Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
766 conjugacy classes of subgroups.
The subgroup lattice has 532 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #60708 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33646 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#2236 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#1952 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13455 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13412 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13003 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #12990 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #12945 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #12941 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #12848 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3495 | — | 2 | 1 | no | — |
| 13 | #3181 | — | 2 | 2 | no | — |
| 14 | #3178 | — | 2 | 2 | no | — |
| 15 | #3177 | — | 2 | 1 | no | — |
| 16 | #3175 | — | 2 | 1 | no | — |
| 17 | #3170 | — | 2 | 1 | no | — |
| 18 | #3168 | — | 2 | 1 | no | — |
| 19 | #3164 | C22 | 1 | 1 | yes | characteristic |
| 20 | #3156 | C22 | 1 | 1 | yes | characteristic |
| 21 | #3137 | — | 2 | 1 | no | — |
| 22 | #3056 | — | 2 | 1 | no | — |
| 23 | #3050 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×#467 | C22 | 1 | 1 | yes | characteristic |
| 25 | #2951 | C22 | 1 | 1 | yes | characteristic |
| 26 | #2950 | C22 | 1 | 1 | yes | characteristic |
| 27 | #2947 | — | 2 | 1 | no | — |
| 28 | C22×#108 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×#428 | — | 2 | 1 | no | — |
| 30 | C22×#107 | C22 | 1 | 1 | yes | characteristic |
| 31 | C2×#405 | C22 | 1 | 1 | yes | characteristic |
| 32 | #2236 | — | 2 | 2 | no | — |
| 33 | #2236 | — | 2 | 2 | no | — |
| 34 | #2236 | — | 2 | 1 | no | — |
| 35 | #2235 | — | 2 | 1 | no | — |
| 36 | #2233 | — | 2 | 1 | no | — |
| 37 | #2232 | — | 2 | 1 | no | — |
| 38 | #2159 | C22 | 1 | 2 | yes | — |
| 39 | C2×#366 | C22 | 1 | 2 | yes | — |
| 40 | C2×#362 | C22 | 1 | 2 | yes | — |
| 41 | C2×#359 | C22 | 1 | 2 | yes | — |
| 42 | #2139 | C22 | 1 | 2 | yes | — |
| 43 | C2×#358 | C22 | 1 | 2 | yes | — |
| 44 | #2112 | C22 | 1 | 1 | yes | characteristic |
| 45 | C2×C22⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 46 | #1952 | — | 2 | 1 | no | — |
| 47 | #1952 | — | 2 | 1 | no | — |
| 48 | #1952 | C22 | 1 | 2 | yes | — |
| 49 | #1952 | C22 | 1 | 2 | yes | — |
| 50 | C2×#343 | — | 2 | 1 | no | — |
| 51 | #1387 | C22 | 1 | 2 | yes | — |
| 52 | #1382 | C22 | 1 | 2 | yes | — |
| 53 | #1379 | C22 | 1 | 2 | yes | — |
| 54 | #1379 | C22 | 1 | 2 | yes | — |
| 55 | C2×#137 | — | 2 | 1 | no | — |
| 56 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 57 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 58 | C22×M16 | — | 2 | 1 | no | — |
| 59 | C22×M16 | D4 | 1 | 1 | yes | characteristic |
| 60 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 61 | #556 | — | 4 | 1 | no | — |
| 62 | #554 | — | 4 | 1 | no | — |
| 63 | #552 | — | 4 | 2 | no | — |
| 64 | #552 | — | 4 | 1 | no | — |
| 65 | #552 | — | 4 | 1 | no | — |
| 66 | #551 | — | 4 | 2 | no | — |
| 67 | #544 | — | 2 | 1 | no | — |
| 68 | #542 | — | 2 | 2 | no | — |
| 69 | #541 | — | 2 | 2 | no | — |
| 70 | #536 | — | 2 | 1 | no | — |
| 71 | #533 | — | 2 | 2 | no | — |
| 72 | #529 | — | 2 | 2 | no | — |
| 73 | #525 | — | 2 | 1 | no | — |
| 74 | C2×C22⋊Q8 | — | 2 | 1 | no | — |
| 75 | #520 | — | 2 | 1 | no | — |
| 76 | #519 | — | 2 | 1 | no | — |
| 77 | C2×#117 | D4 | 1 | 1 | yes | characteristic |
| 78 | C22×C4⋊C4 | — | 2 | 1 | no | — |
| 79 | C2×M32 | C23 | 1 | 2 | yes | — |
| 80 | C22×C16 | C23 | 1 | 2 | yes | — |
| 81 | #501 | — | 2 | 2 | no | — |
| 82 | #500 | — | 2 | 1 | no | — |
| 83 | #499 | — | 2 | 1 | no | — |
| 84 | #497 | — | 2 | 2 | no | — |
| 85 | #495 | — | 2 | 1 | no | — |
| 86 | #494 | — | 2 | 1 | no | — |
| 87 | #491 | — | 2 | 2 | no | — |
| 88 | #489 | — | 2 | 2 | no | — |
| 89 | #488 | — | 2 | 2 | no | — |
| 90 | #487 | — | 2 | 2 | no | — |
| 91 | #470 | — | 2 | 2 | no | — |
| 92 | #468 | — | 2 | 2 | no | — |
| 93 | #467 | — | 2 | 2 | no | — |
| 94 | #467 | — | 2 | 1 | no | — |
| 95 | #467 | — | 2 | 1 | no | — |
| 96 | #467 | D4 | 1 | 1 | yes | characteristic |
| 97 | #467 | D4 | 1 | 1 | yes | characteristic |
| 98 | #467 | D4 | 1 | 2 | yes | — |
| 99 | #467 | D4 | 1 | 2 | yes | — |
| 100 | #466 | D4 | 1 | 1 | yes | characteristic |
| 101 | #466 | D4 | 1 | 1 | yes | characteristic |
| 102 | #465 | — | 2 | 2 | no | — |
| 103 | #465 | — | 2 | 1 | no | — |
| 104 | #465 | — | 2 | 1 | no | — |
| 105 | #441 | — | 2 | 2 | no | — |
| 106 | #441 | — | 2 | 2 | no | — |
| 107 | #440 | — | 2 | 2 | no | — |
| 108 | #440 | — | 2 | 2 | no | — |
| 109 | C4×Q16 | — | 2 | 2 | no | — |
| 110 | C4×Q16 | — | 2 | 1 | no | — |
| 111 | C4×Q16 | — | 2 | 1 | no | — |
| 112 | C4×Q16 | — | 2 | 1 | no | — |
| 113 | C4×SD16 | — | 2 | 2 | no | — |
| 114 | C4×D8 | — | 2 | 1 | no | — |
| 115 | C2×#108 | — | 2 | 1 | no | — |
| 116 | C2×#108 | — | 2 | 4 | no | — |
| 117 | C2×#108 | C23 | 1 | 2 | yes | — |
| 118 | #428 | — | 2 | 1 | no | — |
| 119 | #428 | — | 2 | 2 | no | — |
| 120 | #428 | — | 2 | 2 | no | — |
| 121 | #428 | — | 4 | 1 | no | — |
| 122 | #428 | — | 4 | 1 | no | — |
| 123 | #427 | C23 | 1 | 1 | yes | characteristic |
| 124 | C2×#107 | — | 2 | 1 | no | — |
| 125 | C2×#107 | — | 2 | 1 | no | — |
| 126 | C2×#107 | — | 2 | 1 | no | — |
| 127 | C2×#107 | — | 2 | 1 | no | — |
| 128 | C2×#107 | — | 2 | 1 | no | — |
| 129 | C2×#107 | — | 4 | 1 | no | — |
| 130 | C2×#107 | C23 | 1 | 2 | yes | — |
| 131 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 132 | C2×#106 | — | 4 | 1 | no | — |
| 133 | #419 | — | 2 | 1 | no | — |
| 134 | #417 | — | 2 | 1 | no | — |
| 135 | #416 | — | 2 | 1 | no | — |
| 136 | C2×#103 | — | 2 | 1 | no | — |
| 137 | #405 | — | 2 | 1 | no | — |
| 138 | #405 | — | 2 | 1 | no | — |
| 139 | #405 | D4 | 1 | 1 | yes | characteristic |
| 140 | #405 | D4 | 1 | 1 | yes | characteristic |
| 141 | #405 | C23 | 1 | 1 | yes | characteristic |
| 142 | #405 | C23 | 1 | 1 | yes | characteristic |
| 143 | C2×#97 | — | 2 | 1 | no | — |
| 144 | C2×C4×C8 | — | 2 | 1 | no | — |
| 145 | #367 | — | 2 | 2 | no | — |
| 146 | #366 | — | 2 | 4 | no | — |
| 147 | #366 | — | 2 | 2 | no | — |
| 148 | #362 | — | 2 | 4 | no | — |
| 149 | #359 | — | 2 | 4 | no | — |
| 150 | #358 | — | 2 | 4 | no | — |
| 151 | #358 | — | 2 | 2 | no | — |
| 152 | #357 | — | 2 | 2 | no | — |
| 153 | C22⋊C16 | — | 2 | 1 | no | — |
| 154 | C22⋊C16 | — | 2 | 1 | no | — |
| 155 | C22⋊C16 | — | 2 | 1 | no | — |
| 156 | C22⋊C16 | — | 2 | 1 | no | — |
| 157 | #343 | — | 2 | 2 | no | — |
| 158 | #343 | — | 2 | 2 | no | — |
| 159 | C2×#41 | — | 2 | 1 | no | — |
| 160 | C22×Q8 | — | 2 | 1 | no | — |
| 161 | C23×C4 | — | 4 | 1 | no | — |
| 162 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 163 | #137 | — | 4 | 2 | no | — |
| 164 | #137 | — | 4 | 2 | no | — |
| 165 | #135 | — | 2 | 2 | no | — |
| 166 | #135 | — | 2 | 2 | no | — |
| 167 | C2×Q16 | — | 2 | 2 | no | — |
| 168 | C2×Q16 | — | 2 | 1 | no | — |
| 169 | C2×Q16 | — | 2 | 2 | no | — |
| 170 | C2×Q16 | — | 2 | 2 | no | — |
| 171 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 172 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 173 | C2×SD16 | — | 2 | 2 | no | — |
| 174 | C2×SD16 | — | 2 | 1 | no | — |
| 175 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 176 | C2×M16 | — | 2 | 1 | no | — |
| 177 | C2×M16 | — | 2 | 4 | no | — |
| 178 | C2×M16 | — | 2 | 2 | no | — |
| 179 | C2×M16 | — | 2 | 2 | no | — |
| 180 | C2×M16 | — | 4 | 1 | no | — |
| 181 | C2×M16 | — | 4 | 2 | no | — |
| 182 | C2×M16 | — | 4 | 1 | no | — |
| 183 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 184 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 185 | C22×C8 | — | 2 | 2 | no | — |
| 186 | C22×C8 | — | 2 | 2 | no | — |
| 187 | C22×C8 | — | 2 | 1 | no | — |
| 188 | C22×C8 | — | 4 | 1 | no | — |
| 189 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 190 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 191 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 192 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 193 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 194 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 195 | C4⋊Q8 | — | 2 | 2 | no | — |
| 196 | C4⋊Q8 | — | 2 | 2 | no | — |
| 197 | C4⋊Q8 | — | 4 | 1 | no | — |
| 198 | #126 | — | 4 | 2 | no | — |
| 199 | #126 | — | 4 | 2 | no | — |
| 200 | #126 | — | 4 | 2 | no | — |
| 201 | #126 | — | 4 | 2 | no | — |
| 202 | #125 | — | 2 | 2 | no | — |
| 203 | #125 | — | 4 | 2 | no | — |
| 204 | #125 | — | 4 | 2 | no | — |
| 205 | #125 | — | 4 | 2 | no | — |
| 206 | #125 | — | 4 | 1 | no | — |
| 207 | #124 | — | 2 | 2 | no | — |
| 208 | #124 | — | 4 | 2 | no | — |
| 209 | #123 | — | 4 | 2 | no | — |
| 210 | #123 | — | 4 | 2 | no | — |
| 211 | #123 | — | 4 | 2 | no | — |
| 212 | #123 | — | 4 | 1 | no | — |
| 213 | C22⋊Q8 | — | 2 | 2 | no | — |
| 214 | C22⋊Q8 | — | 2 | 2 | no | — |
| 215 | C22⋊Q8 | — | 2 | 1 | no | — |
| 216 | C22⋊Q8 | — | 2 | 1 | no | — |
| 217 | C22⋊Q8 | — | 4 | 2 | no | — |
| 218 | C22⋊Q8 | — | 4 | 2 | no | — |
| 219 | C22⋊Q8 | — | 4 | 2 | no | — |
| 220 | C22⋊Q8 | — | 4 | 1 | no | — |
| 221 | C22⋊Q8 | — | 4 | 1 | no | — |
| 222 | C22⋊Q8 | — | 4 | 1 | no | — |
| 223 | C22⋊Q8 | — | 4 | 1 | no | — |
| 224 | #121 | — | 2 | 1 | no | — |
| 225 | #121 | — | 2 | 1 | no | — |
| 226 | #120 | — | 4 | 1 | no | — |
| 227 | C4×Q8 | — | 2 | 2 | no | — |
| 228 | C4×Q8 | — | 2 | 2 | no | — |
| 229 | C4×Q8 | — | 2 | 2 | no | — |
| 230 | C4×Q8 | — | 4 | 2 | no | — |
| 231 | C4×Q8 | — | 4 | 1 | no | — |
| 232 | C4×Q8 | — | 4 | 1 | no | — |
| 233 | C4×D4 | — | 2 | 2 | no | — |
| 234 | C4×D4 | — | 4 | 2 | no | — |
| 235 | C4×D4 | — | 4 | 2 | no | — |
| 236 | C4×D4 | — | 4 | 1 | no | — |
| 237 | C4×D4 | — | 4 | 1 | no | — |
| 238 | #117 | — | 2 | 1 | no | — |
| 239 | #117 | — | 2 | 1 | no | — |
| 240 | #117 | — | 2 | 1 | no | — |
| 241 | #117 | — | 4 | 2 | no | — |
| 242 | #117 | — | 4 | 2 | no | — |
| 243 | #117 | — | 4 | 2 | no | — |
| 244 | #117 | — | 4 | 1 | no | — |
| 245 | #117 | D8 | 1 | 1 | yes | characteristic |
| 246 | #117 | D8 | 1 | 1 | yes | characteristic |
| 247 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 248 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 249 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 250 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 251 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 252 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 253 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 254 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 255 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 256 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 257 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 258 | C2×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 259 | C2×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 260 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 261 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 262 | C2×C42 | — | 2 | 1 | no | — |
| 263 | M32 | — | 2 | 2 | no | — |
| 264 | M32 | — | 2 | 2 | no | — |
| 265 | C2×C16 | — | 2 | 2 | no | — |
| 266 | C2×C16 | — | 2 | 2 | no | — |
| 267 | C2×C16 | — | 2 | 2 | no | — |
| 268 | C2×C16 | — | 2 | 2 | no | — |
| 269 | #108 | — | 2 | 4 | no | — |
| 270 | #108 | — | 2 | 2 | no | — |
| 271 | #108 | #41 | 1 | 4 | yes | — |
| 272 | #107 | — | 2 | 2 | no | — |
| 273 | #107 | — | 2 | 1 | no | — |
| 274 | #107 | — | 2 | 2 | no | — |
| 275 | #107 | — | 2 | 2 | no | — |
| 276 | #107 | — | 4 | 1 | no | — |
| 277 | #107 | — | 4 | 1 | no | — |
| 278 | #107 | — | 4 | 2 | no | — |
| 279 | #107 | — | 4 | 2 | no | — |
| 280 | #107 | C2×D4 | 1 | 2 | yes | — |
| 281 | #107 | C2×D4 | 1 | 1 | yes | characteristic |
| 282 | #107 | C2×D4 | 1 | 1 | yes | characteristic |
| 283 | #106 | — | 2 | 2 | no | — |
| 284 | #106 | — | 2 | 1 | no | — |
| 285 | #106 | — | 4 | 2 | no | — |
| 286 | #106 | — | 4 | 2 | no | — |
| 287 | #103 | — | 2 | 2 | no | — |
| 288 | #103 | — | 2 | 2 | no | — |
| 289 | #103 | — | 2 | 2 | no | — |
| 290 | #103 | — | 2 | 2 | no | — |
| 291 | #103 | — | 2 | 2 | no | — |
| 292 | #103 | — | 2 | 1 | no | — |
| 293 | #103 | — | 2 | 1 | no | — |
| 294 | #102 | — | 2 | 2 | no | — |
| 295 | #102 | — | 2 | 1 | no | — |
| 296 | #102 | — | 2 | 1 | no | — |
| 297 | #97 | — | 2 | 1 | no | — |
| 298 | #97 | — | 2 | 1 | no | — |
| 299 | #97 | — | 2 | 1 | no | — |
| 300 | #97 | — | 2 | 1 | no | — |
| 301 | C4×C8 | — | 2 | 1 | no | — |
| 302 | C4×C8 | — | 2 | 1 | no | — |
| 303 | C4×C8 | — | 2 | 1 | no | — |
| 304 | C4×C8 | — | 2 | 1 | no | — |
| 305 | C24 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 306 | #41 | — | 4 | 1 | no | — |
| 307 | #41 | — | 4 | 1 | no | — |
| 308 | #41 | — | 4 | 1 | no | — |
| 309 | #41 | — | 4 | 1 | no | — |
| 310 | C2×Q8 | — | 2 | 2 | no | — |
| 311 | C2×Q8 | — | 2 | 1 | no | — |
| 312 | C2×Q8 | — | 4 | 2 | no | — |
| 313 | C2×Q8 | — | 4 | 1 | no | — |
| 314 | C2×Q8 | — | 4 | 1 | no | — |
| 315 | C2×Q8 | — | 4 | 1 | no | — |
| 316 | C2×Q8 | — | 4 | 1 | no | — |
| 317 | C2×Q8 | — | 4 | 1 | no | — |
| 318 | C2×D4 | — | 2 | 1 | no | — |
| 319 | C2×D4 | — | 4 | 1 | no | — |
| 320 | C2×D4 | — | 4 | 1 | no | — |
| 321 | C2×D4 | — | 4 | 1 | no | — |
| 322 | C22×C4 | — | 2 | 1 | no | — |
| 323 | C22×C4 | — | 2 | 1 | no | — |
| 324 | C22×C4 | — | 2 | 1 | no | — |
| 325 | C22×C4 | — | 2 | 1 | no | — |
| 326 | C22×C4 | — | 2 | 1 | no | — |
| 327 | C22×C4 | — | 2 | 1 | no | — |
| 328 | C22×C4 | — | 4 | 2 | no | — |
| 329 | C22×C4 | — | 4 | 1 | no | — |
| 330 | C22×C4 | — | 4 | 1 | no | — |
| 331 | C22×C4 | — | 4 | 1 | no | — |
| 332 | C22×C4 | — | 4 | 2 | no | — |
| 333 | C22×C4 | — | 8 | 1 | no | — |
| 334 | C22×C4 | — | 8 | 1 | no | — |
| 335 | C22×C4 | — | 16 | 2 | no | — |
| 336 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 337 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 338 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 339 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 340 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 341 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 342 | Q16 | — | 2 | 4 | no | — |
| 343 | Q16 | — | 2 | 2 | no | — |
| 344 | Q16 | — | 4 | 4 | no | — |
| 345 | Q16 | — | 4 | 2 | no | — |
| 346 | SD16 | — | 4 | 4 | no | — |
| 347 | SD16 | — | 4 | 2 | no | — |
| 348 | D8 | — | 2 | 2 | no | — |
| 349 | M16 | — | 2 | 1 | no | — |
| 350 | M16 | — | 2 | 4 | no | — |
| 351 | M16 | — | 2 | 1 | no | — |
| 352 | M16 | — | 2 | 1 | no | — |
| 353 | M16 | — | 4 | 2 | no | — |
| 354 | M16 | — | 4 | 4 | no | — |
| 355 | M16 | — | 4 | 2 | no | — |
| 356 | M16 | #137 | 1 | 2 | yes | — |
| 357 | C2×C8 | — | 2 | 1 | no | — |
| 358 | C2×C8 | — | 2 | 1 | no | — |
| 359 | C2×C8 | — | 2 | 1 | no | — |
| 360 | C2×C8 | — | 2 | 2 | no | — |
| 361 | C2×C8 | — | 2 | 2 | no | — |
| 362 | C2×C8 | — | 2 | 2 | no | — |
| 363 | C2×C8 | — | 2 | 2 | no | — |
| 364 | C2×C8 | — | 2 | 1 | no | — |
| 365 | C2×C8 | — | 4 | 2 | no | — |
| 366 | C2×C8 | — | 4 | 4 | no | — |
| 367 | C2×C8 | — | 4 | 2 | no | — |
| 368 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 369 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 370 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 371 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 372 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 373 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 374 | C4⋊C4 | — | 2 | 1 | no | — |
| 375 | C4⋊C4 | — | 2 | 1 | no | — |
| 376 | C4⋊C4 | — | 2 | 1 | no | — |
| 377 | C4⋊C4 | — | 2 | 2 | no | — |
| 378 | C4⋊C4 | — | 2 | 1 | no | — |
| 379 | C4⋊C4 | — | 4 | 2 | no | — |
| 380 | C4⋊C4 | — | 4 | 2 | no | — |
| 381 | C4⋊C4 | — | 4 | 2 | no | — |
| 382 | C4⋊C4 | — | 4 | 2 | no | — |
| 383 | C4⋊C4 | — | 4 | 2 | no | — |
| 384 | C4⋊C4 | — | 4 | 2 | no | — |
| 385 | C4⋊C4 | — | 4 | 2 | no | — |
| 386 | C4⋊C4 | — | 4 | 2 | no | — |
| 387 | C4⋊C4 | — | 4 | 1 | no | — |
| 388 | C4⋊C4 | — | 4 | 2 | no | — |
| 389 | C4⋊C4 | — | 4 | 2 | no | — |
| 390 | C4⋊C4 | — | 4 | 1 | no | — |
| 391 | C4⋊C4 | — | 4 | 1 | no | — |
| 392 | C4⋊C4 | — | 4 | 2 | no | — |
| 393 | C4⋊C4 | — | 4 | 2 | no | — |
| 394 | C4⋊C4 | — | 8 | 1 | no | — |
| 395 | C4⋊C4 | — | 8 | 1 | no | — |
| 396 | C4⋊C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 397 | C4⋊C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 398 | C4⋊C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 399 | C4⋊C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 400 | C22⋊C4 | — | 2 | 1 | no | — |
| 401 | C22⋊C4 | — | 2 | 1 | no | — |
| 402 | C22⋊C4 | — | 2 | 1 | no | — |
| 403 | C22⋊C4 | — | 2 | 1 | no | — |
| 404 | C22⋊C4 | — | 4 | 2 | no | — |
| 405 | C22⋊C4 | — | 4 | 2 | no | — |
| 406 | C22⋊C4 | — | 4 | 2 | no | — |
| 407 | C22⋊C4 | — | 4 | 1 | no | — |
| 408 | C22⋊C4 | — | 4 | 2 | no | — |
| 409 | C22⋊C4 | — | 4 | 2 | no | — |
| 410 | C22⋊C4 | — | 4 | 1 | no | — |
| 411 | C22⋊C4 | — | 4 | 1 | no | — |
| 412 | C22⋊C4 | — | 4 | 1 | no | — |
| 413 | C22⋊C4 | — | 4 | 1 | no | — |
| 414 | C42 | — | 2 | 1 | no | — |
| 415 | C42 | — | 2 | 1 | no | — |
| 416 | C42 | — | 2 | 1 | no | — |
| 417 | C42 | — | 2 | 1 | no | — |
| 418 | C42 | — | 4 | 2 | no | — |
| 419 | C42 | — | 4 | 2 | no | — |
| 420 | C42 | — | 4 | 2 | no | — |
| 421 | C42 | — | 4 | 1 | no | — |
| 422 | C16 | — | 4 | 2 | no | — |
| 423 | C16 | — | 4 | 2 | no | — |
| 424 | C23 | — | 2 | 1 | no | — |
| 425 | C23 | — | 2 | 2 | no | — |
| 426 | C23 | — | 2 | 1 | no | — |
| 427 | C23 | — | 4 | 1 | no | — |
| 428 | C23 | — | 4 | 1 | no | — |
| 429 | C23 | #447 | 1 | 1 | yes | characteristic |
| 430 | C23 | #447 | 1 | 1 | yes | characteristic |
| 431 | C23 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 432 | Q8 | — | 4 | 1 | no | — |
| 433 | Q8 | — | 4 | 2 | no | — |
| 434 | Q8 | — | 4 | 2 | no | — |
| 435 | Q8 | — | 4 | 1 | no | — |
| 436 | Q8 | — | 8 | 1 | no | — |
| 437 | Q8 | — | 8 | 1 | no | — |
| 438 | Q8 | — | 16 | 2 | no | — |
| 439 | D4 | — | 4 | 1 | no | — |
| 440 | D4 | — | 4 | 1 | no | — |
| 441 | D4 | — | 8 | 1 | no | — |
| 442 | D4 | — | 8 | 2 | no | — |
| 443 | D4 | — | 8 | 1 | no | — |
| 444 | D4 | — | 16 | 1 | no | — |
| 445 | C2×C4 | — | 2 | 1 | no | — |
| 446 | C2×C4 | — | 2 | 1 | no | — |
| 447 | C2×C4 | — | 2 | 1 | no | — |
| 448 | C2×C4 | — | 2 | 1 | no | — |
| 449 | C2×C4 | — | 2 | 1 | no | — |
| 450 | C2×C4 | — | 2 | 1 | no | — |
| 451 | C2×C4 | — | 2 | 1 | no | — |
| 452 | C2×C4 | — | 2 | 1 | no | — |
| 453 | C2×C4 | — | 2 | 1 | no | — |
| 454 | C2×C4 | — | 2 | 1 | no | — |
| 455 | C2×C4 | — | 2 | 1 | no | — |
| 456 | C2×C4 | — | 2 | 1 | no | — |
| 457 | C2×C4 | — | 2 | 1 | no | — |
| 458 | C2×C4 | — | 2 | 1 | no | — |
| 459 | C2×C4 | — | 4 | 2 | no | — |
| 460 | C2×C4 | — | 4 | 1 | no | — |
| 461 | C2×C4 | — | 4 | 1 | no | — |
| 462 | C2×C4 | — | 4 | 2 | no | — |
| 463 | C2×C4 | — | 4 | 1 | no | — |
| 464 | C2×C4 | — | 4 | 2 | no | — |
| 465 | C2×C4 | — | 8 | 1 | no | — |
| 466 | C2×C4 | — | 8 | 1 | no | — |
| 467 | C2×C4 | — | 8 | 1 | no | — |
| 468 | C2×C4 | — | 8 | 1 | no | — |
| 469 | C2×C4 | — | 8 | 1 | no | — |
| 470 | C2×C4 | — | 8 | 2 | no | — |
| 471 | C2×C4 | — | 16 | 2 | no | — |
| 472 | C2×C4 | — | 16 | 1 | no | — |
| 473 | C2×C4 | — | 16 | 2 | no | — |
| 474 | C2×C4 | — | 16 | 2 | no | — |
| 475 | C2×C4 | — | 16 | 2 | no | — |
| 476 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 477 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 478 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 479 | C2×C4 | #525 | 1 | 1 | yes | characteristic |
| 480 | C2×C4 | #536 | 1 | 1 | yes | characteristic |
| 481 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 482 | C2×C4 | C2×#137 | 1 | 1 | yes | characteristic |
| 483 | C2×C4 | #575 | 1 | 1 | yes | characteristic |
| 484 | C8 | — | 2 | 1 | no | — |
| 485 | C8 | — | 2 | 1 | no | — |
| 486 | C8 | — | 2 | 2 | no | — |
| 487 | C8 | — | 4 | 4 | no | — |
| 488 | C8 | — | 4 | 1 | no | — |
| 489 | C8 | #547 | 1 | 2 | yes | — |
| 490 | C8 | #547 | 1 | 2 | yes | — |
| 491 | C22 | — | 2 | 1 | no | — |
| 492 | C22 | — | 2 | 1 | no | — |
| 493 | C22 | — | 2 | 1 | no | — |
| 494 | C22 | — | 2 | 1 | no | — |
| 495 | C22 | — | 2 | 1 | no | — |
| 496 | C22 | — | 2 | 1 | no | — |
| 497 | C22 | — | 2 | 1 | no | — |
| 498 | C22 | — | 4 | 2 | no | — |
| 499 | C22 | — | 4 | 2 | no | — |
| 500 | C22 | — | 8 | 1 | no | — |
| 501 | C22 | — | 8 | 1 | no | — |
| 502 | C22 | — | 8 | 1 | no | — |
| 503 | C22 | #2195 | 1 | 2 | yes | — |
| 504 | C22 | C2×#447 | 1 | 1 | yes | characteristic |
| 505 | C22 | #3112 | 1 | 1 | yes | characteristic |
| 506 | C22 | #3234 | 1 | 1 | yes | characteristic |
| 507 | C4 | — | 2 | 1 | no | — |
| 508 | C4 | — | 2 | 1 | no | — |
| 509 | C4 | — | 4 | 1 | no | — |
| 510 | C4 | — | 4 | 1 | no | — |
| 511 | C4 | — | 4 | 1 | no | — |
| 512 | C4 | — | 4 | 1 | no | — |
| 513 | C4 | — | 8 | 1 | no | — |
| 514 | C4 | — | 8 | 2 | no | — |
| 515 | C4 | — | 8 | 1 | no | — |
| 516 | C4 | — | 8 | 1 | no | — |
| 517 | C4 | — | 8 | 2 | no | — |
| 518 | C4 | — | 16 | 2 | no | — |
| 519 | C4 | #2406 | 1 | 1 | yes | characteristic |
| 520 | C4 | #3007 | 1 | 1 | yes | characteristic |
| 521 | C4 | #3100 | 1 | 1 | yes | characteristic |
| 522 | C4 | #3226 | 1 | 1 | yes | characteristic |
| 523 | C2 | — | 2 | 1 | no | — |
| 524 | C2 | — | 2 | 2 | no | — |
| 525 | C2 | — | 2 | 1 | no | — |
| 526 | C2 | — | 4 | 1 | no | — |
| 527 | C2 | — | 16 | 1 | no | — |
| 528 | C2 | #24060 | 1 | 1 | yes | characteristic |
| 529 | C2 | C2×#2195 | 1 | 1 | yes | characteristic |
| 530 | C2 | #33541 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#2236 | C2 | semidirect |
| 4. C2×#1952 | C2 | semidirect |
| 5. #13455 | C2 | semidirect |
| 6. #13412 | C2 | semidirect |
| 7. #13003 | C2 | semidirect |
| 9. #12945 | C2 | semidirect |
| 10. #12941 | C2 | semidirect |
| 11. #12848 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60708 | 2 |
| #33646 | 2 |
| C2×#2236 | 2 |
| C2×#1952 | 2 |
| #13455 | 2 |
| #13412 | 2 |
| #13003 | 2 |
| #12990 | 2 |
| #12945 | 2 |
| #12941 | 2 |
| #12848 | 2 |