Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
865 conjugacy classes of subgroups.
The subgroup lattice has 403 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#3341 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2224 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#2177 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #28225 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13380 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13304 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13283 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13190 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #12992 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #12921 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #12813 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#555 | — | 2 | 1 | no | — |
| 13 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 14 | #3341 | — | 2 | 4 | no | — |
| 15 | #3341 | — | 2 | 4 | no | — |
| 16 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#482 | — | 2 | 2 | no | — |
| 18 | C2×#476 | — | 2 | 1 | no | — |
| 19 | C2×#466 | C22 | 1 | 2 | yes | — |
| 20 | C2×#466 | C22 | 1 | 2 | yes | — |
| 21 | C2×C4×D8 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C4×D8 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×C8×D4 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×#427 | C22 | 1 | 2 | yes | — |
| 25 | #2535 | — | 2 | 1 | no | — |
| 26 | #2224 | — | 2 | 4 | no | — |
| 27 | #2177 | — | 2 | 4 | no | — |
| 28 | C2×#367 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×#366 | C22 | 1 | 2 | yes | — |
| 30 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 31 | C2×#357 | C22 | 1 | 1 | yes | characteristic |
| 32 | C2×#357 | C22 | 1 | 2 | yes | — |
| 33 | C2×#357 | C22 | 1 | 1 | yes | characteristic |
| 34 | C2×#357 | C22 | 1 | 2 | yes | — |
| 35 | C2×C22⋊C16 | C22 | 1 | 2 | yes | — |
| 36 | C2×C22⋊C16 | C22 | 1 | 2 | yes | — |
| 37 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 38 | #2034 | — | 2 | 2 | no | — |
| 39 | #1846 | C22 | 1 | 1 | yes | characteristic |
| 40 | #1808 | C22 | 1 | 2 | yes | — |
| 41 | #1804 | — | 2 | 1 | no | — |
| 42 | #1380 | C22 | 1 | 2 | yes | — |
| 43 | #1380 | C22 | 1 | 2 | yes | — |
| 44 | #1380 | C22 | 1 | 2 | yes | — |
| 45 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 46 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 47 | C23×C8 | C23 | 1 | 2 | yes | — |
| 48 | #555 | — | 2 | 8 | no | — |
| 49 | #555 | — | 4 | 4 | no | — |
| 50 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 51 | C2×#124 | — | 4 | 2 | no | — |
| 52 | C2×#121 | — | 2 | 2 | no | — |
| 53 | C2×#121 | — | 2 | 2 | no | — |
| 54 | C2×#121 | — | 4 | 1 | no | — |
| 55 | C2×C4×D4 | — | 2 | 1 | no | — |
| 56 | C2×C4×D4 | — | 2 | 1 | no | — |
| 57 | C2×C4×D4 | D4 | 1 | 1 | yes | characteristic |
| 58 | C2×#117 | — | 2 | 2 | no | — |
| 59 | C2×#117 | — | 4 | 1 | no | — |
| 60 | C2×D16 | — | 2 | 2 | no | — |
| 61 | C2×D16 | — | 2 | 2 | no | — |
| 62 | C2×D16 | — | 2 | 2 | no | — |
| 63 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 64 | C22×C16 | C23 | 1 | 2 | yes | — |
| 65 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 66 | #500 | — | 2 | 2 | no | — |
| 67 | #500 | D4 | 1 | 4 | yes | complemented |
| 68 | #482 | — | 2 | 16 | no | — |
| 69 | #476 | — | 2 | 8 | no | — |
| 70 | #466 | — | 2 | 4 | no | — |
| 71 | #466 | — | 2 | 4 | no | — |
| 72 | #466 | — | 2 | 4 | no | — |
| 73 | #466 | — | 4 | 2 | no | — |
| 74 | C4×D8 | — | 2 | 2 | no | — |
| 75 | C4×D8 | — | 2 | 2 | no | — |
| 76 | C4×D8 | — | 2 | 2 | no | — |
| 77 | C4×D8 | D4 | 1 | 4 | yes | complemented |
| 78 | C8×D4 | — | 2 | 2 | no | — |
| 79 | C8×D4 | — | 2 | 2 | no | — |
| 80 | #427 | — | 2 | 4 | no | — |
| 81 | #427 | — | 2 | 4 | no | — |
| 82 | C2×#107 | — | 2 | 1 | no | — |
| 83 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 84 | C2×#107 | C23 | 1 | 2 | yes | — |
| 85 | C2×#107 | C23 | 1 | 2 | yes | — |
| 86 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 87 | C2×#106 | — | 2 | 2 | no | — |
| 88 | C2×#106 | — | 2 | 1 | no | — |
| 89 | C2×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 90 | C2×#102 | — | 2 | 1 | no | — |
| 91 | C2×#102 | — | 2 | 2 | no | — |
| 92 | C2×#102 | — | 2 | 2 | no | — |
| 93 | C2×#102 | — | 2 | 1 | no | — |
| 94 | C2×C22⋊C8 | D4 | 1 | 2 | yes | — |
| 95 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 96 | #397 | — | 4 | 1 | no | — |
| 97 | #392 | — | 4 | 1 | no | — |
| 98 | #386 | — | 4 | 2 | no | — |
| 99 | #382 | — | 2 | 1 | no | — |
| 100 | #380 | — | 2 | 2 | no | — |
| 101 | #367 | — | 2 | 2 | no | — |
| 102 | #366 | — | 2 | 4 | no | — |
| 103 | C4⋊C16 | — | 2 | 2 | no | — |
| 104 | #357 | — | 2 | 2 | no | — |
| 105 | #357 | — | 2 | 2 | no | — |
| 106 | #357 | — | 2 | 4 | no | — |
| 107 | #357 | — | 4 | 2 | no | — |
| 108 | C22⋊C16 | — | 2 | 4 | no | — |
| 109 | C22⋊C16 | — | 4 | 2 | no | — |
| 110 | #346 | — | 2 | 2 | no | — |
| 111 | #340 | — | 2 | 2 | no | — |
| 112 | C22×Q8 | — | 4 | 1 | no | — |
| 113 | C22×D4 | — | 2 | 1 | no | — |
| 114 | C22×D4 | — | 2 | 1 | no | — |
| 115 | C22×D4 | — | 4 | 2 | no | — |
| 116 | C22×D4 | — | 4 | 2 | no | — |
| 117 | C22×D4 | D8 | 1 | 1 | yes | characteristic |
| 118 | C23×C4 | — | 4 | 1 | no | — |
| 119 | C23×C4 | — | 4 | 1 | no | — |
| 120 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 121 | C2×D8 | — | 2 | 1 | no | — |
| 122 | C2×D8 | — | 2 | 1 | no | — |
| 123 | C2×D8 | — | 2 | 1 | no | — |
| 124 | C2×D8 | — | 2 | 1 | no | — |
| 125 | C2×D8 | — | 2 | 1 | no | — |
| 126 | C2×D8 | — | 2 | 1 | no | — |
| 127 | C2×D8 | — | 2 | 1 | no | — |
| 128 | C2×D8 | — | 2 | 1 | no | — |
| 129 | C2×D8 | — | 2 | 1 | no | — |
| 130 | C2×D8 | — | 2 | 1 | no | — |
| 131 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 132 | C2×D8 | #41 | 1 | 2 | yes | — |
| 133 | C22×C8 | — | 2 | 2 | no | — |
| 134 | C22×C8 | — | 2 | 2 | no | — |
| 135 | C22×C8 | — | 2 | 2 | no | — |
| 136 | C22×C8 | — | 2 | 2 | no | — |
| 137 | C22×C8 | — | 2 | 2 | no | — |
| 138 | C22×C8 | — | 2 | 2 | no | — |
| 139 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 140 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 141 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 142 | C4⋊Q8 | — | 2 | 4 | no | — |
| 143 | C4⋊Q8 | — | 4 | 2 | no | — |
| 144 | #124 | — | 4 | 16 | no | — |
| 145 | #121 | — | 2 | 8 | no | — |
| 146 | #121 | — | 4 | 4 | no | — |
| 147 | #121 | — | 4 | 8 | no | — |
| 148 | #121 | — | 4 | 4 | no | — |
| 149 | #121 | — | 8 | 2 | no | — |
| 150 | C4×D4 | — | 2 | 2 | no | — |
| 151 | C4×D4 | — | 2 | 4 | no | — |
| 152 | C4×D4 | — | 2 | 2 | no | — |
| 153 | C4×D4 | — | 4 | 2 | no | — |
| 154 | C4×D4 | — | 4 | 2 | no | — |
| 155 | C4×D4 | — | 4 | 2 | no | — |
| 156 | #117 | — | 2 | 8 | no | — |
| 157 | #117 | — | 4 | 8 | no | — |
| 158 | #117 | — | 4 | 4 | no | — |
| 159 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 160 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 161 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 162 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 163 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 164 | C2×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 165 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 166 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 167 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 168 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 169 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 170 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 171 | C2×C22⋊C4 | D8 | 1 | 2 | yes | — |
| 172 | C2×C42 | — | 4 | 2 | no | — |
| 173 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 174 | D16 | — | 2 | 8 | no | — |
| 175 | C2×C16 | — | 2 | 1 | no | — |
| 176 | C2×C16 | — | 2 | 1 | no | — |
| 177 | C2×C16 | — | 2 | 1 | no | — |
| 178 | C2×C16 | — | 2 | 2 | no | — |
| 179 | C2×C16 | — | 2 | 2 | no | — |
| 180 | C2×C16 | — | 2 | 1 | no | — |
| 181 | C2×C16 | — | 2 | 2 | no | — |
| 182 | C2×C16 | — | 2 | 1 | no | — |
| 183 | C2×C16 | #41 | 1 | 2 | yes | — |
| 184 | #107 | — | 2 | 4 | no | — |
| 185 | #107 | — | 2 | 1 | no | — |
| 186 | #107 | — | 2 | 2 | no | — |
| 187 | #107 | — | 2 | 1 | no | — |
| 188 | #107 | — | 2 | 1 | no | — |
| 189 | #107 | — | 2 | 2 | no | — |
| 190 | #107 | — | 2 | 2 | no | — |
| 191 | #107 | C2×D4 | 1 | 4 | yes | — |
| 192 | #107 | C2×D4 | 1 | 2 | yes | — |
| 193 | #106 | — | 2 | 8 | no | — |
| 194 | #106 | — | 8 | 1 | no | — |
| 195 | C4⋊C8 | — | 2 | 2 | no | — |
| 196 | #102 | — | 2 | 8 | no | — |
| 197 | #102 | — | 2 | 4 | no | — |
| 198 | #102 | — | 4 | 4 | no | — |
| 199 | #102 | — | 4 | 2 | no | — |
| 200 | C22⋊C8 | — | 2 | 4 | no | — |
| 201 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 202 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 203 | #95 | — | 4 | 2 | no | — |
| 204 | #95 | — | 4 | 1 | no | — |
| 205 | #95 | — | 4 | 1 | no | — |
| 206 | C24 | — | 4 | 1 | no | — |
| 207 | C24 | — | 4 | 1 | no | — |
| 208 | C24 | C2×D8 | 1 | 2 | yes | — |
| 209 | C2×Q8 | — | 4 | 4 | no | — |
| 210 | C2×Q8 | — | 16 | 2 | no | — |
| 211 | C2×D4 | — | 2 | 2 | no | — |
| 212 | C2×D4 | — | 2 | 2 | no | — |
| 213 | C2×D4 | — | 2 | 2 | no | — |
| 214 | C2×D4 | — | 4 | 2 | no | — |
| 215 | C2×D4 | — | 4 | 8 | no | — |
| 216 | C2×D4 | — | 4 | 1 | no | — |
| 217 | C2×D4 | — | 4 | 1 | no | — |
| 218 | C2×D4 | — | 4 | 1 | no | — |
| 219 | C2×D4 | — | 4 | 1 | no | — |
| 220 | C2×D4 | — | 4 | 1 | no | — |
| 221 | C2×D4 | — | 4 | 1 | no | — |
| 222 | C2×D4 | — | 4 | 1 | no | — |
| 223 | C2×D4 | — | 4 | 1 | no | — |
| 224 | C2×D4 | — | 4 | 1 | no | — |
| 225 | C2×D4 | — | 4 | 1 | no | — |
| 226 | C2×D4 | — | 8 | 4 | no | — |
| 227 | C2×D4 | — | 8 | 4 | no | — |
| 228 | C2×D4 | — | 16 | 2 | no | — |
| 229 | C2×D4 | — | 16 | 2 | no | — |
| 230 | C2×D4 | — | 16 | 2 | no | — |
| 231 | C22×C4 | — | 2 | 2 | no | — |
| 232 | C22×C4 | — | 2 | 2 | no | — |
| 233 | C22×C4 | — | 2 | 2 | no | — |
| 234 | C22×C4 | — | 2 | 2 | no | — |
| 235 | C22×C4 | — | 2 | 2 | no | — |
| 236 | C22×C4 | — | 2 | 2 | no | — |
| 237 | C22×C4 | — | 4 | 1 | no | — |
| 238 | C22×C4 | — | 4 | 2 | no | — |
| 239 | C22×C4 | — | 4 | 4 | no | — |
| 240 | C22×C4 | — | 4 | 2 | no | — |
| 241 | C22×C4 | — | 4 | 1 | no | — |
| 242 | C22×C4 | — | 8 | 4 | no | — |
| 243 | C22×C4 | — | 8 | 2 | no | — |
| 244 | C22×C4 | — | 16 | 2 | no | — |
| 245 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 246 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 247 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 248 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 249 | D8 | — | 2 | 2 | no | — |
| 250 | D8 | — | 2 | 2 | no | — |
| 251 | D8 | — | 2 | 2 | no | — |
| 252 | D8 | — | 2 | 2 | no | — |
| 253 | D8 | — | 4 | 1 | no | — |
| 254 | D8 | — | 4 | 1 | no | — |
| 255 | D8 | — | 4 | 1 | no | — |
| 256 | D8 | — | 4 | 1 | no | — |
| 257 | C2×C8 | — | 2 | 1 | no | — |
| 258 | C2×C8 | — | 2 | 1 | no | — |
| 259 | C2×C8 | — | 2 | 2 | no | — |
| 260 | C2×C8 | — | 2 | 2 | no | — |
| 261 | C2×C8 | — | 2 | 2 | no | — |
| 262 | C2×C8 | — | 4 | 2 | no | — |
| 263 | C2×C8 | — | 4 | 2 | no | — |
| 264 | C2×C8 | — | 4 | 2 | no | — |
| 265 | C2×C8 | — | 4 | 2 | no | — |
| 266 | C2×C8 | #120 | 1 | 2 | yes | — |
| 267 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 268 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 269 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 270 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 271 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 272 | C2×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 273 | C4⋊C4 | — | 2 | 2 | no | — |
| 274 | C4⋊C4 | — | 2 | 2 | no | — |
| 275 | C4⋊C4 | — | 2 | 4 | no | — |
| 276 | C4⋊C4 | — | 4 | 1 | no | — |
| 277 | C4⋊C4 | — | 4 | 2 | no | — |
| 278 | C4⋊C4 | — | 4 | 4 | no | — |
| 279 | C4⋊C4 | — | 4 | 1 | no | — |
| 280 | C4⋊C4 | — | 4 | 2 | no | — |
| 281 | C4⋊C4 | — | 4 | 1 | no | — |
| 282 | C4⋊C4 | — | 4 | 2 | no | — |
| 283 | C22⋊C4 | — | 2 | 4 | no | — |
| 284 | C22⋊C4 | — | 4 | 8 | no | — |
| 285 | C22⋊C4 | — | 4 | 8 | no | — |
| 286 | C22⋊C4 | — | 4 | 4 | no | — |
| 287 | C22⋊C4 | — | 8 | 4 | no | — |
| 288 | C22⋊C4 | — | 8 | 2 | no | — |
| 289 | C22⋊C4 | — | 16 | 2 | no | — |
| 290 | C42 | — | 4 | 8 | no | — |
| 291 | C42 | #120 | 1 | 2 | yes | — |
| 292 | C42 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 293 | C16 | — | 2 | 2 | no | — |
| 294 | C16 | — | 4 | 1 | no | — |
| 295 | C16 | — | 4 | 2 | no | — |
| 296 | C23 | — | 2 | 2 | no | — |
| 297 | C23 | — | 2 | 2 | no | — |
| 298 | C23 | — | 2 | 2 | no | — |
| 299 | C23 | — | 2 | 2 | no | — |
| 300 | C23 | — | 2 | 2 | no | — |
| 301 | C23 | — | 2 | 2 | no | — |
| 302 | C23 | — | 2 | 2 | no | — |
| 303 | C23 | — | 4 | 2 | no | — |
| 304 | C23 | — | 4 | 2 | no | — |
| 305 | C23 | — | 8 | 1 | no | — |
| 306 | C23 | — | 8 | 1 | no | — |
| 307 | C23 | — | 8 | 2 | no | — |
| 308 | C23 | — | 8 | 2 | no | — |
| 309 | C23 | — | 8 | 2 | no | — |
| 310 | C23 | — | 8 | 2 | no | — |
| 311 | C23 | — | 8 | 1 | no | — |
| 312 | C23 | — | 8 | 1 | no | — |
| 313 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 314 | Q8 | — | 16 | 4 | no | — |
| 315 | D4 | — | 4 | 4 | no | — |
| 316 | D4 | — | 4 | 2 | no | — |
| 317 | D4 | — | 4 | 2 | no | — |
| 318 | D4 | — | 4 | 2 | no | — |
| 319 | D4 | — | 4 | 2 | no | — |
| 320 | D4 | — | 8 | 1 | no | — |
| 321 | D4 | — | 8 | 1 | no | — |
| 322 | D4 | — | 8 | 1 | no | — |
| 323 | D4 | — | 8 | 1 | no | — |
| 324 | D4 | — | 16 | 8 | no | — |
| 325 | D4 | — | 16 | 8 | no | — |
| 326 | C2×C4 | — | 2 | 2 | no | — |
| 327 | C2×C4 | — | 2 | 2 | no | — |
| 328 | C2×C4 | — | 2 | 2 | no | — |
| 329 | C2×C4 | — | 2 | 2 | no | — |
| 330 | C2×C4 | — | 2 | 2 | no | — |
| 331 | C2×C4 | — | 4 | 2 | no | — |
| 332 | C2×C4 | — | 4 | 2 | no | — |
| 333 | C2×C4 | — | 4 | 2 | no | — |
| 334 | C2×C4 | — | 4 | 2 | no | — |
| 335 | C2×C4 | — | 4 | 2 | no | — |
| 336 | C2×C4 | — | 4 | 4 | no | — |
| 337 | C2×C4 | — | 8 | 8 | no | — |
| 338 | C2×C4 | — | 8 | 2 | no | — |
| 339 | C2×C4 | — | 8 | 2 | no | — |
| 340 | C2×C4 | — | 8 | 2 | no | — |
| 341 | C2×C4 | — | 8 | 2 | no | — |
| 342 | C2×C4 | — | 8 | 2 | no | — |
| 343 | C2×C4 | — | 8 | 2 | no | — |
| 344 | C2×C4 | — | 8 | 1 | no | — |
| 345 | C2×C4 | — | 8 | 2 | no | — |
| 346 | C2×C4 | — | 16 | 4 | no | — |
| 347 | C2×C4 | #447 | 1 | 2 | yes | — |
| 348 | C2×C4 | #447 | 1 | 2 | yes | — |
| 349 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 350 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 351 | C2×C4 | #547 | 1 | 1 | yes | characteristic |
| 352 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 353 | C2×C4 | #577 | 1 | 1 | yes | characteristic |
| 354 | C2×C4 | #578 | 1 | 1 | yes | characteristic |
| 355 | C8 | — | 2 | 2 | no | — |
| 356 | C8 | — | 4 | 2 | no | — |
| 357 | C8 | #525 | 1 | 1 | yes | characteristic |
| 358 | C8 | #537 | 1 | 1 | yes | characteristic |
| 359 | C8 | #548 | 1 | 1 | yes | characteristic |
| 360 | C8 | #550 | 1 | 1 | yes | characteristic |
| 361 | C22 | — | 4 | 2 | no | — |
| 362 | C22 | — | 4 | 2 | no | — |
| 363 | C22 | — | 4 | 2 | no | — |
| 364 | C22 | — | 4 | 2 | no | — |
| 365 | C22 | — | 4 | 2 | no | — |
| 366 | C22 | — | 4 | 2 | no | — |
| 367 | C22 | — | 4 | 2 | no | — |
| 368 | C22 | — | 8 | 2 | no | — |
| 369 | C22 | — | 8 | 2 | no | — |
| 370 | C22 | — | 8 | 4 | no | — |
| 371 | C22 | — | 8 | 2 | no | — |
| 372 | C22 | — | 8 | 2 | no | — |
| 373 | C22 | — | 16 | 2 | no | — |
| 374 | C22 | — | 16 | 2 | no | — |
| 375 | C22 | — | 16 | 2 | no | — |
| 376 | C22 | — | 16 | 1 | no | — |
| 377 | C22 | — | 16 | 1 | no | — |
| 378 | C22 | C2×#447 | 1 | 1 | yes | characteristic |
| 379 | C22 | #3294 | 1 | 1 | yes | characteristic |
| 380 | C22 | #3334 | 1 | 1 | yes | characteristic |
| 381 | C22 | #3414 | 1 | 2 | yes | — |
| 382 | C22 | #3415 | 1 | 2 | yes | — |
| 383 | C4 | — | 2 | 2 | no | — |
| 384 | C4 | — | 2 | 2 | no | — |
| 385 | C4 | — | 4 | 2 | no | — |
| 386 | C4 | — | 8 | 2 | no | — |
| 387 | C4 | — | 8 | 4 | no | — |
| 388 | C4 | — | 16 | 2 | no | — |
| 389 | C4 | — | 16 | 1 | no | — |
| 390 | C4 | #2407 | 1 | 1 | yes | characteristic |
| 391 | C4 | #3013 | 1 | 1 | yes | characteristic |
| 392 | C4 | #3330 | 1 | 1 | yes | characteristic |
| 393 | C4 | #3354 | 1 | 1 | yes | characteristic |
| 394 | C2 | — | 8 | 2 | no | — |
| 395 | C2 | — | 16 | 2 | no | — |
| 396 | C2 | — | 16 | 2 | no | — |
| 397 | C2 | #24116 | 1 | 1 | yes | characteristic |
| 398 | C2 | #33519 | 1 | 1 | yes | characteristic |
| 399 | C2 | #33522 | 1 | 1 | yes | characteristic |
| 400 | C2 | #33886 | 1 | 2 | yes | — |
| 401 | C2 | #33914 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| 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 |
|---|---|---|
| 1. C2×#3341 | C2 | semidirect |
| 2. C2×#2224 | C2 | semidirect |
| 3. C2×#2177 | C2 | semidirect |
| 4. #28225 | C2 | semidirect |
| 5. #13380 | C2 | semidirect |
| 6. #13304 | C2 | semidirect |
| 7. #13283 | C2 | semidirect |
| 8. #13190 | C2 | semidirect |
| 9. #12992 | C2 | semidirect |
| 10. #12921 | C2 | semidirect |
| 11. #12813 | C2 | semidirect |
| 67. #500 | D4 | semidirect |
| 77. C4×D8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3341 | 2 |
| C2×#2224 | 2 |
| C2×#2177 | 2 |
| #28225 | 2 |
| #13380 | 2 |
| #13304 | 2 |
| #13283 | 2 |
| #13190 | 2 |
| #12992 | 2 |
| #12921 | 2 |
| #12813 | 2 |