Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5394 conjugacy classes of subgroups.
The subgroup lattice has 235 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #46327 | C2 | 1 | 16 | yes | maximal, complemented |
| 2 | #46020 | C2 | 1 | 16 | yes | maximal, complemented |
| 3 | #46002 | C2 | 1 | 16 | yes | maximal, complemented |
| 4 | C2×#2868 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #36641 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #35674 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #35667 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #35666 | C2 | 1 | 3 | yes | maximal |
| 9 | C22×2−1+4 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#536 | C22 | 1 | 6 | yes | — |
| 11 | C22×C4⋊Q8 | C22 | 1 | 3 | yes | — |
| 12 | C22×C4⋊Q8 | C22 | 1 | 3 | yes | — |
| 13 | C2×#519 | C22 | 1 | 3 | yes | — |
| 14 | #2868 | C22 | 1 | 16 | yes | — |
| 15 | #2863 | C22 | 1 | 8 | yes | — |
| 16 | #2860 | C22 | 1 | 8 | yes | — |
| 17 | #2852 | C22 | 1 | 16 | yes | — |
| 18 | #2843 | C22 | 1 | 16 | yes | — |
| 19 | #2823 | C22 | 1 | 16 | yes | — |
| 20 | #2795 | C22 | 1 | 48 | yes | — |
| 21 | #2792 | C22 | 1 | 48 | yes | — |
| 22 | #2790 | C22 | 1 | 96 | yes | — |
| 23 | #2546 | C22 | 1 | 24 | yes | — |
| 24 | #2545 | C22 | 1 | 24 | yes | — |
| 25 | #2484 | C22 | 1 | 24 | yes | — |
| 26 | #2446 | C22 | 1 | 48 | yes | — |
| 27 | #2434 | C22 | 1 | 24 | yes | — |
| 28 | #2432 | C22 | 1 | 24 | yes | — |
| 29 | #2429 | C22 | 1 | 48 | yes | — |
| 30 | #2413 | C22 | 1 | 48 | yes | — |
| 31 | #2406 | C22 | 1 | 24 | yes | — |
| 32 | #2383 | C22 | 1 | 32 | yes | — |
| 33 | #2310 | C22 | 1 | 3 | yes | — |
| 34 | #2310 | C22 | 1 | 3 | yes | — |
| 35 | C4×#126 | C22 | 1 | 16 | yes | — |
| 36 | #2297 | C22 | 1 | 6 | yes | — |
| 37 | #2296 | C22 | 1 | 6 | yes | — |
| 38 | C2×#384 | C22 | 1 | 8 | yes | — |
| 39 | C2×C43 | C22 | 1 | 1 | yes | characteristic |
| 40 | C2×2−1+4 | — | 4 | 12 | no | — |
| 41 | C22×#41 | C23 | 1 | 2 | yes | — |
| 42 | C22×#41 | C23 | 1 | 4 | yes | — |
| 43 | C22×#41 | C23 | 1 | 4 | yes | — |
| 44 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 45 | C23×Q8 | C23 | 1 | 3 | yes | — |
| 46 | C23×Q8 | C23 | 1 | 4 | yes | — |
| 47 | #536 | — | 4 | 24 | no | — |
| 48 | C2×C4⋊Q8 | — | 4 | 12 | no | — |
| 49 | C2×C4⋊Q8 | — | 4 | 12 | no | — |
| 50 | C2×C4⋊Q8 | C23 | 1 | 24 | yes | — |
| 51 | C2×C4⋊Q8 | C23 | 1 | 24 | yes | — |
| 52 | C2×C4⋊Q8 | C23 | 1 | 12 | yes | — |
| 53 | C2×#126 | C23 | 1 | 16 | yes | — |
| 54 | C2×#124 | C23 | 1 | 12 | yes | — |
| 55 | C2×#123 | C23 | 1 | 24 | yes | — |
| 56 | C2×C22⋊Q8 | C23 | 1 | 24 | yes | — |
| 57 | #519 | — | 4 | 12 | no | — |
| 58 | C2×C4×Q8 | C23 | 1 | 24 | yes | — |
| 59 | C2×#117 | C23 | 1 | 6 | yes | — |
| 60 | C2×#117 | C23 | 1 | 12 | yes | — |
| 61 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 62 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 63 | C22×C42 | C23 | 1 | 4 | yes | — |
| 64 | C22×C42 | C23 | 1 | 3 | yes | — |
| 65 | #400 | C23 | 1 | 48 | yes | — |
| 66 | #398 | C23 | 1 | 48 | yes | — |
| 67 | #398 | C23 | 1 | 96 | yes | — |
| 68 | #396 | C23 | 1 | 96 | yes | — |
| 69 | #395 | C23 | 1 | 48 | yes | — |
| 70 | #394 | C23 | 1 | 48 | yes | — |
| 71 | #391 | C23 | 1 | 24 | yes | — |
| 72 | #391 | C23 | 1 | 24 | yes | — |
| 73 | #391 | C23 | 1 | 12 | yes | — |
| 74 | #390 | C23 | 1 | 12 | yes | — |
| 75 | #389 | C23 | 1 | 24 | yes | — |
| 76 | #389 | C23 | 1 | 96 | yes | — |
| 77 | #388 | C23 | 1 | 48 | yes | — |
| 78 | #388 | C23 | 1 | 96 | yes | — |
| 79 | #387 | C23 | 1 | 48 | yes | — |
| 80 | #387 | C23 | 1 | 96 | yes | — |
| 81 | #386 | C23 | 1 | 24 | yes | — |
| 82 | #385 | C23 | 1 | 24 | yes | — |
| 83 | #385 | C23 | 1 | 96 | yes | — |
| 84 | #384 | C23 | 1 | 16 | yes | — |
| 85 | #384 | C23 | 1 | 48 | yes | — |
| 86 | #383 | C23 | 1 | 16 | yes | — |
| 87 | #381 | C23 | 1 | 12 | yes | — |
| 88 | #381 | C23 | 1 | 12 | yes | — |
| 89 | C4×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 90 | C4×C22⋊C4 | C23 | 1 | 24 | yes | — |
| 91 | C43 | C23 | 1 | 8 | yes | — |
| 92 | 2−1+4 | — | 4 | 64 | no | — |
| 93 | C2×#41 | — | 2 | 24 | no | — |
| 94 | C2×#41 | — | 4 | 24 | no | — |
| 95 | C2×#41 | — | 4 | 24 | no | — |
| 96 | C22×Q8 | — | 2 | 12 | no | — |
| 97 | C22×Q8 | — | 2 | 24 | no | — |
| 98 | C22×Q8 | — | 2 | 12 | no | — |
| 99 | C22×Q8 | — | 4 | 24 | no | — |
| 100 | C22×Q8 | C24 | 1 | 12 | yes | — |
| 101 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 102 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 103 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 104 | C22×D4 | C24 | 1 | 4 | yes | — |
| 105 | C22×D4 | C24 | 1 | 2 | yes | — |
| 106 | C22×D4 | C24 | 1 | 4 | yes | — |
| 107 | C23×C4 | C24 | 1 | 6 | yes | — |
| 108 | C23×C4 | C24 | 1 | 2 | yes | — |
| 109 | C23×C4 | C24 | 1 | 6 | yes | — |
| 110 | C23×C4 | C24 | 1 | 4 | yes | — |
| 111 | C23×C4 | C24 | 1 | 8 | yes | — |
| 112 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 113 | C4⋊Q8 | — | 4 | 48 | no | — |
| 114 | C4⋊Q8 | — | 4 | 48 | no | — |
| 115 | C4⋊Q8 | — | 4 | 24 | no | — |
| 116 | #126 | — | 2 | 64 | no | — |
| 117 | #124 | — | 4 | 24 | no | — |
| 118 | #123 | — | 4 | 48 | no | — |
| 119 | C22⋊Q8 | — | 4 | 48 | no | — |
| 120 | C4×Q8 | — | 4 | 48 | no | — |
| 121 | #117 | — | 2 | 24 | no | — |
| 122 | #117 | — | 4 | 24 | no | — |
| 123 | C2×C4⋊C4 | — | 4 | 24 | no | — |
| 124 | C2×C4⋊C4 | — | 4 | 24 | no | — |
| 125 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 126 | C2×C4⋊C4 | C24 | 1 | 48 | yes | — |
| 127 | C2×C4⋊C4 | C24 | 1 | 48 | yes | — |
| 128 | C2×C4⋊C4 | C24 | 1 | 48 | yes | — |
| 129 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 130 | C2×C4⋊C4 | C24 | 1 | 48 | yes | — |
| 131 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 132 | C2×C22⋊C4 | C24 | 1 | 48 | yes | — |
| 133 | C2×C22⋊C4 | C24 | 1 | 48 | yes | — |
| 134 | C2×C42 | — | 2 | 12 | no | — |
| 135 | C2×C42 | — | 2 | 16 | no | — |
| 136 | C2×C42 | C24 | 1 | 12 | yes | — |
| 137 | C2×C42 | C24 | 1 | 12 | yes | — |
| 138 | C2×C42 | C24 | 1 | 12 | yes | — |
| 139 | C2×C42 | C24 | 1 | 24 | yes | — |
| 140 | C2×C42 | C24 | 1 | 4 | yes | — |
| 141 | #95 | C24 | 1 | 48 | yes | — |
| 142 | #95 | C24 | 1 | 48 | yes | — |
| 143 | #95 | C24 | 1 | 48 | yes | — |
| 144 | #95 | C24 | 1 | 48 | yes | — |
| 145 | C24 | C25 | 1 | 4 | yes | — |
| 146 | C24 | C25 | 1 | 1 | yes | characteristic |
| 147 | #41 | — | 4 | 64 | no | — |
| 148 | #41 | — | 4 | 64 | no | — |
| 149 | #41 | — | 4 | 32 | no | — |
| 150 | C2×Q8 | — | 2 | 24 | no | — |
| 151 | C2×Q8 | — | 2 | 24 | no | — |
| 152 | C2×Q8 | — | 2 | 24 | no | — |
| 153 | C2×Q8 | — | 2 | 24 | no | — |
| 154 | C2×Q8 | — | 2 | 12 | no | — |
| 155 | C2×Q8 | — | 4 | 12 | no | — |
| 156 | C2×Q8 | — | 4 | 64 | no | — |
| 157 | C2×Q8 | — | 4 | 48 | no | — |
| 158 | C2×Q8 | — | 4 | 16 | no | — |
| 159 | C2×D4 | — | 2 | 24 | no | — |
| 160 | C2×D4 | — | 2 | 12 | no | — |
| 161 | C2×D4 | — | 4 | 12 | no | — |
| 162 | C22×C4 | — | 2 | 12 | no | — |
| 163 | C22×C4 | — | 2 | 24 | no | — |
| 164 | C22×C4 | — | 2 | 12 | no | — |
| 165 | C22×C4 | — | 2 | 24 | no | — |
| 166 | C22×C4 | — | 2 | 12 | no | — |
| 167 | C22×C4 | — | 2 | 24 | no | — |
| 168 | C22×C4 | — | 4 | 24 | no | — |
| 169 | C22×C4 | H(5,𝔽2) | 1 | 12 | yes | — |
| 170 | C22×C4 | C25 | 1 | 24 | yes | — |
| 171 | C22×C4 | C25 | 1 | 4 | yes | — |
| 172 | C22×C4 | C25 | 1 | 4 | yes | — |
| 173 | C22×C4 | C25 | 1 | 12 | yes | — |
| 174 | C22×C4 | C25 | 1 | 12 | yes | — |
| 175 | C22×C4 | C25 | 1 | 2 | yes | — |
| 176 | C4⋊C4 | — | 2 | 24 | no | — |
| 177 | C4⋊C4 | — | 2 | 96 | no | — |
| 178 | C4⋊C4 | — | 4 | 48 | no | — |
| 179 | C4⋊C4 | — | 4 | 48 | no | — |
| 180 | C4⋊C4 | — | 4 | 24 | no | — |
| 181 | C4⋊C4 | — | 4 | 48 | no | — |
| 182 | C22⋊C4 | — | 2 | 24 | no | — |
| 183 | C22⋊C4 | — | 2 | 96 | no | — |
| 184 | C22⋊C4 | — | 4 | 48 | no | — |
| 185 | C42 | — | 2 | 24 | no | — |
| 186 | C42 | — | 2 | 24 | no | — |
| 187 | C42 | — | 2 | 24 | no | — |
| 188 | C42 | — | 4 | 24 | no | — |
| 189 | C42 | 2−1+4 | 1 | 16 | yes | complemented |
| 190 | C23 | — | 2 | 4 | no | — |
| 191 | C23 | — | 2 | 12 | no | — |
| 192 | C23 | — | 2 | 16 | no | — |
| 193 | C23 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 194 | C23 | C26 | 1 | 1 | yes | characteristic |
| 195 | Q8 | — | 4 | 16 | no | — |
| 196 | Q8 | — | 4 | 48 | no | — |
| 197 | Q8 | — | 4 | 16 | no | — |
| 198 | Q8 | — | 4 | 8 | no | — |
| 199 | D4 | — | 4 | 16 | no | — |
| 200 | D4 | — | 4 | 16 | no | — |
| 201 | D4 | — | 4 | 8 | no | — |
| 202 | C2×C4 | — | 2 | 48 | no | — |
| 203 | C2×C4 | — | 2 | 12 | no | — |
| 204 | C2×C4 | — | 2 | 12 | no | — |
| 205 | C2×C4 | — | 2 | 24 | no | — |
| 206 | C2×C4 | — | 2 | 12 | no | — |
| 207 | C2×C4 | — | 2 | 24 | no | — |
| 208 | C2×C4 | — | 2 | 12 | no | — |
| 209 | C2×C4 | — | 2 | 12 | no | — |
| 210 | C2×C4 | — | 2 | 48 | no | — |
| 211 | C2×C4 | — | 4 | 32 | no | — |
| 212 | C2×C4 | — | 4 | 24 | no | — |
| 213 | C2×C4 | — | 4 | 8 | no | — |
| 214 | C2×C4 | — | 4 | 16 | no | — |
| 215 | C2×C4 | #560 | 1 | 16 | yes | — |
| 216 | C2×C4 | C2×H(5,𝔽2) | 1 | 12 | yes | — |
| 217 | C2×C4 | C2×2−1+4 | 1 | 24 | yes | — |
| 218 | C22 | — | 2 | 12 | no | — |
| 219 | C22 | — | 4 | 24 | no | — |
| 220 | C22 | — | 4 | 4 | no | — |
| 221 | C22 | C2×#560 | 1 | 4 | yes | — |
| 222 | C22 | C22×H(5,𝔽2) | 1 | 3 | yes | — |
| 223 | C22 | C22×2−1+4 | 1 | 4 | yes | — |
| 224 | C4 | — | 2 | 24 | no | — |
| 225 | C4 | — | 4 | 24 | no | — |
| 226 | C4 | — | 4 | 4 | no | — |
| 227 | C4 | — | 4 | 4 | no | — |
| 228 | C4 | — | 4 | 12 | no | — |
| 229 | C4 | C2×#560 | 1 | 8 | yes | — |
| 230 | C2 | — | 2 | 4 | no | — |
| 231 | C2 | — | 8 | 4 | no | — |
| 232 | C2 | C22×#560 | 1 | 1 | yes | characteristic |
| 233 | C2 | #62833 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #46327 | C2 | semidirect |
| 2. #46020 | C2 | semidirect |
| 3. #46002 | C2 | semidirect |
| 4. C2×#2868 | C2 | semidirect |
| 5. #36641 | C2 | semidirect |
| 6. #35674 | C2 | semidirect |
| 7. #35667 | C2 | semidirect |
| 189. C42 | 2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #46327 | 2 |
| #46020 | 2 |
| #46002 | 2 |
| C2×#2868 | 2 |
| #36641 | 2 |
| #35674 | 2 |
| #35667 | 2 |
| #35666 | 2 |