Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1487 conjugacy classes of subgroups.
The subgroup lattice has 269 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #45375 | C2 | 1 | 2 | yes | maximal |
| 2 | #32468 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #29295 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #28707 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #27658 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #26310 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | D4×#106 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #21722 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #21683 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C2×#499 | C22 | 1 | 2 | yes | — |
| 11 | C2×C8×D4 | C22 | 1 | 4 | yes | — |
| 12 | C2×C8×D4 | C22 | 1 | 2 | yes | — |
| 13 | C22×#106 | C22 | 1 | 4 | yes | — |
| 14 | #2807 | — | 2 | 4 | no | — |
| 15 | #2771 | — | 2 | 4 | no | — |
| 16 | #2600 | — | 2 | 4 | no | — |
| 17 | #2562 | C22 | 1 | 8 | yes | — |
| 18 | #2487 | C22 | 1 | 4 | yes | — |
| 19 | #2486 | C22 | 1 | 4 | yes | — |
| 20 | #2455 | — | 2 | 4 | no | — |
| 21 | #2397 | C22 | 1 | 1 | yes | characteristic |
| 22 | D4×C4⋊C4 | C22 | 1 | 8 | yes | — |
| 23 | #2330 | C22 | 1 | 4 | yes | — |
| 24 | #2088 | C22 | 1 | 8 | yes | — |
| 25 | #2077 | C22 | 1 | 8 | yes | — |
| 26 | #2076 | C22 | 1 | 8 | yes | — |
| 27 | #2075 | C22 | 1 | 8 | yes | — |
| 28 | #1992 | C22 | 1 | 4 | yes | — |
| 29 | #1989 | C22 | 1 | 8 | yes | — |
| 30 | #1955 | C22 | 1 | 4 | yes | — |
| 31 | #1955 | C22 | 1 | 2 | yes | — |
| 32 | #1948 | C22 | 1 | 8 | yes | — |
| 33 | #1943 | C22 | 1 | 4 | yes | — |
| 34 | #1941 | C22 | 1 | 4 | yes | — |
| 35 | #1853 | C22 | 1 | 8 | yes | — |
| 36 | #1853 | C22 | 1 | 8 | yes | — |
| 37 | #1844 | C22 | 1 | 2 | yes | — |
| 38 | #1843 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1824 | C22 | 1 | 4 | yes | — |
| 40 | #1823 | C22 | 1 | 4 | yes | — |
| 41 | #1807 | C22 | 1 | 4 | yes | — |
| 42 | #1807 | C22 | 1 | 8 | yes | — |
| 43 | #1776 | C22 | 1 | 4 | yes | — |
| 44 | C4×#106 | C22 | 1 | 4 | yes | — |
| 45 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 46 | C23×C8 | C23 | 1 | 4 | yes | — |
| 47 | C2×#125 | C23 | 1 | 4 | yes | — |
| 48 | C2×#123 | — | 2 | 4 | no | — |
| 49 | C2×C4×D4 | — | 2 | 8 | no | — |
| 50 | C2×C4×D4 | C23 | 1 | 4 | yes | — |
| 51 | C2×C4×D4 | C23 | 1 | 2 | yes | — |
| 52 | C22×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 53 | C22×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 54 | #499 | — | 2 | 8 | no | — |
| 55 | C8×D4 | — | 2 | 16 | no | — |
| 56 | C8×D4 | — | 4 | 4 | no | — |
| 57 | C2×#107 | C23 | 1 | 4 | yes | — |
| 58 | C2×#106 | — | 4 | 8 | no | — |
| 59 | C2×#106 | C23 | 1 | 4 | yes | — |
| 60 | C2×#106 | C23 | 1 | 4 | yes | — |
| 61 | C2×#106 | C23 | 1 | 8 | yes | — |
| 62 | C2×#106 | C23 | 1 | 4 | yes | — |
| 63 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 64 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 65 | C2×C22⋊C8 | C23 | 1 | 8 | yes | — |
| 66 | C2×C22⋊C8 | C23 | 1 | 4 | yes | — |
| 67 | C2×#97 | C23 | 1 | 2 | yes | — |
| 68 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 69 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 70 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 71 | #400 | — | 2 | 8 | no | — |
| 72 | #400 | — | 2 | 4 | no | — |
| 73 | #399 | — | 2 | 4 | no | — |
| 74 | #399 | D4 | 1 | 4 | yes | — |
| 75 | #398 | — | 2 | 4 | no | — |
| 76 | #397 | — | 2 | 4 | no | — |
| 77 | #394 | — | 2 | 8 | no | — |
| 78 | #392 | D4 | 1 | 4 | yes | — |
| 79 | #389 | — | 2 | 8 | no | — |
| 80 | #389 | — | 2 | 8 | no | — |
| 81 | #389 | C23 | 1 | 8 | yes | — |
| 82 | #388 | — | 2 | 8 | no | — |
| 83 | #388 | — | 2 | 8 | no | — |
| 84 | #387 | — | 2 | 8 | no | — |
| 85 | #386 | — | 2 | 4 | no | — |
| 86 | #385 | — | 2 | 8 | no | — |
| 87 | #385 | — | 2 | 8 | no | — |
| 88 | #385 | — | 2 | 4 | no | — |
| 89 | #384 | C23 | 1 | 4 | yes | — |
| 90 | #381 | — | 2 | 4 | no | — |
| 91 | #380 | C23 | 1 | 4 | yes | — |
| 92 | #380 | C23 | 1 | 4 | yes | — |
| 93 | #380 | C23 | 1 | 8 | yes | — |
| 94 | #380 | C23 | 1 | 8 | yes | — |
| 95 | C4×C4⋊C4 | C23 | 1 | 8 | yes | — |
| 96 | #376 | C23 | 1 | 1 | yes | characteristic |
| 97 | #340 | C23 | 1 | 8 | yes | — |
| 98 | #340 | C23 | 1 | 8 | yes | — |
| 99 | #340 | C23 | 1 | 8 | yes | — |
| 100 | #340 | C23 | 1 | 8 | yes | — |
| 101 | #340 | C23 | 1 | 8 | yes | — |
| 102 | C22×D4 | C2×D4 | 1 | 4 | yes | — |
| 103 | C22×D4 | C2×D4 | 1 | 2 | yes | — |
| 104 | C23×C4 | — | 2 | 4 | no | — |
| 105 | C23×C4 | — | 2 | 4 | no | — |
| 106 | C23×C4 | C24 | 1 | 4 | yes | — |
| 107 | C22×C8 | — | 2 | 16 | no | — |
| 108 | C22×C8 | — | 2 | 8 | no | — |
| 109 | C22×C8 | C24 | 1 | 4 | yes | — |
| 110 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 111 | C22×C8 | C24 | 1 | 2 | yes | — |
| 112 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 113 | #125 | — | 2 | 8 | no | — |
| 114 | #125 | — | 2 | 8 | no | — |
| 115 | #123 | — | 8 | 8 | no | — |
| 116 | C4×D4 | — | 2 | 8 | no | — |
| 117 | C4×D4 | — | 2 | 8 | no | — |
| 118 | C4×D4 | — | 4 | 2 | no | — |
| 119 | C4×D4 | — | 4 | 2 | no | — |
| 120 | C4×D4 | — | 8 | 16 | no | — |
| 121 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 122 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 123 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 124 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 125 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 126 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 127 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 128 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 129 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 130 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 131 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 132 | C2×C4⋊C4 | C24 | 1 | 8 | yes | — |
| 133 | C2×C4⋊C4 | C24 | 1 | 8 | yes | — |
| 134 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 135 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 136 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 137 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 138 | C2×C22⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 139 | C2×C22⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 140 | C2×C42 | — | 2 | 8 | no | — |
| 141 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 142 | C2×C42 | C24 | 1 | 2 | yes | — |
| 143 | #107 | — | 2 | 8 | no | — |
| 144 | #106 | — | 2 | 8 | no | — |
| 145 | #106 | — | 2 | 8 | no | — |
| 146 | #106 | — | 4 | 8 | no | — |
| 147 | #106 | — | 4 | 4 | no | — |
| 148 | C4⋊C8 | — | 2 | 8 | no | — |
| 149 | C4⋊C8 | — | 4 | 2 | no | — |
| 150 | C22⋊C8 | — | 2 | 16 | no | — |
| 151 | C22⋊C8 | — | 4 | 4 | no | — |
| 152 | #97 | — | 2 | 4 | no | — |
| 153 | #97 | — | 4 | 1 | no | — |
| 154 | C4×C8 | — | 2 | 2 | no | — |
| 155 | C4×C8 | #41 | 1 | 8 | yes | — |
| 156 | #95 | — | 2 | 8 | no | — |
| 157 | #95 | — | 2 | 8 | no | — |
| 158 | #95 | — | 2 | 8 | no | — |
| 159 | #95 | — | 2 | 4 | no | — |
| 160 | #95 | — | 2 | 4 | no | — |
| 161 | #95 | — | 2 | 8 | no | — |
| 162 | #95 | C2×D4 | 1 | 4 | yes | — |
| 163 | C24 | C22×D4 | 1 | 4 | yes | — |
| 164 | C2×D4 | — | 2 | 16 | no | — |
| 165 | C2×D4 | — | 2 | 8 | no | — |
| 166 | C2×D4 | — | 2 | 8 | no | — |
| 167 | C2×D4 | — | 4 | 2 | no | — |
| 168 | C22×C4 | — | 2 | 8 | no | — |
| 169 | C22×C4 | — | 2 | 4 | no | — |
| 170 | C22×C4 | — | 2 | 8 | no | — |
| 171 | C22×C4 | — | 2 | 4 | no | — |
| 172 | C22×C4 | — | 2 | 8 | no | — |
| 173 | C22×C4 | — | 2 | 8 | no | — |
| 174 | C22×C4 | — | 4 | 8 | no | — |
| 175 | C22×C4 | — | 8 | 4 | no | — |
| 176 | C22×C4 | — | 8 | 4 | no | — |
| 177 | C22×C4 | — | 8 | 8 | no | — |
| 178 | C22×C4 | — | 8 | 4 | no | — |
| 179 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 180 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 181 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 182 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 183 | C2×C8 | — | 2 | 2 | no | — |
| 184 | C2×C8 | — | 2 | 2 | no | — |
| 185 | C2×C8 | — | 2 | 8 | no | — |
| 186 | C2×C8 | — | 2 | 1 | no | — |
| 187 | C2×C8 | — | 2 | 4 | no | — |
| 188 | C2×C8 | — | 2 | 2 | no | — |
| 189 | C2×C8 | — | 4 | 16 | no | — |
| 190 | C2×C8 | C2×#41 | 1 | 4 | yes | — |
| 191 | C2×C8 | C2×#41 | 1 | 4 | yes | — |
| 192 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 193 | C4⋊C4 | — | 2 | 8 | no | — |
| 194 | C4⋊C4 | — | 2 | 8 | no | — |
| 195 | C4⋊C4 | — | 2 | 8 | no | — |
| 196 | C4⋊C4 | — | 2 | 8 | no | — |
| 197 | C4⋊C4 | — | 2 | 8 | no | — |
| 198 | C4⋊C4 | — | 4 | 2 | no | — |
| 199 | C4⋊C4 | — | 4 | 16 | no | — |
| 200 | C4⋊C4 | — | 8 | 4 | no | — |
| 201 | C4⋊C4 | — | 8 | 4 | no | — |
| 202 | C4⋊C4 | — | 8 | 8 | no | — |
| 203 | C4⋊C4 | — | 8 | 8 | no | — |
| 204 | C22⋊C4 | — | 2 | 16 | no | — |
| 205 | C22⋊C4 | — | 4 | 4 | no | — |
| 206 | C22⋊C4 | — | 4 | 8 | no | — |
| 207 | C22⋊C4 | — | 4 | 16 | no | — |
| 208 | C22⋊C4 | — | 8 | 8 | no | — |
| 209 | C22⋊C4 | — | 8 | 4 | no | — |
| 210 | C42 | — | 2 | 1 | no | — |
| 211 | C42 | — | 2 | 1 | no | — |
| 212 | C42 | — | 8 | 8 | no | — |
| 213 | C42 | C2×#41 | 1 | 4 | yes | — |
| 214 | C42 | 2−1+4 | 1 | 4 | yes | — |
| 215 | C23 | — | 2 | 8 | no | — |
| 216 | C23 | — | 2 | 4 | no | — |
| 217 | C23 | — | 2 | 4 | no | — |
| 218 | C23 | — | 2 | 8 | no | — |
| 219 | C23 | #577 | 1 | 8 | yes | — |
| 220 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 221 | D4 | — | 2 | 32 | no | — |
| 222 | D4 | — | 4 | 8 | no | — |
| 223 | C2×C4 | — | 2 | 8 | no | — |
| 224 | C2×C4 | — | 2 | 8 | no | — |
| 225 | C2×C4 | — | 2 | 2 | no | — |
| 226 | C2×C4 | — | 2 | 4 | no | — |
| 227 | C2×C4 | — | 2 | 4 | no | — |
| 228 | C2×C4 | — | 4 | 4 | no | — |
| 229 | C2×C4 | — | 4 | 8 | no | — |
| 230 | C2×C4 | — | 4 | 4 | no | — |
| 231 | C2×C4 | — | 4 | 8 | no | — |
| 232 | C2×C4 | — | 4 | 8 | no | — |
| 233 | C2×C4 | — | 4 | 8 | no | — |
| 234 | C2×C4 | — | 4 | 8 | no | — |
| 235 | C2×C4 | — | 4 | 8 | no | — |
| 236 | C2×C4 | — | 4 | 8 | no | — |
| 237 | C2×C4 | — | 16 | 8 | no | — |
| 238 | C2×C4 | — | 16 | 8 | no | — |
| 239 | C2×C4 | #547 | 1 | 8 | yes | — |
| 240 | C2×C4 | C22×#41 | 1 | 2 | yes | — |
| 241 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 242 | C2×C4 | C2×2−1+4 | 1 | 2 | yes | — |
| 243 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 244 | C8 | — | 2 | 4 | no | — |
| 245 | C8 | — | 4 | 4 | no | — |
| 246 | C8 | — | 4 | 1 | no | — |
| 247 | C8 | #559 | 1 | 4 | yes | — |
| 248 | C22 | — | 2 | 8 | no | — |
| 249 | C22 | — | 2 | 16 | no | — |
| 250 | C22 | — | 4 | 4 | no | — |
| 251 | C22 | — | 4 | 8 | no | — |
| 252 | C22 | — | 4 | 4 | no | — |
| 253 | C22 | C2×#547 | 1 | 2 | yes | — |
| 254 | C22 | #3496 | 1 | 1 | yes | characteristic |
| 255 | C22 | C2×#577 | 1 | 4 | yes | — |
| 256 | C4 | — | 2 | 8 | no | — |
| 257 | C4 | — | 4 | 2 | no | — |
| 258 | C4 | — | 4 | 4 | no | — |
| 259 | C4 | — | 8 | 8 | no | — |
| 260 | C4 | — | 8 | 8 | no | — |
| 261 | C4 | C2×#559 | 1 | 1 | yes | characteristic |
| 262 | C4 | #3507 | 1 | 2 | yes | — |
| 263 | C4 | #3550 | 1 | 1 | yes | characteristic |
| 264 | C2 | — | 4 | 8 | no | — |
| 265 | C2 | #59827 | 1 | 1 | yes | characteristic |
| 266 | C2 | #61259 | 1 | 4 | yes | — |
| 267 | C2 | #62268 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2397 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #32468 | C2 | semidirect |
| 3. #29295 | C2 | semidirect |
| 4. #28707 | C2 | semidirect |
| 5. #27658 | C2 | semidirect |
| 6. #26310 | C2 | semidirect |
| 7. D4×#106 | C2 | semidirect |
| 8. #21722 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #45375 | 2 |
| #32468 | 2 |
| #29295 | 2 |
| #28707 | 2 |
| #27658 | 2 |
| #26310 | 2 |
| D4×#106 | 2 |
| #21722 | 2 |
| #21683 | 2 |