Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
735 conjugacy classes of subgroups.
The subgroup lattice has 432 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #61370 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33628 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#2175 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #28297 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13455 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13370 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13286 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13267 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #13019 | C2 | 1 | 2 | yes | maximal |
| 10 | #12940 | C2 | 1 | 2 | yes | maximal |
| 11 | #12818 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #3501 | — | 2 | 1 | no | — |
| 13 | C22×SD32 | C22 | 1 | 1 | yes | characteristic |
| 14 | #3389 | — | 2 | 2 | no | — |
| 15 | #3384 | — | 2 | 2 | no | — |
| 16 | #3380 | — | 2 | 1 | no | — |
| 17 | #3380 | — | 2 | 2 | no | — |
| 18 | Q8×D8 | — | 2 | 1 | no | — |
| 19 | #3164 | C22 | 1 | 2 | yes | — |
| 20 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 21 | #3081 | — | 2 | 2 | no | — |
| 22 | #3078 | — | 2 | 1 | no | — |
| 23 | #3077 | — | 2 | 1 | no | — |
| 24 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 25 | #2948 | C22 | 1 | 2 | yes | — |
| 26 | C2×C4×Q16 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×C4×D8 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2229 | — | 2 | 2 | no | — |
| 29 | #2227 | — | 2 | 1 | no | — |
| 30 | #2225 | — | 2 | 1 | no | — |
| 31 | #2175 | — | 2 | 2 | no | — |
| 32 | #2175 | — | 2 | 2 | no | — |
| 33 | #2159 | C22 | 1 | 2 | yes | — |
| 34 | C2×#366 | C22 | 1 | 1 | yes | characteristic |
| 35 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 36 | C2×#361 | C22 | 1 | 2 | yes | — |
| 37 | #2141 | C22 | 1 | 2 | yes | — |
| 38 | C2×#358 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×#357 | C22 | 1 | 1 | yes | characteristic |
| 40 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 41 | #2064 | — | 2 | 2 | no | — |
| 42 | #1979 | C22 | 1 | 2 | yes | — |
| 43 | #1952 | C22 | 1 | 2 | yes | — |
| 44 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 45 | #1801 | — | 2 | 1 | no | — |
| 46 | #1387 | C22 | 1 | 2 | yes | — |
| 47 | #1382 | C22 | 1 | 2 | yes | — |
| 48 | #1382 | C22 | 1 | 2 | yes | — |
| 49 | #1379 | C22 | 1 | 2 | yes | — |
| 50 | #1379 | C22 | 1 | 2 | yes | — |
| 51 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 52 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 53 | #554 | — | 2 | 2 | no | — |
| 54 | #554 | — | 2 | 4 | no | — |
| 55 | #554 | — | 4 | 1 | no | — |
| 56 | #554 | — | 4 | 2 | no | — |
| 57 | D4×Q8 | — | 2 | 2 | no | — |
| 58 | D4×Q8 | — | 4 | 1 | no | — |
| 59 | #544 | — | 2 | 2 | no | — |
| 60 | #543 | — | 4 | 2 | no | — |
| 61 | #533 | — | 2 | 1 | no | — |
| 62 | #533 | D4 | 1 | 1 | yes | characteristic |
| 63 | C2×C22⋊Q8 | — | 4 | 2 | no | — |
| 64 | #518 | — | 2 | 2 | no | — |
| 65 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 66 | C2×C4×D4 | — | 2 | 1 | no | — |
| 67 | C2×SD32 | — | 2 | 2 | no | — |
| 68 | C2×SD32 | — | 2 | 2 | no | — |
| 69 | C2×SD32 | Q8 | 1 | 4 | yes | complemented |
| 70 | C2×M32 | C23 | 1 | 2 | yes | — |
| 71 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 72 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 73 | #501 | — | 2 | 4 | no | — |
| 74 | #500 | — | 2 | 2 | no | — |
| 75 | #500 | D4 | 1 | 1 | yes | characteristic |
| 76 | #500 | D4 | 1 | 1 | yes | characteristic |
| 77 | #499 | — | 2 | 2 | no | — |
| 78 | #499 | — | 2 | 1 | no | — |
| 79 | #499 | D4 | 1 | 2 | yes | — |
| 80 | #498 | — | 2 | 1 | no | — |
| 81 | #479 | — | 4 | 2 | no | — |
| 82 | #478 | — | 2 | 4 | no | — |
| 83 | #478 | — | 2 | 4 | no | — |
| 84 | #478 | — | 2 | 4 | no | — |
| 85 | #477 | — | 4 | 1 | no | — |
| 86 | #476 | — | 2 | 2 | no | — |
| 87 | #476 | — | 2 | 4 | no | — |
| 88 | #475 | — | 4 | 1 | no | — |
| 89 | #474 | — | 2 | 2 | no | — |
| 90 | #474 | — | 2 | 4 | no | — |
| 91 | #446 | — | 2 | 1 | no | — |
| 92 | #446 | — | 2 | 1 | no | — |
| 93 | C8×Q8 | — | 2 | 1 | no | — |
| 94 | C8×Q8 | — | 2 | 1 | no | — |
| 95 | #442 | — | 2 | 4 | no | — |
| 96 | C4×Q16 | — | 2 | 1 | no | — |
| 97 | C4×Q16 | — | 2 | 1 | no | — |
| 98 | C4×Q16 | — | 2 | 1 | no | — |
| 99 | C4×Q16 | — | 2 | 1 | no | — |
| 100 | C4×D8 | — | 2 | 2 | no | — |
| 101 | C4×D8 | — | 2 | 2 | no | — |
| 102 | C4×D8 | — | 2 | 1 | no | — |
| 103 | C4×D8 | — | 2 | 1 | no | — |
| 104 | C4×D8 | D4 | 1 | 2 | yes | — |
| 105 | C4×D8 | D4 | 1 | 2 | yes | — |
| 106 | C2×#108 | — | 2 | 1 | no | — |
| 107 | C2×#108 | C23 | 1 | 2 | yes | — |
| 108 | #428 | — | 2 | 2 | no | — |
| 109 | #428 | — | 2 | 2 | no | — |
| 110 | #427 | — | 2 | 1 | no | — |
| 111 | #427 | C23 | 1 | 2 | yes | — |
| 112 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 113 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 114 | #424 | D4 | 1 | 2 | yes | — |
| 115 | #423 | — | 2 | 1 | no | — |
| 116 | #423 | D4 | 1 | 1 | yes | characteristic |
| 117 | #419 | — | 2 | 2 | no | — |
| 118 | #418 | — | 2 | 2 | no | — |
| 119 | C2×#103 | — | 2 | 1 | no | — |
| 120 | C2×#102 | — | 2 | 1 | no | — |
| 121 | C2×#101 | — | 2 | 2 | no | — |
| 122 | #405 | C23 | 1 | 2 | yes | — |
| 123 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 124 | #367 | — | 2 | 2 | no | — |
| 125 | #366 | — | 2 | 2 | no | — |
| 126 | #366 | — | 2 | 2 | no | — |
| 127 | C4⋊C16 | — | 2 | 1 | no | — |
| 128 | C4⋊C16 | — | 2 | 1 | no | — |
| 129 | #361 | — | 2 | 4 | no | — |
| 130 | #358 | — | 2 | 2 | no | — |
| 131 | #358 | — | 2 | 2 | no | — |
| 132 | #358 | — | 2 | 2 | no | — |
| 133 | #357 | — | 2 | 2 | no | — |
| 134 | #346 | — | 2 | 1 | no | — |
| 135 | #346 | — | 2 | 1 | no | — |
| 136 | #343 | — | 2 | 2 | no | — |
| 137 | C22×Q8 | — | 2 | 1 | no | — |
| 138 | C22×Q8 | — | 4 | 2 | no | — |
| 139 | C22×D4 | — | 2 | 1 | no | — |
| 140 | C23×C4 | — | 4 | 1 | no | — |
| 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×Q8 | 1 | 2 | yes | complemented |
| 147 | C2×D8 | — | 2 | 1 | no | — |
| 148 | C2×D8 | — | 2 | 1 | no | — |
| 149 | C2×D8 | — | 2 | 1 | no | — |
| 150 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 151 | C2×D8 | C2×Q8 | 1 | 2 | yes | — |
| 152 | C2×D8 | #41 | 1 | 2 | yes | — |
| 153 | C2×M16 | — | 2 | 2 | no | — |
| 154 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 155 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 156 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 157 | C4⋊Q8 | — | 2 | 4 | no | — |
| 158 | C4⋊Q8 | — | 2 | 1 | no | — |
| 159 | C4⋊Q8 | — | 2 | 1 | no | — |
| 160 | C4⋊Q8 | — | 4 | 2 | no | — |
| 161 | C4⋊Q8 | D8 | 1 | 1 | yes | characteristic |
| 162 | C4⋊Q8 | D8 | 1 | 1 | yes | characteristic |
| 163 | #125 | — | 2 | 4 | no | — |
| 164 | #125 | — | 2 | 2 | no | — |
| 165 | #125 | — | 4 | 2 | no | — |
| 166 | #125 | D8 | 1 | 2 | yes | — |
| 167 | C22⋊Q8 | — | 2 | 1 | no | — |
| 168 | C22⋊Q8 | — | 2 | 1 | no | — |
| 169 | C22⋊Q8 | — | 4 | 4 | no | — |
| 170 | C22⋊Q8 | — | 4 | 2 | no | — |
| 171 | C22⋊Q8 | — | 4 | 4 | no | — |
| 172 | C22⋊Q8 | — | 4 | 4 | no | — |
| 173 | C22⋊Q8 | — | 4 | 4 | no | — |
| 174 | C22⋊Q8 | — | 4 | 4 | no | — |
| 175 | C22⋊Q8 | — | 4 | 2 | no | — |
| 176 | C22⋊Q8 | — | 4 | 4 | no | — |
| 177 | C22⋊Q8 | — | 4 | 1 | no | — |
| 178 | C22⋊Q8 | — | 4 | 1 | no | — |
| 179 | C4×Q8 | — | 2 | 1 | no | — |
| 180 | C4×Q8 | — | 2 | 1 | no | — |
| 181 | C4×Q8 | — | 4 | 1 | no | — |
| 182 | C4×Q8 | — | 4 | 1 | no | — |
| 183 | C4×Q8 | — | 4 | 1 | no | — |
| 184 | C4×Q8 | — | 4 | 1 | no | — |
| 185 | C4×Q8 | — | 4 | 1 | no | — |
| 186 | C4×Q8 | — | 4 | 1 | no | — |
| 187 | C4×D4 | — | 2 | 4 | no | — |
| 188 | C4×D4 | — | 2 | 4 | no | — |
| 189 | C4×D4 | — | 2 | 2 | no | — |
| 190 | C4×D4 | — | 2 | 2 | no | — |
| 191 | C4×D4 | — | 4 | 2 | no | — |
| 192 | C4×D4 | — | 4 | 2 | no | — |
| 193 | C4×D4 | — | 4 | 1 | no | — |
| 194 | C4×D4 | — | 4 | 1 | no | — |
| 195 | #117 | — | 2 | 2 | no | — |
| 196 | #117 | C2×D4 | 1 | 2 | yes | — |
| 197 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 198 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 199 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 200 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 201 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 202 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 203 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 204 | C2×C42 | — | 4 | 1 | no | — |
| 205 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 206 | SD32 | — | 2 | 8 | no | — |
| 207 | M32 | — | 2 | 2 | no | — |
| 208 | M32 | — | 2 | 2 | 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 | 1 | no | — |
| 214 | C2×C16 | — | 2 | 1 | no | — |
| 215 | C2×C16 | C2×Q8 | 1 | 2 | yes | complemented |
| 216 | #108 | — | 2 | 2 | no | — |
| 217 | #108 | — | 2 | 2 | no | — |
| 218 | #108 | — | 8 | 1 | no | — |
| 219 | #107 | — | 2 | 4 | no | — |
| 220 | #107 | — | 2 | 1 | no | — |
| 221 | #107 | — | 2 | 1 | no | — |
| 222 | #107 | — | 2 | 1 | no | — |
| 223 | #107 | — | 2 | 1 | no | — |
| 224 | #107 | — | 2 | 1 | no | — |
| 225 | #107 | — | 4 | 2 | no | — |
| 226 | #107 | C2×D4 | 1 | 2 | yes | — |
| 227 | #107 | C2×D4 | 1 | 2 | yes | — |
| 228 | #107 | C2×D4 | 1 | 2 | yes | — |
| 229 | #106 | — | 2 | 4 | no | — |
| 230 | #106 | — | 2 | 2 | no | — |
| 231 | #106 | — | 2 | 2 | no | — |
| 232 | #106 | — | 4 | 2 | no | — |
| 233 | C4⋊C8 | — | 2 | 2 | no | — |
| 234 | C4⋊C8 | — | 2 | 2 | no | — |
| 235 | C4⋊C8 | — | 2 | 1 | no | — |
| 236 | C4⋊C8 | — | 2 | 1 | no | — |
| 237 | C4⋊C8 | — | 4 | 1 | no | — |
| 238 | C4⋊C8 | — | 4 | 1 | no | — |
| 239 | #103 | — | 4 | 1 | no | — |
| 240 | #103 | — | 4 | 1 | no | — |
| 241 | #103 | — | 4 | 2 | no | — |
| 242 | #103 | — | 4 | 2 | no | — |
| 243 | #102 | — | 2 | 4 | no | — |
| 244 | #102 | — | 2 | 4 | no | — |
| 245 | #102 | — | 2 | 2 | no | — |
| 246 | #102 | — | 2 | 2 | no | — |
| 247 | #101 | — | 8 | 2 | no | — |
| 248 | #97 | — | 2 | 2 | no | — |
| 249 | C4×C8 | — | 2 | 2 | no | — |
| 250 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 251 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 252 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 253 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 254 | C24 | — | 4 | 1 | no | — |
| 255 | C2×Q8 | — | 2 | 1 | no | — |
| 256 | C2×Q8 | — | 2 | 1 | no | — |
| 257 | C2×Q8 | — | 2 | 2 | no | — |
| 258 | C2×Q8 | — | 4 | 4 | no | — |
| 259 | C2×Q8 | — | 4 | 1 | no | — |
| 260 | C2×Q8 | — | 4 | 1 | no | — |
| 261 | C2×Q8 | — | 4 | 1 | no | — |
| 262 | C2×Q8 | — | 4 | 1 | no | — |
| 263 | C2×Q8 | — | 4 | 1 | no | — |
| 264 | C2×Q8 | — | 4 | 1 | no | — |
| 265 | C2×Q8 | — | 4 | 1 | no | — |
| 266 | C2×Q8 | — | 16 | 4 | no | — |
| 267 | C2×D4 | — | 2 | 2 | no | — |
| 268 | C2×D4 | — | 2 | 2 | no | — |
| 269 | C2×D4 | — | 2 | 2 | no | — |
| 270 | C2×D4 | — | 4 | 1 | no | — |
| 271 | C2×D4 | — | 4 | 1 | no | — |
| 272 | C2×D4 | — | 4 | 1 | no | — |
| 273 | C22×C4 | — | 4 | 4 | no | — |
| 274 | C22×C4 | — | 4 | 4 | no | — |
| 275 | C22×C4 | — | 4 | 1 | no | — |
| 276 | C22×C4 | — | 4 | 1 | no | — |
| 277 | C22×C4 | — | 4 | 2 | no | — |
| 278 | C22×C4 | — | 4 | 2 | no | — |
| 279 | C22×C4 | — | 4 | 1 | no | — |
| 280 | C22×C4 | — | 16 | 2 | no | — |
| 281 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 282 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 283 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 284 | Q16 | — | 2 | 2 | no | — |
| 285 | Q16 | — | 2 | 2 | no | — |
| 286 | Q16 | — | 4 | 1 | no | — |
| 287 | Q16 | — | 4 | 1 | no | — |
| 288 | D8 | — | 2 | 2 | no | — |
| 289 | D8 | — | 2 | 2 | no | — |
| 290 | D8 | — | 2 | 2 | no | — |
| 291 | D8 | C22⋊Q8 | 1 | 4 | yes | — |
| 292 | M16 | — | 2 | 2 | no | — |
| 293 | M16 | — | 2 | 2 | no | — |
| 294 | M16 | — | 4 | 2 | no | — |
| 295 | M16 | — | 4 | 2 | no | — |
| 296 | C2×C8 | — | 2 | 1 | no | — |
| 297 | C2×C8 | — | 2 | 1 | no | — |
| 298 | C2×C8 | — | 2 | 2 | no | — |
| 299 | C2×C8 | — | 4 | 2 | no | — |
| 300 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 301 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 302 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 303 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 304 | C2×C8 | C22×Q8 | 1 | 1 | yes | characteristic |
| 305 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 306 | C2×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 307 | C2×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 308 | C4⋊C4 | — | 2 | 2 | no | — |
| 309 | C4⋊C4 | — | 2 | 1 | no | — |
| 310 | C4⋊C4 | — | 2 | 2 | no | — |
| 311 | C4⋊C4 | — | 2 | 1 | no | — |
| 312 | C4⋊C4 | — | 2 | 2 | no | — |
| 313 | C4⋊C4 | — | 2 | 2 | no | — |
| 314 | C4⋊C4 | — | 2 | 2 | no | — |
| 315 | C4⋊C4 | — | 4 | 4 | no | — |
| 316 | C4⋊C4 | — | 4 | 4 | no | — |
| 317 | C4⋊C4 | — | 4 | 4 | no | — |
| 318 | C4⋊C4 | — | 4 | 1 | no | — |
| 319 | C4⋊C4 | — | 4 | 2 | no | — |
| 320 | C4⋊C4 | — | 4 | 1 | no | — |
| 321 | C4⋊C4 | — | 4 | 2 | no | — |
| 322 | C4⋊C4 | — | 4 | 1 | no | — |
| 323 | C4⋊C4 | — | 4 | 1 | no | — |
| 324 | C4⋊C4 | — | 4 | 1 | no | — |
| 325 | C4⋊C4 | — | 8 | 1 | no | — |
| 326 | C4⋊C4 | — | 8 | 1 | no | — |
| 327 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 328 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 329 | C22⋊C4 | — | 2 | 2 | no | — |
| 330 | C22⋊C4 | — | 4 | 4 | no | — |
| 331 | C22⋊C4 | — | 4 | 4 | no | — |
| 332 | C22⋊C4 | — | 4 | 2 | no | — |
| 333 | C22⋊C4 | — | 4 | 2 | no | — |
| 334 | C22⋊C4 | — | 4 | 2 | no | — |
| 335 | C42 | — | 2 | 2 | no | — |
| 336 | C42 | — | 4 | 2 | no | — |
| 337 | C42 | — | 8 | 1 | no | — |
| 338 | C42 | — | 8 | 1 | no | — |
| 339 | C42 | #120 | 1 | 1 | yes | characteristic |
| 340 | C42 | #120 | 1 | 1 | yes | characteristic |
| 341 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 342 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 343 | C16 | — | 2 | 2 | no | — |
| 344 | C16 | — | 4 | 1 | no | — |
| 345 | C16 | — | 4 | 2 | no | — |
| 346 | C23 | — | 4 | 2 | no | — |
| 347 | C23 | — | 4 | 2 | no | — |
| 348 | C23 | — | 4 | 2 | no | — |
| 349 | C23 | — | 16 | 1 | no | — |
| 350 | C23 | — | 16 | 1 | no | — |
| 351 | C23 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 352 | Q8 | — | 4 | 2 | no | — |
| 353 | Q8 | — | 4 | 2 | no | — |
| 354 | Q8 | — | 4 | 2 | no | — |
| 355 | Q8 | — | 8 | 1 | no | — |
| 356 | Q8 | — | 8 | 2 | no | — |
| 357 | Q8 | — | 8 | 1 | no | — |
| 358 | Q8 | — | 8 | 1 | no | — |
| 359 | Q8 | — | 16 | 1 | no | — |
| 360 | Q8 | — | 16 | 4 | no | — |
| 361 | D4 | — | 2 | 4 | no | — |
| 362 | D4 | — | 4 | 2 | no | — |
| 363 | D4 | — | 4 | 2 | no | — |
| 364 | D4 | — | 4 | 2 | no | — |
| 365 | C2×C4 | — | 2 | 2 | no | — |
| 366 | C2×C4 | — | 2 | 2 | no | — |
| 367 | C2×C4 | — | 4 | 1 | no | — |
| 368 | C2×C4 | — | 4 | 1 | no | — |
| 369 | C2×C4 | — | 4 | 2 | no | — |
| 370 | C2×C4 | — | 4 | 2 | no | — |
| 371 | C2×C4 | — | 4 | 2 | no | — |
| 372 | C2×C4 | — | 4 | 2 | no | — |
| 373 | C2×C4 | — | 8 | 4 | no | — |
| 374 | C2×C4 | — | 8 | 1 | no | — |
| 375 | C2×C4 | — | 8 | 1 | no | — |
| 376 | C2×C4 | — | 8 | 1 | no | — |
| 377 | C2×C4 | — | 16 | 2 | no | — |
| 378 | C2×C4 | — | 16 | 4 | no | — |
| 379 | C2×C4 | — | 16 | 4 | no | — |
| 380 | C2×C4 | — | 16 | 1 | no | — |
| 381 | C2×C4 | — | 16 | 1 | no | — |
| 382 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 383 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 384 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 385 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 386 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 387 | C2×C4 | #548 | 1 | 1 | yes | characteristic |
| 388 | C2×C4 | D4×Q8 | 1 | 1 | yes | characteristic |
| 389 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 390 | C2×C4 | #577 | 1 | 1 | yes | characteristic |
| 391 | C2×C4 | #578 | 1 | 1 | yes | characteristic |
| 392 | C8 | — | 2 | 1 | no | — |
| 393 | C8 | — | 2 | 1 | no | — |
| 394 | C8 | — | 4 | 2 | no | — |
| 395 | C8 | — | 8 | 2 | no | — |
| 396 | C8 | C2×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 397 | C8 | #536 | 1 | 1 | yes | characteristic |
| 398 | C8 | #548 | 1 | 1 | yes | characteristic |
| 399 | C8 | D4×Q8 | 1 | 1 | yes | characteristic |
| 400 | C22 | — | 2 | 1 | no | — |
| 401 | C22 | — | 2 | 1 | no | — |
| 402 | C22 | — | 4 | 4 | no | — |
| 403 | C22 | — | 8 | 2 | no | — |
| 404 | C22 | — | 8 | 2 | no | — |
| 405 | C22 | — | 16 | 2 | no | — |
| 406 | C22 | — | 16 | 2 | no | — |
| 407 | C22 | C2×#447 | 1 | 1 | yes | characteristic |
| 408 | C22 | #3367 | 1 | 1 | yes | characteristic |
| 409 | C22 | #3384 | 1 | 1 | yes | characteristic |
| 410 | C4 | — | 2 | 1 | no | — |
| 411 | C4 | — | 2 | 1 | no | — |
| 412 | C4 | — | 2 | 1 | no | — |
| 413 | C4 | — | 2 | 1 | no | — |
| 414 | C4 | — | 4 | 2 | no | — |
| 415 | C4 | — | 4 | 2 | no | — |
| 416 | C4 | — | 8 | 2 | no | — |
| 417 | C4 | — | 8 | 2 | no | — |
| 418 | C4 | — | 8 | 2 | no | — |
| 419 | C4 | — | 8 | 2 | no | — |
| 420 | C4 | — | 16 | 1 | no | — |
| 421 | C4 | #2409 | 1 | 1 | yes | characteristic |
| 422 | C4 | #3013 | 1 | 1 | yes | characteristic |
| 423 | C4 | #3358 | 1 | 1 | yes | characteristic |
| 424 | C4 | Q8×D8 | 1 | 1 | yes | characteristic |
| 425 | C2 | — | 2 | 1 | no | — |
| 426 | C2 | — | 2 | 1 | no | — |
| 427 | C2 | — | 8 | 4 | no | — |
| 428 | C2 | #24326 | 1 | 1 | yes | characteristic |
| 429 | C2 | #33539 | 1 | 1 | yes | characteristic |
| 430 | C2 | #33542 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33628 | C2 | semidirect |
| 4. #28297 | C2 | semidirect |
| 5. #13455 | C2 | semidirect |
| 6. #13370 | C2 | semidirect |
| 7. #13286 | C2 | semidirect |
| 8. #13267 | C2 | semidirect |
| 69. C2×SD32 | Q8 | semidirect |
| 146. C2×Q16 | C2×Q8 | semidirect |
| 215. C2×C16 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #61370 | 2 |
| #33628 | 2 |
| C2×#2175 | 2 |
| #28297 | 2 |
| #13455 | 2 |
| #13370 | 2 |
| #13286 | 2 |
| #13267 | 2 |
| #13019 | 2 |
| #12940 | 2 |
| #12818 | 2 |