| 331 | C24 | — | 4 | 48 | no | — |
| 332 | C24 | — | 4 | 96 | no | — |
| 333 | C24 | — | 4 | 48 | no | — |
| 334 | C24 | — | 4 | 16 | no | — |
| 335 | C24 | — | 4 | 16 | no | — |
| 336 | C24 | — | 4 | 48 | no | — |
| 337 | C24 | — | 8 | 24 | no | — |
| 338 | C24 | — | 8 | 48 | no | — |
| 339 | C24 | — | 8 | 24 | no | — |
| 340 | C24 | — | 8 | 24 | no | — |
| 341 | C24 | — | 8 | 24 | no | — |
| 342 | C24 | C22×D4 | 1 | 4 | yes | — |
| 343 | C24 | C22×D4 | 1 | 12 | yes | — |
| 344 | C24 | H(5,𝔽2) | 1 | 12 | yes | — |
| 345 | C24 | C25 | 1 | 4 | yes | — |
| 346 | C24 | C25 | 1 | 2 | yes | — |
| 347 | C24 | C25 | 1 | 6 | yes | — |
| 348 | C24 | C25 | 1 | 2 | yes | — |
| 349 | C24 | C25 | 1 | 6 | yes | — |
| 350 | C24 | C25 | 1 | 2 | yes | — |
| 351 | C24 | C25 | 1 | 1 | yes | characteristic |
| 352 | C2×D4 | — | 2 | 24 | no | — |
| 353 | C2×D4 | — | 2 | 48 | no | — |
| 354 | C2×D4 | — | 2 | 24 | no | — |
| 355 | C2×D4 | — | 2 | 48 | no | — |
| 356 | C2×D4 | — | 2 | 48 | no | — |
| 357 | C2×D4 | — | 2 | 24 | no | — |
| 358 | C2×D4 | — | 2 | 24 | no | — |
| 359 | C2×D4 | — | 2 | 24 | no | — |
| 360 | C2×D4 | — | 2 | 48 | no | — |
| 361 | C2×D4 | — | 2 | 24 | no | — |
| 362 | C2×D4 | — | 4 | 48 | no | — |
| 363 | C2×D4 | — | 4 | 48 | no | — |
| 364 | C2×D4 | — | 4 | 12 | no | — |
| 365 | C2×D4 | — | 4 | 12 | no | — |
| 366 | C2×D4 | — | 4 | 12 | no | — |
| 367 | C2×D4 | — | 4 | 12 | no | — |
| 368 | C2×D4 | — | 4 | 24 | no | — |
| 369 | C2×D4 | — | 4 | 12 | no | — |
| 370 | C2×D4 | — | 4 | 12 | no | — |
| 371 | C2×D4 | — | 4 | 12 | no | — |
| 372 | C2×D4 | — | 4 | 12 | no | — |
| 373 | C2×D4 | — | 4 | 12 | no | — |
| 374 | C2×D4 | — | 4 | 12 | no | — |
| 375 | C2×D4 | — | 4 | 24 | no | — |
| 376 | C2×D4 | — | 4 | 48 | no | — |
| 377 | C2×D4 | — | 4 | 96 | no | — |
| 378 | C2×D4 | — | 4 | 48 | no | — |
| 379 | C2×D4 | — | 4 | 384 | no | — |
| 380 | C2×D4 | — | 4 | 384 | no | — |
| 381 | C2×D4 | — | 4 | 48 | no | — |
| 382 | C2×D4 | — | 4 | 48 | no | — |
| 383 | C2×D4 | — | 4 | 24 | no | — |
| 384 | C2×D4 | — | 4 | 24 | no | — |
| 385 | C2×D4 | — | 4 | 24 | no | — |
| 386 | C2×D4 | — | 4 | 24 | no | — |
| 387 | C2×D4 | — | 4 | 24 | no | — |
| 388 | C2×D4 | — | 4 | 128 | no | — |
| 389 | C2×D4 | — | 4 | 32 | no | — |
| 390 | C2×D4 | — | 4 | 32 | no | — |
| 391 | C2×D4 | — | 8 | 48 | no | — |
| 392 | C2×D4 | — | 8 | 48 | no | — |
| 393 | C2×D4 | C22×D4 | 1 | 48 | yes | — |
| 394 | C2×D4 | C22×D4 | 1 | 48 | yes | complemented |
| 395 | C22×C4 | — | 2 | 96 | no | — |
| 396 | C22×C4 | — | 2 | 6 | no | — |
| 397 | C22×C4 | — | 2 | 24 | no | — |
| 398 | C22×C4 | — | 2 | 6 | no | — |
| 399 | C22×C4 | — | 2 | 6 | no | — |
| 400 | C22×C4 | — | 2 | 6 | no | — |
| 401 | C22×C4 | — | 2 | 24 | no | — |
| 402 | C22×C4 | — | 2 | 24 | no | — |
| 403 | C22×C4 | — | 2 | 12 | no | — |
| 404 | C22×C4 | — | 2 | 12 | no | — |
| 405 | C22×C4 | — | 2 | 12 | no | — |
| 406 | C22×C4 | — | 2 | 12 | no | — |
| 407 | C22×C4 | — | 2 | 12 | no | — |
| 408 | C22×C4 | — | 2 | 12 | no | — |
| 409 | C22×C4 | — | 2 | 12 | no | — |
| 410 | C22×C4 | — | 2 | 24 | no | — |
| 411 | C22×C4 | — | 4 | 12 | no | — |
| 412 | C22×C4 | — | 4 | 12 | no | — |
| 413 | C22×C4 | — | 4 | 12 | no | — |
| 414 | C22×C4 | — | 4 | 24 | no | — |
| 415 | C22×C4 | — | 4 | 12 | no | — |
| 416 | C22×C4 | — | 4 | 16 | no | — |
| 417 | C22×C4 | — | 8 | 24 | no | — |
| 418 | C22×C4 | — | 8 | 24 | no | — |
| 419 | C22×C4 | C22×D4 | 1 | 12 | yes | — |
| 420 | C22×C4 | H(5,𝔽2) | 1 | 12 | yes | — |
| 421 | C22×C4 | C25 | 1 | 4 | yes | — |
| 422 | C22×C4 | C25 | 1 | 12 | yes | — |
| 423 | C22×C4 | C25 | 1 | 12 | yes | — |
| 424 | C22×C4 | C25 | 1 | 6 | yes | — |
| 425 | C22×C4 | C25 | 1 | 6 | yes | — |
| 426 | C4⋊C4 | — | 2 | 48 | no | — |
| 427 | C4⋊C4 | — | 2 | 48 | no | — |
| 428 | C22⋊C4 | — | 2 | 24 | no | — |
| 429 | C22⋊C4 | — | 2 | 24 | no | — |
| 430 | C22⋊C4 | — | 2 | 48 | no | — |
| 431 | C22⋊C4 | — | 2 | 48 | no | — |
| 432 | C22⋊C4 | — | 2 | 48 | no | — |
| 433 | C22⋊C4 | — | 4 | 24 | no | — |
| 434 | C22⋊C4 | — | 4 | 12 | no | — |
| 435 | C22⋊C4 | — | 4 | 24 | no | — |
| 436 | C22⋊C4 | — | 4 | 24 | no | — |
| 437 | C22⋊C4 | — | 4 | 24 | no | — |
| 438 | C22⋊C4 | — | 4 | 12 | no | — |
| 439 | C22⋊C4 | — | 4 | 24 | no | — |
| 440 | C42 | — | 2 | 24 | no | — |
| 441 | C42 | C22×D4 | 1 | 16 | yes | complemented |
| 442 | C23 | — | 2 | 2 | no | — |
| 443 | C23 | — | 2 | 6 | no | — |
| 444 | C23 | — | 2 | 24 | no | — |
| 445 | C23 | — | 2 | 24 | no | — |
| 446 | C23 | — | 2 | 6 | no | — |
| 447 | C23 | — | 2 | 6 | no | — |
| 448 | C23 | — | 2 | 12 | no | — |
| 449 | C23 | — | 2 | 6 | no | — |
| 450 | C23 | — | 2 | 12 | no | — |
| 451 | C23 | — | 2 | 6 | no | — |
| 452 | C23 | — | 2 | 24 | no | — |
| 453 | C23 | — | 2 | 48 | no | — |
| 454 | C23 | — | 2 | 24 | no | — |
| 455 | C23 | — | 2 | 4 | no | — |
| 456 | C23 | — | 2 | 2 | no | — |
| 457 | C23 | — | 2 | 6 | no | — |
| 458 | C23 | — | 2 | 12 | no | — |
| 459 | C23 | — | 2 | 6 | no | — |
| 460 | C23 | — | 2 | 6 | no | — |
| 461 | C23 | — | 2 | 12 | no | — |
| 462 | C23 | — | 2 | 12 | no | — |
| 463 | C23 | — | 2 | 6 | no | — |
| 464 | C23 | — | 2 | 12 | no | — |
| 465 | C23 | — | 2 | 6 | no | — |
| 466 | C23 | — | 4 | 48 | no | — |
| 467 | C23 | — | 4 | 96 | no | — |
| 468 | C23 | — | 4 | 48 | no | — |
| 469 | C23 | — | 4 | 4 | no | — |
| 470 | C23 | — | 4 | 12 | no | — |
| 471 | C23 | — | 4 | 12 | no | — |
| 472 | C23 | — | 4 | 24 | no | — |
| 473 | C23 | — | 4 | 12 | no | — |
| 474 | C23 | — | 4 | 16 | no | — |
| 475 | C23 | — | 4 | 12 | no | — |
| 476 | C23 | — | 4 | 12 | no | — |
| 477 | C23 | — | 4 | 24 | no | — |
| 478 | C23 | — | 4 | 24 | no | — |
| 479 | C23 | — | 4 | 4 | no | — |
| 480 | C23 | — | 4 | 8 | no | — |
| 481 | C23 | — | 4 | 12 | no | — |
| 482 | C23 | — | 4 | 16 | no | — |
| 483 | C23 | — | 4 | 48 | no | — |
| 484 | C23 | — | 4 | 24 | no | — |
| 485 | C23 | — | 4 | 24 | no | — |
| 486 | C23 | — | 4 | 12 | no | — |
| 487 | C23 | — | 4 | 12 | no | — |
| 488 | C23 | — | 8 | 24 | no | — |
| 489 | C23 | — | 8 | 48 | no | — |
| 490 | C23 | — | 8 | 24 | no | — |
| 491 | C23 | — | 8 | 24 | no | — |
| 492 | C23 | — | 8 | 192 | no | — |
| 493 | C23 | — | 8 | 64 | no | — |
| 494 | C23 | — | 8 | 24 | no | — |
| 495 | C23 | — | 8 | 64 | no | — |
| 496 | C23 | #560 | 1 | 16 | yes | complemented |
| 497 | C23 | C23×D4 | 1 | 4 | yes | — |
| 498 | C23 | C23×D4 | 1 | 12 | yes | — |
| 499 | C23 | C23×D4 | 1 | 2 | yes | — |
| 500 | C23 | C2×H(5,𝔽2) | 1 | 12 | yes | — |
| 501 | C23 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 502 | C23 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 503 | C23 | C26 | 1 | 1 | yes | characteristic |
| 504 | D4 | — | 4 | 48 | no | — |
| 505 | D4 | — | 4 | 48 | no | — |
| 506 | D4 | — | 4 | 48 | no | — |
| 507 | D4 | — | 4 | 96 | no | — |
| 508 | D4 | — | 4 | 48 | no | — |
| 509 | D4 | — | 4 | 48 | no | — |
| 510 | D4 | — | 4 | 48 | no | — |
| 511 | D4 | — | 4 | 32 | no | — |
| 512 | D4 | — | 4 | 32 | no | — |
| 513 | D4 | — | 8 | 48 | no | — |
| 514 | D4 | — | 8 | 48 | no | — |
| 515 | C2×C4 | — | 2 | 96 | no | — |
| 516 | C2×C4 | — | 2 | 6 | no | — |
| 517 | C2×C4 | — | 2 | 6 | no | — |
| 518 | C2×C4 | — | 2 | 6 | no | — |
| 519 | C2×C4 | — | 2 | 6 | no | — |
| 520 | C2×C4 | — | 2 | 24 | no | — |
| 521 | C2×C4 | — | 2 | 24 | no | — |
| 522 | C2×C4 | — | 2 | 12 | no | — |
| 523 | C2×C4 | — | 2 | 12 | no | — |
| 524 | C2×C4 | — | 2 | 12 | no | — |
| 525 | C2×C4 | — | 2 | 12 | no | — |
| 526 | C2×C4 | — | 2 | 12 | no | — |
| 527 | C2×C4 | — | 2 | 12 | no | — |
| 528 | C2×C4 | — | 2 | 12 | no | — |
| 529 | C2×C4 | — | 4 | 16 | no | — |
| 530 | C2×C4 | — | 4 | 24 | no | — |
| 531 | C2×C4 | — | 4 | 8 | no | — |
| 532 | C2×C4 | — | 4 | 24 | no | — |
| 533 | C2×C4 | — | 8 | 24 | no | — |
| 534 | C2×C4 | — | 8 | 24 | no | — |
| 535 | C2×C4 | C23×D4 | 1 | 12 | yes | — |
| 536 | C2×C4 | C2×H(5,𝔽2) | 1 | 12 | yes | — |
| 537 | C22 | — | 2 | 12 | no | — |
| 538 | C22 | — | 2 | 12 | no | — |
| 539 | C22 | — | 2 | 32 | no | — |
| 540 | C22 | — | 2 | 12 | no | — |
| 541 | C22 | — | 2 | 24 | no | — |
| 542 | C22 | — | 2 | 12 | no | — |
| 543 | C22 | — | 4 | 6 | no | — |
| 544 | C22 | — | 4 | 12 | no | — |
| 545 | C22 | — | 4 | 6 | no | — |
| 546 | C22 | — | 4 | 2 | no | — |
| 547 | C22 | — | 4 | 6 | no | — |
| 548 | C22 | — | 4 | 48 | no | — |
| 549 | C22 | — | 4 | 16 | no | — |
| 550 | C22 | — | 4 | 12 | no | — |
| 551 | C22 | — | 4 | 12 | no | — |
| 552 | C22 | — | 4 | 6 | no | — |
| 553 | C22 | — | 4 | 6 | no | — |
| 554 | C22 | — | 4 | 6 | no | — |
| 555 | C22 | — | 4 | 6 | no | — |
| 556 | C22 | — | 4 | 12 | no | — |
| 557 | C22 | — | 4 | 12 | no | — |
| 558 | C22 | — | 4 | 2 | no | — |
| 559 | C22 | — | 4 | 4 | no | — |
| 560 | C22 | — | 4 | 6 | no | — |
| 561 | C22 | — | 8 | 96 | no | — |
| 562 | C22 | — | 8 | 32 | no | — |
| 563 | C22 | — | 8 | 32 | no | — |
| 564 | C22 | — | 8 | 8 | no | — |
| 565 | C22 | — | 8 | 24 | no | — |
| 566 | C22 | — | 8 | 16 | no | — |
| 567 | C22 | #3484 | 1 | 12 | yes | — |
| 568 | C22 | C2×#560 | 1 | 4 | yes | — |
| 569 | C22 | C2×#560 | 1 | 8 | yes | complemented |
| 570 | C22 | #3543 | 1 | 12 | yes | — |
| 571 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 572 | C22 | C22×H(5,𝔽2) | 1 | 3 | yes | — |
| 573 | C22 | C22×H(5,𝔽2) | 1 | 3 | yes | — |
| 574 | C4 | — | 2 | 12 | no | — |
| 575 | C4 | — | 2 | 12 | no | — |
| 576 | C4 | — | 4 | 12 | no | — |
| 577 | C4 | — | 4 | 12 | no | — |
| 578 | C4 | — | 4 | 4 | no | — |
| 579 | C2 | — | 2 | 8 | no | — |
| 580 | C2 | — | 4 | 12 | no | — |
| 581 | C2 | — | 4 | 4 | no | — |
| 582 | C2 | — | 8 | 4 | no | — |
| 583 | C2 | — | 8 | 6 | no | — |
| 584 | C2 | — | 8 | 2 | no | — |
| 585 | C2 | #46324 | 1 | 8 | yes | direct product factor |
| 586 | C2 | C22×#560 | 1 | 1 | yes | characteristic |
| 587 | C2 | C2×#3543 | 1 | 3 | yes | — |
| 588 | C2 | C2×#3484 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#560 | C2 | semidirect |
| 2. C23×#127 | C2 | semidirect |
| 3. C2×#2709 | C2 | semidirect |
| 4. C2×#2673 | C2 | semidirect |
| 5. C2×#2669 | C2 | semidirect |
| 6. #46324 | C2 | direct (N × H) |
| 7. C2×#2818 | C2 | semidirect |
| 8. C2×#2852 | C2 | semidirect |
| 9. C2×#2405 | C2 | semidirect |
| 10. C2×#2410 | C2 | semidirect |
| 11. C2×#2403 | C2 | semidirect |
| 12. C2×#2380 | C2 | semidirect |
| 15. C24×D4 | C22 | semidirect |
| 17. C2×#560 | C22 | semidirect |
| 21. C22×#127 | C22 | semidirect |
| 22. C22×#127 | C22 | semidirect |
| 23. C22×#127 | C22 | semidirect |
| 24. C22×#126 | C22 | semidirect |
| 25. C22×#123 | C22 | semidirect |
| 26. C22×#123 | C22 | semidirect |
| 27. C22×#121 | C22 | semidirect |
| 28. C22×#121 | C22 | semidirect |
| 30. C22×#120 | C22 | semidirect |
| 31. C22×#120 | C22 | semidirect |
| 32. C23×C22⋊C4 | C22 | semidirect |
| 33. C23×C42 | C22 | semidirect |
| 34. #2852 | C22 | semidirect |
| 35. #2818 | C22 | semidirect |
| 36. #2709 | C22 | semidirect |
| 37. #2673 | C22 | semidirect |
| 38. #2669 | C22 | semidirect |
| 39. #2410 | C22 | semidirect |
| 40. #2405 | C22 | semidirect |
| 41. #2403 | C22 | semidirect |
| 42. C2×#394 | C22 | semidirect |
| 43. C2×#394 | C22 | semidirect |
| 44. C2×#394 | C22 | semidirect |
| 45. C2×#394 | C22 | semidirect |
| 46. C2×#392 | C22 | semidirect |
| 47. C2×#392 | C22 | semidirect |
| 48. C2×#392 | C22 | semidirect |
| 49. C2×#392 | C22 | semidirect |
| 50. #2380 | C22 | semidirect |
| 51. C2×#388 | C22 | semidirect |
| 52. C2×#387 | C22 | semidirect |
| 54. C2×#386 | C22 | semidirect |
| 55. C2×#386 | C22 | semidirect |
| 56. C2×#386 | C22 | semidirect |
| 57. C2×#386 | C22 | semidirect |
| 58. C2×#386 | C22 | semidirect |
| 59. C2×#386 | C22 | semidirect |
| 60. C2×#385 | C22 | semidirect |
| 61. C2×#383 | C22 | semidirect |
| 63. C2×#379 | C22 | semidirect |
| 87. C23×D4 | C23 | semidirect |
| 97. C2×#127 | D4 | semidirect |
| 100. C2×#127 | C23 | semidirect |
| 101. C2×#127 | C23 | semidirect |
| 103. C2×#126 | C23 | semidirect |
| 106. C2×#123 | C23 | semidirect |
| 107. C2×#123 | C23 | semidirect |
| 110. C2×#121 | C23 | semidirect |
| 111. C2×#121 | C23 | semidirect |
| 116. C2×#120 | C23 | semidirect |
| 117. C2×#120 | C23 | semidirect |
| 133. C22×C22⋊C4 | C23 | semidirect |
| 137. C22×C42 | C23 | semidirect |
| 138. #394 | C23 | semidirect |
| 139. #394 | C23 | semidirect |
| 140. #394 | C23 | semidirect |
| 141. #394 | C23 | semidirect |
| 142. #392 | C23 | semidirect |
| 143. #392 | C23 | semidirect |
| 144. #392 | C23 | semidirect |
| 145. #392 | C23 | semidirect |
| 146. #388 | C23 | semidirect |
| 147. #387 | C23 | semidirect |
| 149. #386 | C23 | semidirect |
| 150. #386 | C23 | semidirect |
| 151. #386 | C23 | semidirect |
| 152. #386 | C23 | semidirect |
| 153. #386 | C23 | semidirect |
| 154. #386 | C23 | semidirect |
| 155. #385 | C23 | semidirect |
| 156. #383 | C23 | semidirect |
| 158. #379 | C23 | semidirect |
| 221. C22×D4 | C2×D4 | semidirect |
| 254. #127 | C2×D4 | semidirect |
| 294. C2×C42 | C2×D4 | semidirect |
| 394. C2×D4 | C22×D4 | semidirect |
| 441. C42 | C22×D4 | semidirect |
| 496. C23 | #560 | semidirect |
| 569. C22 | C2×#560 | semidirect |
| 585. C2 | #46324 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#560 | 2 |
| C23×#127 | 2 |
| C2×#2709 | 2 |
| C2×#2673 | 2 |
| C2×#2669 | 2 |
| #46324 | 2 |
| C2×#2818 | 2 |
| C2×#2852 | 2 |
| C2×#2405 | 2 |
| C2×#2410 | 2 |
| C2×#2403 | 2 |
| C2×#2380 | 2 |