Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2254 conjugacy classes of subgroups.
The subgroup lattice has 421 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#3254 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×#3167 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #37641 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #37639 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #33178 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #30254 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #28529 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #28461 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #28420 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #28416 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | #20437 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#533 | C22 | 1 | 2 | yes | — |
| 13 | C2×#532 | C22 | 1 | 2 | yes | — |
| 14 | #3254 | — | 2 | 16 | no | — |
| 15 | #3167 | — | 2 | 16 | no | — |
| 16 | C2×#498 | C22 | 1 | 2 | yes | — |
| 17 | C2×#492 | C22 | 1 | 2 | yes | — |
| 18 | C2×#492 | C22 | 1 | 4 | yes | — |
| 19 | C2×#487 | C22 | 1 | 2 | yes | — |
| 20 | C2×#486 | C22 | 1 | 2 | yes | — |
| 21 | C2×#477 | C22 | 1 | 4 | yes | — |
| 22 | C2×#474 | C22 | 1 | 4 | yes | — |
| 23 | C2×#468 | C22 | 1 | 8 | yes | — |
| 24 | C2×#465 | C22 | 1 | 4 | yes | — |
| 25 | C2×#465 | C22 | 1 | 4 | yes | — |
| 26 | C2×#460 | C22 | 1 | 4 | yes | — |
| 27 | C2×#461 | C22 | 1 | 4 | yes | — |
| 28 | C2×C4×SD16 | C22 | 1 | 4 | yes | — |
| 29 | C2×C4×SD16 | C22 | 1 | 4 | yes | — |
| 30 | C2×#423 | C22 | 1 | 1 | yes | characteristic |
| 31 | C2×#405 | C22 | 1 | 2 | yes | — |
| 32 | C22×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 33 | #2822 | — | 2 | 2 | no | — |
| 34 | #2819 | — | 2 | 2 | no | — |
| 35 | #2625 | — | 2 | 4 | no | — |
| 36 | #2621 | C22 | 1 | 4 | yes | — |
| 37 | #2619 | C22 | 1 | 4 | yes | — |
| 38 | #2615 | — | 2 | 4 | no | — |
| 39 | #2582 | — | 2 | 2 | no | — |
| 40 | #2568 | C22 | 1 | 4 | yes | — |
| 41 | #2563 | C22 | 1 | 4 | yes | — |
| 42 | #2546 | C22 | 1 | 4 | yes | — |
| 43 | #2541 | C22 | 1 | 4 | yes | — |
| 44 | #2537 | — | 2 | 2 | no | — |
| 45 | #2285 | C22 | 1 | 1 | yes | characteristic |
| 46 | #2097 | C22 | 1 | 8 | yes | — |
| 47 | #2092 | C22 | 1 | 8 | yes | — |
| 48 | #2014 | C22 | 1 | 8 | yes | — |
| 49 | #2013 | C22 | 1 | 8 | yes | — |
| 50 | #1973 | C22 | 1 | 8 | yes | — |
| 51 | #1968 | C22 | 1 | 8 | yes | — |
| 52 | #1961 | C22 | 1 | 4 | yes | — |
| 53 | #1960 | C22 | 1 | 4 | yes | — |
| 54 | #1846 | C22 | 1 | 4 | yes | — |
| 55 | #1770 | C22 | 1 | 4 | yes | — |
| 56 | C8×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 57 | #1729 | C22 | 1 | 2 | yes | — |
| 58 | C22×#41 | — | 2 | 2 | no | — |
| 59 | C22×SD16 | C23 | 1 | 4 | yes | — |
| 60 | C22×SD16 | C23 | 1 | 4 | yes | — |
| 61 | C22×SD16 | C23 | 1 | 8 | yes | — |
| 62 | C22×M16 | C23 | 1 | 2 | yes | — |
| 63 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 64 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 65 | #533 | — | 2 | 8 | no | — |
| 66 | #533 | — | 2 | 8 | no | — |
| 67 | #532 | — | 2 | 8 | no | — |
| 68 | #532 | — | 2 | 8 | no | — |
| 69 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 70 | C2×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 71 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 72 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 73 | C2×#127 | C23 | 1 | 2 | yes | — |
| 74 | C2×#127 | C23 | 1 | 4 | yes | — |
| 75 | C2×#125 | — | 2 | 2 | no | — |
| 76 | C2×#124 | — | 2 | 2 | no | — |
| 77 | C2×C22⋊Q8 | — | 2 | 2 | no | — |
| 78 | C2×C22⋊Q8 | C23 | 1 | 4 | yes | — |
| 79 | C2×#121 | — | 2 | 2 | no | — |
| 80 | C2×#121 | — | 2 | 8 | no | — |
| 81 | C2×#121 | C23 | 1 | 4 | yes | — |
| 82 | C2×C4×Q8 | — | 2 | 2 | no | — |
| 83 | C2×C4×Q8 | C23 | 1 | 4 | yes | — |
| 84 | C2×C4×D4 | — | 2 | 2 | no | — |
| 85 | C2×C4×D4 | C23 | 1 | 4 | yes | — |
| 86 | C2×#117 | — | 2 | 2 | no | — |
| 87 | C2×#117 | C23 | 1 | 2 | yes | — |
| 88 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 89 | #498 | — | 2 | 8 | no | — |
| 90 | #492 | — | 2 | 8 | no | — |
| 91 | #492 | — | 4 | 8 | no | — |
| 92 | #487 | — | 2 | 8 | no | — |
| 93 | #486 | — | 2 | 8 | no | — |
| 94 | #477 | — | 2 | 16 | no | — |
| 95 | #474 | — | 2 | 16 | no | — |
| 96 | #468 | — | 2 | 32 | no | — |
| 97 | #465 | — | 2 | 16 | no | — |
| 98 | #465 | — | 2 | 16 | no | — |
| 99 | #461 | — | 2 | 16 | no | — |
| 100 | #460 | — | 2 | 16 | no | — |
| 101 | C4×SD16 | — | 2 | 16 | no | — |
| 102 | C4×SD16 | — | 2 | 16 | no | — |
| 103 | C2×#107 | C23 | 1 | 8 | yes | — |
| 104 | C2×#106 | C23 | 1 | 4 | yes | — |
| 105 | C2×#106 | C23 | 1 | 4 | yes | — |
| 106 | #423 | D4 | 1 | 8 | yes | complemented |
| 107 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 108 | C2×#103 | C23 | 1 | 8 | yes | — |
| 109 | C2×#103 | C23 | 1 | 4 | yes | — |
| 110 | C2×#103 | C23 | 1 | 4 | yes | — |
| 111 | C2×#102 | C23 | 1 | 4 | yes | — |
| 112 | C2×#102 | C23 | 1 | 8 | yes | — |
| 113 | C2×#102 | C23 | 1 | 4 | yes | — |
| 114 | #405 | — | 4 | 4 | no | — |
| 115 | C2×#97 | C23 | 1 | 4 | yes | — |
| 116 | C2×C4×C8 | D4 | 1 | 8 | yes | complemented |
| 117 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 118 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 119 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 120 | #400 | — | 2 | 8 | no | — |
| 121 | #398 | — | 2 | 8 | no | — |
| 122 | #398 | — | 2 | 8 | no | — |
| 123 | #397 | — | 2 | 8 | no | — |
| 124 | #394 | — | 2 | 8 | no | — |
| 125 | #393 | — | 2 | 8 | no | — |
| 126 | #392 | — | 2 | 8 | no | — |
| 127 | #392 | — | 2 | 8 | no | — |
| 128 | #391 | — | 2 | 8 | no | — |
| 129 | #391 | C23 | 1 | 8 | yes | — |
| 130 | #390 | C23 | 1 | 8 | yes | — |
| 131 | #389 | — | 2 | 8 | no | — |
| 132 | #384 | D4 | 1 | 2 | yes | — |
| 133 | #382 | D4 | 1 | 2 | yes | — |
| 134 | #382 | C23 | 1 | 4 | yes | — |
| 135 | #382 | C23 | 1 | 4 | yes | — |
| 136 | #380 | — | 2 | 8 | no | — |
| 137 | #380 | D4 | 1 | 4 | yes | — |
| 138 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 139 | #336 | C23 | 1 | 8 | yes | — |
| 140 | C2×#41 | — | 4 | 8 | no | — |
| 141 | C2×#41 | — | 8 | 8 | no | — |
| 142 | C22×Q8 | — | 2 | 2 | no | — |
| 143 | C22×Q8 | — | 2 | 2 | no | — |
| 144 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 145 | C22×D4 | — | 2 | 4 | no | — |
| 146 | C22×D4 | — | 2 | 2 | no | — |
| 147 | C22×D4 | — | 2 | 2 | no | — |
| 148 | C22×D4 | — | 2 | 8 | no | — |
| 149 | C22×D4 | C24 | 1 | 4 | yes | — |
| 150 | C23×C4 | — | 2 | 4 | no | — |
| 151 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 152 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 153 | C2×SD16 | — | 2 | 8 | no | — |
| 154 | C2×SD16 | — | 2 | 8 | no | — |
| 155 | C2×SD16 | — | 2 | 16 | no | — |
| 156 | C2×SD16 | — | 2 | 8 | no | — |
| 157 | C2×SD16 | — | 2 | 8 | no | — |
| 158 | C2×SD16 | — | 4 | 4 | no | — |
| 159 | C2×SD16 | — | 4 | 4 | no | — |
| 160 | C2×SD16 | — | 4 | 8 | no | — |
| 161 | C2×SD16 | — | 4 | 8 | no | — |
| 162 | C2×M16 | — | 4 | 2 | no | — |
| 163 | C2×M16 | — | 4 | 2 | no | — |
| 164 | C2×M16 | C2×D4 | 1 | 8 | yes | — |
| 165 | C22×C8 | — | 2 | 2 | no | — |
| 166 | C22×C8 | — | 2 | 2 | no | — |
| 167 | C22×C8 | — | 4 | 1 | no | — |
| 168 | C22×C8 | — | 4 | 1 | no | — |
| 169 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 170 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 171 | C22×C8 | C24 | 1 | 2 | yes | — |
| 172 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 173 | C22×C8 | C24 | 1 | 4 | yes | — |
| 174 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 175 | C4⋊Q8 | — | 2 | 4 | no | — |
| 176 | C4⋊Q8 | — | 2 | 4 | no | — |
| 177 | C4⋊Q8 | — | 2 | 4 | no | — |
| 178 | C4⋊Q8 | — | 2 | 4 | no | — |
| 179 | C4⋊Q8 | — | 2 | 4 | no | — |
| 180 | C4⋊Q8 | — | 2 | 4 | no | — |
| 181 | C4⋊Q8 | — | 4 | 4 | no | — |
| 182 | C4⋊Q8 | — | 4 | 4 | no | — |
| 183 | #127 | — | 2 | 4 | no | — |
| 184 | #127 | — | 2 | 4 | no | — |
| 185 | #127 | — | 4 | 4 | no | — |
| 186 | #127 | — | 4 | 4 | no | — |
| 187 | #125 | — | 4 | 8 | no | — |
| 188 | #124 | — | 4 | 8 | no | — |
| 189 | C22⋊Q8 | — | 2 | 8 | no | — |
| 190 | C22⋊Q8 | — | 2 | 8 | no | — |
| 191 | C22⋊Q8 | — | 4 | 8 | no | — |
| 192 | #121 | — | 2 | 8 | no | — |
| 193 | #121 | — | 2 | 8 | no | — |
| 194 | #121 | — | 4 | 8 | no | — |
| 195 | #121 | — | 8 | 16 | no | — |
| 196 | C4×Q8 | — | 2 | 8 | no | — |
| 197 | C4×Q8 | — | 2 | 8 | no | — |
| 198 | C4×Q8 | — | 4 | 8 | no | — |
| 199 | C4×D4 | — | 2 | 8 | no | — |
| 200 | C4×D4 | — | 2 | 8 | no | — |
| 201 | C4×D4 | — | 4 | 8 | no | — |
| 202 | #117 | — | 4 | 2 | no | — |
| 203 | #117 | — | 4 | 2 | no | — |
| 204 | #117 | — | 4 | 8 | no | — |
| 205 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 206 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 207 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 208 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 209 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 210 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 211 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 212 | C2×C4⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 213 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 214 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 215 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 216 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 217 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 218 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 219 | C2×C22⋊C4 | — | 2 | 16 | no | — |
| 220 | C2×C22⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 221 | C2×C42 | — | 2 | 4 | no | — |
| 222 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 223 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 224 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 225 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 226 | C2×C42 | C24 | 1 | 4 | yes | — |
| 227 | C2×C42 | C24 | 1 | 2 | yes | — |
| 228 | #107 | — | 2 | 16 | no | — |
| 229 | #106 | — | 2 | 8 | no | — |
| 230 | #106 | — | 2 | 8 | no | — |
| 231 | C4⋊C8 | C2×D4 | 1 | 16 | yes | — |
| 232 | #103 | — | 2 | 8 | no | — |
| 233 | #103 | — | 2 | 8 | no | — |
| 234 | #103 | — | 2 | 16 | no | — |
| 235 | #102 | — | 2 | 8 | no | — |
| 236 | #102 | — | 2 | 8 | no | — |
| 237 | #102 | — | 2 | 16 | no | — |
| 238 | #97 | — | 4 | 4 | no | — |
| 239 | C4×C8 | — | 2 | 8 | no | — |
| 240 | C4×C8 | C2×D4 | 1 | 8 | yes | — |
| 241 | C4×C8 | C2×D4 | 1 | 8 | yes | — |
| 242 | #95 | — | 2 | 8 | no | — |
| 243 | #95 | — | 2 | 8 | no | — |
| 244 | #95 | — | 2 | 16 | no | — |
| 245 | #95 | C2×D4 | 1 | 8 | yes | — |
| 246 | C24 | — | 2 | 4 | no | — |
| 247 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 248 | #41 | — | 8 | 32 | no | — |
| 249 | C2×Q8 | — | 2 | 4 | no | — |
| 250 | C2×Q8 | — | 2 | 4 | no | — |
| 251 | C2×Q8 | — | 2 | 4 | no | — |
| 252 | C2×Q8 | — | 2 | 4 | no | — |
| 253 | C2×Q8 | — | 2 | 4 | no | — |
| 254 | C2×Q8 | — | 2 | 4 | no | — |
| 255 | C2×Q8 | — | 4 | 4 | no | — |
| 256 | C2×Q8 | — | 4 | 4 | no | — |
| 257 | C2×Q8 | — | 4 | 8 | no | — |
| 258 | C2×Q8 | — | 8 | 4 | no | — |
| 259 | C2×D4 | — | 2 | 4 | no | — |
| 260 | C2×D4 | — | 2 | 4 | no | — |
| 261 | C2×D4 | — | 2 | 4 | no | — |
| 262 | C2×D4 | — | 2 | 4 | no | — |
| 263 | C2×D4 | — | 2 | 4 | no | — |
| 264 | C2×D4 | — | 2 | 4 | no | — |
| 265 | C2×D4 | — | 4 | 8 | no | — |
| 266 | C2×D4 | — | 4 | 4 | no | — |
| 267 | C2×D4 | — | 4 | 4 | no | — |
| 268 | C2×D4 | — | 4 | 8 | no | — |
| 269 | C2×D4 | — | 8 | 4 | no | — |
| 270 | C2×D4 | — | 8 | 16 | no | — |
| 271 | C2×D4 | — | 8 | 8 | no | — |
| 272 | C2×D4 | — | 8 | 8 | no | — |
| 273 | C22×C4 | — | 2 | 8 | no | — |
| 274 | C22×C4 | — | 2 | 1 | no | — |
| 275 | C22×C4 | — | 2 | 1 | no | — |
| 276 | C22×C4 | — | 2 | 1 | no | — |
| 277 | C22×C4 | — | 2 | 1 | no | — |
| 278 | C22×C4 | — | 2 | 4 | no | — |
| 279 | C22×C4 | — | 2 | 4 | no | — |
| 280 | C22×C4 | — | 2 | 4 | no | — |
| 281 | C22×C4 | — | 4 | 8 | no | — |
| 282 | C22×C4 | — | 4 | 16 | no | — |
| 283 | C22×C4 | #127 | 1 | 8 | yes | — |
| 284 | C22×C4 | C22×D4 | 1 | 8 | yes | — |
| 285 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 286 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 287 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 288 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 289 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 290 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 291 | SD16 | — | 4 | 16 | no | — |
| 292 | SD16 | — | 4 | 16 | no | — |
| 293 | SD16 | — | 8 | 16 | no | — |
| 294 | M16 | — | 4 | 8 | no | — |
| 295 | C2×C8 | — | 2 | 4 | no | — |
| 296 | C2×C8 | — | 2 | 1 | no | — |
| 297 | C2×C8 | — | 2 | 1 | no | — |
| 298 | C2×C8 | — | 2 | 2 | no | — |
| 299 | C2×C8 | — | 2 | 2 | no | — |
| 300 | C2×C8 | — | 2 | 4 | no | — |
| 301 | C2×C8 | — | 2 | 8 | no | — |
| 302 | C2×C8 | — | 4 | 4 | no | — |
| 303 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 304 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 305 | C2×C8 | C22×D4 | 1 | 8 | yes | — |
| 306 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 307 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 308 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 309 | C4⋊C4 | — | 2 | 16 | no | — |
| 310 | C4⋊C4 | — | 2 | 4 | no | — |
| 311 | C4⋊C4 | — | 2 | 4 | no | — |
| 312 | C4⋊C4 | — | 2 | 4 | no | — |
| 313 | C4⋊C4 | — | 2 | 4 | no | — |
| 314 | C4⋊C4 | — | 4 | 4 | no | — |
| 315 | C4⋊C4 | — | 4 | 4 | no | — |
| 316 | C4⋊C4 | — | 4 | 4 | no | — |
| 317 | C4⋊C4 | — | 4 | 4 | no | — |
| 318 | C4⋊C4 | — | 4 | 8 | no | — |
| 319 | C4⋊C4 | — | 4 | 8 | no | — |
| 320 | C4⋊C4 | — | 4 | 4 | no | — |
| 321 | C4⋊C4 | — | 8 | 8 | no | — |
| 322 | C22⋊C4 | — | 4 | 4 | no | — |
| 323 | C22⋊C4 | — | 4 | 8 | no | — |
| 324 | C22⋊C4 | — | 4 | 8 | no | — |
| 325 | C22⋊C4 | — | 4 | 8 | no | — |
| 326 | C22⋊C4 | — | 8 | 16 | no | — |
| 327 | C42 | — | 2 | 4 | no | — |
| 328 | C42 | — | 2 | 4 | no | — |
| 329 | C42 | — | 4 | 8 | no | — |
| 330 | C42 | #127 | 1 | 4 | yes | — |
| 331 | C42 | #127 | 1 | 4 | yes | — |
| 332 | C42 | C22×D4 | 1 | 4 | yes | — |
| 333 | C42 | C22×D4 | 1 | 4 | yes | — |
| 334 | C23 | — | 2 | 1 | no | — |
| 335 | C23 | — | 2 | 1 | no | — |
| 336 | C23 | — | 2 | 4 | no | — |
| 337 | C23 | — | 4 | 4 | no | — |
| 338 | C23 | — | 4 | 4 | no | — |
| 339 | C23 | — | 4 | 4 | no | — |
| 340 | C23 | — | 4 | 8 | no | — |
| 341 | C23 | — | 4 | 8 | no | — |
| 342 | C23 | C2×#127 | 1 | 2 | yes | — |
| 343 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 344 | Q8 | — | 4 | 4 | no | — |
| 345 | Q8 | — | 4 | 4 | no | — |
| 346 | Q8 | — | 4 | 4 | no | — |
| 347 | Q8 | — | 4 | 4 | no | — |
| 348 | Q8 | — | 8 | 8 | no | — |
| 349 | Q8 | — | 8 | 8 | no | — |
| 350 | D4 | — | 4 | 4 | no | — |
| 351 | D4 | — | 4 | 4 | no | — |
| 352 | D4 | — | 4 | 4 | no | — |
| 353 | D4 | — | 4 | 4 | no | — |
| 354 | D4 | — | 8 | 8 | no | — |
| 355 | D4 | — | 8 | 16 | no | — |
| 356 | D4 | — | 8 | 8 | no | — |
| 357 | D4 | — | 16 | 16 | no | — |
| 358 | C2×C4 | — | 2 | 1 | no | — |
| 359 | C2×C4 | — | 2 | 1 | no | — |
| 360 | C2×C4 | — | 2 | 4 | no | — |
| 361 | C2×C4 | — | 2 | 16 | no | — |
| 362 | C2×C4 | — | 2 | 2 | no | — |
| 363 | C2×C4 | — | 2 | 2 | no | — |
| 364 | C2×C4 | — | 2 | 2 | no | — |
| 365 | C2×C4 | — | 2 | 2 | no | — |
| 366 | C2×C4 | — | 2 | 8 | no | — |
| 367 | C2×C4 | — | 2 | 16 | no | — |
| 368 | C2×C4 | — | 4 | 4 | no | — |
| 369 | C2×C4 | — | 4 | 4 | no | — |
| 370 | C2×C4 | — | 4 | 4 | no | — |
| 371 | C2×C4 | — | 4 | 4 | no | — |
| 372 | C2×C4 | — | 4 | 4 | no | — |
| 373 | C2×C4 | — | 4 | 8 | no | — |
| 374 | C2×C4 | — | 4 | 8 | no | — |
| 375 | C2×C4 | — | 8 | 16 | no | — |
| 376 | C2×C4 | C2×#127 | 1 | 2 | yes | — |
| 377 | C2×C4 | C2×#127 | 1 | 4 | yes | — |
| 378 | C2×C4 | C2×#127 | 1 | 4 | yes | — |
| 379 | C2×C4 | #577 | 1 | 8 | yes | — |
| 380 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 381 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 382 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 383 | C2×C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 384 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 385 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 386 | C8 | — | 2 | 2 | no | — |
| 387 | C8 | — | 2 | 4 | no | — |
| 388 | C8 | — | 2 | 8 | no | — |
| 389 | C8 | #537 | 1 | 4 | yes | — |
| 390 | C22 | — | 2 | 2 | no | — |
| 391 | C22 | — | 2 | 4 | no | — |
| 392 | C22 | — | 4 | 2 | no | — |
| 393 | C22 | — | 4 | 2 | no | — |
| 394 | C22 | — | 8 | 4 | no | — |
| 395 | C22 | — | 8 | 8 | no | — |
| 396 | C22 | — | 8 | 8 | no | — |
| 397 | C22 | — | 8 | 4 | no | — |
| 398 | C22 | — | 8 | 4 | no | — |
| 399 | C22 | C22×#127 | 1 | 1 | yes | characteristic |
| 400 | C22 | #3496 | 1 | 1 | yes | characteristic |
| 401 | C22 | #3497 | 1 | 1 | yes | characteristic |
| 402 | C22 | C2×#577 | 1 | 4 | yes | — |
| 403 | C4 | — | 2 | 2 | no | — |
| 404 | C4 | — | 2 | 4 | no | — |
| 405 | C4 | — | 4 | 8 | no | — |
| 406 | C4 | — | 8 | 4 | no | — |
| 407 | C4 | — | 8 | 4 | no | — |
| 408 | C4 | — | 8 | 4 | no | — |
| 409 | C4 | #3154 | 1 | 8 | yes | — |
| 410 | C4 | C2×#537 | 1 | 1 | yes | characteristic |
| 411 | C4 | #3449 | 1 | 1 | yes | characteristic |
| 412 | C4 | #3496 | 1 | 1 | yes | characteristic |
| 413 | C4 | #3497 | 1 | 1 | yes | characteristic |
| 414 | C2 | — | 4 | 2 | no | — |
| 415 | C2 | — | 16 | 4 | no | — |
| 416 | C2 | #35166 | 1 | 1 | yes | characteristic |
| 417 | C2 | C2×#3154 | 1 | 2 | yes | — |
| 418 | C2 | #62326 | 1 | 2 | yes | — |
| 419 | C2 | #62371 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37639 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3254 | C2 | semidirect |
| 2. C2×#3167 | C2 | semidirect |
| 3. #37641 | C2 | semidirect |
| 5. #33178 | C2 | semidirect |
| 6. #30254 | C2 | semidirect |
| 7. #28529 | C2 | semidirect |
| 8. #28461 | C2 | semidirect |
| 9. #28420 | C2 | semidirect |
| 10. #28416 | C2 | semidirect |
| 11. #20437 | C2 | semidirect |
| 106. #423 | D4 | semidirect |
| 116. C2×C4×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3254 | 2 |
| C2×#3167 | 2 |
| #37641 | 2 |
| #37639 | 2 |
| #33178 | 2 |
| #30254 | 2 |
| #28529 | 2 |
| #28461 | 2 |
| #28420 | 2 |
| #28416 | 2 |
| #20437 | 2 |