Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
10578 conjugacy classes of subgroups.
The subgroup lattice has 261 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×#123 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×#2724 | C2 | 1 | 48 | yes | maximal, direct product factor |
| 3 | C22×#398 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C22×#397 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C22×#390 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C22×#389 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C22×#385 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | C24×D4 | C22 | 1 | 1 | yes | characteristic |
| 9 | C25×C4 | C22 | 1 | 2 | yes | — |
| 10 | C22×#123 | — | 2 | 32 | no | — |
| 11 | C22×#123 | C22 | 1 | 48 | yes | — |
| 12 | C23×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 13 | C23×C4⋊C4 | C22 | 1 | 2 | yes | — |
| 14 | C23×C4⋊C4 | C22 | 1 | 2 | yes | — |
| 15 | C23×C4⋊C4 | C22 | 1 | 2 | yes | — |
| 16 | C23×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 17 | C23×C22⋊C4 | C22 | 1 | 4 | yes | — |
| 18 | C23×C22⋊C4 | C22 | 1 | 2 | yes | — |
| 19 | C23×C22⋊C4 | C22 | 1 | 4 | yes | — |
| 20 | C23×C42 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2724 | C22 | 1 | 256 | yes | direct product factor |
| 22 | C2×#398 | C22 | 1 | 48 | yes | complemented |
| 23 | C2×#397 | C22 | 1 | 96 | yes | complemented |
| 24 | C2×#390 | C22 | 1 | 24 | yes | — |
| 25 | C2×#389 | C22 | 1 | 48 | yes | complemented |
| 26 | C2×#385 | C22 | 1 | 96 | yes | complemented |
| 27 | C22×#95 | C22 | 1 | 4 | yes | — |
| 28 | C22×#95 | C22 | 1 | 4 | yes | — |
| 29 | C22×#95 | C22 | 1 | 2 | yes | — |
| 30 | C26 | C23 | 1 | 2 | yes | — |
| 31 | C23×D4 | — | 4 | 4 | no | — |
| 32 | C23×D4 | — | 4 | 4 | no | — |
| 33 | C23×D4 | D4 | 1 | 16 | yes | — |
| 34 | C23×D4 | C23 | 1 | 12 | yes | — |
| 35 | C24×C4 | — | 2 | 16 | no | — |
| 36 | C24×C4 | — | 4 | 8 | no | — |
| 37 | C24×C4 | — | 4 | 8 | no | — |
| 38 | C24×C4 | C23 | 1 | 4 | yes | — |
| 39 | C24×C4 | C23 | 1 | 4 | yes | — |
| 40 | C24×C4 | C23 | 1 | 2 | yes | — |
| 41 | C24×C4 | C23 | 1 | 1 | yes | characteristic |
| 42 | C24×C4 | C23 | 1 | 2 | yes | — |
| 43 | C24×C4 | C23 | 1 | 24 | yes | — |
| 44 | C2×#123 | — | 2 | 384 | no | — |
| 45 | C2×#123 | C23 | 1 | 128 | yes | — |
| 46 | C22×C4⋊C4 | — | 2 | 32 | no | — |
| 47 | C22×C4⋊C4 | — | 2 | 16 | no | — |
| 48 | C22×C4⋊C4 | — | 4 | 4 | no | — |
| 49 | C22×C4⋊C4 | — | 4 | 8 | no | — |
| 50 | C22×C4⋊C4 | — | 4 | 8 | no | — |
| 51 | C22×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 52 | C22×C4⋊C4 | C23 | 1 | 48 | yes | — |
| 53 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 54 | C22×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 55 | C22×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 56 | C22×C22⋊C4 | — | 2 | 16 | no | — |
| 57 | C22×C22⋊C4 | — | 2 | 32 | no | — |
| 58 | C22×C22⋊C4 | — | 2 | 32 | no | — |
| 59 | C22×C22⋊C4 | C23 | 1 | 24 | yes | — |
| 60 | C22×C22⋊C4 | C23 | 1 | 48 | yes | — |
| 61 | C22×C22⋊C4 | C23 | 1 | 48 | yes | — |
| 62 | C22×C42 | — | 4 | 4 | no | — |
| 63 | C22×C42 | C23 | 1 | 12 | yes | — |
| 64 | #398 | C23 | 1 | 128 | yes | complemented |
| 65 | #397 | C23 | 1 | 256 | yes | complemented |
| 66 | #390 | C23 | 1 | 64 | yes | — |
| 67 | #389 | C23 | 1 | 128 | yes | complemented |
| 68 | #385 | C23 | 1 | 256 | yes | complemented |
| 69 | C2×#95 | C23 | 1 | 24 | yes | — |
| 70 | C2×#95 | C23 | 1 | 48 | yes | — |
| 71 | C2×#95 | C23 | 1 | 48 | yes | — |
| 72 | C25 | — | 2 | 16 | no | — |
| 73 | C25 | — | 2 | 8 | no | — |
| 74 | C25 | — | 2 | 8 | no | — |
| 75 | C25 | — | 2 | 16 | no | — |
| 76 | C25 | C2×D4 | 1 | 16 | yes | — |
| 77 | C25 | C24 | 1 | 12 | yes | — |
| 78 | C25 | C24 | 1 | 1 | yes | characteristic |
| 79 | C22×D4 | — | 4 | 64 | no | — |
| 80 | C22×D4 | — | 4 | 24 | no | — |
| 81 | C22×D4 | — | 4 | 24 | no | — |
| 82 | C22×D4 | C2×D4 | 1 | 96 | yes | — |
| 83 | C22×D4 | C24 | 1 | 16 | yes | — |
| 84 | C23×C4 | — | 2 | 4 | no | — |
| 85 | C23×C4 | — | 2 | 8 | no | — |
| 86 | C23×C4 | — | 2 | 4 | no | — |
| 87 | C23×C4 | — | 2 | 8 | no | — |
| 88 | C23×C4 | — | 2 | 16 | no | — |
| 89 | C23×C4 | — | 2 | 16 | no | — |
| 90 | C23×C4 | — | 2 | 96 | no | — |
| 91 | C23×C4 | — | 2 | 16 | no | — |
| 92 | C23×C4 | — | 2 | 16 | no | — |
| 93 | C23×C4 | — | 2 | 8 | no | — |
| 94 | C23×C4 | — | 2 | 16 | no | — |
| 95 | C23×C4 | — | 2 | 16 | no | — |
| 96 | C23×C4 | — | 2 | 16 | no | — |
| 97 | C23×C4 | — | 4 | 48 | no | — |
| 98 | C23×C4 | — | 4 | 48 | no | — |
| 99 | C23×C4 | — | 4 | 128 | no | — |
| 100 | C23×C4 | C2×D4 | 1 | 8 | yes | — |
| 101 | C23×C4 | #41 | 1 | 16 | yes | — |
| 102 | C23×C4 | C24 | 1 | 24 | yes | — |
| 103 | C23×C4 | C24 | 1 | 12 | yes | — |
| 104 | C23×C4 | C24 | 1 | 24 | yes | — |
| 105 | C23×C4 | C24 | 1 | 6 | yes | — |
| 106 | C23×C4 | C24 | 1 | 12 | yes | — |
| 107 | C23×C4 | C24 | 1 | 32 | yes | — |
| 108 | #123 | — | 2 | 512 | no | — |
| 109 | C2×C4⋊C4 | — | 2 | 192 | no | — |
| 110 | C2×C4⋊C4 | — | 2 | 96 | no | — |
| 111 | C2×C4⋊C4 | — | 4 | 48 | no | — |
| 112 | C2×C4⋊C4 | — | 4 | 24 | no | — |
| 113 | C2×C4⋊C4 | — | 4 | 48 | no | — |
| 114 | C2×C4⋊C4 | C24 | 1 | 64 | yes | — |
| 115 | C2×C4⋊C4 | C24 | 1 | 32 | yes | — |
| 116 | C2×C4⋊C4 | C24 | 1 | 32 | yes | — |
| 117 | C2×C4⋊C4 | C24 | 1 | 32 | yes | — |
| 118 | C2×C4⋊C4 | C24 | 1 | 16 | yes | — |
| 119 | C2×C22⋊C4 | — | 2 | 96 | no | — |
| 120 | C2×C22⋊C4 | — | 2 | 192 | no | — |
| 121 | C2×C22⋊C4 | — | 2 | 192 | no | — |
| 122 | C2×C22⋊C4 | C24 | 1 | 64 | yes | — |
| 123 | C2×C22⋊C4 | C24 | 1 | 32 | yes | — |
| 124 | C2×C22⋊C4 | C24 | 1 | 64 | yes | — |
| 125 | C2×C42 | — | 4 | 24 | no | — |
| 126 | C2×C42 | C24 | 1 | 16 | yes | — |
| 127 | #95 | C24 | 1 | 64 | yes | — |
| 128 | #95 | C24 | 1 | 64 | yes | — |
| 129 | #95 | C24 | 1 | 32 | yes | — |
| 130 | C24 | — | 2 | 64 | no | — |
| 131 | C24 | — | 2 | 128 | no | — |
| 132 | C24 | — | 2 | 48 | no | — |
| 133 | C24 | — | 2 | 24 | no | — |
| 134 | C24 | — | 2 | 24 | no | — |
| 135 | C24 | — | 2 | 48 | no | — |
| 136 | C24 | — | 4 | 64 | no | — |
| 137 | C24 | — | 4 | 32 | no | — |
| 138 | C24 | — | 4 | 32 | no | — |
| 139 | C24 | C22×D4 | 1 | 48 | yes | — |
| 140 | C24 | C22×D4 | 1 | 4 | yes | — |
| 141 | C24 | C2×#41 | 1 | 8 | yes | — |
| 142 | C24 | H(5,𝔽2) | 1 | 4 | yes | — |
| 143 | C24 | H(5,𝔽2) | 1 | 4 | yes | — |
| 144 | C24 | 2−1+4 | 1 | 8 | yes | — |
| 145 | C24 | C25 | 1 | 8 | yes | — |
| 146 | C24 | C25 | 1 | 3 | yes | — |
| 147 | C2×D4 | — | 4 | 192 | no | — |
| 148 | C2×D4 | — | 4 | 16 | no | — |
| 149 | C2×D4 | — | 4 | 16 | no | — |
| 150 | C2×D4 | C22×D4 | 1 | 64 | yes | — |
| 151 | C22×C4 | — | 2 | 32 | no | — |
| 152 | C22×C4 | — | 2 | 64 | no | — |
| 153 | C22×C4 | — | 2 | 24 | no | — |
| 154 | C22×C4 | — | 2 | 12 | no | — |
| 155 | C22×C4 | — | 2 | 12 | no | — |
| 156 | C22×C4 | — | 2 | 24 | no | — |
| 157 | C22×C4 | — | 2 | 48 | no | — |
| 158 | C22×C4 | — | 2 | 48 | no | — |
| 159 | C22×C4 | — | 2 | 64 | no | — |
| 160 | C22×C4 | — | 2 | 48 | no | — |
| 161 | C22×C4 | — | 2 | 48 | no | — |
| 162 | C22×C4 | — | 2 | 24 | no | — |
| 163 | C22×C4 | — | 2 | 48 | no | — |
| 164 | C22×C4 | — | 2 | 48 | no | — |
| 165 | C22×C4 | — | 2 | 48 | no | — |
| 166 | C22×C4 | — | 4 | 32 | no | — |
| 167 | C22×C4 | — | 4 | 32 | no | — |
| 168 | C22×C4 | — | 4 | 384 | no | — |
| 169 | C22×C4 | — | 4 | 64 | no | — |
| 170 | C22×C4 | — | 4 | 64 | no | — |
| 171 | C22×C4 | — | 4 | 32 | no | — |
| 172 | C22×C4 | C22×D4 | 1 | 24 | yes | — |
| 173 | C22×C4 | C2×#41 | 1 | 48 | yes | — |
| 174 | C22×C4 | C25 | 1 | 16 | yes | — |
| 175 | C22×C4 | C25 | 1 | 8 | yes | — |
| 176 | C22×C4 | C25 | 1 | 16 | yes | — |
| 177 | C22×C4 | C25 | 1 | 4 | yes | — |
| 178 | C22×C4 | C25 | 1 | 8 | yes | — |
| 179 | C4⋊C4 | — | 2 | 128 | no | — |
| 180 | C4⋊C4 | — | 2 | 64 | no | — |
| 181 | C4⋊C4 | — | 4 | 32 | no | — |
| 182 | C4⋊C4 | — | 4 | 16 | no | — |
| 183 | C4⋊C4 | — | 4 | 32 | no | — |
| 184 | C22⋊C4 | — | 2 | 64 | no | — |
| 185 | C22⋊C4 | — | 2 | 128 | no | — |
| 186 | C22⋊C4 | — | 2 | 128 | no | — |
| 187 | C42 | — | 4 | 16 | no | — |
| 188 | C23 | — | 2 | 16 | no | — |
| 189 | C23 | — | 2 | 8 | no | — |
| 190 | C23 | — | 2 | 8 | no | — |
| 191 | C23 | — | 2 | 16 | no | — |
| 192 | C23 | — | 2 | 96 | no | — |
| 193 | C23 | — | 2 | 192 | no | — |
| 194 | C23 | — | 4 | 96 | no | — |
| 195 | C23 | — | 4 | 48 | no | — |
| 196 | C23 | — | 4 | 256 | no | — |
| 197 | C23 | — | 4 | 48 | no | — |
| 198 | C23 | #534 | 1 | 16 | yes | — |
| 199 | C23 | #539 | 1 | 32 | yes | — |
| 200 | C23 | #547 | 1 | 32 | yes | — |
| 201 | C23 | #559 | 1 | 16 | yes | — |
| 202 | C23 | #563 | 1 | 16 | yes | — |
| 203 | C23 | C23×D4 | 1 | 16 | yes | — |
| 204 | C23 | C23×D4 | 1 | 6 | yes | — |
| 205 | C23 | C22×#41 | 1 | 12 | yes | — |
| 206 | C23 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 207 | C23 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 208 | C23 | C2×2−1+4 | 1 | 12 | yes | — |
| 209 | C23 | C26 | 1 | 1 | yes | characteristic |
| 210 | D4 | — | 4 | 64 | no | — |
| 211 | C2×C4 | — | 2 | 8 | no | — |
| 212 | C2×C4 | — | 2 | 4 | no | — |
| 213 | C2×C4 | — | 2 | 4 | no | — |
| 214 | C2×C4 | — | 2 | 8 | no | — |
| 215 | C2×C4 | — | 2 | 48 | no | — |
| 216 | C2×C4 | — | 2 | 96 | no | — |
| 217 | C2×C4 | — | 2 | 16 | no | — |
| 218 | C2×C4 | — | 2 | 16 | no | — |
| 219 | C2×C4 | — | 2 | 16 | no | — |
| 220 | C2×C4 | — | 2 | 16 | no | — |
| 221 | C2×C4 | — | 2 | 8 | no | — |
| 222 | C2×C4 | — | 2 | 16 | no | — |
| 223 | C2×C4 | — | 2 | 16 | no | — |
| 224 | C2×C4 | — | 2 | 16 | no | — |
| 225 | C2×C4 | — | 4 | 128 | no | — |
| 226 | C2×C4 | — | 4 | 96 | no | — |
| 227 | C2×C4 | — | 4 | 96 | no | — |
| 228 | C2×C4 | — | 4 | 48 | no | — |
| 229 | C2×C4 | C23×D4 | 1 | 8 | yes | — |
| 230 | C2×C4 | C22×#41 | 1 | 16 | yes | — |
| 231 | C22 | — | 2 | 16 | no | — |
| 232 | C22 | — | 2 | 32 | no | — |
| 233 | C22 | — | 4 | 192 | no | — |
| 234 | C22 | — | 4 | 16 | no | — |
| 235 | C22 | — | 4 | 8 | no | — |
| 236 | C22 | — | 4 | 8 | no | — |
| 237 | C22 | #2724 | 1 | 64 | yes | direct product factor |
| 238 | C22 | C2×#534 | 1 | 12 | yes | — |
| 239 | C22 | C2×#539 | 1 | 24 | yes | — |
| 240 | C22 | C2×#547 | 1 | 24 | yes | — |
| 241 | C22 | C2×#559 | 1 | 12 | yes | — |
| 242 | C22 | C2×#563 | 1 | 12 | yes | — |
| 243 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 244 | C22 | C23×#41 | 1 | 2 | yes | — |
| 245 | C22 | C22×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 246 | C22 | C22×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 247 | C22 | C22×2−1+4 | 1 | 2 | yes | — |
| 248 | C4 | — | 2 | 8 | no | — |
| 249 | C4 | — | 2 | 16 | no | — |
| 250 | C4 | — | 4 | 16 | no | — |
| 251 | C4 | — | 4 | 16 | no | — |
| 252 | C4 | — | 4 | 8 | no | — |
| 253 | C2 | — | 4 | 16 | no | — |
| 254 | C2 | C2×#2724 | 1 | 24 | yes | direct product factor |
| 255 | C2 | C22×#534 | 1 | 1 | yes | characteristic |
| 256 | C2 | C22×#539 | 1 | 2 | yes | — |
| 257 | C2 | C22×#563 | 1 | 1 | yes | characteristic |
| 258 | C2 | C22×#547 | 1 | 2 | yes | — |
| 259 | C2 | C22×#559 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2724 | C2 | direct (N × H) |
| 3. C22×#398 | C2 | semidirect |
| 4. C22×#397 | C2 | semidirect |
| 6. C22×#389 | C2 | semidirect |
| 7. C22×#385 | C2 | semidirect |
| 21. #2724 | C22 | direct (N × H) |
| 22. C2×#398 | C22 | semidirect |
| 23. C2×#397 | C22 | semidirect |
| 25. C2×#389 | C22 | semidirect |
| 26. C2×#385 | C22 | semidirect |
| 64. #398 | C23 | semidirect |
| 65. #397 | C23 | semidirect |
| 67. #389 | C23 | semidirect |
| 68. #385 | C23 | semidirect |
| 237. C22 | #2724 | direct (N × H) |
| 254. C2 | C2×#2724 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#123 | 2 |
| C2×#2724 | 2 |
| C22×#398 | 2 |
| C22×#397 | 2 |
| C22×#390 | 2 |
| C22×#389 | 2 |
| C22×#385 | 2 |