Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1065 conjugacy classes of subgroups.
The subgroup lattice has 686 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#3416 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33722 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#2249 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#2223 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#2195 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#2192 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #25804 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #25803 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #13339 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #13336 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #13332 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #13332 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | #13246 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | #13241 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | #12614 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C2×#578 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#576 | C22 | 1 | 1 | yes | characteristic |
| 18 | #3416 | — | 2 | 4 | no | — |
| 19 | #3416 | — | 2 | 4 | no | — |
| 20 | C2×#510 | — | 2 | 1 | no | — |
| 21 | C2×#510 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#510 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | C2×#509 | — | 2 | 1 | no | — |
| 24 | C2×#509 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#509 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#508 | — | 2 | 1 | no | — |
| 27 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 28 | C22×Q32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 29 | C22×SD32 | — | 2 | 1 | no | — |
| 30 | C22×SD32 | C22 | 1 | 1 | yes | characteristic |
| 31 | C22×SD32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 32 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×#504 | C22 | 1 | 1 | yes | characteristic |
| 34 | #3156 | C22 | 1 | 1 | yes | characteristic |
| 35 | #3155 | C22 | 1 | 1 | yes | characteristic, complemented |
| 36 | C2×#495 | C22 | 1 | 1 | yes | characteristic |
| 37 | C2×#495 | C22 | 1 | 1 | yes | characteristic, complemented |
| 38 | C2×#473 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×#472 | C22 | 1 | 1 | yes | characteristic |
| 40 | C2×#454 | — | 2 | 1 | no | — |
| 41 | C2×#455 | — | 2 | 1 | no | — |
| 42 | #3010 | — | 2 | 1 | no | — |
| 43 | #3009 | — | 2 | 1 | no | — |
| 44 | #2251 | — | 2 | 1 | no | — |
| 45 | #2250 | — | 2 | 1 | no | — |
| 46 | #2249 | — | 2 | 1 | no | — |
| 47 | #2249 | — | 2 | 4 | no | — |
| 48 | #2248 | — | 2 | 1 | no | — |
| 49 | #2223 | — | 2 | 4 | no | — |
| 50 | #2195 | — | 2 | 4 | no | — |
| 51 | #2192 | — | 2 | 4 | no | — |
| 52 | C2×#368 | C22 | 1 | 1 | yes | characteristic, complemented |
| 53 | C2×#362 | C22 | 1 | 1 | yes | characteristic, complemented |
| 54 | C2×#362 | C22 | 1 | 1 | yes | characteristic, complemented |
| 55 | C2×#361 | C22 | 1 | 1 | yes | characteristic |
| 56 | C2×#361 | C22 | 1 | 1 | yes | characteristic, complemented |
| 57 | C2×#359 | C22 | 1 | 1 | yes | characteristic |
| 58 | C2×#359 | C22 | 1 | 1 | yes | characteristic, complemented |
| 59 | C2×#350 | C22 | 1 | 1 | yes | characteristic |
| 60 | #2108 | C22 | 1 | 1 | yes | characteristic, complemented |
| 61 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 62 | #2019 | — | 2 | 1 | no | — |
| 63 | #2019 | — | 2 | 1 | no | — |
| 64 | #2004 | — | 2 | 1 | no | — |
| 65 | #2003 | — | 2 | 1 | no | — |
| 66 | #1979 | C22 | 1 | 1 | yes | characteristic, complemented |
| 67 | #1978 | C22 | 1 | 1 | yes | characteristic |
| 68 | #1974 | C22 | 1 | 1 | yes | characteristic |
| 69 | #1974 | C22 | 1 | 1 | yes | characteristic, complemented |
| 70 | #1764 | C22 | 1 | 1 | yes | characteristic |
| 71 | #1379 | C22 | 1 | 1 | yes | characteristic, complemented |
| 72 | #1376 | C22 | 1 | 1 | yes | characteristic, complemented |
| 73 | C2×2−1+4 | — | 2 | 1 | no | — |
| 74 | C2×H(5,𝔽2) | — | 2 | 1 | no | — |
| 75 | #578 | — | 2 | 2 | no | — |
| 76 | #578 | — | 2 | 2 | no | — |
| 77 | #578 | — | 2 | 2 | no | — |
| 78 | #578 | — | 4 | 1 | no | — |
| 79 | #576 | — | 2 | 2 | no | — |
| 80 | #576 | — | 2 | 2 | no | — |
| 81 | #576 | — | 2 | 2 | no | — |
| 82 | #576 | — | 4 | 1 | no | — |
| 83 | C2×#137 | — | 2 | 1 | no | — |
| 84 | C2×#137 | — | 2 | 1 | no | — |
| 85 | C2×#137 | — | 2 | 1 | no | — |
| 86 | C2×#137 | — | 4 | 1 | no | — |
| 87 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 88 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 89 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 90 | C2×AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 91 | C2×#135 | — | 2 | 1 | no | — |
| 92 | C2×#135 | — | 2 | 1 | no | — |
| 93 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 94 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 95 | C22×Q16 | — | 2 | 1 | no | — |
| 96 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 97 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 98 | C22×SD16 | — | 2 | 1 | no | — |
| 99 | C22×SD16 | — | 2 | 1 | no | — |
| 100 | C22×D8 | — | 2 | 1 | no | — |
| 101 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 102 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 103 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 104 | #536 | — | 2 | 1 | no | — |
| 105 | #535 | — | 2 | 1 | no | — |
| 106 | C2×#124 | — | 2 | 1 | no | — |
| 107 | C2×#124 | — | 2 | 1 | no | — |
| 108 | #510 | — | 2 | 8 | no | — |
| 109 | #510 | — | 2 | 2 | no | — |
| 110 | #510 | — | 2 | 2 | no | — |
| 111 | #510 | — | 2 | 2 | no | — |
| 112 | #510 | — | 4 | 1 | no | — |
| 113 | #509 | — | 2 | 8 | no | — |
| 114 | #509 | — | 2 | 2 | no | — |
| 115 | #509 | — | 2 | 2 | no | — |
| 116 | #509 | — | 2 | 2 | no | — |
| 117 | #509 | — | 4 | 1 | no | — |
| 118 | #508 | — | 2 | 8 | no | — |
| 119 | #508 | — | 2 | 2 | no | — |
| 120 | #508 | — | 2 | 2 | no | — |
| 121 | C2×Q32 | — | 2 | 1 | no | — |
| 122 | C2×Q32 | — | 2 | 2 | no | — |
| 123 | C2×Q32 | — | 2 | 1 | no | — |
| 124 | C2×Q32 | — | 2 | 2 | no | — |
| 125 | C2×Q32 | — | 4 | 1 | no | — |
| 126 | C2×Q32 | D4 | 1 | 1 | yes | characteristic |
| 127 | C2×SD32 | — | 2 | 8 | no | — |
| 128 | C2×SD32 | — | 2 | 1 | no | — |
| 129 | C2×SD32 | — | 2 | 1 | no | — |
| 130 | C2×SD32 | — | 2 | 1 | no | — |
| 131 | C2×SD32 | — | 2 | 1 | no | — |
| 132 | C2×SD32 | — | 2 | 1 | no | — |
| 133 | C2×SD32 | — | 2 | 1 | no | — |
| 134 | C2×SD32 | — | 2 | 2 | no | — |
| 135 | C2×SD32 | — | 2 | 2 | no | — |
| 136 | C2×SD32 | — | 2 | 1 | no | — |
| 137 | C2×SD32 | — | 2 | 2 | no | — |
| 138 | C2×SD32 | — | 2 | 1 | no | — |
| 139 | C2×SD32 | D4 | 1 | 1 | yes | characteristic |
| 140 | C2×SD32 | D4 | 1 | 1 | yes | characteristic |
| 141 | C2×SD32 | D4 | 1 | 4 | yes | complemented |
| 142 | C2×D16 | — | 2 | 1 | no | — |
| 143 | C2×D16 | — | 2 | 2 | no | — |
| 144 | C2×D16 | — | 2 | 1 | no | — |
| 145 | C2×D16 | — | 2 | 2 | no | — |
| 146 | C2×D16 | — | 4 | 1 | no | — |
| 147 | C2×D16 | D4 | 1 | 1 | yes | characteristic |
| 148 | #504 | — | 2 | 2 | no | — |
| 149 | #504 | D4 | 1 | 4 | yes | complemented |
| 150 | C2×M32 | — | 2 | 1 | no | — |
| 151 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 152 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 153 | C22×C16 | — | 2 | 1 | no | — |
| 154 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 155 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 156 | #497 | — | 4 | 1 | no | — |
| 157 | #496 | — | 4 | 1 | no | — |
| 158 | #495 | — | 2 | 2 | no | — |
| 159 | #495 | — | 2 | 1 | no | — |
| 160 | #495 | — | 2 | 2 | no | — |
| 161 | #495 | — | 2 | 1 | no | — |
| 162 | #495 | — | 4 | 1 | no | — |
| 163 | #495 | — | 4 | 1 | no | — |
| 164 | #494 | — | 2 | 1 | no | — |
| 165 | #493 | — | 2 | 1 | no | — |
| 166 | #473 | — | 2 | 2 | no | — |
| 167 | #473 | — | 2 | 2 | no | — |
| 168 | #472 | — | 2 | 2 | no | — |
| 169 | #472 | — | 2 | 2 | no | — |
| 170 | #455 | — | 4 | 4 | no | — |
| 171 | #454 | — | 4 | 4 | no | — |
| 172 | #452 | — | 4 | 2 | no | — |
| 173 | #450 | — | 4 | 2 | no | — |
| 174 | #449 | — | 4 | 2 | no | — |
| 175 | #448 | — | 4 | 2 | no | — |
| 176 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 177 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 178 | C2×#104 | — | 2 | 1 | no | — |
| 179 | C2×#104 | — | 2 | 1 | no | — |
| 180 | C2×#104 | D4 | 1 | 1 | yes | characteristic |
| 181 | #419 | — | 2 | 1 | no | — |
| 182 | #418 | — | 2 | 1 | no | — |
| 183 | #416 | D4 | 1 | 1 | yes | characteristic |
| 184 | C2×#101 | — | 2 | 1 | no | — |
| 185 | C2×#101 | — | 2 | 1 | no | — |
| 186 | C2×#100 | — | 2 | 1 | no | — |
| 187 | C2×#100 | — | 2 | 1 | no | — |
| 188 | #408 | — | 2 | 1 | no | — |
| 189 | #408 | — | 2 | 1 | no | — |
| 190 | #405 | C23 | 1 | 1 | yes | characteristic |
| 191 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 192 | #368 | — | 2 | 2 | no | — |
| 193 | #362 | — | 2 | 2 | no | — |
| 194 | #362 | — | 2 | 2 | no | — |
| 195 | #361 | — | 2 | 2 | no | — |
| 196 | #361 | — | 2 | 2 | no | — |
| 197 | #359 | — | 2 | 2 | no | — |
| 198 | #359 | — | 2 | 2 | no | — |
| 199 | #350 | D4 | 1 | 4 | yes | complemented |
| 200 | #346 | — | 2 | 1 | no | — |
| 201 | C4×C16 | — | 2 | 2 | no | — |
| 202 | C4×C16 | — | 2 | 1 | no | — |
| 203 | #341 | D4 | 1 | 1 | yes | characteristic |
| 204 | #339 | D4 | 1 | 1 | yes | characteristic |
| 205 | 2−1+4 | — | 4 | 2 | no | — |
| 206 | 2−1+4 | — | 4 | 2 | no | — |
| 207 | 2−1+4 | — | 4 | 2 | no | — |
| 208 | 2−1+4 | — | 8 | 1 | no | — |
| 209 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 210 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 211 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 212 | H(5,𝔽2) | — | 8 | 1 | no | — |
| 213 | C2×#41 | — | 2 | 1 | no | — |
| 214 | C2×#41 | — | 2 | 1 | no | — |
| 215 | C2×#41 | — | 4 | 1 | no | — |
| 216 | C2×#41 | — | 4 | 1 | no | — |
| 217 | C2×#41 | — | 4 | 1 | no | — |
| 218 | C2×#41 | — | 4 | 1 | no | — |
| 219 | C2×#41 | — | 8 | 1 | no | — |
| 220 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 221 | C22×Q8 | — | 2 | 1 | no | — |
| 222 | C22×Q8 | — | 2 | 1 | no | — |
| 223 | C22×Q8 | — | 4 | 1 | no | — |
| 224 | C22×Q8 | — | 4 | 1 | no | — |
| 225 | C22×D4 | — | 2 | 1 | no | — |
| 226 | C22×D4 | — | 2 | 1 | no | — |
| 227 | C22×D4 | — | 4 | 1 | no | — |
| 228 | C22×D4 | — | 4 | 1 | no | — |
| 229 | C22×D4 | — | 8 | 1 | no | — |
| 230 | #137 | — | 4 | 4 | no | — |
| 231 | #137 | — | 4 | 2 | no | — |
| 232 | #137 | — | 4 | 4 | no | — |
| 233 | #137 | — | 4 | 2 | no | — |
| 234 | #137 | — | 4 | 2 | no | — |
| 235 | #137 | — | 4 | 2 | no | — |
| 236 | #137 | — | 4 | 2 | no | — |
| 237 | #137 | — | 8 | 1 | no | — |
| 238 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 239 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 240 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 241 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 242 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 243 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 244 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 245 | AGL(1,ℤ/8) | — | 8 | 1 | no | — |
| 246 | #135 | — | 2 | 4 | no | — |
| 247 | #135 | — | 2 | 4 | no | — |
| 248 | #135 | — | 2 | 1 | no | — |
| 249 | #135 | — | 2 | 1 | no | — |
| 250 | #135 | — | 2 | 1 | no | — |
| 251 | #135 | — | 2 | 1 | no | — |
| 252 | #135 | — | 2 | 1 | no | — |
| 253 | #135 | — | 2 | 1 | no | — |
| 254 | #135 | — | 2 | 1 | no | — |
| 255 | #135 | — | 2 | 1 | no | — |
| 256 | #135 | — | 4 | 2 | no | — |
| 257 | #135 | — | 4 | 2 | no | — |
| 258 | C2×Q16 | — | 2 | 4 | no | — |
| 259 | C2×Q16 | — | 2 | 1 | no | — |
| 260 | C2×Q16 | — | 2 | 1 | no | — |
| 261 | C2×Q16 | — | 2 | 1 | no | — |
| 262 | C2×Q16 | — | 2 | 1 | no | — |
| 263 | C2×Q16 | — | 2 | 1 | no | — |
| 264 | C2×Q16 | — | 2 | 1 | no | — |
| 265 | C2×Q16 | — | 2 | 1 | no | — |
| 266 | C2×Q16 | — | 2 | 1 | no | — |
| 267 | C2×Q16 | — | 2 | 1 | no | — |
| 268 | C2×Q16 | — | 2 | 1 | no | — |
| 269 | C2×Q16 | — | 2 | 1 | no | — |
| 270 | C2×Q16 | — | 4 | 2 | no | — |
| 271 | C2×Q16 | — | 4 | 1 | no | — |
| 272 | C2×Q16 | — | 4 | 1 | no | — |
| 273 | C2×Q16 | — | 4 | 1 | no | — |
| 274 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 275 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 276 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 277 | C2×SD16 | — | 2 | 1 | no | — |
| 278 | C2×SD16 | — | 2 | 1 | no | — |
| 279 | C2×SD16 | — | 4 | 1 | no | — |
| 280 | C2×SD16 | — | 4 | 1 | no | — |
| 281 | C2×SD16 | — | 4 | 2 | no | — |
| 282 | C2×SD16 | — | 4 | 1 | no | — |
| 283 | C2×SD16 | — | 4 | 2 | no | — |
| 284 | C2×SD16 | — | 4 | 1 | no | — |
| 285 | C2×SD16 | — | 4 | 1 | no | — |
| 286 | C2×SD16 | — | 4 | 1 | no | — |
| 287 | C2×SD16 | — | 4 | 2 | no | — |
| 288 | C2×SD16 | — | 4 | 2 | no | — |
| 289 | C2×SD16 | — | 8 | 1 | no | — |
| 290 | C2×SD16 | — | 8 | 1 | no | — |
| 291 | C2×D8 | — | 2 | 4 | no | — |
| 292 | C2×D8 | — | 2 | 1 | no | — |
| 293 | C2×D8 | — | 2 | 1 | no | — |
| 294 | C2×D8 | — | 2 | 1 | no | — |
| 295 | C2×D8 | — | 2 | 1 | no | — |
| 296 | C2×D8 | — | 2 | 1 | no | — |
| 297 | C2×D8 | — | 2 | 1 | no | — |
| 298 | C2×D8 | — | 2 | 1 | no | — |
| 299 | C2×D8 | — | 2 | 1 | no | — |
| 300 | C2×D8 | — | 2 | 1 | no | — |
| 301 | C2×D8 | — | 2 | 1 | no | — |
| 302 | C2×D8 | — | 2 | 1 | no | — |
| 303 | C2×D8 | — | 4 | 2 | no | — |
| 304 | C2×D8 | — | 4 | 1 | no | — |
| 305 | C2×D8 | — | 4 | 1 | no | — |
| 306 | C2×D8 | — | 4 | 1 | no | — |
| 307 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 308 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 309 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 310 | #131 | — | 2 | 1 | no | — |
| 311 | #131 | — | 2 | 1 | no | — |
| 312 | #131 | C2×D4 | 1 | 2 | yes | — |
| 313 | #131 | C2×D4 | 1 | 2 | yes | — |
| 314 | C2×M16 | — | 2 | 1 | no | — |
| 315 | C2×M16 | — | 2 | 1 | no | — |
| 316 | C2×M16 | — | 2 | 1 | no | — |
| 317 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 318 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 319 | C22×C8 | — | 2 | 1 | no | — |
| 320 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 321 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 322 | C4⋊Q8 | — | 4 | 1 | no | — |
| 323 | #127 | — | 4 | 1 | no | — |
| 324 | #124 | — | 4 | 2 | no | — |
| 325 | #124 | — | 4 | 1 | no | — |
| 326 | #124 | — | 4 | 2 | no | — |
| 327 | #124 | — | 4 | 1 | no | — |
| 328 | #124 | — | 4 | 2 | no | — |
| 329 | #124 | — | 4 | 2 | no | — |
| 330 | #123 | — | 8 | 1 | no | — |
| 331 | C22⋊Q8 | — | 4 | 2 | no | — |
| 332 | #121 | — | 4 | 2 | no | — |
| 333 | #120 | — | 8 | 1 | no | — |
| 334 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 335 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 336 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 337 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 338 | Q32 | — | 2 | 4 | no | — |
| 339 | Q32 | — | 2 | 2 | no | — |
| 340 | Q32 | — | 4 | 4 | no | — |
| 341 | Q32 | — | 4 | 2 | no | — |
| 342 | SD32 | — | 2 | 4 | no | — |
| 343 | SD32 | — | 2 | 4 | no | — |
| 344 | SD32 | — | 2 | 2 | no | — |
| 345 | SD32 | — | 2 | 4 | no | — |
| 346 | SD32 | — | 2 | 4 | no | — |
| 347 | SD32 | — | 2 | 2 | no | — |
| 348 | SD32 | — | 4 | 2 | no | — |
| 349 | SD32 | — | 4 | 2 | no | — |
| 350 | SD32 | — | 4 | 2 | no | — |
| 351 | SD32 | — | 4 | 2 | no | — |
| 352 | D16 | — | 2 | 4 | no | — |
| 353 | D16 | — | 2 | 2 | no | — |
| 354 | D16 | — | 4 | 4 | no | — |
| 355 | D16 | — | 4 | 2 | no | — |
| 356 | M32 | — | 2 | 1 | no | — |
| 357 | M32 | — | 2 | 1 | no | — |
| 358 | M32 | — | 2 | 4 | no | — |
| 359 | M32 | C2×D4 | 1 | 2 | yes | — |
| 360 | M32 | C2×D4 | 1 | 2 | yes | — |
| 361 | C2×C16 | — | 2 | 1 | no | — |
| 362 | C2×C16 | — | 2 | 1 | no | — |
| 363 | C2×C16 | — | 2 | 1 | no | — |
| 364 | C2×C16 | — | 2 | 1 | no | — |
| 365 | C2×C16 | — | 2 | 4 | no | — |
| 366 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 367 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 368 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 369 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 370 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 371 | #108 | — | 2 | 1 | no | — |
| 372 | #108 | — | 2 | 1 | no | — |
| 373 | #108 | — | 2 | 1 | no | — |
| 374 | #108 | — | 2 | 1 | no | — |
| 375 | #104 | — | 4 | 2 | no | — |
| 376 | #104 | — | 4 | 2 | no | — |
| 377 | #104 | D8 | 1 | 4 | yes | — |
| 378 | #103 | — | 4 | 2 | no | — |
| 379 | #103 | D8 | 1 | 2 | yes | — |
| 380 | #102 | — | 4 | 2 | no | — |
| 381 | #102 | D8 | 1 | 2 | yes | — |
| 382 | #101 | — | 4 | 2 | no | — |
| 383 | #101 | — | 4 | 2 | no | — |
| 384 | #100 | — | 4 | 2 | no | — |
| 385 | #100 | — | 4 | 2 | no | — |
| 386 | C22⋊C8 | — | 4 | 2 | no | — |
| 387 | C22⋊C8 | — | 4 | 2 | no | — |
| 388 | #97 | — | 2 | 1 | no | — |
| 389 | C4×C8 | — | 2 | 1 | no | — |
| 390 | C4×C8 | — | 2 | 1 | no | — |
| 391 | C4×C8 | — | 2 | 1 | no | — |
| 392 | C24 | — | 4 | 1 | no | — |
| 393 | C24 | — | 4 | 1 | no | — |
| 394 | C24 | — | 8 | 1 | no | — |
| 395 | #41 | — | 2 | 1 | no | — |
| 396 | #41 | — | 2 | 1 | no | — |
| 397 | #41 | — | 4 | 4 | no | — |
| 398 | #41 | — | 4 | 4 | no | — |
| 399 | #41 | — | 4 | 1 | no | — |
| 400 | #41 | — | 4 | 1 | no | — |
| 401 | #41 | — | 4 | 1 | no | — |
| 402 | #41 | — | 4 | 1 | no | — |
| 403 | #41 | — | 4 | 1 | no | — |
| 404 | #41 | — | 4 | 1 | no | — |
| 405 | #41 | — | 4 | 1 | no | — |
| 406 | #41 | — | 4 | 1 | no | — |
| 407 | #41 | — | 8 | 2 | no | — |
| 408 | #41 | — | 8 | 2 | no | — |
| 409 | #41 | — | 8 | 4 | no | — |
| 410 | #41 | — | 8 | 2 | no | — |
| 411 | #41 | — | 8 | 4 | no | — |
| 412 | #41 | — | 8 | 2 | no | — |
| 413 | #41 | — | 8 | 2 | no | — |
| 414 | #41 | — | 8 | 2 | no | — |
| 415 | #41 | #127 | 1 | 2 | yes | — |
| 416 | #41 | C2×D8 | 1 | 2 | yes | — |
| 417 | C2×Q8 | — | 2 | 1 | no | — |
| 418 | C2×Q8 | — | 2 | 2 | no | — |
| 419 | C2×Q8 | — | 2 | 1 | no | — |
| 420 | C2×Q8 | — | 4 | 4 | no | — |
| 421 | C2×Q8 | — | 4 | 1 | no | — |
| 422 | C2×Q8 | — | 4 | 1 | no | — |
| 423 | C2×Q8 | — | 4 | 1 | no | — |
| 424 | C2×Q8 | — | 4 | 1 | no | — |
| 425 | C2×Q8 | — | 4 | 1 | no | — |
| 426 | C2×Q8 | — | 4 | 1 | no | — |
| 427 | C2×Q8 | — | 4 | 1 | no | — |
| 428 | C2×Q8 | — | 4 | 1 | no | — |
| 429 | C2×Q8 | — | 4 | 1 | no | — |
| 430 | C2×Q8 | — | 4 | 1 | no | — |
| 431 | C2×Q8 | — | 4 | 1 | no | — |
| 432 | C2×Q8 | — | 8 | 1 | no | — |
| 433 | C2×Q8 | — | 8 | 2 | no | — |
| 434 | C2×Q8 | — | 8 | 2 | no | — |
| 435 | C2×Q8 | — | 8 | 2 | no | — |
| 436 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 437 | C2×D4 | — | 2 | 1 | no | — |
| 438 | C2×D4 | — | 2 | 2 | no | — |
| 439 | C2×D4 | — | 2 | 1 | no | — |
| 440 | C2×D4 | — | 2 | 1 | no | — |
| 441 | C2×D4 | — | 4 | 4 | no | — |
| 442 | C2×D4 | — | 4 | 1 | no | — |
| 443 | C2×D4 | — | 4 | 1 | no | — |
| 444 | C2×D4 | — | 4 | 1 | no | — |
| 445 | C2×D4 | — | 4 | 1 | no | — |
| 446 | C2×D4 | — | 4 | 1 | no | — |
| 447 | C2×D4 | — | 4 | 1 | no | — |
| 448 | C2×D4 | — | 4 | 1 | no | — |
| 449 | C2×D4 | — | 4 | 1 | no | — |
| 450 | C2×D4 | — | 4 | 1 | no | — |
| 451 | C2×D4 | — | 4 | 1 | no | — |
| 452 | C2×D4 | — | 4 | 1 | no | — |
| 453 | C2×D4 | — | 4 | 1 | no | — |
| 454 | C2×D4 | — | 4 | 1 | no | — |
| 455 | C2×D4 | — | 8 | 2 | no | — |
| 456 | C2×D4 | — | 8 | 1 | no | — |
| 457 | C2×D4 | — | 8 | 2 | no | — |
| 458 | C2×D4 | — | 8 | 1 | no | — |
| 459 | C2×D4 | — | 8 | 2 | no | — |
| 460 | C2×D4 | — | 8 | 4 | no | — |
| 461 | C2×D4 | — | 8 | 4 | no | — |
| 462 | C2×D4 | — | 8 | 1 | no | — |
| 463 | C2×D4 | — | 8 | 2 | no | — |
| 464 | C2×D4 | C2×D8 | 1 | 1 | yes | characteristic |
| 465 | C22×C4 | — | 2 | 1 | no | — |
| 466 | C22×C4 | — | 2 | 1 | no | — |
| 467 | C22×C4 | — | 4 | 1 | no | — |
| 468 | C22×C4 | — | 4 | 1 | no | — |
| 469 | C22×C4 | — | 4 | 1 | no | — |
| 470 | C22×C4 | — | 4 | 1 | no | — |
| 471 | C22×C4 | — | 8 | 1 | no | — |
| 472 | C22×C4 | — | 8 | 1 | no | — |
| 473 | C22×C4 | — | 8 | 1 | no | — |
| 474 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 475 | Q16 | — | 2 | 1 | no | — |
| 476 | Q16 | — | 2 | 2 | no | — |
| 477 | Q16 | — | 2 | 1 | no | — |
| 478 | Q16 | — | 2 | 1 | no | — |
| 479 | Q16 | — | 2 | 2 | no | — |
| 480 | Q16 | — | 2 | 1 | no | — |
| 481 | Q16 | — | 4 | 4 | no | — |
| 482 | Q16 | — | 4 | 1 | no | — |
| 483 | Q16 | — | 4 | 1 | no | — |
| 484 | Q16 | — | 4 | 1 | no | — |
| 485 | Q16 | — | 4 | 1 | no | — |
| 486 | Q16 | — | 4 | 1 | no | — |
| 487 | Q16 | — | 4 | 1 | no | — |
| 488 | Q16 | — | 8 | 2 | no | — |
| 489 | Q16 | — | 8 | 2 | no | — |
| 490 | SD16 | — | 4 | 4 | no | — |
| 491 | SD16 | — | 4 | 4 | no | — |
| 492 | SD16 | — | 4 | 1 | no | — |
| 493 | SD16 | — | 4 | 1 | no | — |
| 494 | SD16 | — | 4 | 1 | no | — |
| 495 | SD16 | — | 4 | 1 | no | — |
| 496 | SD16 | — | 8 | 4 | no | — |
| 497 | SD16 | — | 8 | 2 | no | — |
| 498 | SD16 | — | 8 | 4 | no | — |
| 499 | SD16 | — | 8 | 2 | no | — |
| 500 | SD16 | — | 8 | 2 | no | — |
| 501 | SD16 | — | 8 | 2 | no | — |
| 502 | D8 | — | 2 | 1 | no | — |
| 503 | D8 | — | 2 | 2 | no | — |
| 504 | D8 | — | 2 | 1 | no | — |
| 505 | D8 | — | 2 | 2 | no | — |
| 506 | D8 | — | 2 | 1 | no | — |
| 507 | D8 | — | 2 | 1 | no | — |
| 508 | D8 | — | 4 | 4 | no | — |
| 509 | D8 | — | 4 | 1 | no | — |
| 510 | D8 | — | 4 | 1 | no | — |
| 511 | D8 | — | 4 | 1 | no | — |
| 512 | D8 | — | 4 | 1 | no | — |
| 513 | D8 | — | 4 | 1 | no | — |
| 514 | D8 | — | 4 | 1 | no | — |
| 515 | D8 | — | 8 | 2 | no | — |
| 516 | D8 | — | 8 | 2 | no | — |
| 517 | M16 | — | 2 | 1 | no | — |
| 518 | M16 | — | 2 | 1 | no | — |
| 519 | M16 | — | 2 | 2 | no | — |
| 520 | M16 | — | 2 | 2 | no | — |
| 521 | M16 | — | 4 | 1 | no | — |
| 522 | M16 | — | 4 | 1 | no | — |
| 523 | M16 | — | 4 | 1 | no | — |
| 524 | M16 | — | 4 | 1 | no | — |
| 525 | M16 | #127 | 1 | 2 | yes | — |
| 526 | M16 | C2×D8 | 1 | 2 | yes | — |
| 527 | C2×C8 | — | 2 | 1 | no | — |
| 528 | C2×C8 | — | 2 | 1 | no | — |
| 529 | C2×C8 | — | 2 | 1 | no | — |
| 530 | C2×C8 | — | 2 | 1 | no | — |
| 531 | C2×C8 | — | 2 | 2 | no | — |
| 532 | C2×C8 | — | 2 | 2 | no | — |
| 533 | C2×C8 | — | 4 | 1 | no | — |
| 534 | C2×C8 | — | 4 | 1 | no | — |
| 535 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 536 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 537 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 538 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 539 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 540 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 541 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 542 | C4⋊C4 | — | 8 | 2 | no | — |
| 543 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 544 | C22⋊C4 | — | 2 | 1 | no | — |
| 545 | C22⋊C4 | — | 8 | 4 | no | — |
| 546 | C22⋊C4 | — | 8 | 2 | no | — |
| 547 | C22⋊C4 | — | 8 | 4 | no | — |
| 548 | C22⋊C4 | — | 8 | 2 | no | — |
| 549 | C42 | — | 2 | 1 | no | — |
| 550 | C42 | — | 2 | 1 | no | — |
| 551 | C42 | C2×D8 | 1 | 2 | yes | — |
| 552 | C16 | — | 2 | 2 | no | — |
| 553 | C16 | — | 2 | 2 | no | — |
| 554 | C16 | — | 2 | 2 | no | — |
| 555 | C16 | #120 | 1 | 2 | yes | — |
| 556 | C16 | #120 | 1 | 2 | yes | — |
| 557 | C23 | — | 2 | 1 | no | — |
| 558 | C23 | — | 4 | 1 | no | — |
| 559 | C23 | — | 4 | 1 | no | — |
| 560 | C23 | — | 4 | 2 | no | — |
| 561 | C23 | — | 8 | 4 | no | — |
| 562 | C23 | — | 8 | 1 | no | — |
| 563 | C23 | — | 8 | 1 | no | — |
| 564 | C23 | — | 8 | 1 | no | — |
| 565 | C23 | — | 8 | 1 | no | — |
| 566 | C23 | — | 8 | 1 | no | — |
| 567 | C23 | — | 16 | 2 | no | — |
| 568 | C23 | — | 16 | 2 | no | — |
| 569 | C23 | — | 32 | 2 | no | — |
| 570 | C23 | #576 | 1 | 1 | yes | characteristic |
| 571 | Q8 | — | 2 | 1 | no | — |
| 572 | Q8 | — | 4 | 1 | no | — |
| 573 | Q8 | — | 4 | 2 | no | — |
| 574 | Q8 | — | 4 | 1 | no | — |
| 575 | Q8 | — | 4 | 2 | no | — |
| 576 | Q8 | — | 4 | 1 | no | — |
| 577 | Q8 | — | 4 | 1 | no | — |
| 578 | Q8 | — | 8 | 4 | no | — |
| 579 | Q8 | — | 8 | 1 | no | — |
| 580 | Q8 | — | 8 | 1 | no | — |
| 581 | Q8 | — | 8 | 1 | no | — |
| 582 | Q8 | — | 8 | 1 | no | — |
| 583 | Q8 | — | 8 | 1 | no | — |
| 584 | Q8 | — | 8 | 1 | no | — |
| 585 | Q8 | #493 | 1 | 2 | yes | — |
| 586 | D4 | — | 2 | 2 | no | — |
| 587 | D4 | — | 2 | 2 | no | — |
| 588 | D4 | — | 2 | 1 | no | — |
| 589 | D4 | — | 4 | 2 | no | — |
| 590 | D4 | — | 4 | 1 | no | — |
| 591 | D4 | — | 4 | 1 | no | — |
| 592 | D4 | — | 4 | 1 | no | — |
| 593 | D4 | — | 4 | 2 | no | — |
| 594 | D4 | — | 4 | 1 | no | — |
| 595 | D4 | — | 8 | 4 | no | — |
| 596 | D4 | — | 8 | 4 | no | — |
| 597 | D4 | — | 8 | 4 | no | — |
| 598 | D4 | — | 8 | 1 | no | — |
| 599 | D4 | — | 8 | 1 | no | — |
| 600 | D4 | — | 8 | 1 | no | — |
| 601 | D4 | — | 8 | 1 | no | — |
| 602 | D4 | — | 8 | 1 | no | — |
| 603 | D4 | — | 8 | 1 | no | — |
| 604 | D4 | — | 8 | 1 | no | — |
| 605 | D4 | — | 8 | 1 | no | — |
| 606 | D4 | — | 8 | 1 | no | — |
| 607 | D4 | — | 8 | 1 | no | — |
| 608 | D4 | — | 32 | 2 | no | — |
| 609 | D4 | #493 | 1 | 2 | yes | — |
| 610 | C2×C4 | — | 2 | 1 | no | — |
| 611 | C2×C4 | — | 2 | 2 | no | — |
| 612 | C2×C4 | — | 2 | 2 | no | — |
| 613 | C2×C4 | — | 2 | 2 | no | — |
| 614 | C2×C4 | — | 2 | 2 | no | — |
| 615 | C2×C4 | — | 4 | 2 | no | — |
| 616 | C2×C4 | — | 4 | 2 | no | — |
| 617 | C2×C4 | — | 4 | 1 | no | — |
| 618 | C2×C4 | — | 4 | 1 | no | — |
| 619 | C2×C4 | — | 8 | 4 | no | — |
| 620 | C2×C4 | — | 8 | 4 | no | — |
| 621 | C2×C4 | — | 8 | 4 | no | — |
| 622 | C2×C4 | — | 8 | 1 | no | — |
| 623 | C2×C4 | — | 8 | 1 | no | — |
| 624 | C2×C4 | — | 8 | 1 | no | — |
| 625 | C2×C4 | — | 8 | 1 | no | — |
| 626 | C2×C4 | — | 8 | 1 | no | — |
| 627 | C2×C4 | — | 8 | 1 | no | — |
| 628 | C2×C4 | — | 8 | 1 | no | — |
| 629 | C2×C4 | — | 8 | 1 | no | — |
| 630 | C2×C4 | — | 8 | 1 | no | — |
| 631 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 632 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 633 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 634 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 635 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 636 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 637 | C8 | — | 2 | 2 | no | — |
| 638 | C8 | — | 2 | 2 | no | — |
| 639 | C8 | — | 2 | 2 | no | — |
| 640 | C8 | — | 8 | 1 | no | — |
| 641 | C8 | — | 8 | 1 | no | — |
| 642 | C8 | C2×#120 | 1 | 1 | yes | characteristic |
| 643 | C8 | #535 | 1 | 1 | yes | characteristic |
| 644 | C8 | D42 | 1 | 1 | yes | characteristic |
| 645 | C8 | D42 | 1 | 1 | yes | characteristic |
| 646 | C22 | — | 2 | 2 | no | — |
| 647 | C22 | — | 2 | 2 | no | — |
| 648 | C22 | — | 4 | 2 | no | — |
| 649 | C22 | — | 8 | 2 | no | — |
| 650 | C22 | — | 8 | 2 | no | — |
| 651 | C22 | — | 8 | 1 | no | — |
| 652 | C22 | — | 8 | 1 | no | — |
| 653 | C22 | — | 8 | 2 | no | — |
| 654 | C22 | — | 16 | 2 | no | — |
| 655 | C22 | — | 16 | 2 | no | — |
| 656 | C22 | — | 16 | 2 | no | — |
| 657 | C22 | — | 16 | 2 | no | — |
| 658 | C22 | — | 16 | 1 | no | — |
| 659 | C22 | — | 32 | 1 | no | — |
| 660 | C22 | — | 32 | 1 | no | — |
| 661 | C22 | — | 32 | 4 | no | — |
| 662 | C22 | #3155 | 1 | 1 | yes | characteristic |
| 663 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 664 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 665 | C4 | — | 2 | 2 | no | — |
| 666 | C4 | — | 4 | 2 | no | — |
| 667 | C4 | — | 4 | 2 | no | — |
| 668 | C4 | — | 8 | 1 | no | — |
| 669 | C4 | — | 8 | 2 | no | — |
| 670 | C4 | — | 8 | 1 | no | — |
| 671 | C4 | — | 16 | 1 | no | — |
| 672 | C4 | #2433 | 1 | 1 | yes | characteristic |
| 673 | C4 | C2×#493 | 1 | 1 | yes | characteristic |
| 674 | C4 | D4×D8 | 1 | 1 | yes | characteristic |
| 675 | C4 | D4×D8 | 1 | 1 | yes | characteristic |
| 676 | C2 | — | 2 | 1 | no | — |
| 677 | C2 | — | 2 | 1 | no | — |
| 678 | C2 | — | 4 | 2 | no | — |
| 679 | C2 | — | 16 | 1 | no | — |
| 680 | C2 | — | 16 | 1 | no | — |
| 681 | C2 | — | 16 | 2 | no | — |
| 682 | C2 | — | 32 | 1 | no | — |
| 683 | C2 | #25779 | 1 | 1 | yes | characteristic |
| 684 | C2 | #33857 | 1 | 2 | yes | — |
| 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. C2×#3416 | C2 | semidirect |
| 2. #33722 | C2 | semidirect |
| 3. C2×#2249 | C2 | semidirect |
| 4. C2×#2223 | C2 | semidirect |
| 5. C2×#2195 | C2 | semidirect |
| 6. C2×#2192 | C2 | semidirect |
| 7. #25804 | C2 | semidirect |
| 8. #25803 | C2 | semidirect |
| 9. #13339 | C2 | semidirect |
| 10. #13336 | C2 | semidirect |
| 11. #13332 | C2 | semidirect |
| 12. #13332 | C2 | semidirect |
| 13. #13246 | C2 | semidirect |
| 14. #13241 | C2 | semidirect |
| 15. #12614 | C2 | semidirect |
| 22. C2×#510 | C22 | semidirect |
| 28. C22×Q32 | C22 | semidirect |
| 31. C22×SD32 | C22 | semidirect |
| 35. #3155 | C22 | semidirect |
| 37. C2×#495 | C22 | semidirect |
| 52. C2×#368 | C22 | semidirect |
| 53. C2×#362 | C22 | semidirect |
| 54. C2×#362 | C22 | semidirect |
| 56. C2×#361 | C22 | semidirect |
| 58. C2×#359 | C22 | semidirect |
| 60. #2108 | C22 | semidirect |
| 61. C2×C4×C16 | C22 | semidirect |
| 66. #1979 | C22 | semidirect |
| 69. #1974 | C22 | semidirect |
| 71. #1379 | C22 | semidirect |
| 72. #1376 | C22 | semidirect |
| 141. C2×SD32 | D4 | semidirect |
| 149. #504 | D4 | semidirect |
| 199. #350 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3416 | 2 |
| #33722 | 2 |
| C2×#2249 | 2 |
| C2×#2223 | 2 |
| C2×#2195 | 2 |
| C2×#2192 | 2 |
| #25804 | 2 |
| #25803 | 2 |
| #13339 | 2 |
| #13336 | 2 |
| #13332 | 2 |
| #13246 | 2 |
| #13241 | 2 |
| #12614 | 2 |