Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2396 conjugacy classes of subgroups.
The subgroup lattice has 457 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #52958 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #36082 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #32886 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #32717 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #31260 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #31065 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #30543 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #27252 | C2 | 1 | 6 | yes | maximal, complemented |
| 9 | #25896 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | #23859 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | #23311 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C22×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 13 | C23×Q16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C2×#548 | C22 | 1 | 3 | yes | — |
| 15 | C2×#489 | C22 | 1 | 3 | yes | complemented |
| 16 | C2×#477 | C22 | 1 | 3 | yes | complemented |
| 17 | C2×#452 | C22 | 1 | 6 | yes | — |
| 18 | C2×#448 | C22 | 1 | 3 | yes | complemented |
| 19 | C2×#450 | C22 | 1 | 6 | yes | — |
| 20 | #2835 | — | 2 | 1 | no | — |
| 21 | #2832 | — | 2 | 3 | no | — |
| 22 | #2811 | — | 2 | 1 | no | — |
| 23 | #2712 | — | 2 | 3 | no | — |
| 24 | #2616 | — | 2 | 1 | no | — |
| 25 | #2616 | C22 | 1 | 2 | yes | — |
| 26 | #2616 | C22 | 1 | 1 | yes | characteristic |
| 27 | #2547 | C22 | 1 | 3 | yes | complemented |
| 28 | #2492 | C22 | 1 | 6 | yes | complemented |
| 29 | #2435 | — | 2 | 3 | no | — |
| 30 | #2435 | C22 | 1 | 3 | yes | — |
| 31 | #2409 | C22 | 1 | 3 | yes | complemented |
| 32 | #2408 | — | 2 | 3 | no | — |
| 33 | #2408 | C22 | 1 | 3 | yes | — |
| 34 | #2388 | — | 2 | 1 | no | — |
| 35 | #2388 | C22 | 1 | 1 | yes | characteristic |
| 36 | #2341 | C22 | 1 | 3 | yes | — |
| 37 | #2075 | C22 | 1 | 6 | yes | complemented |
| 38 | #2055 | C22 | 1 | 6 | yes | complemented |
| 39 | #2036 | C22 | 1 | 6 | yes | complemented |
| 40 | #2036 | C22 | 1 | 6 | yes | complemented |
| 41 | #2033 | C22 | 1 | 6 | yes | complemented |
| 42 | #2023 | C22 | 1 | 6 | yes | complemented |
| 43 | #2015 | C22 | 1 | 6 | yes | complemented |
| 44 | #2001 | C22 | 1 | 6 | yes | complemented |
| 45 | #1983 | C22 | 1 | 3 | yes | complemented |
| 46 | #1973 | C22 | 1 | 6 | yes | complemented |
| 47 | #1955 | C22 | 1 | 3 | yes | complemented |
| 48 | #1925 | C22 | 1 | 6 | yes | complemented |
| 49 | #1924 | C22 | 1 | 6 | yes | complemented |
| 50 | #1893 | C22 | 1 | 3 | yes | complemented |
| 51 | #1878 | C22 | 1 | 6 | yes | complemented |
| 52 | #1869 | C22 | 1 | 3 | yes | complemented |
| 53 | #1788 | C22 | 1 | 6 | yes | — |
| 54 | #1787 | C22 | 1 | 3 | yes | complemented |
| 55 | #1786 | C22 | 1 | 6 | yes | — |
| 56 | #1780 | C22 | 1 | 2 | yes | — |
| 57 | #1763 | C22 | 1 | 3 | yes | complemented |
| 58 | C2×H(5,𝔽2) | — | 4 | 6 | no | — |
| 59 | C2×H(5,𝔽2) | — | 4 | 6 | no | — |
| 60 | C22×#41 | — | 2 | 1 | no | — |
| 61 | C22×#41 | C23 | 1 | 3 | yes | — |
| 62 | C22×#41 | C23 | 1 | 1 | yes | characteristic |
| 63 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 64 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 65 | C23×D4 | — | 2 | 3 | no | — |
| 66 | C23×D4 | C23 | 1 | 3 | yes | — |
| 67 | C22×Q16 | — | 4 | 6 | no | — |
| 68 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 69 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 70 | C22×SD16 | C23 | 1 | 6 | yes | — |
| 71 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 72 | #548 | — | 4 | 6 | no | — |
| 73 | #548 | — | 4 | 6 | no | — |
| 74 | C2×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 75 | C2×#124 | — | 2 | 3 | no | — |
| 76 | C2×#124 | — | 2 | 3 | no | — |
| 77 | C2×#124 | C23 | 1 | 3 | yes | — |
| 78 | C2×C22⋊Q8 | — | 2 | 3 | no | — |
| 79 | C2×C22⋊Q8 | C23 | 1 | 6 | yes | — |
| 80 | C2×C22⋊Q8 | C23 | 1 | 3 | yes | — |
| 81 | C2×#121 | — | 2 | 3 | no | — |
| 82 | C2×#121 | C23 | 1 | 3 | yes | — |
| 83 | C2×#120 | — | 2 | 3 | no | — |
| 84 | C2×C4×Q8 | C23 | 1 | 3 | yes | — |
| 85 | C2×C4×D4 | — | 2 | 3 | no | — |
| 86 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 87 | #489 | — | 4 | 6 | no | — |
| 88 | #477 | — | 4 | 6 | no | — |
| 89 | #452 | — | 4 | 12 | no | — |
| 90 | #450 | — | 4 | 12 | no | — |
| 91 | #448 | — | 4 | 6 | no | — |
| 92 | C2×#107 | C23 | 1 | 6 | yes | — |
| 93 | C2×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 94 | C2×#103 | C23 | 1 | 6 | yes | — |
| 95 | C2×#103 | C23 | 1 | 6 | yes | — |
| 96 | C2×#103 | C23 | 1 | 6 | yes | — |
| 97 | C2×#102 | C23 | 1 | 6 | yes | — |
| 98 | C2×#102 | C23 | 1 | 6 | yes | — |
| 99 | C2×#102 | C23 | 1 | 6 | yes | — |
| 100 | C2×C22⋊C8 | C23 | 1 | 6 | yes | — |
| 101 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 102 | C2×C22⋊C8 | C23 | 1 | 6 | yes | — |
| 103 | C2×#97 | C23 | 1 | 3 | yes | — |
| 104 | #400 | — | 2 | 6 | no | — |
| 105 | #397 | — | 2 | 3 | no | — |
| 106 | #397 | — | 2 | 3 | no | — |
| 107 | #395 | — | 2 | 2 | no | — |
| 108 | #395 | — | 2 | 1 | no | — |
| 109 | #395 | — | 2 | 3 | no | — |
| 110 | #395 | D4 | 1 | 1 | yes | characteristic |
| 111 | #394 | — | 2 | 6 | no | — |
| 112 | #394 | — | 2 | 3 | no | — |
| 113 | #394 | — | 2 | 3 | no | — |
| 114 | #394 | — | 2 | 3 | no | — |
| 115 | #394 | — | 2 | 3 | no | — |
| 116 | #394 | D4 | 1 | 3 | yes | — |
| 117 | #393 | — | 2 | 3 | no | — |
| 118 | #393 | — | 2 | 6 | no | — |
| 119 | #393 | D4 | 1 | 1 | yes | characteristic |
| 120 | #392 | — | 2 | 3 | no | — |
| 121 | #392 | D4 | 1 | 3 | yes | — |
| 122 | #391 | — | 2 | 3 | no | — |
| 123 | #391 | — | 2 | 6 | no | — |
| 124 | #391 | C23 | 1 | 6 | yes | — |
| 125 | #390 | — | 2 | 6 | no | — |
| 126 | #390 | C23 | 1 | 6 | yes | — |
| 127 | #388 | — | 2 | 6 | no | — |
| 128 | #386 | — | 2 | 3 | no | — |
| 129 | #385 | — | 2 | 3 | no | — |
| 130 | #380 | — | 2 | 3 | no | — |
| 131 | #380 | C23 | 1 | 3 | yes | — |
| 132 | #380 | C23 | 1 | 6 | yes | — |
| 133 | #380 | C23 | 1 | 3 | yes | — |
| 134 | #379 | — | 2 | 6 | no | — |
| 135 | #379 | — | 2 | 3 | no | — |
| 136 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 137 | C4×C22⋊C4 | — | 2 | 3 | no | — |
| 138 | C4×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 139 | #340 | C23 | 1 | 6 | yes | — |
| 140 | #340 | C23 | 1 | 6 | yes | — |
| 141 | #336 | C23 | 1 | 6 | yes | — |
| 142 | C25 | — | 2 | 3 | no | — |
| 143 | H(5,𝔽2) | — | 4 | 48 | no | — |
| 144 | H(5,𝔽2) | — | 4 | 16 | no | — |
| 145 | C2×#41 | — | 2 | 6 | no | — |
| 146 | C2×#41 | — | 2 | 6 | no | — |
| 147 | C2×#41 | — | 2 | 12 | no | — |
| 148 | C2×#41 | — | 4 | 3 | no | — |
| 149 | C2×#41 | — | 4 | 3 | no | — |
| 150 | C2×#41 | — | 4 | 6 | no | — |
| 151 | C2×#41 | — | 4 | 12 | no | — |
| 152 | C22×Q8 | — | 2 | 1 | no | — |
| 153 | C22×Q8 | — | 2 | 3 | no | — |
| 154 | C22×Q8 | — | 2 | 1 | no | — |
| 155 | C22×Q8 | — | 2 | 6 | no | — |
| 156 | C22×Q8 | — | 4 | 3 | no | — |
| 157 | C22×Q8 | — | 4 | 3 | no | — |
| 158 | C22×Q8 | — | 4 | 3 | no | — |
| 159 | C22×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 160 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 161 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 162 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 163 | C22×D4 | — | 2 | 3 | no | — |
| 164 | C22×D4 | — | 2 | 12 | no | — |
| 165 | C22×D4 | — | 2 | 3 | no | — |
| 166 | C22×D4 | — | 2 | 3 | no | — |
| 167 | C22×D4 | — | 4 | 6 | no | — |
| 168 | C22×D4 | — | 4 | 3 | no | — |
| 169 | C22×D4 | — | 4 | 3 | no | — |
| 170 | C22×D4 | — | 4 | 12 | no | — |
| 171 | C22×D4 | — | 4 | 24 | no | — |
| 172 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 173 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 174 | C22×D4 | C24 | 1 | 3 | yes | — |
| 175 | C23×C4 | — | 2 | 3 | no | — |
| 176 | C23×C4 | — | 2 | 1 | no | — |
| 177 | C23×C4 | — | 2 | 1 | no | — |
| 178 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 179 | C23×C4 | C24 | 1 | 3 | yes | — |
| 180 | C2×Q16 | — | 2 | 12 | no | — |
| 181 | C2×Q16 | — | 4 | 6 | no | — |
| 182 | C2×Q16 | — | 4 | 16 | no | — |
| 183 | C2×SD16 | — | 2 | 12 | no | — |
| 184 | C2×SD16 | — | 4 | 6 | no | — |
| 185 | C2×SD16 | — | 4 | 6 | no | — |
| 186 | C22×C8 | — | 4 | 3 | no | — |
| 187 | C22×C8 | C24 | 1 | 2 | yes | — |
| 188 | C22×C8 | C24 | 1 | 6 | yes | — |
| 189 | C4⋊Q8 | — | 4 | 3 | no | — |
| 190 | C4⋊Q8 | — | 4 | 3 | no | — |
| 191 | #124 | — | 2 | 6 | no | — |
| 192 | #124 | — | 4 | 3 | no | — |
| 193 | #124 | — | 4 | 12 | no | — |
| 194 | #124 | — | 8 | 6 | no | — |
| 195 | C22⋊Q8 | — | 4 | 3 | no | — |
| 196 | C22⋊Q8 | — | 4 | 3 | no | — |
| 197 | C22⋊Q8 | — | 4 | 12 | no | — |
| 198 | C22⋊Q8 | — | 4 | 6 | no | — |
| 199 | C22⋊Q8 | — | 4 | 6 | no | — |
| 200 | #121 | — | 2 | 6 | no | — |
| 201 | #121 | — | 4 | 3 | no | — |
| 202 | #121 | — | 4 | 12 | no | — |
| 203 | #120 | — | 8 | 6 | no | — |
| 204 | C4×Q8 | — | 2 | 6 | no | — |
| 205 | C4×Q8 | — | 4 | 3 | no | — |
| 206 | C4×D4 | — | 4 | 3 | no | — |
| 207 | C4×D4 | — | 4 | 3 | no | — |
| 208 | C4×D4 | — | 4 | 12 | no | — |
| 209 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 210 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 211 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 212 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 213 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 214 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 215 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 216 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 217 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 218 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 219 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 220 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 221 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 222 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 223 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 224 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 225 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 226 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 227 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 228 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 229 | C2×C42 | — | 2 | 3 | no | — |
| 230 | C2×C42 | — | 2 | 3 | no | — |
| 231 | C2×C42 | C24 | 1 | 3 | yes | — |
| 232 | #107 | — | 4 | 6 | no | — |
| 233 | C4⋊C8 | — | 4 | 3 | no | — |
| 234 | #103 | — | 2 | 12 | no | — |
| 235 | #103 | — | 4 | 6 | no | — |
| 236 | #103 | — | 4 | 6 | no | — |
| 237 | #102 | — | 2 | 12 | no | — |
| 238 | #102 | — | 4 | 6 | no | — |
| 239 | #102 | — | 4 | 6 | no | — |
| 240 | C22⋊C8 | — | 4 | 3 | no | — |
| 241 | C22⋊C8 | — | 4 | 6 | no | — |
| 242 | C22⋊C8 | — | 4 | 6 | no | — |
| 243 | #97 | — | 4 | 3 | no | — |
| 244 | #95 | — | 2 | 3 | no | — |
| 245 | #95 | — | 2 | 3 | no | — |
| 246 | #95 | — | 2 | 3 | no | — |
| 247 | #95 | — | 2 | 6 | no | — |
| 248 | #95 | — | 2 | 6 | no | — |
| 249 | #95 | — | 2 | 6 | no | — |
| 250 | #95 | C2×D4 | 1 | 3 | yes | — |
| 251 | #95 | C2×D4 | 1 | 3 | yes | — |
| 252 | C24 | — | 2 | 3 | no | — |
| 253 | C24 | — | 4 | 12 | no | — |
| 254 | C24 | — | 4 | 6 | no | — |
| 255 | C24 | — | 8 | 3 | no | — |
| 256 | C24 | — | 8 | 6 | no | — |
| 257 | C24 | — | 8 | 3 | no | — |
| 258 | C24 | C22×D4 | 1 | 3 | yes | — |
| 259 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 260 | #41 | — | 4 | 24 | no | — |
| 261 | #41 | — | 4 | 12 | no | — |
| 262 | #41 | — | 4 | 32 | no | — |
| 263 | #41 | — | 4 | 4 | no | — |
| 264 | #41 | — | 4 | 12 | no | — |
| 265 | #41 | — | 4 | 12 | no | — |
| 266 | C2×Q8 | — | 2 | 3 | no | — |
| 267 | C2×Q8 | — | 2 | 3 | no | — |
| 268 | C2×Q8 | — | 2 | 6 | no | — |
| 269 | C2×Q8 | — | 2 | 3 | no | — |
| 270 | C2×Q8 | — | 2 | 6 | no | — |
| 271 | C2×Q8 | — | 4 | 3 | no | — |
| 272 | C2×Q8 | — | 4 | 6 | no | — |
| 273 | C2×Q8 | — | 4 | 6 | no | — |
| 274 | C2×Q8 | — | 4 | 4 | no | — |
| 275 | C2×Q8 | — | 4 | 4 | no | — |
| 276 | C2×Q8 | — | 8 | 3 | no | — |
| 277 | C2×Q8 | — | 8 | 3 | no | — |
| 278 | C2×Q8 | — | 8 | 3 | no | — |
| 279 | C2×Q8 | — | 8 | 6 | no | — |
| 280 | C2×Q8 | — | 8 | 6 | no | — |
| 281 | C2×Q8 | H(5,𝔽2) | 1 | 6 | yes | — |
| 282 | C2×D4 | — | 2 | 12 | no | — |
| 283 | C2×D4 | — | 2 | 6 | no | — |
| 284 | C2×D4 | — | 2 | 6 | no | — |
| 285 | C2×D4 | — | 2 | 6 | no | — |
| 286 | C2×D4 | — | 2 | 3 | no | — |
| 287 | C2×D4 | — | 4 | 3 | no | — |
| 288 | C2×D4 | — | 4 | 6 | no | — |
| 289 | C2×D4 | — | 4 | 12 | no | — |
| 290 | C2×D4 | — | 4 | 12 | no | — |
| 291 | C2×D4 | — | 4 | 6 | no | — |
| 292 | C2×D4 | — | 4 | 3 | no | — |
| 293 | C2×D4 | — | 4 | 3 | no | — |
| 294 | C2×D4 | — | 4 | 12 | no | — |
| 295 | C2×D4 | — | 4 | 24 | no | — |
| 296 | C2×D4 | — | 4 | 96 | no | — |
| 297 | C2×D4 | — | 4 | 12 | no | — |
| 298 | C2×D4 | — | 4 | 6 | no | — |
| 299 | C2×D4 | — | 4 | 6 | no | — |
| 300 | C2×D4 | — | 8 | 3 | no | — |
| 301 | C2×D4 | H(5,𝔽2) | 1 | 6 | yes | — |
| 302 | C22×C4 | — | 2 | 1 | no | — |
| 303 | C22×C4 | — | 2 | 12 | no | — |
| 304 | C22×C4 | — | 2 | 3 | no | — |
| 305 | C22×C4 | — | 2 | 6 | no | — |
| 306 | C22×C4 | — | 2 | 6 | no | — |
| 307 | C22×C4 | — | 2 | 3 | no | — |
| 308 | C22×C4 | — | 2 | 3 | no | — |
| 309 | C22×C4 | — | 2 | 3 | no | — |
| 310 | C22×C4 | — | 2 | 6 | no | — |
| 311 | C22×C4 | — | 2 | 2 | no | — |
| 312 | C22×C4 | — | 2 | 3 | no | — |
| 313 | C22×C4 | — | 2 | 1 | no | — |
| 314 | C22×C4 | — | 4 | 12 | no | — |
| 315 | C22×C4 | — | 4 | 6 | no | — |
| 316 | C22×C4 | — | 8 | 3 | no | — |
| 317 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 318 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 319 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 320 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 321 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 322 | Q16 | — | 4 | 8 | no | — |
| 323 | Q16 | — | 4 | 8 | no | — |
| 324 | SD16 | — | 8 | 12 | no | — |
| 325 | C2×C8 | — | 2 | 6 | no | — |
| 326 | C2×C8 | — | 2 | 6 | no | — |
| 327 | C2×C8 | — | 2 | 6 | no | — |
| 328 | C2×C8 | — | 2 | 6 | no | — |
| 329 | C2×C8 | — | 4 | 4 | no | — |
| 330 | C4⋊C4 | — | 2 | 6 | no | — |
| 331 | C4⋊C4 | — | 2 | 6 | no | — |
| 332 | C4⋊C4 | — | 2 | 3 | no | — |
| 333 | C4⋊C4 | — | 2 | 3 | no | — |
| 334 | C4⋊C4 | — | 4 | 3 | no | — |
| 335 | C4⋊C4 | — | 4 | 6 | no | — |
| 336 | C4⋊C4 | — | 4 | 6 | no | — |
| 337 | C4⋊C4 | — | 8 | 6 | no | — |
| 338 | C4⋊C4 | — | 8 | 3 | no | — |
| 339 | C22⋊C4 | — | 4 | 6 | no | — |
| 340 | C22⋊C4 | — | 4 | 3 | no | — |
| 341 | C22⋊C4 | — | 4 | 3 | no | — |
| 342 | C22⋊C4 | — | 4 | 6 | no | — |
| 343 | C22⋊C4 | — | 4 | 6 | no | — |
| 344 | C22⋊C4 | — | 4 | 6 | no | — |
| 345 | C22⋊C4 | — | 4 | 6 | no | — |
| 346 | C22⋊C4 | — | 4 | 6 | no | — |
| 347 | C22⋊C4 | — | 8 | 6 | no | — |
| 348 | C22⋊C4 | — | 8 | 6 | no | — |
| 349 | C22⋊C4 | — | 8 | 3 | no | — |
| 350 | C22⋊C4 | — | 8 | 6 | no | — |
| 351 | C22⋊C4 | — | 8 | 6 | no | — |
| 352 | C42 | — | 2 | 3 | no | — |
| 353 | C42 | — | 2 | 3 | no | — |
| 354 | C42 | — | 4 | 6 | no | — |
| 355 | C42 | — | 8 | 3 | no | — |
| 356 | C23 | — | 2 | 3 | no | — |
| 357 | C23 | — | 2 | 3 | no | — |
| 358 | C23 | — | 2 | 6 | no | — |
| 359 | C23 | — | 2 | 3 | no | — |
| 360 | C23 | — | 2 | 3 | no | — |
| 361 | C23 | — | 2 | 6 | no | — |
| 362 | C23 | — | 2 | 6 | no | — |
| 363 | C23 | — | 2 | 3 | no | — |
| 364 | C23 | — | 4 | 24 | no | — |
| 365 | C23 | — | 4 | 6 | no | — |
| 366 | C23 | — | 4 | 3 | no | — |
| 367 | C23 | — | 4 | 6 | no | — |
| 368 | C23 | — | 4 | 3 | no | — |
| 369 | C23 | — | 16 | 12 | no | — |
| 370 | C23 | — | 16 | 12 | no | — |
| 371 | C23 | — | 16 | 6 | no | — |
| 372 | C23 | — | 16 | 6 | no | — |
| 373 | C23 | #576 | 1 | 2 | yes | — |
| 374 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 375 | Q8 | — | 2 | 6 | no | — |
| 376 | Q8 | — | 4 | 4 | no | — |
| 377 | Q8 | — | 4 | 1 | no | — |
| 378 | Q8 | — | 4 | 2 | no | — |
| 379 | Q8 | — | 4 | 6 | no | — |
| 380 | Q8 | — | 4 | 3 | no | — |
| 381 | Q8 | — | 8 | 4 | no | — |
| 382 | Q8 | — | 8 | 4 | no | — |
| 383 | Q8 | — | 16 | 6 | no | — |
| 384 | Q8 | H(3,𝔽4) | 1 | 4 | yes | — |
| 385 | D4 | — | 2 | 12 | no | — |
| 386 | D4 | — | 4 | 24 | no | — |
| 387 | D4 | — | 4 | 24 | no | — |
| 388 | D4 | — | 4 | 12 | no | — |
| 389 | D4 | — | 4 | 3 | no | — |
| 390 | D4 | — | 4 | 3 | no | — |
| 391 | D4 | — | 8 | 6 | no | — |
| 392 | D4 | — | 8 | 12 | no | — |
| 393 | C2×C4 | — | 2 | 6 | no | — |
| 394 | C2×C4 | — | 2 | 3 | no | — |
| 395 | C2×C4 | — | 2 | 6 | no | — |
| 396 | C2×C4 | — | 2 | 3 | no | — |
| 397 | C2×C4 | — | 2 | 3 | no | — |
| 398 | C2×C4 | — | 2 | 3 | no | — |
| 399 | C2×C4 | — | 2 | 3 | no | — |
| 400 | C2×C4 | — | 2 | 6 | no | — |
| 401 | C2×C4 | — | 2 | 3 | no | — |
| 402 | C2×C4 | — | 4 | 24 | no | — |
| 403 | C2×C4 | — | 4 | 1 | no | — |
| 404 | C2×C4 | — | 4 | 3 | no | — |
| 405 | C2×C4 | — | 4 | 3 | no | — |
| 406 | C2×C4 | — | 4 | 6 | no | — |
| 407 | C2×C4 | — | 4 | 3 | no | — |
| 408 | C2×C4 | — | 4 | 3 | no | — |
| 409 | C2×C4 | — | 4 | 6 | no | — |
| 410 | C2×C4 | — | 4 | 3 | no | — |
| 411 | C2×C4 | — | 4 | 3 | no | — |
| 412 | C2×C4 | — | 4 | 6 | no | — |
| 413 | C2×C4 | — | 4 | 6 | no | — |
| 414 | C2×C4 | — | 4 | 6 | no | — |
| 415 | C2×C4 | — | 4 | 3 | no | — |
| 416 | C2×C4 | — | 8 | 4 | no | — |
| 417 | C2×C4 | — | 8 | 4 | no | — |
| 418 | C2×C4 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 419 | C2×C4 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 420 | C8 | — | 4 | 2 | no | — |
| 421 | C8 | — | 4 | 6 | no | — |
| 422 | C22 | — | 2 | 6 | no | — |
| 423 | C22 | — | 4 | 1 | no | — |
| 424 | C22 | — | 4 | 3 | no | — |
| 425 | C22 | — | 4 | 6 | no | — |
| 426 | C22 | — | 4 | 3 | no | — |
| 427 | C22 | — | 4 | 3 | no | — |
| 428 | C22 | — | 4 | 6 | no | — |
| 429 | C22 | — | 4 | 6 | no | — |
| 430 | C22 | — | 4 | 3 | no | — |
| 431 | C22 | — | 4 | 6 | no | — |
| 432 | C22 | — | 4 | 6 | no | — |
| 433 | C22 | — | 8 | 6 | no | — |
| 434 | C22 | — | 8 | 6 | no | — |
| 435 | C22 | — | 16 | 24 | no | — |
| 436 | C22 | #3484 | 1 | 3 | yes | — |
| 437 | C22 | C2×#576 | 1 | 1 | yes | characteristic |
| 438 | C22 | #3586 | 1 | 3 | yes | — |
| 439 | C4 | — | 2 | 4 | no | — |
| 440 | C4 | — | 4 | 3 | no | — |
| 441 | C4 | — | 4 | 1 | no | — |
| 442 | C4 | — | 4 | 3 | no | — |
| 443 | C4 | — | 4 | 3 | no | — |
| 444 | C4 | — | 8 | 1 | no | — |
| 445 | C4 | — | 8 | 2 | no | — |
| 446 | C4 | — | 8 | 3 | no | — |
| 447 | C4 | — | 8 | 6 | no | — |
| 448 | C4 | C2×H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 449 | C4 | #3532 | 1 | 3 | yes | — |
| 450 | C2 | — | 4 | 2 | no | — |
| 451 | C2 | — | 8 | 3 | no | — |
| 452 | C2 | — | 8 | 6 | no | — |
| 453 | C2 | #45439 | 1 | 1 | yes | characteristic |
| 454 | C2 | #61624 | 1 | 3 | yes | — |
| 455 | C2 | #61674 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36082 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #52958 | C2 | semidirect |
| 3. #32886 | C2 | semidirect |
| 4. #32717 | C2 | semidirect |
| 5. #31260 | C2 | semidirect |
| 6. #31065 | C2 | semidirect |
| 7. #30543 | C2 | semidirect |
| 8. #27252 | C2 | semidirect |
| 9. #25896 | C2 | semidirect |
| 10. #23859 | C2 | semidirect |
| 11. #23311 | C2 | semidirect |
| 13. C23×Q16 | C22 | semidirect |
| 15. C2×#489 | C22 | semidirect |
| 16. C2×#477 | C22 | semidirect |
| 18. C2×#448 | C22 | semidirect |
| 27. #2547 | C22 | semidirect |
| 28. #2492 | C22 | semidirect |
| 31. #2409 | C22 | semidirect |
| 37. #2075 | C22 | semidirect |
| 38. #2055 | C22 | semidirect |
| 39. #2036 | C22 | semidirect |
| 40. #2036 | C22 | semidirect |
| 41. #2033 | C22 | semidirect |
| 42. #2023 | C22 | semidirect |
| 43. #2015 | C22 | semidirect |
| 44. #2001 | C22 | semidirect |
| 45. #1983 | C22 | semidirect |
| 46. #1973 | C22 | semidirect |
| 47. #1955 | C22 | semidirect |
| 48. #1925 | C22 | semidirect |
| 49. #1924 | C22 | semidirect |
| 50. #1893 | C22 | semidirect |
| 51. #1878 | C22 | semidirect |
| 52. #1869 | C22 | semidirect |
| 54. #1787 | C22 | semidirect |
| 57. #1763 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #52958 | 2 |
| #36082 | 2 |
| #32886 | 2 |
| #32717 | 2 |
| #31260 | 2 |
| #31065 | 2 |
| #30543 | 2 |
| #27252 | 2 |
| #25896 | 2 |
| #23859 | 2 |
| #23311 | 2 |