Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5854 conjugacy classes of subgroups.
The subgroup lattice has 246 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×C4×Q8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×#389 | C2 | 1 | 3 | yes | maximal |
| 3 | #15202 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 4 | C2×#1446 | C2 | 1 | 2 | yes | maximal |
| 5 | C2×#1444 | C2 | 1 | 6 | yes | maximal |
| 6 | C2×C4⋊#95 | C2 | 1 | 3 | yes | maximal |
| 7 | C24×Q8 | C22 | 1 | 1 | yes | characteristic |
| 8 | C22×C4×Q8 | — | 2 | 24 | no | — |
| 9 | C22×C4×Q8 | C22 | 1 | 8 | yes | — |
| 10 | C23×C4⋊C4 | C22 | 1 | 3 | yes | — |
| 11 | C23×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 12 | C23×C4⋊C4 | C22 | 1 | 3 | yes | — |
| 13 | C23×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 14 | C23×C42 | C22 | 1 | 3 | yes | — |
| 15 | C2×#389 | — | 2 | 24 | no | — |
| 16 | C2×#389 | C22 | 1 | 24 | yes | — |
| 17 | C22×#95 | C22 | 1 | 6 | yes | — |
| 18 | C22×#95 | C22 | 1 | 6 | yes | — |
| 19 | C22×#95 | C22 | 1 | 6 | yes | — |
| 20 | #1446 | C22 | 1 | 16 | yes | — |
| 21 | #1444 | C22 | 1 | 48 | yes | — |
| 22 | C4⋊#95 | C22 | 1 | 24 | yes | — |
| 23 | C23×Q8 | — | 2 | 12 | no | — |
| 24 | C23×Q8 | — | 2 | 12 | no | — |
| 25 | C23×Q8 | Q8 | 1 | 8 | yes | — |
| 26 | C23×Q8 | C23 | 1 | 4 | yes | — |
| 27 | C24×C4 | C23 | 1 | 2 | yes | — |
| 28 | C24×C4 | C23 | 1 | 3 | yes | — |
| 29 | C24×C4 | C23 | 1 | 6 | yes | — |
| 30 | C24×C4 | C23 | 1 | 3 | yes | — |
| 31 | C24×C4 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×C4×Q8 | — | 2 | 128 | no | — |
| 33 | C2×C4×Q8 | — | 2 | 96 | no | — |
| 34 | C22×C4⋊C4 | — | 2 | 24 | no | — |
| 35 | C22×C4⋊C4 | — | 2 | 12 | no | — |
| 36 | C22×C4⋊C4 | — | 2 | 12 | no | — |
| 37 | C22×C4⋊C4 | — | 2 | 12 | no | — |
| 38 | C22×C4⋊C4 | — | 2 | 24 | no | — |
| 39 | C22×C4⋊C4 | — | 2 | 12 | no | — |
| 40 | C22×C4⋊C4 | — | 4 | 24 | no | — |
| 41 | C22×C4⋊C4 | Q8 | 1 | 24 | yes | complemented |
| 42 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 43 | C22×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 44 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 45 | C22×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 46 | C22×C42 | — | 2 | 24 | no | — |
| 47 | C22×C42 | D4 | 1 | 24 | yes | — |
| 48 | C22×C42 | C23 | 1 | 12 | yes | — |
| 49 | #389 | — | 2 | 96 | no | — |
| 50 | C2×#95 | — | 2 | 24 | no | — |
| 51 | C2×#95 | — | 2 | 24 | no | — |
| 52 | C2×#95 | — | 4 | 12 | no | — |
| 53 | C2×#95 | C23 | 1 | 24 | yes | — |
| 54 | C2×#95 | C23 | 1 | 24 | yes | — |
| 55 | C2×#95 | C23 | 1 | 24 | yes | — |
| 56 | C25 | C24 | 1 | 1 | yes | characteristic |
| 57 | C22×Q8 | — | 2 | 32 | no | — |
| 58 | C22×Q8 | — | 2 | 96 | no | — |
| 59 | C22×Q8 | — | 2 | 96 | no | — |
| 60 | C22×Q8 | — | 2 | 24 | no | — |
| 61 | C22×Q8 | — | 2 | 24 | no | — |
| 62 | C22×Q8 | C2×Q8 | 1 | 16 | yes | — |
| 63 | C23×C4 | — | 2 | 12 | no | — |
| 64 | C23×C4 | — | 2 | 12 | no | — |
| 65 | C23×C4 | — | 2 | 12 | no | — |
| 66 | C23×C4 | — | 2 | 12 | no | — |
| 67 | C23×C4 | — | 2 | 12 | no | — |
| 68 | C23×C4 | — | 2 | 24 | no | — |
| 69 | C23×C4 | — | 2 | 12 | no | — |
| 70 | C23×C4 | — | 2 | 24 | no | — |
| 71 | C23×C4 | — | 2 | 6 | no | — |
| 72 | C23×C4 | — | 2 | 6 | no | — |
| 73 | C23×C4 | — | 2 | 12 | no | — |
| 74 | C23×C4 | — | 2 | 12 | no | — |
| 75 | C23×C4 | — | 2 | 4 | no | — |
| 76 | C23×C4 | C2×D4 | 1 | 12 | yes | — |
| 77 | C23×C4 | C2×D4 | 1 | 12 | yes | — |
| 78 | C23×C4 | C2×D4 | 1 | 12 | yes | — |
| 79 | C23×C4 | C2×Q8 | 1 | 24 | yes | — |
| 80 | C23×C4 | C2×Q8 | 1 | 12 | yes | — |
| 81 | C23×C4 | C2×Q8 | 1 | 12 | yes | — |
| 82 | C23×C4 | #41 | 1 | 12 | yes | — |
| 83 | C23×C4 | #41 | 1 | 4 | yes | — |
| 84 | C23×C4 | C24 | 1 | 4 | yes | — |
| 85 | C23×C4 | C24 | 1 | 6 | yes | — |
| 86 | C23×C4 | C24 | 1 | 12 | yes | — |
| 87 | C23×C4 | C24 | 1 | 6 | yes | — |
| 88 | C23×C4 | C24 | 1 | 2 | yes | — |
| 89 | C4×Q8 | — | 2 | 256 | no | — |
| 90 | C2×C4⋊C4 | — | 2 | 96 | no | — |
| 91 | C2×C4⋊C4 | — | 2 | 96 | no | — |
| 92 | C2×C4⋊C4 | — | 2 | 48 | no | — |
| 93 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 94 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 95 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 96 | C2×C4⋊C4 | — | 2 | 48 | no | — |
| 97 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 98 | C2×C4⋊C4 | — | 4 | 48 | no | — |
| 99 | C2×C4⋊C4 | — | 4 | 96 | no | — |
| 100 | C2×C4⋊C4 | — | 4 | 16 | no | — |
| 101 | C2×C4⋊C4 | C2×Q8 | 1 | 48 | yes | complemented |
| 102 | C2×C42 | — | 2 | 96 | no | — |
| 103 | C2×C42 | — | 2 | 48 | no | — |
| 104 | C2×C42 | C2×D4 | 1 | 48 | yes | — |
| 105 | #95 | — | 2 | 48 | no | — |
| 106 | #95 | — | 2 | 48 | no | — |
| 107 | #95 | — | 4 | 24 | no | — |
| 108 | C24 | C22×D4 | 1 | 6 | yes | — |
| 109 | C24 | C22×Q8 | 1 | 2 | yes | — |
| 110 | C24 | C22×Q8 | 1 | 6 | yes | — |
| 111 | C24 | C2×#41 | 1 | 2 | yes | — |
| 112 | C24 | C2×#41 | 1 | 6 | yes | — |
| 113 | C24 | H(5,𝔽2) | 1 | 6 | yes | — |
| 114 | C24 | 2−1+4 | 1 | 2 | yes | — |
| 115 | C24 | C25 | 1 | 1 | yes | characteristic |
| 116 | C2×Q8 | — | 2 | 32 | no | — |
| 117 | C2×Q8 | — | 2 | 256 | no | — |
| 118 | C2×Q8 | — | 2 | 96 | no | — |
| 119 | C2×Q8 | — | 2 | 96 | no | — |
| 120 | C22×C4 | — | 2 | 24 | no | — |
| 121 | C22×C4 | — | 2 | 48 | no | — |
| 122 | C22×C4 | — | 2 | 24 | no | — |
| 123 | C22×C4 | — | 2 | 48 | no | — |
| 124 | C22×C4 | — | 2 | 12 | no | — |
| 125 | C22×C4 | — | 2 | 12 | no | — |
| 126 | C22×C4 | — | 2 | 24 | no | — |
| 127 | C22×C4 | — | 2 | 48 | no | — |
| 128 | C22×C4 | — | 2 | 24 | no | — |
| 129 | C22×C4 | — | 2 | 24 | no | — |
| 130 | C22×C4 | — | 2 | 96 | no | — |
| 131 | C22×C4 | — | 2 | 12 | no | — |
| 132 | C22×C4 | — | 2 | 12 | no | — |
| 133 | C22×C4 | — | 2 | 24 | no | — |
| 134 | C22×C4 | — | 2 | 12 | no | — |
| 135 | C22×C4 | — | 2 | 6 | no | — |
| 136 | C22×C4 | — | 2 | 24 | no | — |
| 137 | C22×C4 | — | 2 | 12 | no | — |
| 138 | C22×C4 | — | 2 | 6 | no | — |
| 139 | C22×C4 | — | 2 | 12 | no | — |
| 140 | C22×C4 | — | 2 | 12 | no | — |
| 141 | C22×C4 | — | 2 | 4 | no | — |
| 142 | C22×C4 | — | 4 | 24 | no | — |
| 143 | C22×C4 | — | 4 | 24 | no | — |
| 144 | C22×C4 | — | 4 | 24 | no | — |
| 145 | C22×C4 | — | 4 | 24 | no | — |
| 146 | C22×C4 | — | 4 | 48 | no | — |
| 147 | C22×C4 | — | 4 | 8 | no | — |
| 148 | C22×C4 | — | 4 | 24 | no | — |
| 149 | C22×C4 | — | 4 | 24 | no | — |
| 150 | C22×C4 | #120 | 1 | 16 | yes | — |
| 151 | C22×C4 | C22⋊Q8 | 1 | 48 | yes | — |
| 152 | C22×C4 | C22⋊Q8 | 1 | 48 | yes | — |
| 153 | C22×C4 | C4⋊Q8 | 1 | 48 | yes | — |
| 154 | C22×C4 | C22×D4 | 1 | 12 | yes | — |
| 155 | C22×C4 | C22×D4 | 1 | 12 | yes | — |
| 156 | C22×C4 | C22×D4 | 1 | 12 | yes | — |
| 157 | C22×C4 | C22×Q8 | 1 | 24 | yes | — |
| 158 | C22×C4 | C22×Q8 | 1 | 12 | yes | — |
| 159 | C22×C4 | C22×Q8 | 1 | 12 | yes | — |
| 160 | C22×C4 | C2×#41 | 1 | 12 | yes | — |
| 161 | C22×C4 | C2×#41 | 1 | 4 | yes | — |
| 162 | C4⋊C4 | — | 2 | 96 | no | — |
| 163 | C4⋊C4 | — | 2 | 96 | no | — |
| 164 | C4⋊C4 | — | 4 | 96 | no | — |
| 165 | C4⋊C4 | — | 4 | 16 | no | — |
| 166 | C42 | — | 2 | 96 | no | — |
| 167 | C23 | C2×#120 | 1 | 4 | yes | — |
| 168 | C23 | C2×C22⋊Q8 | 1 | 12 | yes | — |
| 169 | C23 | C2×C22⋊Q8 | 1 | 12 | yes | — |
| 170 | C23 | #525 | 1 | 12 | yes | — |
| 171 | C23 | C2×C4⋊Q8 | 1 | 12 | yes | — |
| 172 | C23 | #535 | 1 | 12 | yes | — |
| 173 | C23 | #537 | 1 | 12 | yes | — |
| 174 | C23 | #554 | 1 | 24 | yes | — |
| 175 | C23 | #557 | 1 | 4 | yes | — |
| 176 | C23 | #557 | 1 | 12 | yes | — |
| 177 | C23 | Q82 | 1 | 12 | yes | — |
| 178 | C23 | #559 | 1 | 12 | yes | — |
| 179 | C23 | C23×D4 | 1 | 3 | yes | — |
| 180 | C23 | C23×Q8 | 1 | 1 | yes | characteristic |
| 181 | C23 | C23×Q8 | 1 | 3 | yes | — |
| 182 | C23 | C22×#41 | 1 | 1 | yes | characteristic |
| 183 | C23 | C22×#41 | 1 | 3 | yes | — |
| 184 | C23 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 185 | C23 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 186 | Q8 | — | 2 | 128 | no | — |
| 187 | C2×C4 | — | 2 | 96 | no | — |
| 188 | C2×C4 | — | 2 | 12 | no | — |
| 189 | C2×C4 | — | 2 | 24 | no | — |
| 190 | C2×C4 | — | 2 | 12 | no | — |
| 191 | C2×C4 | — | 2 | 24 | no | — |
| 192 | C2×C4 | — | 2 | 12 | no | — |
| 193 | C2×C4 | — | 2 | 24 | no | — |
| 194 | C2×C4 | — | 2 | 12 | no | — |
| 195 | C2×C4 | — | 2 | 12 | no | — |
| 196 | C2×C4 | — | 2 | 48 | no | — |
| 197 | C2×C4 | — | 4 | 96 | no | — |
| 198 | C2×C4 | — | 4 | 48 | no | — |
| 199 | C2×C4 | — | 4 | 12 | no | — |
| 200 | C2×C4 | — | 4 | 12 | no | — |
| 201 | C2×C4 | — | 4 | 12 | no | — |
| 202 | C2×C4 | — | 4 | 12 | no | — |
| 203 | C2×C4 | — | 4 | 24 | no | — |
| 204 | C2×C4 | — | 4 | 4 | no | — |
| 205 | C2×C4 | — | 4 | 12 | no | — |
| 206 | C2×C4 | — | 4 | 12 | no | — |
| 207 | C2×C4 | — | 8 | 16 | no | — |
| 208 | C2×C4 | #395 | 1 | 64 | yes | — |
| 209 | C2×C4 | C2×#120 | 1 | 8 | yes | — |
| 210 | C2×C4 | C2×C22⋊Q8 | 1 | 24 | yes | — |
| 211 | C2×C4 | C2×C22⋊Q8 | 1 | 24 | yes | — |
| 212 | C2×C4 | C2×C4⋊Q8 | 1 | 24 | yes | — |
| 213 | C22 | C2×#395 | 1 | 8 | yes | — |
| 214 | C22 | #2567 | 1 | 24 | yes | — |
| 215 | C22 | #2580 | 1 | 24 | yes | — |
| 216 | C22 | #2582 | 1 | 24 | yes | — |
| 217 | C22 | #2584 | 1 | 24 | yes | — |
| 218 | C22 | #2615 | 1 | 8 | yes | — |
| 219 | C22 | #2848 | 1 | 8 | yes | — |
| 220 | C22 | C22×#120 | 1 | 1 | yes | characteristic |
| 221 | C22 | C22×C22⋊Q8 | 1 | 3 | yes | — |
| 222 | C22 | C22×C22⋊Q8 | 1 | 3 | yes | — |
| 223 | C22 | C2×#525 | 1 | 3 | yes | — |
| 224 | C22 | C22×C4⋊Q8 | 1 | 3 | yes | — |
| 225 | C22 | C2×#535 | 1 | 3 | yes | — |
| 226 | C22 | C2×#537 | 1 | 3 | yes | — |
| 227 | C22 | C2×#554 | 1 | 6 | yes | — |
| 228 | C22 | C2×#557 | 1 | 3 | yes | — |
| 229 | C22 | C2×#557 | 1 | 1 | yes | characteristic |
| 230 | C22 | C2×Q82 | 1 | 3 | yes | — |
| 231 | C22 | C2×#559 | 1 | 3 | yes | — |
| 232 | C4 | — | 2 | 24 | no | — |
| 233 | C4 | — | 4 | 12 | no | — |
| 234 | C4 | — | 4 | 24 | no | — |
| 235 | C4 | — | 8 | 4 | no | — |
| 236 | C4 | C2×#395 | 1 | 16 | yes | — |
| 237 | C2 | #15202 | 1 | 16 | yes | direct product factor |
| 238 | C2 | C22×#395 | 1 | 1 | yes | characteristic |
| 239 | C2 | C2×#2615 | 1 | 1 | yes | characteristic |
| 240 | C2 | C2×#2580 | 1 | 3 | yes | — |
| 241 | C2 | C2×#2584 | 1 | 3 | yes | — |
| 242 | C2 | C2×#2567 | 1 | 3 | yes | — |
| 243 | C2 | C2×#2582 | 1 | 3 | yes | — |
| 244 | C2 | C2×#2848 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #15202 | C2 | direct (N × H) |
| 41. C22×C4⋊C4 | Q8 | semidirect |
| 101. C2×C4⋊C4 | C2×Q8 | semidirect |
| 237. C2 | #15202 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×Q8 | 2 |
| C22×#389 | 2 |
| #15202 | 2 |
| C2×#1446 | 2 |
| C2×#1444 | 2 |
| C2×C4⋊#95 | 2 |