Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1950 conjugacy classes of subgroups.
The subgroup lattice has 245 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#2645 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#2276 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #33093 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 4 | C2×#1928 | C2 | 1 | 3 | yes | maximal |
| 5 | C2×#1918 | C2 | 1 | 3 | yes | maximal |
| 6 | C2×#1878 | C2 | 1 | 3 | yes | maximal |
| 7 | C2×#1863 | C2 | 1 | 3 | yes | maximal |
| 8 | C2×#1840 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C23×Q16 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×C4×Q8 | C22 | 1 | 3 | yes | — |
| 11 | C22×#107 | C22 | 1 | 3 | yes | — |
| 12 | C22×#103 | C22 | 1 | 3 | yes | — |
| 13 | C22×#103 | C22 | 1 | 3 | yes | — |
| 14 | #2645 | — | 2 | 4 | no | — |
| 15 | #2645 | C22 | 1 | 8 | yes | — |
| 16 | C2×#401 | — | 2 | 1 | no | — |
| 17 | C2×#400 | — | 2 | 3 | no | — |
| 18 | C2×#391 | C22 | 1 | 3 | yes | — |
| 19 | C2×#389 | C22 | 1 | 3 | yes | — |
| 20 | #2276 | — | 2 | 4 | no | — |
| 21 | #2276 | C22 | 1 | 8 | yes | — |
| 22 | C2×C4×C4⋊C4 | — | 2 | 3 | no | — |
| 23 | C2×C4×C4⋊C4 | C22 | 1 | 3 | yes | — |
| 24 | C2×#376 | — | 2 | 1 | no | — |
| 25 | C2×#376 | C22 | 1 | 1 | yes | characteristic |
| 26 | #1928 | C22 | 1 | 24 | yes | — |
| 27 | #1918 | C22 | 1 | 24 | yes | — |
| 28 | #1878 | C22 | 1 | 24 | yes | — |
| 29 | #1863 | C22 | 1 | 24 | yes | — |
| 30 | #1840 | C22 | 1 | 8 | yes | — |
| 31 | C2×#340 | C22 | 1 | 3 | yes | — |
| 32 | C2×#340 | C22 | 1 | 3 | yes | — |
| 33 | C2×#336 | C22 | 1 | 3 | yes | — |
| 34 | C2×#336 | C22 | 1 | 3 | yes | — |
| 35 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 36 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 37 | C22×Q16 | — | 2 | 12 | no | — |
| 38 | C22×Q16 | C23 | 1 | 4 | yes | — |
| 39 | C23×C8 | C23 | 1 | 3 | yes | — |
| 40 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×C4×Q8 | — | 2 | 12 | no | — |
| 42 | C2×C4×Q8 | — | 2 | 12 | no | — |
| 43 | C2×C4×Q8 | — | 2 | 6 | no | — |
| 44 | C2×C4×Q8 | C23 | 1 | 12 | yes | — |
| 45 | C22×C4⋊C4 | — | 2 | 3 | no | — |
| 46 | C22×C4⋊C4 | — | 2 | 3 | no | — |
| 47 | C22×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 48 | C22×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 49 | C22×C42 | — | 2 | 3 | no | — |
| 50 | C22×C42 | — | 2 | 3 | no | — |
| 51 | C22×C42 | C23 | 1 | 3 | yes | — |
| 52 | C2×#107 | — | 2 | 12 | no | — |
| 53 | C2×#107 | C23 | 1 | 12 | yes | — |
| 54 | C2×#103 | — | 2 | 12 | no | — |
| 55 | C2×#103 | — | 2 | 12 | no | — |
| 56 | C2×#103 | C23 | 1 | 12 | yes | — |
| 57 | C2×#103 | C23 | 1 | 12 | yes | — |
| 58 | #401 | — | 2 | 8 | no | — |
| 59 | #400 | — | 2 | 24 | no | — |
| 60 | #391 | — | 2 | 6 | no | — |
| 61 | #391 | C23 | 1 | 12 | yes | — |
| 62 | #389 | — | 2 | 6 | no | — |
| 63 | #389 | C23 | 1 | 12 | yes | — |
| 64 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 65 | C4×C4⋊C4 | — | 2 | 24 | no | — |
| 66 | C4×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 67 | #376 | — | 2 | 8 | no | — |
| 68 | #376 | D4 | 1 | 4 | yes | — |
| 69 | #376 | C23 | 1 | 4 | yes | — |
| 70 | C2×#95 | — | 2 | 3 | no | — |
| 71 | C2×#95 | — | 2 | 3 | no | — |
| 72 | C2×#95 | — | 2 | 3 | no | — |
| 73 | C2×#95 | — | 2 | 3 | no | — |
| 74 | C2×#95 | D4 | 1 | 3 | yes | — |
| 75 | C2×#95 | D4 | 1 | 1 | yes | characteristic |
| 76 | #340 | C23 | 1 | 12 | yes | — |
| 77 | #340 | C23 | 1 | 12 | yes | — |
| 78 | #336 | C23 | 1 | 12 | yes | — |
| 79 | #336 | C23 | 1 | 12 | yes | — |
| 80 | C22×Q8 | — | 2 | 1 | no | — |
| 81 | C22×Q8 | — | 2 | 1 | no | — |
| 82 | C22×Q8 | — | 2 | 6 | no | — |
| 83 | C22×Q8 | — | 2 | 6 | no | — |
| 84 | C22×Q8 | — | 2 | 6 | no | — |
| 85 | C22×Q8 | #41 | 1 | 12 | yes | — |
| 86 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 87 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 88 | C23×C4 | — | 2 | 1 | no | — |
| 89 | C23×C4 | — | 2 | 3 | no | — |
| 90 | C23×C4 | — | 2 | 3 | no | — |
| 91 | C23×C4 | — | 2 | 1 | no | — |
| 92 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 93 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 94 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 95 | C2×Q16 | — | 2 | 24 | no | — |
| 96 | C2×Q16 | — | 4 | 16 | no | — |
| 97 | C22×C8 | — | 2 | 6 | no | — |
| 98 | C22×C8 | — | 2 | 6 | no | — |
| 99 | C22×C8 | — | 2 | 6 | no | — |
| 100 | C22×C8 | — | 2 | 6 | no | — |
| 101 | C22×C8 | C24 | 1 | 2 | yes | — |
| 102 | C22×C8 | C24 | 1 | 6 | yes | — |
| 103 | C4×Q8 | — | 2 | 48 | no | — |
| 104 | C4×Q8 | — | 2 | 24 | no | — |
| 105 | C4×Q8 | — | 2 | 24 | no | — |
| 106 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 107 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 108 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 109 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 110 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 111 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 112 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 113 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 114 | C2×C4⋊C4 | — | 8 | 6 | no | — |
| 115 | C2×C4⋊C4 | #41 | 1 | 12 | yes | — |
| 116 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 117 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 118 | C2×C42 | — | 2 | 24 | no | — |
| 119 | C2×C42 | — | 2 | 12 | no | — |
| 120 | C2×C42 | — | 2 | 6 | no | — |
| 121 | C2×C42 | — | 2 | 6 | no | — |
| 122 | C2×C42 | — | 2 | 12 | no | — |
| 123 | C2×C42 | — | 8 | 6 | no | — |
| 124 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 125 | C2×C42 | C24 | 1 | 6 | yes | — |
| 126 | #107 | — | 2 | 24 | no | — |
| 127 | #103 | — | 2 | 24 | no | — |
| 128 | #103 | — | 2 | 24 | no | — |
| 129 | #95 | — | 2 | 12 | no | — |
| 130 | #95 | — | 2 | 12 | no | — |
| 131 | #95 | — | 2 | 12 | no | — |
| 132 | #95 | — | 2 | 12 | no | — |
| 133 | #95 | C2×D4 | 1 | 4 | yes | — |
| 134 | #95 | C2×D4 | 1 | 12 | yes | — |
| 135 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 136 | C2×Q8 | — | 2 | 12 | no | — |
| 137 | C2×Q8 | — | 2 | 24 | no | — |
| 138 | C2×Q8 | — | 2 | 6 | no | — |
| 139 | C2×Q8 | — | 2 | 6 | no | — |
| 140 | C2×Q8 | — | 2 | 6 | no | — |
| 141 | C2×Q8 | — | 4 | 6 | no | — |
| 142 | C2×Q8 | — | 4 | 4 | no | — |
| 143 | C2×Q8 | — | 4 | 12 | no | — |
| 144 | C2×Q8 | #126 | 1 | 16 | yes | — |
| 145 | C2×Q8 | C2×#41 | 1 | 12 | yes | — |
| 146 | C22×C4 | — | 2 | 12 | no | — |
| 147 | C22×C4 | — | 2 | 2 | no | — |
| 148 | C22×C4 | — | 2 | 6 | no | — |
| 149 | C22×C4 | — | 2 | 6 | no | — |
| 150 | C22×C4 | — | 2 | 12 | no | — |
| 151 | C22×C4 | — | 2 | 6 | no | — |
| 152 | C22×C4 | — | 2 | 6 | no | — |
| 153 | C22×C4 | — | 2 | 12 | no | — |
| 154 | C22×C4 | — | 2 | 6 | no | — |
| 155 | C22×C4 | — | 2 | 2 | no | — |
| 156 | C22×C4 | — | 2 | 6 | no | — |
| 157 | C22×C4 | — | 4 | 6 | no | — |
| 158 | C22×C4 | — | 4 | 6 | no | — |
| 159 | C22×C4 | — | 4 | 6 | no | — |
| 160 | C22×C4 | — | 4 | 6 | no | — |
| 161 | C22×C4 | — | 4 | 6 | no | — |
| 162 | C22×C4 | — | 4 | 6 | no | — |
| 163 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 164 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 165 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 166 | C22×C4 | C2×#41 | 1 | 6 | yes | — |
| 167 | C22×C4 | C2×#41 | 1 | 6 | yes | — |
| 168 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 169 | Q16 | — | 4 | 16 | no | — |
| 170 | C2×C8 | — | 2 | 6 | no | — |
| 171 | C2×C8 | — | 2 | 6 | no | — |
| 172 | C2×C8 | — | 2 | 6 | no | — |
| 173 | C2×C8 | — | 2 | 6 | no | — |
| 174 | C2×C8 | — | 4 | 4 | no | — |
| 175 | C2×C8 | — | 4 | 12 | no | — |
| 176 | C4⋊C4 | — | 2 | 48 | no | — |
| 177 | C4⋊C4 | — | 2 | 12 | no | — |
| 178 | C4⋊C4 | — | 2 | 6 | no | — |
| 179 | C4⋊C4 | — | 2 | 6 | no | — |
| 180 | C4⋊C4 | — | 2 | 6 | no | — |
| 181 | C4⋊C4 | — | 4 | 6 | no | — |
| 182 | C4⋊C4 | — | 8 | 12 | no | — |
| 183 | C4⋊C4 | C2×#41 | 1 | 12 | yes | — |
| 184 | C42 | — | 2 | 48 | no | — |
| 185 | C42 | — | 2 | 12 | no | — |
| 186 | C42 | — | 2 | 6 | no | — |
| 187 | C42 | — | 2 | 6 | no | — |
| 188 | C42 | — | 8 | 12 | no | — |
| 189 | C23 | #525 | 1 | 6 | yes | — |
| 190 | C23 | #576 | 1 | 2 | yes | — |
| 191 | C23 | #578 | 1 | 6 | yes | — |
| 192 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 193 | Q8 | — | 2 | 16 | no | — |
| 194 | Q8 | — | 2 | 12 | no | — |
| 195 | Q8 | — | 4 | 2 | no | — |
| 196 | Q8 | — | 4 | 6 | no | — |
| 197 | Q8 | — | 8 | 4 | no | — |
| 198 | Q8 | C2×#126 | 1 | 8 | yes | — |
| 199 | C2×C4 | — | 2 | 16 | no | — |
| 200 | C2×C4 | — | 2 | 12 | no | — |
| 201 | C2×C4 | — | 2 | 6 | no | — |
| 202 | C2×C4 | — | 2 | 6 | no | — |
| 203 | C2×C4 | — | 2 | 12 | no | — |
| 204 | C2×C4 | — | 2 | 6 | no | — |
| 205 | C2×C4 | — | 2 | 6 | no | — |
| 206 | C2×C4 | — | 2 | 12 | no | — |
| 207 | C2×C4 | — | 4 | 12 | no | — |
| 208 | C2×C4 | — | 4 | 24 | no | — |
| 209 | C2×C4 | — | 4 | 12 | no | — |
| 210 | C2×C4 | — | 4 | 6 | no | — |
| 211 | C2×C4 | — | 4 | 6 | no | — |
| 212 | C2×C4 | — | 4 | 6 | no | — |
| 213 | C2×C4 | — | 4 | 6 | no | — |
| 214 | C2×C4 | — | 4 | 6 | no | — |
| 215 | C2×C4 | — | 4 | 6 | no | — |
| 216 | C2×C4 | — | 8 | 4 | no | — |
| 217 | C2×C4 | — | 8 | 12 | no | — |
| 218 | C2×C4 | #525 | 1 | 6 | yes | — |
| 219 | C2×C4 | C2×#126 | 1 | 4 | yes | — |
| 220 | C2×C4 | #529 | 1 | 12 | yes | — |
| 221 | C2×C4 | C22×#41 | 1 | 3 | yes | — |
| 222 | C2×C4 | C22×#41 | 1 | 3 | yes | — |
| 223 | C8 | — | 4 | 2 | no | — |
| 224 | C8 | — | 4 | 6 | no | — |
| 225 | C22 | #2852 | 1 | 4 | yes | — |
| 226 | C22 | #3124 | 1 | 12 | yes | — |
| 227 | C22 | #3127 | 1 | 12 | yes | — |
| 228 | C22 | C2×#525 | 1 | 3 | yes | — |
| 229 | C22 | C2×#576 | 1 | 1 | yes | characteristic |
| 230 | C22 | C2×#578 | 1 | 3 | yes | — |
| 231 | C4 | — | 2 | 8 | no | — |
| 232 | C4 | — | 4 | 6 | no | — |
| 233 | C4 | — | 4 | 12 | no | — |
| 234 | C4 | — | 4 | 6 | no | — |
| 235 | C4 | — | 8 | 2 | no | — |
| 236 | C4 | — | 8 | 6 | no | — |
| 237 | C4 | #2852 | 1 | 4 | yes | — |
| 238 | C4 | C22×#126 | 1 | 1 | yes | characteristic |
| 239 | C4 | C2×#529 | 1 | 3 | yes | — |
| 240 | C2 | #33093 | 1 | 8 | yes | direct product factor |
| 241 | C2 | C2×#2852 | 1 | 1 | yes | characteristic |
| 242 | C2 | C2×#3124 | 1 | 3 | yes | — |
| 243 | C2 | C2×#3127 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #33093 | C2 | direct (N × H) |
| 240. C2 | #33093 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2645 | 2 |
| C2×#2276 | 2 |
| #33093 | 2 |
| C2×#1928 | 2 |
| C2×#1918 | 2 |
| C2×#1878 | 2 |
| C2×#1863 | 2 |
| C2×#1840 | 2 |