Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1316 conjugacy classes of subgroups.
The subgroup lattice has 486 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×D4×D8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#2207 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#2187 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×C4×SD32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #28228 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13381 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13305 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #13202 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #13189 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #12811 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #12731 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | C23×D8 | C22 | 1 | 2 | yes | — |
| 13 | C2×D42 | — | 2 | 1 | no | — |
| 14 | C22×SD32 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×SD32 | C22 | 1 | 2 | yes | complemented |
| 16 | D4×D8 | — | 2 | 4 | no | — |
| 17 | D4×D8 | — | 2 | 4 | no | — |
| 18 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#467 | C22 | 1 | 2 | yes | complemented |
| 20 | C2×#466 | C22 | 1 | 2 | yes | — |
| 21 | C2×#459 | — | 2 | 1 | no | — |
| 22 | C2×#447 | — | 2 | 2 | no | — |
| 23 | C2×C4×Q16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×C4×D8 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×C8×D4 | C22 | 1 | 1 | yes | characteristic |
| 26 | #2538 | — | 2 | 1 | no | — |
| 27 | #2207 | — | 2 | 4 | no | — |
| 28 | #2187 | — | 2 | 8 | no | — |
| 29 | C4×SD32 | — | 2 | 4 | no | — |
| 30 | C2×#367 | C22 | 1 | 1 | yes | characteristic, complemented |
| 31 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 32 | C2×#358 | C22 | 1 | 1 | yes | characteristic, complemented |
| 33 | C2×#357 | C22 | 1 | 2 | yes | — |
| 34 | C2×#357 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×#357 | C22 | 1 | 2 | yes | complemented |
| 36 | C2×#357 | C22 | 1 | 2 | yes | complemented |
| 37 | C2×#357 | C22 | 1 | 2 | yes | complemented |
| 38 | C2×C22⋊C16 | C22 | 1 | 2 | yes | complemented |
| 39 | C2×C22⋊C16 | C22 | 1 | 2 | yes | complemented |
| 40 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 41 | #2036 | — | 2 | 2 | no | — |
| 42 | #1845 | C22 | 1 | 1 | yes | characteristic, complemented |
| 43 | #1808 | C22 | 1 | 2 | yes | complemented |
| 44 | #1803 | — | 2 | 1 | no | — |
| 45 | #1380 | C22 | 1 | 2 | yes | complemented |
| 46 | C23×D4 | — | 2 | 2 | no | — |
| 47 | C23×D4 | — | 4 | 1 | no | — |
| 48 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 49 | C22×D8 | — | 2 | 4 | no | — |
| 50 | C22×D8 | — | 2 | 4 | no | — |
| 51 | C22×D8 | — | 2 | 4 | no | — |
| 52 | C22×D8 | — | 2 | 4 | no | — |
| 53 | C22×D8 | — | 2 | 2 | no | — |
| 54 | C22×D8 | — | 2 | 1 | no | — |
| 55 | C22×D8 | — | 4 | 2 | no | — |
| 56 | C22×D8 | — | 4 | 2 | no | — |
| 57 | C22×D8 | C23 | 1 | 2 | yes | — |
| 58 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 59 | C23×C8 | C23 | 1 | 2 | yes | — |
| 60 | D42 | — | 2 | 8 | no | — |
| 61 | D42 | — | 4 | 4 | no | — |
| 62 | C2×#127 | — | 2 | 1 | no | — |
| 63 | C2×C22⋊Q8 | — | 2 | 2 | no | — |
| 64 | C2×#121 | — | 2 | 2 | no | — |
| 65 | C2×#121 | — | 4 | 1 | no | — |
| 66 | C2×#120 | — | 4 | 2 | no | — |
| 67 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 68 | C2×C4×D4 | — | 2 | 1 | no | — |
| 69 | C2×C4×D4 | D4 | 1 | 1 | yes | characteristic |
| 70 | C2×SD32 | — | 2 | 2 | no | — |
| 71 | C2×SD32 | — | 2 | 2 | no | — |
| 72 | C2×SD32 | — | 2 | 4 | no | — |
| 73 | C2×SD32 | — | 2 | 4 | no | — |
| 74 | C2×SD32 | — | 4 | 2 | no | — |
| 75 | C2×SD32 | D4 | 1 | 4 | yes | complemented |
| 76 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 77 | C22×C16 | C23 | 1 | 2 | yes | — |
| 78 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 79 | #493 | — | 2 | 2 | no | — |
| 80 | #493 | D4 | 1 | 4 | yes | — |
| 81 | #467 | — | 2 | 4 | no | — |
| 82 | #467 | — | 4 | 2 | no | — |
| 83 | #466 | — | 2 | 4 | no | — |
| 84 | #466 | — | 2 | 4 | no | — |
| 85 | #459 | — | 2 | 8 | no | — |
| 86 | #447 | — | 2 | 16 | no | — |
| 87 | C4×Q16 | — | 2 | 2 | no | — |
| 88 | C4×Q16 | — | 2 | 2 | no | — |
| 89 | C4×D8 | — | 2 | 2 | no | — |
| 90 | C4×D8 | D4 | 1 | 4 | yes | — |
| 91 | C8×D4 | — | 2 | 2 | no | — |
| 92 | C8×D4 | — | 2 | 2 | no | — |
| 93 | C2×#107 | — | 2 | 1 | no | — |
| 94 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 95 | C2×#107 | C23 | 1 | 2 | yes | — |
| 96 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 97 | C2×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 98 | C2×#103 | — | 2 | 2 | no | — |
| 99 | C2×#103 | — | 2 | 1 | no | — |
| 100 | C2×#102 | — | 2 | 1 | no | — |
| 101 | C2×#102 | — | 2 | 2 | no | — |
| 102 | C2×C22⋊C8 | D4 | 1 | 2 | yes | — |
| 103 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 104 | #396 | — | 4 | 2 | no | — |
| 105 | #390 | — | 4 | 1 | no | — |
| 106 | #389 | — | 4 | 1 | no | — |
| 107 | #384 | — | 2 | 1 | no | — |
| 108 | #380 | — | 2 | 2 | no | — |
| 109 | #367 | — | 2 | 2 | no | — |
| 110 | C4⋊C16 | — | 2 | 2 | no | — |
| 111 | #358 | — | 2 | 2 | no | — |
| 112 | #357 | — | 2 | 2 | no | — |
| 113 | #357 | — | 2 | 4 | no | — |
| 114 | #357 | — | 2 | 4 | no | — |
| 115 | #357 | — | 2 | 4 | no | — |
| 116 | #357 | — | 4 | 2 | no | — |
| 117 | C22⋊C16 | — | 2 | 4 | no | — |
| 118 | C22⋊C16 | — | 4 | 2 | no | — |
| 119 | C4×C16 | — | 2 | 2 | no | — |
| 120 | #340 | — | 2 | 2 | no | — |
| 121 | C25 | — | 4 | 2 | no | — |
| 122 | C22×Q8 | — | 2 | 1 | no | — |
| 123 | C22×D4 | — | 2 | 8 | no | — |
| 124 | C22×D4 | — | 2 | 1 | no | — |
| 125 | C22×D4 | — | 2 | 2 | no | — |
| 126 | C22×D4 | — | 4 | 4 | no | — |
| 127 | C22×D4 | — | 4 | 4 | no | — |
| 128 | C22×D4 | — | 4 | 4 | no | — |
| 129 | C22×D4 | — | 4 | 8 | no | — |
| 130 | C22×D4 | — | 4 | 2 | no | — |
| 131 | C22×D4 | — | 4 | 2 | no | — |
| 132 | C22×D4 | — | 4 | 1 | no | — |
| 133 | C22×D4 | — | 4 | 1 | no | — |
| 134 | C22×D4 | — | 8 | 2 | no | — |
| 135 | C22×D4 | — | 8 | 2 | no | — |
| 136 | C22×D4 | — | 8 | 4 | no | — |
| 137 | C22×D4 | — | 8 | 4 | no | — |
| 138 | C22×D4 | D8 | 1 | 1 | yes | characteristic |
| 139 | C23×C4 | — | 4 | 1 | no | — |
| 140 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 141 | C2×Q16 | — | 2 | 1 | no | — |
| 142 | C2×Q16 | — | 2 | 1 | no | — |
| 143 | C2×Q16 | — | 2 | 1 | no | — |
| 144 | C2×Q16 | — | 2 | 1 | no | — |
| 145 | C2×Q16 | — | 2 | 1 | no | — |
| 146 | C2×Q16 | C2×D4 | 1 | 2 | yes | complemented |
| 147 | C2×D8 | — | 2 | 8 | no | — |
| 148 | C2×D8 | — | 2 | 4 | no | — |
| 149 | C2×D8 | — | 2 | 1 | no | — |
| 150 | C2×D8 | — | 2 | 2 | no | — |
| 151 | C2×D8 | — | 2 | 2 | no | — |
| 152 | C2×D8 | — | 2 | 2 | no | — |
| 153 | C2×D8 | — | 2 | 1 | no | — |
| 154 | C2×D8 | — | 2 | 1 | no | — |
| 155 | C2×D8 | — | 2 | 2 | no | — |
| 156 | C2×D8 | — | 2 | 2 | no | — |
| 157 | C2×D8 | — | 4 | 4 | no | — |
| 158 | C2×D8 | — | 4 | 4 | no | — |
| 159 | C2×D8 | — | 4 | 8 | no | — |
| 160 | C2×D8 | — | 4 | 8 | no | — |
| 161 | C2×D8 | — | 4 | 8 | no | — |
| 162 | C2×D8 | — | 4 | 8 | no | — |
| 163 | C2×D8 | — | 8 | 1 | no | — |
| 164 | C2×D8 | — | 8 | 1 | no | — |
| 165 | C2×D8 | C2×D4 | 1 | 4 | yes | — |
| 166 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 167 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 168 | C2×D8 | #41 | 1 | 2 | yes | — |
| 169 | C22×C8 | — | 2 | 2 | no | — |
| 170 | C22×C8 | — | 2 | 2 | no | — |
| 171 | C22×C8 | — | 2 | 2 | no | — |
| 172 | C22×C8 | — | 2 | 2 | no | — |
| 173 | C22×C8 | — | 2 | 2 | no | — |
| 174 | C22×C8 | — | 2 | 2 | no | — |
| 175 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 176 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 177 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 178 | #127 | — | 2 | 4 | no | — |
| 179 | #127 | — | 4 | 2 | no | — |
| 180 | C22⋊Q8 | — | 4 | 4 | no | — |
| 181 | C22⋊Q8 | — | 8 | 2 | no | — |
| 182 | #121 | — | 2 | 8 | no | — |
| 183 | #121 | — | 4 | 8 | no | — |
| 184 | #121 | — | 4 | 4 | no | — |
| 185 | #120 | — | 4 | 16 | no | — |
| 186 | C4×Q8 | — | 4 | 2 | no | — |
| 187 | C4×Q8 | — | 4 | 2 | no | — |
| 188 | C4×D4 | — | 2 | 2 | no | — |
| 189 | C4×D4 | — | 2 | 4 | no | — |
| 190 | C4×D4 | — | 2 | 2 | no | — |
| 191 | C4×D4 | — | 4 | 2 | no | — |
| 192 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 193 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 194 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 195 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 196 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 197 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 198 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 199 | C2×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 200 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 201 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 202 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 203 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 204 | C2×C22⋊C4 | D8 | 1 | 2 | yes | — |
| 205 | C2×C42 | — | 4 | 1 | no | — |
| 206 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 207 | SD32 | — | 2 | 8 | no | — |
| 208 | SD32 | — | 4 | 8 | no | — |
| 209 | C2×C16 | — | 2 | 1 | no | — |
| 210 | C2×C16 | — | 2 | 1 | no | — |
| 211 | C2×C16 | — | 2 | 1 | no | — |
| 212 | C2×C16 | — | 2 | 2 | no | — |
| 213 | C2×C16 | — | 2 | 2 | no | — |
| 214 | C2×C16 | — | 2 | 1 | no | — |
| 215 | C2×C16 | — | 2 | 2 | no | — |
| 216 | C2×C16 | — | 2 | 1 | no | — |
| 217 | C2×C16 | C2×D4 | 1 | 2 | yes | complemented |
| 218 | #107 | — | 2 | 1 | no | — |
| 219 | #107 | — | 2 | 1 | no | — |
| 220 | #107 | — | 2 | 2 | no | — |
| 221 | #107 | — | 2 | 1 | no | — |
| 222 | #107 | — | 2 | 2 | no | — |
| 223 | #107 | — | 8 | 1 | no | — |
| 224 | #107 | C2×D4 | 1 | 2 | yes | — |
| 225 | C4⋊C8 | — | 2 | 2 | no | — |
| 226 | #103 | — | 4 | 4 | no | — |
| 227 | #103 | — | 4 | 2 | no | — |
| 228 | #102 | — | 2 | 8 | no | — |
| 229 | #102 | — | 2 | 4 | no | — |
| 230 | C22⋊C8 | — | 2 | 4 | no | — |
| 231 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 232 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 233 | #95 | — | 4 | 2 | no | — |
| 234 | C24 | — | 4 | 8 | no | — |
| 235 | C24 | — | 4 | 1 | no | — |
| 236 | C24 | — | 4 | 2 | no | — |
| 237 | C24 | — | 8 | 4 | no | — |
| 238 | C24 | — | 8 | 4 | no | — |
| 239 | C24 | — | 8 | 4 | no | — |
| 240 | C24 | — | 8 | 4 | no | — |
| 241 | C24 | — | 16 | 2 | no | — |
| 242 | C24 | — | 16 | 2 | no | — |
| 243 | C24 | C2×D8 | 1 | 2 | yes | — |
| 244 | C2×Q8 | — | 2 | 2 | no | — |
| 245 | C2×Q8 | — | 4 | 1 | no | — |
| 246 | C2×Q8 | — | 4 | 1 | no | — |
| 247 | C2×Q8 | — | 4 | 1 | no | — |
| 248 | C2×Q8 | — | 4 | 1 | no | — |
| 249 | C2×Q8 | — | 4 | 1 | no | — |
| 250 | C2×D4 | — | 2 | 2 | no | — |
| 251 | C2×D4 | — | 2 | 4 | no | — |
| 252 | C2×D4 | — | 2 | 2 | no | — |
| 253 | C2×D4 | — | 2 | 2 | no | — |
| 254 | C2×D4 | — | 2 | 2 | no | — |
| 255 | C2×D4 | — | 2 | 2 | no | — |
| 256 | C2×D4 | — | 4 | 8 | no | — |
| 257 | C2×D4 | — | 4 | 16 | no | — |
| 258 | C2×D4 | — | 4 | 8 | no | — |
| 259 | C2×D4 | — | 4 | 2 | no | — |
| 260 | C2×D4 | — | 4 | 2 | no | — |
| 261 | C2×D4 | — | 4 | 1 | no | — |
| 262 | C2×D4 | — | 4 | 2 | no | — |
| 263 | C2×D4 | — | 4 | 1 | no | — |
| 264 | C2×D4 | — | 4 | 2 | no | — |
| 265 | C2×D4 | — | 4 | 1 | no | — |
| 266 | C2×D4 | — | 4 | 2 | no | — |
| 267 | C2×D4 | — | 4 | 4 | no | — |
| 268 | C2×D4 | — | 4 | 4 | no | — |
| 269 | C2×D4 | — | 8 | 4 | no | — |
| 270 | C2×D4 | — | 8 | 4 | no | — |
| 271 | C2×D4 | — | 8 | 8 | no | — |
| 272 | C2×D4 | — | 8 | 8 | no | — |
| 273 | C2×D4 | — | 8 | 8 | no | — |
| 274 | C2×D4 | — | 8 | 32 | no | — |
| 275 | C2×D4 | — | 8 | 4 | no | — |
| 276 | C2×D4 | — | 8 | 4 | no | — |
| 277 | C2×D4 | — | 8 | 2 | no | — |
| 278 | C2×D4 | — | 8 | 2 | no | — |
| 279 | C2×D4 | — | 16 | 1 | no | — |
| 280 | C2×D4 | — | 16 | 1 | no | — |
| 281 | C2×D4 | SD32 | 1 | 4 | yes | complemented |
| 282 | C22×C4 | — | 2 | 2 | no | — |
| 283 | C22×C4 | — | 2 | 2 | no | — |
| 284 | C22×C4 | — | 2 | 2 | no | — |
| 285 | C22×C4 | — | 2 | 2 | no | — |
| 286 | C22×C4 | — | 2 | 2 | no | — |
| 287 | C22×C4 | — | 2 | 2 | no | — |
| 288 | C22×C4 | — | 4 | 1 | no | — |
| 289 | C22×C4 | — | 4 | 1 | no | — |
| 290 | C22×C4 | — | 4 | 4 | no | — |
| 291 | C22×C4 | — | 4 | 1 | no | — |
| 292 | C22×C4 | — | 4 | 2 | no | — |
| 293 | C22×C4 | — | 8 | 2 | no | — |
| 294 | C22×C4 | — | 8 | 2 | no | — |
| 295 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 296 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 297 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 298 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 299 | Q16 | — | 2 | 2 | no | — |
| 300 | Q16 | — | 2 | 2 | no | — |
| 301 | Q16 | — | 4 | 1 | no | — |
| 302 | Q16 | — | 4 | 1 | no | — |
| 303 | D8 | — | 2 | 2 | no | — |
| 304 | D8 | — | 2 | 2 | no | — |
| 305 | D8 | — | 2 | 4 | no | — |
| 306 | D8 | — | 2 | 2 | no | — |
| 307 | D8 | — | 2 | 4 | no | — |
| 308 | D8 | — | 4 | 16 | no | — |
| 309 | D8 | — | 4 | 2 | no | — |
| 310 | D8 | — | 4 | 2 | no | — |
| 311 | D8 | — | 8 | 4 | no | — |
| 312 | D8 | #121 | 1 | 4 | yes | — |
| 313 | C2×C8 | — | 2 | 1 | no | — |
| 314 | C2×C8 | — | 2 | 1 | no | — |
| 315 | C2×C8 | — | 2 | 2 | no | — |
| 316 | C2×C8 | — | 2 | 2 | no | — |
| 317 | C2×C8 | — | 2 | 2 | no | — |
| 318 | C2×C8 | — | 4 | 2 | no | — |
| 319 | C2×C8 | — | 4 | 2 | no | — |
| 320 | C2×C8 | — | 4 | 2 | no | — |
| 321 | C2×C8 | — | 4 | 2 | no | — |
| 322 | C2×C8 | #120 | 1 | 2 | yes | — |
| 323 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 324 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 325 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 326 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 327 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 328 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 329 | C4⋊C4 | — | 2 | 2 | no | — |
| 330 | C4⋊C4 | — | 2 | 2 | no | — |
| 331 | C4⋊C4 | — | 4 | 1 | no | — |
| 332 | C4⋊C4 | — | 4 | 1 | no | — |
| 333 | C4⋊C4 | — | 4 | 2 | no | — |
| 334 | C4⋊C4 | — | 4 | 1 | no | — |
| 335 | C4⋊C4 | — | 4 | 2 | no | — |
| 336 | C4⋊C4 | — | 8 | 4 | no | — |
| 337 | C4⋊C4 | — | 8 | 2 | no | — |
| 338 | C4⋊C4 | — | 8 | 2 | no | — |
| 339 | C4⋊C4 | — | 16 | 1 | no | — |
| 340 | C22⋊C4 | — | 2 | 4 | no | — |
| 341 | C22⋊C4 | — | 4 | 8 | no | — |
| 342 | C22⋊C4 | — | 4 | 4 | no | — |
| 343 | C22⋊C4 | — | 8 | 4 | no | — |
| 344 | C22⋊C4 | — | 16 | 2 | no | — |
| 345 | C42 | — | 8 | 2 | no | — |
| 346 | C42 | #120 | 1 | 2 | yes | — |
| 347 | C42 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 348 | C16 | — | 2 | 2 | no | — |
| 349 | C16 | — | 4 | 1 | no | — |
| 350 | C16 | — | 4 | 2 | no | — |
| 351 | C23 | — | 2 | 2 | no | — |
| 352 | C23 | — | 2 | 2 | no | — |
| 353 | C23 | — | 2 | 2 | no | — |
| 354 | C23 | — | 2 | 2 | no | — |
| 355 | C23 | — | 2 | 2 | no | — |
| 356 | C23 | — | 2 | 2 | no | — |
| 357 | C23 | — | 4 | 2 | no | — |
| 358 | C23 | — | 4 | 2 | no | — |
| 359 | C23 | — | 4 | 2 | no | — |
| 360 | C23 | — | 4 | 4 | no | — |
| 361 | C23 | — | 8 | 16 | no | — |
| 362 | C23 | — | 8 | 16 | no | — |
| 363 | C23 | — | 8 | 1 | no | — |
| 364 | C23 | — | 8 | 1 | no | — |
| 365 | C23 | — | 8 | 1 | no | — |
| 366 | C23 | — | 8 | 1 | no | — |
| 367 | C23 | — | 8 | 2 | no | — |
| 368 | C23 | — | 8 | 2 | no | — |
| 369 | C23 | — | 8 | 2 | no | — |
| 370 | C23 | — | 8 | 2 | no | — |
| 371 | C23 | — | 8 | 2 | no | — |
| 372 | C23 | — | 8 | 2 | no | — |
| 373 | C23 | — | 16 | 8 | no | — |
| 374 | C23 | — | 16 | 8 | no | — |
| 375 | C23 | — | 16 | 8 | no | — |
| 376 | C23 | — | 16 | 8 | no | — |
| 377 | C23 | — | 16 | 8 | no | — |
| 378 | C23 | C2×SD32 | 1 | 4 | yes | — |
| 379 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 380 | Q8 | — | 4 | 2 | no | — |
| 381 | Q8 | — | 4 | 2 | no | — |
| 382 | Q8 | — | 8 | 1 | no | — |
| 383 | Q8 | — | 8 | 1 | no | — |
| 384 | D4 | — | 2 | 4 | no | — |
| 385 | D4 | — | 2 | 8 | no | — |
| 386 | D4 | — | 4 | 4 | no | — |
| 387 | D4 | — | 4 | 2 | no | — |
| 388 | D4 | — | 4 | 4 | no | — |
| 389 | D4 | — | 4 | 2 | no | — |
| 390 | D4 | — | 4 | 2 | no | — |
| 391 | D4 | — | 8 | 16 | no | — |
| 392 | D4 | — | 8 | 16 | no | — |
| 393 | D4 | — | 8 | 8 | no | — |
| 394 | D4 | — | 8 | 2 | no | — |
| 395 | D4 | — | 8 | 2 | no | — |
| 396 | D4 | — | 16 | 4 | no | — |
| 397 | C2×C4 | — | 2 | 1 | no | — |
| 398 | C2×C4 | — | 2 | 1 | no | — |
| 399 | C2×C4 | — | 2 | 2 | no | — |
| 400 | C2×C4 | — | 2 | 2 | no | — |
| 401 | C2×C4 | — | 2 | 2 | no | — |
| 402 | C2×C4 | — | 4 | 2 | no | — |
| 403 | C2×C4 | — | 4 | 2 | no | — |
| 404 | C2×C4 | — | 4 | 2 | no | — |
| 405 | C2×C4 | — | 4 | 2 | no | — |
| 406 | C2×C4 | — | 4 | 2 | no | — |
| 407 | C2×C4 | — | 4 | 2 | no | — |
| 408 | C2×C4 | — | 8 | 1 | no | — |
| 409 | C2×C4 | — | 8 | 1 | no | — |
| 410 | C2×C4 | — | 8 | 1 | no | — |
| 411 | C2×C4 | — | 8 | 1 | no | — |
| 412 | C2×C4 | — | 8 | 8 | no | — |
| 413 | C2×C4 | — | 8 | 2 | no | — |
| 414 | C2×C4 | — | 8 | 1 | no | — |
| 415 | C2×C4 | — | 8 | 2 | no | — |
| 416 | C2×C4 | — | 8 | 1 | no | — |
| 417 | C2×C4 | — | 8 | 1 | no | — |
| 418 | C2×C4 | — | 8 | 2 | no | — |
| 419 | C2×C4 | #447 | 1 | 2 | yes | — |
| 420 | C2×C4 | #447 | 1 | 2 | yes | — |
| 421 | C2×C4 | C2×SD32 | 1 | 2 | yes | — |
| 422 | C2×C4 | #509 | 1 | 2 | yes | — |
| 423 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 424 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 425 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 426 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 427 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 428 | C2×C4 | #577 | 1 | 1 | yes | characteristic |
| 429 | C8 | — | 2 | 2 | no | — |
| 430 | C8 | — | 4 | 2 | no | — |
| 431 | C8 | C2×#121 | 1 | 1 | yes | characteristic |
| 432 | C8 | #535 | 1 | 1 | yes | characteristic |
| 433 | C8 | D42 | 1 | 1 | yes | characteristic |
| 434 | C8 | #546 | 1 | 1 | yes | characteristic |
| 435 | C22 | — | 2 | 4 | no | — |
| 436 | C22 | — | 2 | 4 | no | — |
| 437 | C22 | — | 2 | 4 | no | — |
| 438 | C22 | — | 4 | 2 | no | — |
| 439 | C22 | — | 4 | 2 | no | — |
| 440 | C22 | — | 4 | 2 | no | — |
| 441 | C22 | — | 4 | 2 | no | — |
| 442 | C22 | — | 4 | 4 | no | — |
| 443 | C22 | — | 8 | 2 | no | — |
| 444 | C22 | — | 8 | 2 | no | — |
| 445 | C22 | — | 8 | 2 | no | — |
| 446 | C22 | — | 8 | 2 | no | — |
| 447 | C22 | — | 8 | 2 | no | — |
| 448 | C22 | — | 8 | 2 | no | — |
| 449 | C22 | — | 8 | 4 | no | — |
| 450 | C22 | — | 8 | 4 | no | — |
| 451 | C22 | — | 8 | 4 | no | — |
| 452 | C22 | — | 16 | 32 | no | — |
| 453 | C22 | — | 16 | 2 | no | — |
| 454 | C22 | — | 16 | 2 | no | — |
| 455 | C22 | — | 16 | 2 | no | — |
| 456 | C22 | — | 16 | 2 | no | — |
| 457 | C22 | C2×#447 | 1 | 1 | yes | characteristic |
| 458 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 459 | C22 | #3294 | 1 | 1 | yes | characteristic |
| 460 | C22 | C22×SD32 | 1 | 1 | yes | characteristic |
| 461 | C22 | C2×#509 | 1 | 1 | yes | characteristic |
| 462 | C22 | #3415 | 1 | 1 | yes | characteristic |
| 463 | C22 | #3416 | 1 | 1 | yes | characteristic |
| 464 | C4 | — | 2 | 2 | no | — |
| 465 | C4 | — | 4 | 2 | no | — |
| 466 | C4 | — | 8 | 2 | no | — |
| 467 | C4 | — | 8 | 2 | no | — |
| 468 | C4 | — | 16 | 2 | no | — |
| 469 | C4 | — | 16 | 1 | no | — |
| 470 | C4 | #2187 | 1 | 4 | yes | complemented |
| 471 | C4 | #2408 | 1 | 1 | yes | characteristic |
| 472 | C4 | #3012 | 1 | 1 | yes | characteristic |
| 473 | C4 | #3272 | 1 | 1 | yes | characteristic |
| 474 | C4 | #3325 | 1 | 1 | yes | characteristic |
| 475 | C2 | — | 4 | 4 | no | — |
| 476 | C2 | — | 8 | 4 | no | — |
| 477 | C2 | — | 16 | 4 | no | — |
| 478 | C2 | #24236 | 1 | 1 | yes | characteristic |
| 479 | C2 | C2×#2187 | 1 | 1 | yes | characteristic |
| 480 | C2 | #33526 | 1 | 1 | yes | characteristic |
| 481 | C2 | #33840 | 1 | 1 | yes | characteristic |
| 482 | C2 | D4×SD32 | 1 | 1 | yes | characteristic |
| 483 | C2 | #33862 | 1 | 1 | yes | characteristic |
| 484 | C2 | #33891 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×D4×D8 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2207 | C2 | semidirect |
| 3. C2×#2187 | C2 | semidirect |
| 4. C2×C4×SD32 | C2 | semidirect |
| 5. #28228 | C2 | semidirect |
| 6. #13381 | C2 | semidirect |
| 7. #13305 | C2 | semidirect |
| 8. #13202 | C2 | semidirect |
| 9. #13189 | C2 | semidirect |
| 10. #12811 | C2 | semidirect |
| 11. #12731 | C2 | semidirect |
| 15. C22×SD32 | C22 | semidirect |
| 19. C2×#467 | C22 | semidirect |
| 23. C2×C4×Q16 | C22 | semidirect |
| 30. C2×#367 | C22 | semidirect |
| 31. C2×C4⋊C16 | C22 | semidirect |
| 32. C2×#358 | C22 | semidirect |
| 35. C2×#357 | C22 | semidirect |
| 36. C2×#357 | C22 | semidirect |
| 37. C2×#357 | C22 | semidirect |
| 38. C2×C22⋊C16 | C22 | semidirect |
| 39. C2×C22⋊C16 | C22 | semidirect |
| 40. C2×C4×C16 | C22 | semidirect |
| 42. #1845 | C22 | semidirect |
| 43. #1808 | C22 | semidirect |
| 45. #1380 | C22 | semidirect |
| 75. C2×SD32 | D4 | semidirect |
| 146. C2×Q16 | C2×D4 | semidirect |
| 217. C2×C16 | C2×D4 | semidirect |
| 281. C2×D4 | SD32 | semidirect |
| 470. C4 | #2187 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×D8 | 2 |
| C2×#2207 | 2 |
| C2×#2187 | 2 |
| C2×C4×SD32 | 2 |
| #28228 | 2 |
| #13381 | 2 |
| #13305 | 2 |
| #13202 | 2 |
| #13189 | 2 |
| #12811 | 2 |
| #12731 | 2 |