Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
784 conjugacy classes of subgroups.
The subgroup lattice has 646 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #33722 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2212 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#2192 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #33526 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #25804 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #25441 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #13445 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13412 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #13336 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #13326 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #13314 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #13308 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | #13144 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | #13122 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | #13011 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C2×#576 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#510 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×#509 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#509 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 21 | C22×SD32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 23 | #3156 | C22 | 1 | 1 | yes | characteristic |
| 24 | #3155 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | C2×#495 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#473 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#472 | C22 | 1 | 1 | yes | characteristic |
| 28 | #3062 | — | 2 | 1 | no | — |
| 29 | #3061 | C22 | 1 | 1 | yes | characteristic |
| 30 | C2×#455 | — | 2 | 1 | no | — |
| 31 | #3009 | — | 2 | 1 | no | — |
| 32 | C2×#443 | C22 | 1 | 1 | yes | characteristic, complemented |
| 33 | #2251 | — | 2 | 1 | no | — |
| 34 | #2250 | — | 2 | 1 | no | — |
| 35 | #2249 | — | 2 | 1 | no | — |
| 36 | #2248 | — | 2 | 1 | no | — |
| 37 | #2212 | — | 2 | 4 | no | — |
| 38 | #2192 | — | 2 | 4 | no | — |
| 39 | #2189 | — | 2 | 1 | no | — |
| 40 | #2187 | — | 2 | 1 | no | — |
| 41 | #2186 | — | 2 | 1 | no | — |
| 42 | #2184 | — | 2 | 1 | no | — |
| 43 | C2×#364 | C22 | 1 | 1 | yes | characteristic, complemented |
| 44 | C2×#362 | C22 | 1 | 1 | yes | characteristic, complemented |
| 45 | C2×#361 | C22 | 1 | 1 | yes | characteristic |
| 46 | C2×#361 | C22 | 1 | 1 | yes | characteristic, complemented |
| 47 | C2×#359 | C22 | 1 | 1 | yes | characteristic |
| 48 | C2×#359 | C22 | 1 | 1 | yes | characteristic, complemented |
| 49 | #2139 | C22 | 1 | 1 | yes | characteristic |
| 50 | C2×#358 | C22 | 1 | 1 | yes | characteristic, complemented |
| 51 | C2×#357 | C22 | 1 | 1 | yes | characteristic |
| 52 | C2×#350 | C22 | 1 | 1 | yes | characteristic |
| 53 | #2113 | C22 | 1 | 1 | yes | characteristic |
| 54 | #2108 | C22 | 1 | 1 | yes | characteristic, complemented |
| 55 | #2037 | — | 2 | 1 | no | — |
| 56 | #2019 | — | 2 | 1 | no | — |
| 57 | #2006 | — | 2 | 1 | no | — |
| 58 | #2003 | — | 2 | 1 | no | — |
| 59 | #1978 | C22 | 1 | 1 | yes | characteristic |
| 60 | #1974 | C22 | 1 | 1 | yes | characteristic |
| 61 | #1970 | C22 | 1 | 1 | yes | characteristic, complemented |
| 62 | #1952 | C22 | 1 | 1 | yes | characteristic, complemented |
| 63 | #1907 | — | 2 | 1 | no | — |
| 64 | #1764 | C22 | 1 | 1 | yes | characteristic |
| 65 | #1387 | C22 | 1 | 1 | yes | characteristic, complemented |
| 66 | #1381 | C22 | 1 | 1 | yes | characteristic, complemented |
| 67 | #1379 | C22 | 1 | 1 | yes | characteristic, complemented |
| 68 | #1374 | C22 | 1 | 1 | yes | characteristic, complemented |
| 69 | C2×H(5,𝔽2) | — | 2 | 1 | no | — |
| 70 | #576 | — | 2 | 2 | no | — |
| 71 | #576 | — | 4 | 1 | no | — |
| 72 | #576 | — | 4 | 1 | no | — |
| 73 | #576 | — | 4 | 1 | no | — |
| 74 | C2×#137 | — | 2 | 1 | no | — |
| 75 | C2×#137 | — | 2 | 1 | no | — |
| 76 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 77 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 78 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 79 | C2×AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 80 | C2×#135 | — | 2 | 1 | no | — |
| 81 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 82 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 83 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 84 | C22×SD16 | — | 2 | 1 | no | — |
| 85 | C22×D8 | — | 2 | 1 | no | — |
| 86 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 87 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 88 | C2×#131 | — | 2 | 1 | no | — |
| 89 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 90 | #537 | — | 2 | 1 | no | — |
| 91 | #536 | — | 2 | 1 | no | — |
| 92 | #535 | — | 2 | 1 | no | — |
| 93 | C2×#124 | — | 2 | 1 | no | — |
| 94 | #510 | — | 2 | 2 | no | — |
| 95 | #510 | — | 4 | 1 | no | — |
| 96 | #509 | — | 2 | 2 | no | — |
| 97 | #509 | — | 2 | 2 | no | — |
| 98 | #509 | — | 4 | 1 | no | — |
| 99 | #509 | — | 4 | 1 | no | — |
| 100 | #508 | — | 4 | 1 | no | — |
| 101 | #508 | — | 4 | 1 | no | — |
| 102 | C2×Q32 | — | 2 | 1 | no | — |
| 103 | C2×Q32 | D4 | 1 | 1 | yes | characteristic |
| 104 | C2×SD32 | — | 2 | 1 | no | — |
| 105 | C2×SD32 | — | 2 | 1 | no | — |
| 106 | C2×SD32 | — | 2 | 1 | no | — |
| 107 | C2×SD32 | — | 2 | 1 | no | — |
| 108 | C2×SD32 | — | 2 | 1 | no | — |
| 109 | C2×SD32 | — | 4 | 1 | no | — |
| 110 | C2×SD32 | — | 4 | 1 | no | — |
| 111 | C2×SD32 | D4 | 1 | 1 | yes | characteristic |
| 112 | C2×SD32 | D4 | 1 | 1 | yes | characteristic |
| 113 | C2×D16 | — | 2 | 2 | no | — |
| 114 | C2×D16 | — | 2 | 1 | no | — |
| 115 | C2×D16 | — | 2 | 1 | no | — |
| 116 | C2×D16 | — | 4 | 1 | no | — |
| 117 | C2×D16 | — | 4 | 1 | no | — |
| 118 | C2×D16 | D4 | 1 | 1 | yes | characteristic |
| 119 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 120 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 121 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 122 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 123 | #497 | — | 4 | 1 | no | — |
| 124 | #496 | — | 4 | 1 | no | — |
| 125 | #495 | — | 2 | 1 | no | — |
| 126 | #495 | — | 2 | 2 | no | — |
| 127 | #495 | — | 2 | 1 | no | — |
| 128 | #495 | — | 4 | 1 | no | — |
| 129 | #494 | — | 2 | 1 | no | — |
| 130 | #493 | — | 2 | 1 | no | — |
| 131 | #473 | — | 4 | 1 | no | — |
| 132 | #473 | — | 4 | 1 | no | — |
| 133 | #472 | — | 2 | 2 | no | — |
| 134 | #472 | — | 4 | 1 | no | — |
| 135 | #469 | — | 4 | 1 | no | — |
| 136 | #468 | — | 4 | 1 | no | — |
| 137 | #468 | — | 4 | 1 | no | — |
| 138 | #467 | — | 2 | 1 | no | — |
| 139 | #466 | — | 2 | 1 | no | — |
| 140 | #466 | — | 4 | 1 | no | — |
| 141 | #465 | — | 4 | 1 | no | — |
| 142 | #455 | — | 4 | 4 | no | — |
| 143 | #452 | — | 4 | 1 | no | — |
| 144 | #452 | — | 4 | 1 | no | — |
| 145 | #450 | — | 4 | 1 | no | — |
| 146 | #450 | — | 4 | 1 | no | — |
| 147 | #443 | — | 2 | 2 | no | — |
| 148 | #443 | — | 4 | 1 | no | — |
| 149 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 150 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 151 | #428 | — | 2 | 1 | no | — |
| 152 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 153 | C2×#104 | — | 2 | 1 | no | — |
| 154 | C2×#104 | — | 2 | 1 | no | — |
| 155 | C2×#104 | D4 | 1 | 1 | yes | characteristic |
| 156 | #418 | — | 2 | 1 | no | — |
| 157 | #417 | — | 2 | 1 | no | — |
| 158 | #416 | D4 | 1 | 1 | yes | characteristic |
| 159 | C2×#102 | — | 2 | 1 | no | — |
| 160 | C2×#101 | — | 2 | 1 | no | — |
| 161 | C2×#100 | — | 2 | 1 | no | — |
| 162 | C2×#100 | — | 2 | 1 | no | — |
| 163 | #408 | — | 2 | 1 | no | — |
| 164 | #405 | C23 | 1 | 1 | yes | characteristic |
| 165 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 166 | #390 | — | 2 | 1 | no | — |
| 167 | #364 | — | 2 | 2 | no | — |
| 168 | #362 | — | 2 | 2 | no | — |
| 169 | #361 | — | 4 | 1 | no | — |
| 170 | #361 | D4 | 1 | 4 | yes | complemented |
| 171 | #359 | — | 2 | 2 | no | — |
| 172 | #359 | — | 4 | 1 | no | — |
| 173 | #358 | — | 2 | 1 | no | — |
| 174 | #358 | — | 4 | 1 | no | — |
| 175 | #357 | — | 2 | 1 | no | — |
| 176 | #357 | — | 2 | 1 | no | — |
| 177 | #357 | — | 2 | 1 | no | — |
| 178 | #350 | — | 2 | 2 | no | — |
| 179 | C22⋊C16 | — | 2 | 1 | no | — |
| 180 | C22⋊C16 | — | 2 | 1 | no | — |
| 181 | #346 | — | 2 | 1 | no | — |
| 182 | C4×C16 | — | 2 | 1 | no | — |
| 183 | #343 | — | 2 | 1 | no | — |
| 184 | #341 | D4 | 1 | 1 | yes | characteristic |
| 185 | #339 | D4 | 1 | 1 | yes | characteristic |
| 186 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 187 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 188 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 189 | H(5,𝔽2) | — | 8 | 1 | no | — |
| 190 | C2×#41 | — | 2 | 1 | no | — |
| 191 | C2×#41 | — | 2 | 1 | no | — |
| 192 | C2×#41 | — | 4 | 1 | no | — |
| 193 | C2×#41 | — | 4 | 1 | no | — |
| 194 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 195 | C22×Q8 | — | 2 | 1 | no | — |
| 196 | C22×D4 | — | 2 | 1 | no | — |
| 197 | C22×D4 | — | 2 | 1 | no | — |
| 198 | C22×D4 | — | 4 | 1 | no | — |
| 199 | C22×D4 | — | 4 | 1 | no | — |
| 200 | C22×D4 | — | 8 | 1 | no | — |
| 201 | #137 | — | 4 | 2 | no | — |
| 202 | #137 | — | 4 | 2 | no | — |
| 203 | #137 | — | 4 | 2 | no | — |
| 204 | #137 | — | 8 | 1 | no | — |
| 205 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 206 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 207 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 208 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 209 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 210 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 211 | AGL(1,ℤ/8) | — | 8 | 1 | no | — |
| 212 | AGL(1,ℤ/8) | — | 8 | 1 | no | — |
| 213 | #135 | — | 2 | 1 | no | — |
| 214 | #135 | — | 2 | 1 | no | — |
| 215 | #135 | — | 2 | 1 | no | — |
| 216 | #135 | — | 2 | 1 | no | — |
| 217 | #135 | — | 2 | 1 | no | — |
| 218 | #135 | — | 2 | 1 | no | — |
| 219 | #135 | — | 2 | 1 | no | — |
| 220 | #135 | — | 2 | 1 | no | — |
| 221 | #135 | — | 4 | 2 | no | — |
| 222 | #135 | — | 4 | 2 | no | — |
| 223 | C2×Q16 | — | 2 | 1 | no | — |
| 224 | C2×Q16 | — | 2 | 1 | no | — |
| 225 | C2×Q16 | — | 2 | 1 | no | — |
| 226 | C2×Q16 | — | 2 | 1 | no | — |
| 227 | C2×Q16 | — | 2 | 1 | no | — |
| 228 | C2×Q16 | — | 2 | 1 | no | — |
| 229 | C2×Q16 | — | 4 | 1 | no | — |
| 230 | C2×Q16 | — | 4 | 1 | no | — |
| 231 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 232 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 233 | C2×SD16 | — | 2 | 1 | no | — |
| 234 | C2×SD16 | — | 2 | 1 | no | — |
| 235 | C2×SD16 | — | 4 | 1 | no | — |
| 236 | C2×SD16 | — | 4 | 2 | no | — |
| 237 | C2×SD16 | — | 4 | 1 | no | — |
| 238 | C2×SD16 | — | 4 | 1 | no | — |
| 239 | C2×SD16 | — | 4 | 1 | no | — |
| 240 | C2×SD16 | — | 4 | 1 | no | — |
| 241 | C2×SD16 | — | 4 | 1 | no | — |
| 242 | C2×SD16 | — | 4 | 1 | no | — |
| 243 | C2×SD16 | — | 8 | 1 | no | — |
| 244 | C2×D8 | — | 2 | 1 | no | — |
| 245 | C2×D8 | — | 2 | 1 | no | — |
| 246 | C2×D8 | — | 2 | 1 | no | — |
| 247 | C2×D8 | — | 2 | 1 | no | — |
| 248 | C2×D8 | — | 2 | 1 | no | — |
| 249 | C2×D8 | — | 2 | 1 | no | — |
| 250 | C2×D8 | — | 2 | 1 | no | — |
| 251 | C2×D8 | — | 2 | 1 | no | — |
| 252 | C2×D8 | — | 2 | 1 | no | — |
| 253 | C2×D8 | — | 2 | 1 | no | — |
| 254 | C2×D8 | — | 2 | 1 | no | — |
| 255 | C2×D8 | — | 4 | 1 | no | — |
| 256 | C2×D8 | — | 4 | 1 | no | — |
| 257 | C2×D8 | — | 4 | 2 | no | — |
| 258 | C2×D8 | — | 4 | 2 | no | — |
| 259 | C2×D8 | — | 4 | 1 | no | — |
| 260 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 261 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 262 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 263 | #131 | — | 2 | 1 | no | — |
| 264 | #131 | — | 2 | 1 | no | — |
| 265 | #131 | — | 2 | 1 | no | — |
| 266 | #131 | — | 2 | 1 | no | — |
| 267 | #131 | — | 4 | 2 | no | — |
| 268 | #131 | — | 8 | 1 | no | — |
| 269 | C2×M16 | — | 2 | 1 | no | — |
| 270 | C2×M16 | — | 2 | 1 | no | — |
| 271 | C2×M16 | — | 2 | 1 | no | — |
| 272 | C2×M16 | — | 4 | 1 | no | — |
| 273 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 274 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 275 | C22×C8 | — | 2 | 1 | no | — |
| 276 | C22×C8 | — | 4 | 1 | no | — |
| 277 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 278 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 279 | C4⋊Q8 | — | 4 | 1 | no | — |
| 280 | #127 | — | 4 | 1 | no | — |
| 281 | #124 | — | 4 | 2 | no | — |
| 282 | #124 | — | 4 | 1 | no | — |
| 283 | #124 | — | 4 | 1 | no | — |
| 284 | #124 | — | 4 | 2 | no | — |
| 285 | #123 | — | 8 | 1 | no | — |
| 286 | C22⋊Q8 | — | 4 | 1 | no | — |
| 287 | C22⋊Q8 | — | 4 | 1 | no | — |
| 288 | C22⋊Q8 | — | 4 | 1 | no | — |
| 289 | C22⋊Q8 | — | 4 | 1 | no | — |
| 290 | #121 | — | 4 | 1 | no | — |
| 291 | #121 | — | 4 | 1 | no | — |
| 292 | #121 | — | 4 | 1 | no | — |
| 293 | #121 | — | 4 | 1 | no | — |
| 294 | #121 | — | 8 | 1 | no | — |
| 295 | #120 | — | 8 | 1 | no | — |
| 296 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 297 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 298 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 299 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 300 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 301 | Q32 | — | 4 | 1 | no | — |
| 302 | Q32 | — | 4 | 2 | no | — |
| 303 | SD32 | — | 4 | 1 | no | — |
| 304 | SD32 | — | 4 | 1 | no | — |
| 305 | SD32 | — | 4 | 2 | no | — |
| 306 | SD32 | — | 4 | 2 | no | — |
| 307 | SD32 | — | 4 | 2 | no | — |
| 308 | SD32 | — | 4 | 2 | no | — |
| 309 | SD32 | — | 4 | 2 | no | — |
| 310 | D16 | — | 4 | 1 | no | — |
| 311 | D16 | — | 4 | 2 | no | — |
| 312 | D16 | — | 4 | 2 | no | — |
| 313 | D16 | — | 4 | 4 | no | — |
| 314 | M32 | — | 2 | 1 | no | — |
| 315 | M32 | — | 2 | 1 | no | — |
| 316 | M32 | — | 2 | 1 | no | — |
| 317 | M32 | C2×D4 | 1 | 2 | yes | — |
| 318 | C2×C16 | — | 2 | 1 | no | — |
| 319 | C2×C16 | — | 2 | 1 | no | — |
| 320 | C2×C16 | — | 2 | 1 | no | — |
| 321 | C2×C16 | — | 2 | 1 | no | — |
| 322 | C2×C16 | — | 2 | 1 | no | — |
| 323 | C2×C16 | — | 2 | 1 | no | — |
| 324 | C2×C16 | — | 2 | 1 | no | — |
| 325 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 326 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 327 | #108 | — | 2 | 1 | no | — |
| 328 | #108 | — | 2 | 1 | no | — |
| 329 | #108 | — | 2 | 1 | no | — |
| 330 | #108 | C2×D4 | 1 | 2 | yes | — |
| 331 | #107 | — | 2 | 1 | no | — |
| 332 | #107 | — | 2 | 1 | no | — |
| 333 | #107 | — | 4 | 1 | no | — |
| 334 | #106 | — | 4 | 1 | no | — |
| 335 | #104 | — | 2 | 2 | no | — |
| 336 | #104 | — | 4 | 2 | no | — |
| 337 | #104 | — | 4 | 2 | no | — |
| 338 | #103 | — | 4 | 1 | no | — |
| 339 | #103 | D8 | 1 | 1 | yes | characteristic |
| 340 | #103 | D8 | 1 | 1 | yes | characteristic |
| 341 | #102 | — | 4 | 1 | no | — |
| 342 | #102 | — | 4 | 1 | no | — |
| 343 | #102 | — | 4 | 1 | no | — |
| 344 | #102 | — | 4 | 1 | no | — |
| 345 | #102 | — | 4 | 1 | no | — |
| 346 | #102 | D8 | 1 | 1 | yes | characteristic |
| 347 | #102 | D8 | 1 | 1 | yes | characteristic |
| 348 | #101 | — | 4 | 2 | no | — |
| 349 | #100 | — | 4 | 2 | no | — |
| 350 | #100 | — | 4 | 2 | no | — |
| 351 | C22⋊C8 | — | 4 | 1 | no | — |
| 352 | C22⋊C8 | — | 4 | 1 | no | — |
| 353 | #97 | — | 2 | 1 | no | — |
| 354 | C4×C8 | — | 2 | 1 | no | — |
| 355 | C4×C8 | — | 2 | 1 | no | — |
| 356 | C4×C8 | — | 2 | 1 | no | — |
| 357 | C24 | — | 4 | 1 | no | — |
| 358 | C24 | — | 4 | 1 | no | — |
| 359 | C24 | — | 8 | 1 | no | — |
| 360 | #41 | — | 2 | 1 | no | — |
| 361 | #41 | — | 2 | 1 | no | — |
| 362 | #41 | — | 2 | 1 | no | — |
| 363 | #41 | — | 2 | 1 | no | — |
| 364 | #41 | — | 4 | 4 | no | — |
| 365 | #41 | — | 4 | 1 | no | — |
| 366 | #41 | — | 4 | 1 | no | — |
| 367 | #41 | — | 4 | 1 | no | — |
| 368 | #41 | — | 4 | 1 | no | — |
| 369 | #41 | — | 4 | 1 | no | — |
| 370 | #41 | — | 4 | 1 | no | — |
| 371 | #41 | — | 4 | 1 | no | — |
| 372 | #41 | — | 4 | 1 | no | — |
| 373 | #41 | — | 8 | 2 | no | — |
| 374 | #41 | — | 8 | 2 | no | — |
| 375 | #41 | — | 8 | 2 | no | — |
| 376 | C2×Q8 | — | 2 | 1 | no | — |
| 377 | C2×Q8 | — | 2 | 1 | no | — |
| 378 | C2×Q8 | — | 4 | 1 | no | — |
| 379 | C2×Q8 | — | 4 | 1 | no | — |
| 380 | C2×Q8 | — | 4 | 1 | no | — |
| 381 | C2×Q8 | — | 4 | 1 | no | — |
| 382 | C2×Q8 | — | 4 | 1 | no | — |
| 383 | C2×Q8 | — | 4 | 1 | no | — |
| 384 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 385 | C2×D4 | — | 2 | 1 | no | — |
| 386 | C2×D4 | — | 2 | 1 | no | — |
| 387 | C2×D4 | — | 2 | 2 | no | — |
| 388 | C2×D4 | — | 2 | 1 | no | — |
| 389 | C2×D4 | — | 4 | 4 | no | — |
| 390 | C2×D4 | — | 4 | 1 | no | — |
| 391 | C2×D4 | — | 4 | 1 | no | — |
| 392 | C2×D4 | — | 4 | 1 | no | — |
| 393 | C2×D4 | — | 4 | 1 | no | — |
| 394 | C2×D4 | — | 4 | 1 | no | — |
| 395 | C2×D4 | — | 4 | 1 | no | — |
| 396 | C2×D4 | — | 4 | 1 | no | — |
| 397 | C2×D4 | — | 4 | 1 | no | — |
| 398 | C2×D4 | — | 4 | 1 | no | — |
| 399 | C2×D4 | — | 4 | 1 | no | — |
| 400 | C2×D4 | — | 4 | 1 | no | — |
| 401 | C2×D4 | — | 4 | 1 | no | — |
| 402 | C2×D4 | — | 4 | 1 | no | — |
| 403 | C2×D4 | — | 8 | 1 | no | — |
| 404 | C2×D4 | — | 8 | 1 | no | — |
| 405 | C2×D4 | — | 8 | 1 | no | — |
| 406 | C2×D4 | — | 8 | 2 | no | — |
| 407 | C2×D4 | — | 8 | 1 | no | — |
| 408 | C2×D4 | — | 8 | 2 | no | — |
| 409 | C2×D4 | — | 8 | 2 | no | — |
| 410 | C2×D4 | — | 8 | 4 | no | — |
| 411 | C2×D4 | — | 8 | 4 | no | — |
| 412 | C2×D4 | — | 8 | 1 | no | — |
| 413 | C2×D4 | C2×D8 | 1 | 1 | yes | characteristic |
| 414 | C22×C4 | — | 2 | 1 | no | — |
| 415 | C22×C4 | — | 2 | 1 | no | — |
| 416 | C22×C4 | — | 4 | 1 | no | — |
| 417 | C22×C4 | — | 4 | 1 | no | — |
| 418 | C22×C4 | — | 4 | 1 | no | — |
| 419 | C22×C4 | — | 4 | 1 | no | — |
| 420 | C22×C4 | — | 8 | 1 | no | — |
| 421 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 422 | Q16 | — | 2 | 1 | no | — |
| 423 | Q16 | — | 2 | 1 | no | — |
| 424 | Q16 | — | 2 | 1 | no | — |
| 425 | Q16 | — | 2 | 1 | no | — |
| 426 | Q16 | — | 4 | 1 | no | — |
| 427 | Q16 | — | 4 | 1 | no | — |
| 428 | Q16 | — | 4 | 1 | no | — |
| 429 | Q16 | — | 4 | 1 | no | — |
| 430 | Q16 | — | 8 | 2 | no | — |
| 431 | Q16 | — | 8 | 2 | no | — |
| 432 | SD16 | — | 4 | 4 | no | — |
| 433 | SD16 | — | 4 | 1 | no | — |
| 434 | SD16 | — | 4 | 1 | no | — |
| 435 | SD16 | — | 4 | 1 | no | — |
| 436 | SD16 | — | 4 | 1 | no | — |
| 437 | SD16 | — | 8 | 2 | no | — |
| 438 | SD16 | — | 8 | 2 | no | — |
| 439 | SD16 | — | 8 | 2 | no | — |
| 440 | SD16 | — | 8 | 2 | no | — |
| 441 | SD16 | — | 8 | 2 | no | — |
| 442 | SD16 | — | 8 | 2 | no | — |
| 443 | D8 | — | 2 | 1 | no | — |
| 444 | D8 | — | 2 | 1 | no | — |
| 445 | D8 | — | 2 | 2 | no | — |
| 446 | D8 | — | 2 | 1 | no | — |
| 447 | D8 | — | 2 | 1 | no | — |
| 448 | D8 | — | 2 | 2 | no | — |
| 449 | D8 | — | 4 | 4 | no | — |
| 450 | D8 | — | 4 | 1 | no | — |
| 451 | D8 | — | 4 | 1 | no | — |
| 452 | D8 | — | 4 | 1 | no | — |
| 453 | D8 | — | 4 | 1 | no | — |
| 454 | D8 | — | 4 | 1 | no | — |
| 455 | D8 | — | 4 | 1 | no | — |
| 456 | D8 | — | 8 | 2 | no | — |
| 457 | D8 | — | 8 | 2 | no | — |
| 458 | M16 | — | 2 | 1 | no | — |
| 459 | M16 | — | 2 | 1 | no | — |
| 460 | M16 | — | 2 | 1 | no | — |
| 461 | M16 | — | 2 | 2 | no | — |
| 462 | M16 | — | 4 | 1 | no | — |
| 463 | M16 | — | 4 | 1 | no | — |
| 464 | M16 | — | 4 | 1 | no | — |
| 465 | M16 | — | 4 | 1 | no | — |
| 466 | M16 | — | 4 | 1 | no | — |
| 467 | M16 | — | 8 | 2 | no | — |
| 468 | M16 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 469 | C2×C8 | — | 2 | 1 | no | — |
| 470 | C2×C8 | — | 2 | 1 | no | — |
| 471 | C2×C8 | — | 2 | 1 | no | — |
| 472 | C2×C8 | — | 2 | 1 | no | — |
| 473 | C2×C8 | — | 4 | 1 | no | — |
| 474 | C2×C8 | — | 4 | 1 | no | — |
| 475 | C2×C8 | — | 4 | 1 | no | — |
| 476 | C2×C8 | — | 4 | 1 | no | — |
| 477 | C2×C8 | — | 8 | 2 | no | — |
| 478 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 479 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 480 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 481 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 482 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 483 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 484 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 485 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 486 | C4⋊C4 | — | 4 | 1 | no | — |
| 487 | C4⋊C4 | — | 4 | 1 | no | — |
| 488 | C4⋊C4 | — | 8 | 1 | no | — |
| 489 | C4⋊C4 | — | 8 | 1 | no | — |
| 490 | C4⋊C4 | — | 8 | 1 | no | — |
| 491 | C4⋊C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 492 | C4⋊C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 493 | C22⋊C4 | — | 2 | 1 | no | — |
| 494 | C22⋊C4 | — | 8 | 1 | no | — |
| 495 | C22⋊C4 | — | 8 | 1 | no | — |
| 496 | C22⋊C4 | — | 8 | 1 | no | — |
| 497 | C22⋊C4 | — | 8 | 1 | no | — |
| 498 | C22⋊C4 | — | 8 | 1 | no | — |
| 499 | C22⋊C4 | — | 8 | 1 | no | — |
| 500 | C22⋊C4 | — | 8 | 4 | no | — |
| 501 | C22⋊C4 | — | 16 | 2 | no | — |
| 502 | C42 | — | 2 | 1 | no | — |
| 503 | C42 | — | 2 | 1 | no | — |
| 504 | C42 | — | 2 | 1 | no | — |
| 505 | C16 | — | 2 | 1 | no | — |
| 506 | C16 | — | 2 | 1 | no | — |
| 507 | C16 | — | 2 | 2 | no | — |
| 508 | C16 | — | 4 | 1 | no | — |
| 509 | C16 | — | 4 | 1 | no | — |
| 510 | C23 | — | 2 | 1 | no | — |
| 511 | C23 | — | 4 | 1 | no | — |
| 512 | C23 | — | 4 | 1 | no | — |
| 513 | C23 | — | 4 | 2 | no | — |
| 514 | C23 | — | 8 | 4 | no | — |
| 515 | C23 | — | 8 | 1 | no | — |
| 516 | C23 | — | 8 | 1 | no | — |
| 517 | C23 | — | 8 | 1 | no | — |
| 518 | C23 | — | 8 | 1 | no | — |
| 519 | C23 | — | 8 | 1 | no | — |
| 520 | C23 | — | 16 | 1 | no | — |
| 521 | C23 | — | 16 | 1 | no | — |
| 522 | C23 | — | 16 | 1 | no | — |
| 523 | C23 | — | 16 | 1 | no | — |
| 524 | C23 | — | 32 | 1 | no | — |
| 525 | C23 | — | 32 | 1 | no | — |
| 526 | C23 | #576 | 1 | 1 | yes | characteristic |
| 527 | Q8 | — | 2 | 1 | no | — |
| 528 | Q8 | — | 2 | 1 | no | — |
| 529 | Q8 | — | 4 | 1 | no | — |
| 530 | Q8 | — | 4 | 1 | no | — |
| 531 | Q8 | — | 4 | 1 | no | — |
| 532 | Q8 | — | 4 | 1 | no | — |
| 533 | Q8 | — | 8 | 1 | no | — |
| 534 | Q8 | — | 8 | 1 | no | — |
| 535 | Q8 | — | 8 | 1 | no | — |
| 536 | Q8 | — | 8 | 1 | no | — |
| 537 | D4 | — | 2 | 1 | no | — |
| 538 | D4 | — | 2 | 1 | no | — |
| 539 | D4 | — | 4 | 1 | no | — |
| 540 | D4 | — | 4 | 1 | no | — |
| 541 | D4 | — | 4 | 1 | no | — |
| 542 | D4 | — | 4 | 2 | no | — |
| 543 | D4 | — | 4 | 1 | no | — |
| 544 | D4 | — | 4 | 1 | no | — |
| 545 | D4 | — | 4 | 1 | no | — |
| 546 | D4 | — | 4 | 2 | no | — |
| 547 | D4 | — | 8 | 4 | no | — |
| 548 | D4 | — | 8 | 4 | no | — |
| 549 | D4 | — | 8 | 1 | no | — |
| 550 | D4 | — | 8 | 1 | no | — |
| 551 | D4 | — | 8 | 1 | no | — |
| 552 | D4 | — | 8 | 1 | no | — |
| 553 | D4 | — | 8 | 1 | no | — |
| 554 | D4 | — | 8 | 1 | no | — |
| 555 | D4 | — | 8 | 1 | no | — |
| 556 | D4 | — | 8 | 1 | no | — |
| 557 | D4 | — | 8 | 1 | no | — |
| 558 | D4 | — | 8 | 1 | no | — |
| 559 | D4 | — | 32 | 1 | no | — |
| 560 | D4 | — | 32 | 1 | no | — |
| 561 | D4 | — | 32 | 1 | no | — |
| 562 | C2×C4 | — | 2 | 1 | no | — |
| 563 | C2×C4 | — | 2 | 1 | no | — |
| 564 | C2×C4 | — | 2 | 1 | no | — |
| 565 | C2×C4 | — | 4 | 1 | no | — |
| 566 | C2×C4 | — | 4 | 1 | no | — |
| 567 | C2×C4 | — | 4 | 1 | no | — |
| 568 | C2×C4 | — | 4 | 1 | no | — |
| 569 | C2×C4 | — | 4 | 1 | no | — |
| 570 | C2×C4 | — | 4 | 1 | no | — |
| 571 | C2×C4 | — | 4 | 1 | no | — |
| 572 | C2×C4 | — | 8 | 4 | no | — |
| 573 | C2×C4 | — | 8 | 1 | no | — |
| 574 | C2×C4 | — | 8 | 1 | no | — |
| 575 | C2×C4 | — | 8 | 1 | no | — |
| 576 | C2×C4 | — | 8 | 1 | no | — |
| 577 | C2×C4 | — | 8 | 1 | no | — |
| 578 | C2×C4 | — | 8 | 1 | no | — |
| 579 | C2×C4 | — | 8 | 1 | no | — |
| 580 | C2×C4 | — | 8 | 1 | no | — |
| 581 | C2×C4 | — | 16 | 1 | no | — |
| 582 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 583 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 584 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 585 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 586 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 587 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 588 | C8 | — | 2 | 2 | no | — |
| 589 | C8 | — | 2 | 1 | no | — |
| 590 | C8 | — | 2 | 1 | no | — |
| 591 | C8 | — | 4 | 1 | no | — |
| 592 | C8 | — | 4 | 2 | no | — |
| 593 | C8 | — | 8 | 1 | no | — |
| 594 | C8 | #534 | 1 | 1 | yes | characteristic |
| 595 | C8 | #535 | 1 | 1 | yes | characteristic |
| 596 | C8 | D42 | 1 | 1 | yes | characteristic |
| 597 | C8 | #560 | 1 | 1 | yes | characteristic |
| 598 | C22 | — | 2 | 1 | no | — |
| 599 | C22 | — | 2 | 1 | no | — |
| 600 | C22 | — | 4 | 1 | no | — |
| 601 | C22 | — | 4 | 1 | no | — |
| 602 | C22 | — | 4 | 1 | no | — |
| 603 | C22 | — | 8 | 1 | no | — |
| 604 | C22 | — | 8 | 1 | no | — |
| 605 | C22 | — | 8 | 1 | no | — |
| 606 | C22 | — | 8 | 1 | no | — |
| 607 | C22 | — | 8 | 1 | no | — |
| 608 | C22 | — | 8 | 2 | no | — |
| 609 | C22 | — | 8 | 1 | no | — |
| 610 | C22 | — | 16 | 2 | no | — |
| 611 | C22 | — | 16 | 2 | no | — |
| 612 | C22 | — | 16 | 1 | no | — |
| 613 | C22 | — | 16 | 1 | no | — |
| 614 | C22 | — | 16 | 1 | no | — |
| 615 | C22 | — | 16 | 1 | no | — |
| 616 | C22 | — | 16 | 1 | no | — |
| 617 | C22 | — | 32 | 1 | no | — |
| 618 | C22 | — | 32 | 1 | no | — |
| 619 | C22 | — | 32 | 4 | no | — |
| 620 | C22 | #3220 | 1 | 1 | yes | characteristic |
| 621 | C22 | #3274 | 1 | 1 | yes | characteristic |
| 622 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 623 | C4 | — | 4 | 1 | no | — |
| 624 | C4 | — | 4 | 1 | no | — |
| 625 | C4 | — | 4 | 1 | no | — |
| 626 | C4 | — | 8 | 1 | no | — |
| 627 | C4 | — | 8 | 1 | no | — |
| 628 | C4 | — | 8 | 1 | no | — |
| 629 | C4 | — | 16 | 1 | no | — |
| 630 | C4 | — | 16 | 1 | no | — |
| 631 | C4 | #2669 | 1 | 1 | yes | characteristic |
| 632 | C4 | #3205 | 1 | 1 | yes | characteristic |
| 633 | C4 | #3264 | 1 | 1 | yes | characteristic |
| 634 | C4 | D4×D8 | 1 | 1 | yes | characteristic |
| 635 | C2 | — | 2 | 1 | no | — |
| 636 | C2 | — | 2 | 1 | no | — |
| 637 | C2 | — | 8 | 1 | no | — |
| 638 | C2 | — | 16 | 1 | no | — |
| 639 | C2 | — | 16 | 1 | no | — |
| 640 | C2 | — | 16 | 2 | no | — |
| 641 | C2 | — | 32 | 1 | no | — |
| 642 | C2 | #30444 | 1 | 1 | yes | characteristic |
| 643 | C2 | #33854 | 1 | 1 | yes | characteristic |
| 644 | C2 | #33857 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33722 | C2 | semidirect |
| 2. C2×#2212 | C2 | semidirect |
| 3. C2×#2192 | C2 | semidirect |
| 4. #33526 | C2 | semidirect |
| 5. #25804 | C2 | semidirect |
| 6. #25441 | C2 | semidirect |
| 7. #13445 | C2 | semidirect |
| 8. #13412 | C2 | semidirect |
| 9. #13336 | C2 | semidirect |
| 10. #13326 | C2 | semidirect |
| 11. #13314 | C2 | semidirect |
| 12. #13308 | C2 | semidirect |
| 13. #13144 | C2 | semidirect |
| 14. #13122 | C2 | semidirect |
| 15. #13011 | C2 | semidirect |
| 17. C2×#510 | C22 | semidirect |
| 21. C22×SD32 | C22 | semidirect |
| 24. #3155 | C22 | semidirect |
| 32. C2×#443 | C22 | semidirect |
| 43. C2×#364 | C22 | semidirect |
| 44. C2×#362 | C22 | semidirect |
| 46. C2×#361 | C22 | semidirect |
| 48. C2×#359 | C22 | semidirect |
| 50. C2×#358 | C22 | semidirect |
| 54. #2108 | C22 | semidirect |
| 61. #1970 | C22 | semidirect |
| 62. #1952 | C22 | semidirect |
| 65. #1387 | C22 | semidirect |
| 66. #1381 | C22 | semidirect |
| 67. #1379 | C22 | semidirect |
| 68. #1374 | C22 | semidirect |
| 170. #361 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33722 | 2 |
| C2×#2212 | 2 |
| C2×#2192 | 2 |
| #33526 | 2 |
| #25804 | 2 |
| #25441 | 2 |
| #13445 | 2 |
| #13412 | 2 |
| #13336 | 2 |
| #13326 | 2 |
| #13314 | 2 |
| #13308 | 2 |
| #13144 | 2 |
| #13122 | 2 |
| #13011 | 2 |