Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5382 conjugacy classes of subgroups.
The subgroup lattice has 274 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #52287 | C2 | 1 | 32 | yes | maximal, direct product factor |
| 2 | C2×#2811 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#2806 | C2 | 1 | 6 | yes | maximal |
| 4 | C2×#2792 | C2 | 1 | 6 | yes | maximal |
| 5 | C2×#2734 | C2 | 1 | 3 | yes | maximal |
| 6 | C2×#2399 | C2 | 1 | 6 | yes | maximal |
| 7 | C2×C4×C4⋊Q8 | C2 | 1 | 3 | yes | maximal |
| 8 | C2×#2295 | C2 | 1 | 6 | yes | maximal |
| 9 | C22×C4⋊Q8 | C22 | 1 | 3 | yes | — |
| 10 | C22×C4×Q8 | C22 | 1 | 2 | yes | — |
| 11 | C22×C4×Q8 | C22 | 1 | 3 | yes | — |
| 12 | C22×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×C4×Q8 | C22 | 1 | 2 | yes | — |
| 14 | #2811 | C22 | 1 | 16 | yes | — |
| 15 | #2806 | C22 | 1 | 96 | yes | — |
| 16 | #2792 | C22 | 1 | 96 | yes | — |
| 17 | #2734 | C22 | 1 | 48 | yes | — |
| 18 | #2399 | C22 | 1 | 96 | yes | — |
| 19 | C2×#398 | C22 | 1 | 12 | yes | — |
| 20 | C2×#398 | C22 | 1 | 6 | yes | — |
| 21 | C2×#398 | C22 | 1 | 6 | yes | — |
| 22 | C2×#395 | C22 | 1 | 6 | yes | — |
| 23 | C2×#395 | C22 | 1 | 6 | yes | — |
| 24 | C2×#395 | C22 | 1 | 12 | yes | — |
| 25 | C4×C4⋊Q8 | C22 | 1 | 48 | yes | — |
| 26 | #2295 | C22 | 1 | 96 | yes | — |
| 27 | C2×#391 | C22 | 1 | 6 | yes | — |
| 28 | C2×#391 | C22 | 1 | 3 | yes | — |
| 29 | C2×#391 | C22 | 1 | 6 | yes | — |
| 30 | C2×#391 | C22 | 1 | 3 | yes | — |
| 31 | C2×#389 | C22 | 1 | 6 | yes | — |
| 32 | C2×#389 | C22 | 1 | 12 | yes | — |
| 33 | C2×#389 | C22 | 1 | 6 | yes | — |
| 34 | C2×#387 | C22 | 1 | 12 | yes | — |
| 35 | C2×#387 | C22 | 1 | 12 | yes | — |
| 36 | C2×#384 | C22 | 1 | 3 | yes | — |
| 37 | C2×C4×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 38 | C2×C4×C4⋊C4 | C22 | 1 | 12 | yes | — |
| 39 | C2×#376 | C22 | 1 | 6 | yes | — |
| 40 | C2×C43 | C22 | 1 | 3 | yes | — |
| 41 | C23×Q8 | C23 | 1 | 3 | yes | — |
| 42 | C23×Q8 | C23 | 1 | 2 | yes | — |
| 43 | C23×Q8 | C23 | 1 | 2 | yes | — |
| 44 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×C4⋊Q8 | — | 2 | 24 | no | — |
| 46 | C2×C4⋊Q8 | C23 | 1 | 24 | yes | — |
| 47 | C2×C4×Q8 | — | 4 | 8 | no | — |
| 48 | C2×C4×Q8 | — | 4 | 12 | no | — |
| 49 | C2×C4×Q8 | — | 4 | 4 | no | — |
| 50 | C2×C4×Q8 | Q8 | 1 | 32 | yes | complemented |
| 51 | C2×C4×Q8 | C23 | 1 | 8 | yes | — |
| 52 | C2×C4×Q8 | C23 | 1 | 24 | yes | — |
| 53 | C2×C4×Q8 | C23 | 1 | 16 | yes | — |
| 54 | C2×C4×Q8 | C23 | 1 | 16 | yes | — |
| 55 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 56 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 57 | C22×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 58 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 59 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 60 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 61 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 62 | C22×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 63 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 64 | C22×C42 | C23 | 1 | 3 | yes | — |
| 65 | C22×C42 | C23 | 1 | 3 | yes | — |
| 66 | C22×C42 | C23 | 1 | 6 | yes | — |
| 67 | C22×C42 | C23 | 1 | 6 | yes | — |
| 68 | C22×C42 | C23 | 1 | 3 | yes | — |
| 69 | C22×C42 | C23 | 1 | 6 | yes | — |
| 70 | #398 | C23 | 1 | 48 | yes | — |
| 71 | #398 | C23 | 1 | 96 | yes | — |
| 72 | #398 | C23 | 1 | 48 | yes | — |
| 73 | #395 | C23 | 1 | 96 | yes | — |
| 74 | #395 | C23 | 1 | 48 | yes | — |
| 75 | #395 | C23 | 1 | 48 | yes | — |
| 76 | #391 | C23 | 1 | 24 | yes | — |
| 77 | #391 | C23 | 1 | 24 | yes | — |
| 78 | #391 | C23 | 1 | 48 | yes | — |
| 79 | #391 | C23 | 1 | 48 | yes | — |
| 80 | #389 | C23 | 1 | 96 | yes | — |
| 81 | #389 | C23 | 1 | 48 | yes | — |
| 82 | #389 | C23 | 1 | 48 | yes | — |
| 83 | #387 | C23 | 1 | 96 | yes | — |
| 84 | #387 | C23 | 1 | 96 | yes | — |
| 85 | #384 | C23 | 1 | 24 | yes | — |
| 86 | C4×C4⋊C4 | C23 | 1 | 96 | yes | — |
| 87 | C4×C4⋊C4 | C23 | 1 | 48 | yes | — |
| 88 | #376 | C23 | 1 | 48 | yes | — |
| 89 | C2×#95 | C23 | 1 | 12 | yes | — |
| 90 | C2×#95 | C23 | 1 | 12 | yes | — |
| 91 | C2×#95 | C23 | 1 | 6 | yes | — |
| 92 | C2×#95 | C23 | 1 | 12 | yes | — |
| 93 | C2×#95 | C23 | 1 | 6 | yes | — |
| 94 | C2×#95 | C23 | 1 | 12 | yes | — |
| 95 | C43 | C23 | 1 | 24 | yes | — |
| 96 | C22×Q8 | — | 2 | 12 | no | — |
| 97 | C22×Q8 | — | 4 | 2 | no | — |
| 98 | C22×Q8 | — | 4 | 6 | no | — |
| 99 | C22×Q8 | — | 4 | 4 | no | — |
| 100 | C22×Q8 | — | 4 | 4 | no | — |
| 101 | C22×Q8 | — | 4 | 4 | no | — |
| 102 | C22×Q8 | — | 4 | 2 | no | — |
| 103 | C22×Q8 | — | 4 | 6 | no | — |
| 104 | C22×Q8 | — | 4 | 8 | no | — |
| 105 | C22×Q8 | C2×Q8 | 1 | 16 | yes | — |
| 106 | C22×Q8 | #41 | 1 | 8 | yes | — |
| 107 | C22×Q8 | C24 | 1 | 12 | yes | — |
| 108 | C22×Q8 | C24 | 1 | 8 | yes | — |
| 109 | C22×Q8 | C24 | 1 | 8 | yes | — |
| 110 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 111 | C23×C4 | C24 | 1 | 3 | yes | — |
| 112 | C23×C4 | C24 | 1 | 6 | yes | — |
| 113 | C23×C4 | C24 | 1 | 6 | yes | — |
| 114 | C23×C4 | C24 | 1 | 6 | yes | — |
| 115 | C23×C4 | C24 | 1 | 6 | yes | — |
| 116 | C23×C4 | C24 | 1 | 3 | yes | — |
| 117 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 118 | C4⋊Q8 | — | 2 | 96 | no | — |
| 119 | C4×Q8 | — | 4 | 16 | no | — |
| 120 | C4×Q8 | — | 4 | 48 | no | — |
| 121 | C4×Q8 | — | 4 | 32 | no | — |
| 122 | C4×Q8 | C2×Q8 | 1 | 128 | yes | complemented |
| 123 | C2×C4⋊C4 | — | 2 | 48 | no | — |
| 124 | C2×C4⋊C4 | — | 4 | 24 | no | — |
| 125 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 126 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 127 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 128 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 129 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 130 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 131 | C2×C4⋊C4 | C2×Q8 | 1 | 48 | yes | — |
| 132 | C2×C4⋊C4 | C24 | 1 | 48 | yes | — |
| 133 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 134 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 135 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 136 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 137 | C2×C4⋊C4 | C24 | 1 | 48 | yes | — |
| 138 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 139 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 140 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 141 | C2×C42 | — | 2 | 12 | no | — |
| 142 | C2×C42 | — | 2 | 12 | no | — |
| 143 | C2×C42 | — | 2 | 24 | no | — |
| 144 | C2×C42 | — | 4 | 12 | no | — |
| 145 | C2×C42 | — | 4 | 6 | no | — |
| 146 | C2×C42 | C2×Q8 | 1 | 48 | yes | — |
| 147 | C2×C42 | C24 | 1 | 12 | yes | — |
| 148 | C2×C42 | C24 | 1 | 24 | yes | — |
| 149 | C2×C42 | C24 | 1 | 12 | yes | — |
| 150 | C2×C42 | C24 | 1 | 24 | yes | — |
| 151 | C2×C42 | C24 | 1 | 12 | yes | — |
| 152 | C2×C42 | C24 | 1 | 24 | yes | — |
| 153 | #95 | C24 | 1 | 24 | yes | — |
| 154 | #95 | C24 | 1 | 48 | yes | — |
| 155 | #95 | C24 | 1 | 24 | yes | — |
| 156 | #95 | C24 | 1 | 48 | yes | — |
| 157 | #95 | C24 | 1 | 48 | yes | — |
| 158 | #95 | C24 | 1 | 48 | yes | — |
| 159 | C24 | C25 | 1 | 1 | yes | characteristic |
| 160 | C2×Q8 | — | 2 | 24 | no | — |
| 161 | C2×Q8 | — | 4 | 4 | no | — |
| 162 | C2×Q8 | — | 4 | 12 | no | — |
| 163 | C2×Q8 | — | 4 | 8 | no | — |
| 164 | C2×Q8 | — | 4 | 32 | no | — |
| 165 | C2×Q8 | — | 4 | 16 | no | — |
| 166 | C2×Q8 | — | 4 | 8 | no | — |
| 167 | C2×Q8 | — | 4 | 8 | no | — |
| 168 | C2×Q8 | — | 4 | 4 | no | — |
| 169 | C2×Q8 | — | 4 | 12 | no | — |
| 170 | C2×Q8 | — | 4 | 16 | no | — |
| 171 | C2×Q8 | — | 4 | 48 | no | — |
| 172 | C2×Q8 | — | 4 | 32 | no | — |
| 173 | C2×Q8 | C22×Q8 | 1 | 32 | yes | — |
| 174 | C2×Q8 | C2×#41 | 1 | 16 | yes | — |
| 175 | C22×C4 | — | 2 | 12 | no | — |
| 176 | C22×C4 | — | 2 | 6 | no | — |
| 177 | C22×C4 | — | 2 | 12 | no | — |
| 178 | C22×C4 | — | 2 | 12 | no | — |
| 179 | C22×C4 | — | 2 | 12 | no | — |
| 180 | C22×C4 | — | 2 | 6 | no | — |
| 181 | C22×C4 | — | 2 | 12 | no | — |
| 182 | C22×C4 | — | 2 | 6 | no | — |
| 183 | C22×C4 | — | 2 | 12 | no | — |
| 184 | C22×C4 | — | 2 | 12 | no | — |
| 185 | C22×C4 | — | 2 | 12 | no | — |
| 186 | C22×C4 | — | 2 | 12 | no | — |
| 187 | C22×C4 | — | 2 | 24 | no | — |
| 188 | C22×C4 | C22×Q8 | 1 | 24 | yes | — |
| 189 | C22×C4 | C22×Q8 | 1 | 24 | yes | — |
| 190 | C22×C4 | C22×Q8 | 1 | 8 | yes | — |
| 191 | C22×C4 | C2×#41 | 1 | 12 | yes | — |
| 192 | C22×C4 | H(5,𝔽2) | 1 | 4 | yes | — |
| 193 | C22×C4 | C25 | 1 | 12 | yes | — |
| 194 | C22×C4 | C25 | 1 | 12 | yes | — |
| 195 | C22×C4 | C25 | 1 | 6 | yes | — |
| 196 | C22×C4 | C25 | 1 | 12 | yes | — |
| 197 | C22×C4 | C25 | 1 | 12 | yes | — |
| 198 | C22×C4 | C25 | 1 | 6 | yes | — |
| 199 | C22×C4 | C25 | 1 | 2 | yes | — |
| 200 | C4⋊C4 | — | 2 | 96 | no | — |
| 201 | C4⋊C4 | — | 4 | 48 | no | — |
| 202 | C4⋊C4 | — | 4 | 24 | no | — |
| 203 | C4⋊C4 | — | 4 | 24 | no | — |
| 204 | C4⋊C4 | — | 4 | 12 | no | — |
| 205 | C4⋊C4 | — | 4 | 12 | no | — |
| 206 | C4⋊C4 | — | 4 | 24 | no | — |
| 207 | C4⋊C4 | — | 4 | 24 | no | — |
| 208 | C4⋊C4 | C22×Q8 | 1 | 96 | yes | — |
| 209 | C42 | — | 2 | 24 | no | — |
| 210 | C42 | — | 2 | 24 | no | — |
| 211 | C42 | — | 2 | 48 | no | — |
| 212 | C42 | — | 4 | 24 | no | — |
| 213 | C42 | — | 4 | 12 | no | — |
| 214 | C42 | C22×Q8 | 1 | 96 | yes | — |
| 215 | C23 | C23×Q8 | 1 | 4 | yes | — |
| 216 | C23 | C22×#41 | 1 | 2 | yes | — |
| 217 | C23 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 218 | C23 | #585 | 1 | 2 | yes | — |
| 219 | C23 | #585 | 1 | 4 | yes | — |
| 220 | C23 | C26 | 1 | 1 | yes | characteristic |
| 221 | Q8 | — | 4 | 32 | no | — |
| 222 | Q8 | — | 4 | 16 | no | — |
| 223 | Q8 | — | 4 | 48 | no | — |
| 224 | Q8 | — | 4 | 32 | no | — |
| 225 | C2×C4 | — | 2 | 12 | no | — |
| 226 | C2×C4 | — | 2 | 6 | no | — |
| 227 | C2×C4 | — | 2 | 12 | no | — |
| 228 | C2×C4 | — | 2 | 12 | no | — |
| 229 | C2×C4 | — | 2 | 12 | no | — |
| 230 | C2×C4 | — | 2 | 6 | no | — |
| 231 | C2×C4 | — | 2 | 48 | no | — |
| 232 | C2×C4 | — | 2 | 24 | no | — |
| 233 | C2×C4 | — | 2 | 48 | no | — |
| 234 | C2×C4 | — | 2 | 12 | no | — |
| 235 | C2×C4 | — | 2 | 6 | no | — |
| 236 | C2×C4 | — | 2 | 12 | no | — |
| 237 | C2×C4 | — | 2 | 12 | no | — |
| 238 | C2×C4 | — | 2 | 12 | no | — |
| 239 | C2×C4 | — | 2 | 12 | no | — |
| 240 | C2×C4 | — | 2 | 24 | no | — |
| 241 | C2×C4 | — | 4 | 24 | no | — |
| 242 | C2×C4 | — | 4 | 12 | no | — |
| 243 | C2×C4 | — | 4 | 24 | no | — |
| 244 | C2×C4 | Q82 | 1 | 16 | yes | — |
| 245 | C2×C4 | C23×Q8 | 1 | 24 | yes | — |
| 246 | C2×C4 | C23×Q8 | 1 | 24 | yes | — |
| 247 | C2×C4 | C23×Q8 | 1 | 8 | yes | — |
| 248 | C2×C4 | C22×#41 | 1 | 12 | yes | — |
| 249 | C2×C4 | C2×H(5,𝔽2) | 1 | 4 | yes | — |
| 250 | C22 | #3458 | 1 | 8 | yes | — |
| 251 | C22 | #3459 | 1 | 4 | yes | — |
| 252 | C22 | C2×Q82 | 1 | 4 | yes | — |
| 253 | C22 | Q8×#41 | 1 | 8 | yes | — |
| 254 | C22 | #3529 | 1 | 4 | yes | — |
| 255 | C22 | C24×Q8 | 1 | 2 | yes | — |
| 256 | C22 | C23×#41 | 1 | 1 | yes | characteristic |
| 257 | C22 | C22×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 258 | C22 | C2×#585 | 1 | 1 | yes | characteristic |
| 259 | C22 | C2×#585 | 1 | 2 | yes | — |
| 260 | C4 | — | 2 | 12 | no | — |
| 261 | C4 | — | 2 | 24 | no | — |
| 262 | C4 | — | 2 | 24 | no | — |
| 263 | C4 | — | 4 | 12 | no | — |
| 264 | C4 | — | 4 | 6 | no | — |
| 265 | C4 | — | 4 | 12 | no | — |
| 266 | C4 | C2×Q82 | 1 | 8 | yes | — |
| 267 | C2 | #52287 | 1 | 8 | yes | direct product factor |
| 268 | C2 | C2×#3459 | 1 | 1 | yes | characteristic |
| 269 | C2 | C2×#3458 | 1 | 2 | yes | — |
| 270 | C2 | C2×#3529 | 1 | 1 | yes | characteristic |
| 271 | C2 | C22×Q82 | 1 | 1 | yes | characteristic |
| 272 | C2 | C2×Q8×#41 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| socle, prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #52287 | C2 | direct (N × H) |
| 50. C2×C4×Q8 | Q8 | semidirect |
| 122. C4×Q8 | C2×Q8 | semidirect |
| 267. C2 | #52287 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #52287 | 2 |
| C2×#2811 | 2 |
| C2×#2806 | 2 |
| C2×#2792 | 2 |
| C2×#2734 | 2 |
| C2×#2399 | 2 |
| C2×C4×C4⋊Q8 | 2 |
| C2×#2295 | 2 |