Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1780 conjugacy classes of subgroups.
The subgroup lattice has 550 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #45509 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #45479 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #32930 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #32929 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #32708 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #32706 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #32119 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #32117 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #31269 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | #31266 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | #28714 | C2 | 1 | 3 | yes | maximal, complemented |
| 12 | #27799 | C2 | 1 | 3 | yes | maximal, complemented |
| 13 | #27798 | C2 | 1 | 3 | yes | maximal, complemented |
| 14 | #24275 | C2 | 1 | 3 | yes | maximal, complemented |
| 15 | #21440 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C22×AGL(1,ℤ/8) | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C24×C8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×H(3,𝔽4) | — | 2 | 1 | no | — |
| 19 | C2×#501 | C22 | 1 | 3 | yes | complemented |
| 20 | C2×#466 | C22 | 1 | 3 | yes | — |
| 21 | C2×#465 | C22 | 1 | 3 | yes | — |
| 22 | C2×#435 | C22 | 1 | 3 | yes | — |
| 23 | #2848 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #2847 | C22 | 1 | 1 | yes | characteristic |
| 25 | #2847 | C22 | 1 | 1 | yes | characteristic |
| 26 | #2847 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | #2812 | — | 2 | 1 | no | — |
| 28 | #2812 | — | 2 | 1 | no | — |
| 29 | #2782 | — | 2 | 3 | no | — |
| 30 | #2753 | — | 2 | 3 | no | — |
| 31 | #2712 | — | 2 | 3 | no | — |
| 32 | #2684 | — | 2 | 3 | no | — |
| 33 | #2562 | C22 | 1 | 3 | yes | — |
| 34 | #2547 | C22 | 1 | 3 | yes | — |
| 35 | #2491 | C22 | 1 | 3 | yes | — |
| 36 | #2491 | C22 | 1 | 3 | yes | — |
| 37 | #2490 | C22 | 1 | 3 | yes | complemented |
| 38 | #2489 | C22 | 1 | 3 | yes | complemented |
| 39 | #2408 | C22 | 1 | 3 | yes | — |
| 40 | #2407 | C22 | 1 | 3 | yes | — |
| 41 | #2375 | — | 2 | 1 | no | — |
| 42 | C4×#120 | C22 | 1 | 1 | yes | characteristic |
| 43 | #2077 | C22 | 1 | 3 | yes | complemented |
| 44 | #2077 | C22 | 1 | 3 | yes | complemented |
| 45 | #2076 | C22 | 1 | 3 | yes | complemented |
| 46 | #2075 | C22 | 1 | 3 | yes | complemented |
| 47 | #2055 | C22 | 1 | 3 | yes | complemented |
| 48 | #2054 | C22 | 1 | 3 | yes | complemented |
| 49 | #2035 | C22 | 1 | 3 | yes | complemented |
| 50 | #2034 | C22 | 1 | 3 | yes | — |
| 51 | #2033 | C22 | 1 | 3 | yes | — |
| 52 | #2033 | C22 | 1 | 3 | yes | complemented |
| 53 | #2025 | C22 | 1 | 3 | yes | complemented |
| 54 | #2025 | C22 | 1 | 3 | yes | complemented |
| 55 | #2024 | C22 | 1 | 3 | yes | complemented |
| 56 | #2023 | C22 | 1 | 3 | yes | complemented |
| 57 | #2000 | C22 | 1 | 3 | yes | — |
| 58 | #1999 | C22 | 1 | 3 | yes | — |
| 59 | #1989 | C22 | 1 | 3 | yes | complemented |
| 60 | #1987 | C22 | 1 | 3 | yes | complemented |
| 61 | #1973 | C22 | 1 | 3 | yes | complemented |
| 62 | #1972 | C22 | 1 | 3 | yes | complemented |
| 63 | #1928 | C22 | 1 | 3 | yes | complemented |
| 64 | #1926 | C22 | 1 | 3 | yes | complemented |
| 65 | #1926 | C22 | 1 | 3 | yes | complemented |
| 66 | #1925 | C22 | 1 | 3 | yes | complemented |
| 67 | #1893 | C22 | 1 | 3 | yes | — |
| 68 | #1893 | C22 | 1 | 3 | yes | — |
| 69 | #1893 | C22 | 1 | 3 | yes | complemented |
| 70 | #1892 | C22 | 1 | 3 | yes | complemented |
| 71 | #1869 | C22 | 1 | 3 | yes | — |
| 72 | #1850 | C22 | 1 | 3 | yes | complemented |
| 73 | #1809 | C22 | 1 | 3 | yes | complemented |
| 74 | #1808 | C22 | 1 | 3 | yes | complemented |
| 75 | #1807 | C22 | 1 | 3 | yes | complemented |
| 76 | #1788 | C22 | 1 | 3 | yes | complemented |
| 77 | #1786 | C22 | 1 | 3 | yes | complemented |
| 78 | #1779 | C22 | 1 | 1 | yes | characteristic, complemented |
| 79 | #1776 | C22 | 1 | 3 | yes | complemented |
| 80 | #1752 | C22 | 1 | 3 | yes | complemented |
| 81 | C22×#41 | C23 | 1 | 1 | yes | characteristic |
| 82 | C23×D4 | C23 | 1 | 1 | yes | characteristic |
| 83 | C24×C4 | C23 | 1 | 1 | yes | characteristic |
| 84 | C2×AGL(1,ℤ/8) | — | 4 | 6 | no | — |
| 85 | C22×SD16 | C23 | 1 | 1 | yes | characteristic |
| 86 | C22×SD16 | C23 | 1 | 1 | yes | characteristic |
| 87 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 88 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 89 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 90 | C23×C8 | — | 4 | 6 | no | — |
| 91 | C23×C8 | C23 | 1 | 3 | yes | — |
| 92 | C23×C8 | C23 | 1 | 3 | yes | — |
| 93 | H(3,𝔽4) | — | 4 | 8 | no | — |
| 94 | C2×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 95 | C2×#125 | C23 | 1 | 3 | yes | — |
| 96 | C2×#124 | — | 2 | 3 | no | — |
| 97 | C2×#124 | — | 2 | 3 | no | — |
| 98 | C2×#124 | — | 2 | 3 | no | — |
| 99 | C2×C22⋊Q8 | C23 | 1 | 3 | yes | — |
| 100 | C2×#121 | C23 | 1 | 3 | yes | — |
| 101 | C2×#121 | C23 | 1 | 3 | yes | — |
| 102 | C2×#121 | C23 | 1 | 3 | yes | — |
| 103 | C2×#120 | — | 2 | 1 | no | — |
| 104 | C2×#120 | — | 2 | 1 | no | — |
| 105 | C2×#120 | — | 2 | 1 | no | — |
| 106 | C2×#120 | — | 2 | 3 | no | — |
| 107 | C2×#120 | D4 | 1 | 1 | yes | characteristic |
| 108 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 109 | #501 | — | 4 | 6 | no | — |
| 110 | #466 | — | 4 | 6 | no | — |
| 111 | #465 | — | 4 | 6 | no | — |
| 112 | #435 | — | 4 | 6 | no | — |
| 113 | C2×#107 | C23 | 1 | 3 | yes | — |
| 114 | C2×#107 | C23 | 1 | 3 | yes | — |
| 115 | C2×#106 | C23 | 1 | 3 | yes | — |
| 116 | C2×#106 | C23 | 1 | 3 | yes | — |
| 117 | C2×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 118 | C2×#103 | C23 | 1 | 3 | yes | — |
| 119 | C2×#103 | C23 | 1 | 3 | yes | — |
| 120 | C2×#102 | C23 | 1 | 3 | yes | — |
| 121 | C2×#102 | C23 | 1 | 3 | yes | — |
| 122 | C2×#102 | C23 | 1 | 3 | yes | — |
| 123 | C2×#102 | C23 | 1 | 3 | yes | — |
| 124 | C2×#102 | C23 | 1 | 3 | yes | — |
| 125 | C2×#102 | C23 | 1 | 3 | yes | — |
| 126 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 127 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 128 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 129 | C2×#97 | C23 | 1 | 3 | yes | — |
| 130 | #400 | — | 2 | 3 | no | — |
| 131 | #400 | — | 2 | 3 | no | — |
| 132 | #397 | — | 2 | 3 | no | — |
| 133 | #397 | — | 2 | 3 | no | — |
| 134 | #397 | — | 2 | 3 | no | — |
| 135 | #397 | — | 2 | 3 | no | — |
| 136 | #394 | — | 2 | 3 | no | — |
| 137 | #394 | — | 2 | 3 | no | — |
| 138 | #394 | — | 2 | 3 | no | — |
| 139 | #394 | — | 2 | 3 | no | — |
| 140 | #394 | — | 2 | 3 | no | — |
| 141 | #394 | — | 2 | 3 | no | — |
| 142 | #393 | — | 2 | 3 | no | — |
| 143 | #393 | — | 2 | 3 | no | — |
| 144 | #392 | — | 2 | 3 | no | — |
| 145 | #392 | — | 2 | 3 | no | — |
| 146 | #391 | — | 2 | 3 | no | — |
| 147 | #391 | C23 | 1 | 3 | yes | — |
| 148 | #390 | — | 2 | 3 | no | — |
| 149 | #390 | C23 | 1 | 3 | yes | — |
| 150 | #390 | C23 | 1 | 3 | yes | — |
| 151 | #390 | C23 | 1 | 3 | yes | — |
| 152 | #388 | — | 2 | 3 | no | — |
| 153 | #388 | — | 2 | 3 | no | — |
| 154 | #388 | — | 2 | 3 | no | — |
| 155 | #388 | — | 2 | 3 | no | — |
| 156 | #388 | — | 2 | 3 | no | — |
| 157 | #388 | — | 2 | 3 | no | — |
| 158 | #386 | — | 2 | 3 | no | — |
| 159 | #386 | — | 2 | 3 | no | — |
| 160 | #386 | — | 2 | 3 | no | — |
| 161 | #386 | — | 2 | 3 | no | — |
| 162 | #386 | D4 | 1 | 3 | yes | — |
| 163 | #385 | — | 2 | 3 | no | — |
| 164 | #385 | — | 2 | 3 | no | — |
| 165 | #385 | D4 | 1 | 3 | yes | — |
| 166 | #382 | — | 2 | 3 | no | — |
| 167 | #380 | C23 | 1 | 3 | yes | — |
| 168 | #380 | C23 | 1 | 3 | yes | — |
| 169 | #380 | C23 | 1 | 3 | yes | — |
| 170 | #380 | C23 | 1 | 3 | yes | — |
| 171 | #380 | C23 | 1 | 3 | yes | — |
| 172 | #380 | C23 | 1 | 3 | yes | — |
| 173 | #379 | — | 2 | 1 | no | — |
| 174 | #379 | — | 2 | 3 | no | — |
| 175 | #379 | D4 | 1 | 1 | yes | characteristic |
| 176 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 177 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 178 | C4×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 179 | #340 | C23 | 1 | 3 | yes | — |
| 180 | #340 | C23 | 1 | 3 | yes | — |
| 181 | #340 | C23 | 1 | 3 | yes | — |
| 182 | #340 | C23 | 1 | 3 | yes | — |
| 183 | #340 | C23 | 1 | 3 | yes | — |
| 184 | #340 | C23 | 1 | 3 | yes | — |
| 185 | #340 | C23 | 1 | 3 | yes | — |
| 186 | #340 | C23 | 1 | 3 | yes | — |
| 187 | #336 | C23 | 1 | 3 | yes | — |
| 188 | #336 | C23 | 1 | 3 | yes | — |
| 189 | C25 | — | 2 | 1 | no | — |
| 190 | C25 | C2×D4 | 1 | 1 | yes | characteristic |
| 191 | C2×#41 | — | 4 | 3 | no | — |
| 192 | C2×#41 | — | 4 | 3 | no | — |
| 193 | C22×Q8 | — | 2 | 3 | no | — |
| 194 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 195 | C22×D4 | — | 2 | 3 | no | — |
| 196 | C22×D4 | — | 2 | 3 | no | — |
| 197 | C22×D4 | — | 2 | 3 | no | — |
| 198 | C22×D4 | — | 2 | 1 | no | — |
| 199 | C22×D4 | — | 2 | 1 | no | — |
| 200 | C22×D4 | — | 4 | 3 | no | — |
| 201 | C22×D4 | — | 4 | 3 | no | — |
| 202 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 203 | C22×D4 | C24 | 1 | 1 | yes | characteristic |
| 204 | C22×D4 | C24 | 1 | 1 | yes | characteristic |
| 205 | C22×D4 | C24 | 1 | 1 | yes | characteristic |
| 206 | C23×C4 | — | 2 | 1 | no | — |
| 207 | C23×C4 | — | 4 | 3 | no | — |
| 208 | C23×C4 | — | 4 | 3 | no | — |
| 209 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 210 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 211 | C23×C4 | C24 | 1 | 3 | yes | — |
| 212 | AGL(1,ℤ/8) | — | 4 | 16 | no | — |
| 213 | C2×SD16 | — | 2 | 6 | no | — |
| 214 | C2×SD16 | — | 4 | 3 | no | — |
| 215 | C2×D8 | — | 2 | 6 | no | — |
| 216 | C2×D8 | — | 4 | 3 | no | — |
| 217 | C2×M16 | — | 2 | 6 | no | — |
| 218 | C22×C8 | — | 2 | 6 | no | — |
| 219 | C22×C8 | — | 2 | 6 | no | — |
| 220 | C22×C8 | — | 2 | 6 | no | — |
| 221 | C22×C8 | — | 4 | 3 | no | — |
| 222 | C22×C8 | — | 4 | 3 | no | — |
| 223 | C22×C8 | — | 4 | 3 | no | — |
| 224 | C22×C8 | — | 4 | 16 | no | — |
| 225 | C22×C8 | C24 | 1 | 3 | yes | — |
| 226 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 227 | C22×C8 | C24 | 1 | 3 | yes | — |
| 228 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 229 | C4⋊Q8 | — | 4 | 3 | no | — |
| 230 | C4⋊Q8 | — | 4 | 3 | no | — |
| 231 | #125 | — | 4 | 3 | no | — |
| 232 | #125 | — | 4 | 3 | no | — |
| 233 | #124 | — | 4 | 12 | no | — |
| 234 | #124 | — | 4 | 12 | no | — |
| 235 | #124 | — | 4 | 12 | no | — |
| 236 | C22⋊Q8 | — | 4 | 3 | no | — |
| 237 | C22⋊Q8 | — | 4 | 3 | no | — |
| 238 | #121 | — | 4 | 3 | no | — |
| 239 | #121 | — | 4 | 3 | no | — |
| 240 | #121 | — | 4 | 3 | no | — |
| 241 | #121 | — | 4 | 3 | no | — |
| 242 | #121 | — | 4 | 3 | no | — |
| 243 | #121 | — | 4 | 3 | no | — |
| 244 | #120 | — | 2 | 4 | no | — |
| 245 | #120 | — | 4 | 12 | no | — |
| 246 | #120 | — | 4 | 4 | no | — |
| 247 | #120 | — | 4 | 4 | no | — |
| 248 | #120 | — | 4 | 4 | no | — |
| 249 | C4×D4 | — | 4 | 3 | no | — |
| 250 | C4×D4 | — | 4 | 3 | no | — |
| 251 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 252 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 253 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 254 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 255 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 256 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 257 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 258 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 259 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 260 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 261 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 262 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 263 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 264 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 265 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 266 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 267 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 268 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 269 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 270 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 271 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 272 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 273 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 274 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 275 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 276 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 277 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 278 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 279 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 280 | C2×C42 | — | 2 | 3 | no | — |
| 281 | C2×C42 | — | 2 | 3 | no | — |
| 282 | C2×C42 | C24 | 1 | 3 | yes | — |
| 283 | #107 | — | 2 | 6 | no | — |
| 284 | #107 | — | 4 | 3 | no | — |
| 285 | #106 | — | 2 | 6 | no | — |
| 286 | #106 | — | 4 | 3 | no | — |
| 287 | C4⋊C8 | — | 4 | 3 | no | — |
| 288 | #103 | — | 4 | 3 | no | — |
| 289 | #103 | — | 4 | 3 | no | — |
| 290 | #102 | — | 4 | 3 | no | — |
| 291 | #102 | — | 4 | 3 | no | — |
| 292 | #102 | — | 4 | 3 | no | — |
| 293 | #102 | — | 4 | 3 | no | — |
| 294 | #102 | — | 4 | 3 | no | — |
| 295 | #102 | — | 4 | 3 | no | — |
| 296 | C22⋊C8 | — | 4 | 3 | no | — |
| 297 | C22⋊C8 | — | 4 | 3 | no | — |
| 298 | C22⋊C8 | — | 4 | 3 | no | — |
| 299 | #97 | — | 2 | 6 | no | — |
| 300 | #95 | — | 2 | 3 | no | — |
| 301 | #95 | — | 2 | 3 | no | — |
| 302 | #95 | — | 2 | 3 | no | — |
| 303 | #95 | — | 2 | 3 | no | — |
| 304 | #95 | — | 2 | 3 | no | — |
| 305 | #95 | — | 2 | 3 | no | — |
| 306 | #95 | — | 2 | 3 | no | — |
| 307 | #95 | — | 2 | 3 | no | — |
| 308 | #95 | — | 2 | 3 | no | — |
| 309 | #95 | — | 2 | 3 | no | — |
| 310 | #95 | C2×D4 | 1 | 3 | yes | — |
| 311 | #95 | C2×D4 | 1 | 3 | yes | — |
| 312 | C24 | — | 2 | 1 | no | — |
| 313 | C24 | — | 2 | 1 | no | — |
| 314 | C24 | — | 2 | 1 | no | — |
| 315 | C24 | — | 4 | 3 | no | — |
| 316 | C24 | — | 4 | 3 | no | — |
| 317 | C24 | — | 4 | 2 | no | — |
| 318 | C24 | — | 8 | 3 | no | — |
| 319 | C24 | — | 8 | 3 | no | — |
| 320 | C24 | AGL(1,ℤ/8) | 1 | 4 | yes | complemented |
| 321 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 322 | C24 | C22×D4 | 1 | 3 | yes | — |
| 323 | #41 | — | 4 | 4 | no | — |
| 324 | #41 | — | 8 | 6 | no | — |
| 325 | C2×Q8 | — | 2 | 3 | no | — |
| 326 | C2×Q8 | — | 2 | 3 | no | — |
| 327 | C2×Q8 | — | 4 | 6 | no | — |
| 328 | C2×Q8 | — | 8 | 3 | no | — |
| 329 | C2×Q8 | — | 8 | 3 | no | — |
| 330 | C2×D4 | — | 2 | 6 | no | — |
| 331 | C2×D4 | — | 2 | 3 | no | — |
| 332 | C2×D4 | — | 2 | 3 | no | — |
| 333 | C2×D4 | — | 2 | 3 | no | — |
| 334 | C2×D4 | — | 2 | 3 | no | — |
| 335 | C2×D4 | — | 2 | 3 | no | — |
| 336 | C2×D4 | — | 2 | 3 | no | — |
| 337 | C2×D4 | — | 4 | 3 | no | — |
| 338 | C2×D4 | — | 4 | 3 | no | — |
| 339 | C2×D4 | — | 4 | 4 | no | — |
| 340 | C2×D4 | — | 4 | 6 | no | — |
| 341 | C2×D4 | — | 4 | 6 | no | — |
| 342 | C2×D4 | — | 4 | 6 | no | — |
| 343 | C2×D4 | — | 8 | 3 | no | — |
| 344 | C2×D4 | — | 8 | 6 | no | — |
| 345 | C2×D4 | — | 8 | 3 | no | — |
| 346 | C2×D4 | — | 8 | 3 | no | — |
| 347 | C2×D4 | — | 8 | 3 | no | — |
| 348 | C2×D4 | — | 8 | 3 | no | — |
| 349 | C2×D4 | — | 8 | 3 | no | — |
| 350 | C2×D4 | — | 8 | 3 | no | — |
| 351 | C2×D4 | — | 8 | 3 | no | — |
| 352 | C22×C4 | — | 2 | 3 | no | — |
| 353 | C22×C4 | — | 2 | 3 | no | — |
| 354 | C22×C4 | — | 2 | 3 | no | — |
| 355 | C22×C4 | — | 2 | 3 | no | — |
| 356 | C22×C4 | — | 2 | 3 | no | — |
| 357 | C22×C4 | — | 2 | 3 | no | — |
| 358 | C22×C4 | — | 2 | 3 | no | — |
| 359 | C22×C4 | — | 2 | 3 | no | — |
| 360 | C22×C4 | — | 2 | 3 | no | — |
| 361 | C22×C4 | — | 2 | 1 | no | — |
| 362 | C22×C4 | — | 2 | 3 | no | — |
| 363 | C22×C4 | — | 2 | 3 | no | — |
| 364 | C22×C4 | — | 2 | 3 | no | — |
| 365 | C22×C4 | — | 4 | 3 | no | — |
| 366 | C22×C4 | — | 4 | 3 | no | — |
| 367 | C22×C4 | — | 4 | 3 | no | — |
| 368 | C22×C4 | — | 4 | 4 | no | — |
| 369 | C22×C4 | — | 4 | 12 | no | — |
| 370 | C22×C4 | — | 8 | 3 | no | — |
| 371 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 372 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 373 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 374 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 375 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 376 | SD16 | — | 4 | 4 | no | — |
| 377 | SD16 | — | 4 | 4 | no | — |
| 378 | D8 | — | 4 | 4 | no | — |
| 379 | D8 | — | 4 | 4 | no | — |
| 380 | M16 | — | 4 | 4 | no | — |
| 381 | C2×C8 | — | 2 | 3 | no | — |
| 382 | C2×C8 | — | 2 | 3 | no | — |
| 383 | C2×C8 | — | 2 | 3 | no | — |
| 384 | C2×C8 | — | 2 | 3 | no | — |
| 385 | C2×C8 | — | 2 | 3 | no | — |
| 386 | C2×C8 | — | 2 | 3 | no | — |
| 387 | C2×C8 | — | 2 | 3 | no | — |
| 388 | C2×C8 | — | 4 | 12 | no | — |
| 389 | C2×C8 | — | 4 | 12 | no | — |
| 390 | C2×C8 | H(5,𝔽2) | 1 | 6 | yes | — |
| 391 | C4⋊C4 | — | 2 | 3 | no | — |
| 392 | C4⋊C4 | — | 2 | 3 | no | — |
| 393 | C4⋊C4 | — | 2 | 3 | no | — |
| 394 | C4⋊C4 | — | 2 | 3 | no | — |
| 395 | C4⋊C4 | — | 2 | 3 | no | — |
| 396 | C4⋊C4 | — | 2 | 3 | no | — |
| 397 | C4⋊C4 | — | 2 | 3 | no | — |
| 398 | C4⋊C4 | — | 2 | 3 | no | — |
| 399 | C4⋊C4 | — | 4 | 3 | no | — |
| 400 | C4⋊C4 | — | 8 | 3 | no | — |
| 401 | C4⋊C4 | — | 8 | 3 | no | — |
| 402 | C4⋊C4 | — | 8 | 3 | no | — |
| 403 | C4⋊C4 | — | 8 | 3 | no | — |
| 404 | C22⋊C4 | — | 2 | 6 | no | — |
| 405 | C22⋊C4 | — | 4 | 3 | no | — |
| 406 | C22⋊C4 | — | 4 | 3 | no | — |
| 407 | C22⋊C4 | — | 4 | 3 | no | — |
| 408 | C22⋊C4 | — | 4 | 6 | no | — |
| 409 | C22⋊C4 | — | 4 | 6 | no | — |
| 410 | C22⋊C4 | — | 4 | 6 | no | — |
| 411 | C22⋊C4 | — | 4 | 6 | no | — |
| 412 | C22⋊C4 | — | 4 | 6 | no | — |
| 413 | C22⋊C4 | — | 4 | 6 | no | — |
| 414 | C22⋊C4 | — | 8 | 3 | no | — |
| 415 | C22⋊C4 | — | 8 | 3 | no | — |
| 416 | C22⋊C4 | — | 8 | 3 | no | — |
| 417 | C22⋊C4 | — | 8 | 3 | no | — |
| 418 | C22⋊C4 | — | 8 | 3 | no | — |
| 419 | C22⋊C4 | — | 8 | 3 | no | — |
| 420 | C22⋊C4 | — | 8 | 3 | no | — |
| 421 | C22⋊C4 | — | 8 | 3 | no | — |
| 422 | C22⋊C4 | — | 8 | 3 | no | — |
| 423 | C22⋊C4 | — | 8 | 3 | no | — |
| 424 | C42 | — | 2 | 3 | no | — |
| 425 | C42 | — | 2 | 3 | no | — |
| 426 | C42 | — | 4 | 6 | no | — |
| 427 | C42 | — | 8 | 3 | no | — |
| 428 | C23 | — | 2 | 3 | no | — |
| 429 | C23 | — | 2 | 3 | no | — |
| 430 | C23 | — | 2 | 3 | no | — |
| 431 | C23 | — | 2 | 3 | no | — |
| 432 | C23 | — | 2 | 3 | no | — |
| 433 | C23 | — | 2 | 3 | no | — |
| 434 | C23 | — | 2 | 3 | no | — |
| 435 | C23 | — | 2 | 3 | no | — |
| 436 | C23 | — | 2 | 1 | no | — |
| 437 | C23 | — | 4 | 12 | no | — |
| 438 | C23 | — | 4 | 4 | no | — |
| 439 | C23 | — | 4 | 12 | no | — |
| 440 | C23 | — | 4 | 3 | no | — |
| 441 | C23 | — | 4 | 3 | no | — |
| 442 | C23 | — | 4 | 3 | no | — |
| 443 | C23 | — | 4 | 3 | no | — |
| 444 | C23 | — | 4 | 3 | no | — |
| 445 | C23 | — | 4 | 3 | no | — |
| 446 | C23 | — | 4 | 1 | no | — |
| 447 | C23 | — | 4 | 1 | no | — |
| 448 | C23 | — | 4 | 1 | no | — |
| 449 | C23 | — | 8 | 4 | no | — |
| 450 | C23 | — | 16 | 6 | no | — |
| 451 | C23 | — | 16 | 6 | no | — |
| 452 | C23 | C2×AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 453 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 454 | Q8 | — | 4 | 1 | no | — |
| 455 | Q8 | — | 4 | 3 | no | — |
| 456 | Q8 | — | 16 | 6 | no | — |
| 457 | D4 | — | 4 | 1 | no | — |
| 458 | D4 | — | 4 | 1 | no | — |
| 459 | D4 | — | 4 | 1 | no | — |
| 460 | D4 | — | 4 | 3 | no | — |
| 461 | D4 | — | 4 | 3 | no | — |
| 462 | D4 | — | 4 | 3 | no | — |
| 463 | D4 | — | 8 | 6 | no | — |
| 464 | D4 | — | 8 | 4 | no | — |
| 465 | D4 | — | 8 | 4 | no | — |
| 466 | D4 | — | 16 | 6 | no | — |
| 467 | D4 | — | 16 | 6 | no | — |
| 468 | D4 | — | 16 | 6 | no | — |
| 469 | C2×C4 | — | 2 | 3 | no | — |
| 470 | C2×C4 | — | 2 | 3 | no | — |
| 471 | C2×C4 | — | 2 | 3 | no | — |
| 472 | C2×C4 | — | 2 | 3 | no | — |
| 473 | C2×C4 | — | 2 | 3 | no | — |
| 474 | C2×C4 | — | 2 | 3 | no | — |
| 475 | C2×C4 | — | 2 | 3 | no | — |
| 476 | C2×C4 | — | 2 | 3 | no | — |
| 477 | C2×C4 | — | 2 | 3 | no | — |
| 478 | C2×C4 | — | 2 | 3 | no | — |
| 479 | C2×C4 | — | 4 | 12 | no | — |
| 480 | C2×C4 | — | 4 | 3 | no | — |
| 481 | C2×C4 | — | 4 | 1 | no | — |
| 482 | C2×C4 | — | 4 | 3 | no | — |
| 483 | C2×C4 | — | 4 | 3 | no | — |
| 484 | C2×C4 | — | 4 | 3 | no | — |
| 485 | C2×C4 | — | 4 | 3 | no | — |
| 486 | C2×C4 | — | 4 | 3 | no | — |
| 487 | C2×C4 | — | 4 | 3 | no | — |
| 488 | C2×C4 | — | 4 | 3 | no | — |
| 489 | C2×C4 | — | 4 | 3 | no | — |
| 490 | C2×C4 | — | 4 | 3 | no | — |
| 491 | C2×C4 | — | 4 | 3 | no | — |
| 492 | C2×C4 | — | 4 | 3 | no | — |
| 493 | C2×C4 | — | 4 | 3 | no | — |
| 494 | C2×C4 | — | 4 | 3 | no | — |
| 495 | C2×C4 | — | 4 | 3 | no | — |
| 496 | C2×C4 | — | 4 | 3 | no | — |
| 497 | C2×C4 | — | 4 | 3 | no | — |
| 498 | C2×C4 | — | 4 | 3 | no | — |
| 499 | C2×C4 | — | 8 | 4 | no | — |
| 500 | C2×C4 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 501 | C2×C4 | #585 | 1 | 3 | yes | — |
| 502 | C8 | — | 4 | 3 | no | — |
| 503 | C8 | — | 4 | 1 | no | — |
| 504 | C8 | — | 4 | 3 | no | — |
| 505 | C8 | H(3,𝔽4) | 1 | 4 | yes | complemented |
| 506 | C22 | — | 2 | 6 | no | — |
| 507 | C22 | — | 2 | 6 | no | — |
| 508 | C22 | — | 2 | 6 | no | — |
| 509 | C22 | — | 4 | 16 | no | — |
| 510 | C22 | — | 4 | 3 | no | — |
| 511 | C22 | — | 4 | 3 | no | — |
| 512 | C22 | — | 4 | 3 | no | — |
| 513 | C22 | — | 4 | 3 | no | — |
| 514 | C22 | — | 4 | 3 | no | — |
| 515 | C22 | — | 4 | 3 | no | — |
| 516 | C22 | — | 4 | 1 | no | — |
| 517 | C22 | — | 8 | 3 | no | — |
| 518 | C22 | — | 8 | 3 | no | — |
| 519 | C22 | — | 8 | 3 | no | — |
| 520 | C22 | — | 8 | 3 | no | — |
| 521 | C22 | — | 8 | 3 | no | — |
| 522 | C22 | — | 8 | 3 | no | — |
| 523 | C22 | — | 8 | 1 | no | — |
| 524 | C22 | — | 8 | 1 | no | — |
| 525 | C22 | — | 8 | 1 | no | — |
| 526 | C22 | — | 16 | 8 | no | — |
| 527 | C22 | #3496 | 1 | 3 | yes | — |
| 528 | C22 | C22×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 529 | C22 | #3584 | 1 | 3 | yes | — |
| 530 | C4 | — | 4 | 3 | no | — |
| 531 | C4 | — | 4 | 3 | no | — |
| 532 | C4 | — | 4 | 3 | no | — |
| 533 | C4 | — | 4 | 1 | no | — |
| 534 | C4 | — | 8 | 3 | no | — |
| 535 | C4 | — | 8 | 3 | no | — |
| 536 | C4 | — | 8 | 3 | no | — |
| 537 | C4 | — | 8 | 3 | no | — |
| 538 | C4 | — | 8 | 1 | no | — |
| 539 | C4 | C2×H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 540 | C4 | #3540 | 1 | 3 | yes | — |
| 541 | C2 | — | 4 | 6 | no | — |
| 542 | C2 | — | 8 | 1 | no | — |
| 543 | C2 | — | 16 | 1 | no | — |
| 544 | C2 | — | 16 | 1 | no | — |
| 545 | C2 | — | 16 | 1 | no | — |
| 546 | C2 | #45479 | 1 | 1 | yes | characteristic |
| 547 | C2 | #62253 | 1 | 3 | yes | — |
| 548 | C2 | #62282 | 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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #45509 | C2 | semidirect |
| 2. #45479 | C2 | semidirect |
| 3. #32930 | C2 | semidirect |
| 4. #32929 | C2 | semidirect |
| 5. #32708 | C2 | semidirect |
| 6. #32706 | C2 | semidirect |
| 7. #32119 | C2 | semidirect |
| 8. #32117 | C2 | semidirect |
| 9. #31269 | C2 | semidirect |
| 10. #31266 | C2 | semidirect |
| 11. #28714 | C2 | semidirect |
| 12. #27799 | C2 | semidirect |
| 13. #27798 | C2 | semidirect |
| 14. #24275 | C2 | semidirect |
| 15. #21440 | C2 | semidirect |
| 16. C22×AGL(1,ℤ/8) | C22 | semidirect |
| 17. C24×C8 | C22 | semidirect |
| 19. C2×#501 | C22 | semidirect |
| 23. #2848 | C22 | semidirect |
| 26. #2847 | C22 | semidirect |
| 37. #2490 | C22 | semidirect |
| 38. #2489 | C22 | semidirect |
| 43. #2077 | C22 | semidirect |
| 44. #2077 | C22 | semidirect |
| 45. #2076 | C22 | semidirect |
| 46. #2075 | C22 | semidirect |
| 47. #2055 | C22 | semidirect |
| 48. #2054 | C22 | semidirect |
| 49. #2035 | C22 | semidirect |
| 52. #2033 | C22 | semidirect |
| 53. #2025 | C22 | semidirect |
| 54. #2025 | C22 | semidirect |
| 55. #2024 | C22 | semidirect |
| 56. #2023 | C22 | semidirect |
| 59. #1989 | C22 | semidirect |
| 60. #1987 | C22 | semidirect |
| 61. #1973 | C22 | semidirect |
| 62. #1972 | C22 | semidirect |
| 63. #1928 | C22 | semidirect |
| 64. #1926 | C22 | semidirect |
| 65. #1926 | C22 | semidirect |
| 66. #1925 | C22 | semidirect |
| 69. #1893 | C22 | semidirect |
| 70. #1892 | C22 | semidirect |
| 72. #1850 | C22 | semidirect |
| 73. #1809 | C22 | semidirect |
| 74. #1808 | C22 | semidirect |
| 75. #1807 | C22 | semidirect |
| 76. #1788 | C22 | semidirect |
| 77. #1786 | C22 | semidirect |
| 78. #1779 | C22 | semidirect |
| 79. #1776 | C22 | semidirect |
| 80. #1752 | C22 | semidirect |
| 320. C24 | AGL(1,ℤ/8) | semidirect |
| 505. C8 | H(3,𝔽4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #45509 | 2 |
| #45479 | 2 |
| #32930 | 2 |
| #32929 | 2 |
| #32708 | 2 |
| #32706 | 2 |
| #32119 | 2 |
| #32117 | 2 |
| #31269 | 2 |
| #31266 | 2 |
| #28714 | 2 |
| #27799 | 2 |
| #27798 | 2 |
| #24275 | 2 |
| #21440 | 2 |