Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1235 conjugacy classes of subgroups.
The subgroup lattice has 226 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #45427 | C2 | 1 | 2 | yes | maximal |
| 2 | #32691 | C2 | 1 | 4 | yes | maximal |
| 3 | #29294 | C2 | 1 | 2 | yes | maximal |
| 4 | #28811 | C2 | 1 | 4 | yes | maximal |
| 5 | #28678 | C2 | 1 | 4 | yes | maximal |
| 6 | #26459 | C2 | 1 | 8 | yes | maximal |
| 7 | #23159 | C2 | 1 | 4 | yes | maximal |
| 8 | #21722 | C2 | 1 | 2 | yes | maximal |
| 9 | #21715 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C2×#499 | C22 | 1 | 4 | yes | — |
| 11 | C2×#499 | C22 | 1 | 2 | yes | — |
| 12 | C2×C8×Q8 | C22 | 1 | 2 | yes | — |
| 13 | #2792 | — | 2 | 8 | no | — |
| 14 | #2584 | C22 | 1 | 8 | yes | — |
| 15 | #2552 | C22 | 1 | 8 | yes | — |
| 16 | #2507 | — | 2 | 8 | no | — |
| 17 | #2399 | C22 | 1 | 1 | yes | characteristic |
| 18 | #2370 | C22 | 1 | 8 | yes | — |
| 19 | #2330 | C22 | 1 | 4 | yes | — |
| 20 | #2087 | C22 | 1 | 4 | yes | — |
| 21 | #2087 | C22 | 1 | 4 | yes | — |
| 22 | #2086 | C22 | 1 | 4 | yes | — |
| 23 | #2086 | C22 | 1 | 4 | yes | — |
| 24 | #2059 | C22 | 1 | 8 | yes | — |
| 25 | #2059 | C22 | 1 | 8 | yes | — |
| 26 | #1992 | C22 | 1 | 4 | yes | — |
| 27 | #1989 | C22 | 1 | 8 | yes | — |
| 28 | #1989 | C22 | 1 | 8 | yes | — |
| 29 | #1985 | C22 | 1 | 2 | yes | — |
| 30 | #1947 | C22 | 1 | 8 | yes | — |
| 31 | #1943 | C22 | 1 | 4 | yes | — |
| 32 | #1941 | C22 | 1 | 4 | yes | — |
| 33 | #1941 | C22 | 1 | 8 | yes | — |
| 34 | #1940 | C22 | 1 | 4 | yes | — |
| 35 | #1921 | C22 | 1 | 8 | yes | — |
| 36 | #1918 | C22 | 1 | 8 | yes | — |
| 37 | #1844 | C22 | 1 | 2 | yes | — |
| 38 | #1843 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1776 | C22 | 1 | 4 | yes | — |
| 40 | C4×#106 | C22 | 1 | 4 | yes | — |
| 41 | C4×#106 | C22 | 1 | 4 | yes | — |
| 42 | C8×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 43 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 44 | C2×#125 | C23 | 1 | 4 | yes | — |
| 45 | C2×#125 | C23 | 1 | 8 | yes | — |
| 46 | C2×C4×Q8 | C23 | 1 | 2 | yes | — |
| 47 | #499 | — | 4 | 8 | no | — |
| 48 | #499 | Q8 | 1 | 16 | yes | complemented |
| 49 | C8×Q8 | — | 4 | 4 | no | — |
| 50 | C2×#107 | C23 | 1 | 4 | yes | — |
| 51 | C2×#107 | C23 | 1 | 4 | yes | — |
| 52 | C2×#107 | C23 | 1 | 4 | yes | — |
| 53 | C2×#106 | C23 | 1 | 4 | yes | — |
| 54 | C2×#106 | C23 | 1 | 4 | yes | — |
| 55 | C2×#106 | C23 | 1 | 4 | yes | — |
| 56 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 57 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 58 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 59 | C2×#97 | C23 | 1 | 2 | yes | — |
| 60 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 62 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 63 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 64 | #398 | — | 2 | 8 | no | — |
| 65 | #398 | — | 2 | 8 | no | — |
| 66 | #398 | — | 2 | 8 | no | — |
| 67 | #398 | D4 | 1 | 4 | yes | — |
| 68 | #395 | — | 2 | 8 | no | — |
| 69 | #395 | D4 | 1 | 4 | yes | — |
| 70 | #389 | — | 2 | 8 | no | — |
| 71 | #389 | — | 2 | 8 | no | — |
| 72 | #389 | — | 2 | 8 | no | — |
| 73 | #389 | — | 2 | 8 | no | — |
| 74 | #389 | — | 2 | 8 | no | — |
| 75 | #389 | C23 | 1 | 8 | yes | — |
| 76 | #389 | C23 | 1 | 8 | yes | — |
| 77 | #389 | C23 | 1 | 8 | yes | — |
| 78 | #387 | — | 2 | 8 | no | — |
| 79 | #387 | — | 2 | 8 | no | — |
| 80 | #387 | — | 2 | 8 | no | — |
| 81 | #384 | — | 2 | 8 | no | — |
| 82 | #384 | C23 | 1 | 4 | yes | — |
| 83 | C4×C4⋊C4 | — | 2 | 8 | no | — |
| 84 | C4×C4⋊C4 | C23 | 1 | 8 | yes | — |
| 85 | C4×C4⋊C4 | C23 | 1 | 8 | yes | — |
| 86 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 87 | #376 | C23 | 1 | 1 | yes | characteristic |
| 88 | #340 | C23 | 1 | 8 | yes | — |
| 89 | #340 | C23 | 1 | 8 | yes | — |
| 90 | #340 | C23 | 1 | 8 | yes | — |
| 91 | #340 | C23 | 1 | 4 | yes | — |
| 92 | #340 | C23 | 1 | 4 | yes | — |
| 93 | #340 | C23 | 1 | 8 | yes | — |
| 94 | #336 | C23 | 1 | 8 | yes | — |
| 95 | C22×Q8 | C2×D4 | 1 | 2 | yes | — |
| 96 | C22×C8 | C24 | 1 | 4 | yes | — |
| 97 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 98 | C22×C8 | C24 | 1 | 2 | yes | — |
| 99 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 100 | #125 | — | 2 | 8 | no | — |
| 101 | #125 | — | 4 | 8 | no | — |
| 102 | #125 | — | 4 | 8 | no | — |
| 103 | #125 | C2×Q8 | 1 | 16 | yes | — |
| 104 | C4×Q8 | — | 4 | 2 | no | — |
| 105 | C4×Q8 | — | 4 | 2 | no | — |
| 106 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 107 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 108 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 109 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 110 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 111 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 112 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 113 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 114 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 115 | C2×C4⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 116 | C2×C4⋊C4 | C24 | 1 | 8 | yes | — |
| 117 | C2×C4⋊C4 | C24 | 1 | 8 | yes | — |
| 118 | C2×C42 | — | 2 | 8 | no | — |
| 119 | C2×C42 | — | 2 | 8 | no | — |
| 120 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 121 | C2×C42 | C24 | 1 | 4 | yes | — |
| 122 | C2×C42 | C24 | 1 | 2 | yes | — |
| 123 | #107 | — | 4 | 4 | no | — |
| 124 | #107 | — | 4 | 4 | no | — |
| 125 | #107 | C2×Q8 | 1 | 16 | yes | — |
| 126 | #106 | — | 2 | 8 | no | — |
| 127 | #106 | — | 4 | 4 | no | — |
| 128 | #106 | C2×Q8 | 1 | 16 | yes | — |
| 129 | C4⋊C8 | — | 2 | 8 | no | — |
| 130 | C4⋊C8 | — | 4 | 2 | no | — |
| 131 | C4⋊C8 | — | 4 | 4 | no | — |
| 132 | #97 | — | 2 | 4 | no | — |
| 133 | #97 | — | 4 | 1 | no | — |
| 134 | C4×C8 | — | 2 | 2 | no | — |
| 135 | C4×C8 | — | 2 | 8 | no | — |
| 136 | C4×C8 | C2×Q8 | 1 | 8 | yes | — |
| 137 | #95 | — | 2 | 8 | no | — |
| 138 | #95 | — | 2 | 8 | no | — |
| 139 | #95 | — | 2 | 8 | no | — |
| 140 | #95 | — | 2 | 8 | no | — |
| 141 | #95 | — | 2 | 8 | no | — |
| 142 | #95 | C2×D4 | 1 | 4 | yes | — |
| 143 | #95 | C2×D4 | 1 | 4 | yes | — |
| 144 | #95 | C2×D4 | 1 | 4 | yes | — |
| 145 | C2×Q8 | — | 4 | 2 | no | — |
| 146 | C2×Q8 | — | 4 | 4 | no | — |
| 147 | C22×C4 | — | 2 | 8 | no | — |
| 148 | C22×C4 | — | 2 | 8 | no | — |
| 149 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 150 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 151 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 152 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 153 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 154 | C2×C8 | — | 2 | 2 | no | — |
| 155 | C2×C8 | — | 2 | 2 | no | — |
| 156 | C2×C8 | — | 2 | 1 | no | — |
| 157 | C2×C8 | — | 2 | 4 | no | — |
| 158 | C2×C8 | — | 2 | 2 | no | — |
| 159 | C2×C8 | — | 2 | 8 | no | — |
| 160 | C2×C8 | C22×Q8 | 1 | 4 | yes | — |
| 161 | C2×C8 | C22×Q8 | 1 | 4 | yes | — |
| 162 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 163 | C4⋊C4 | — | 2 | 32 | no | — |
| 164 | C4⋊C4 | — | 2 | 8 | no | — |
| 165 | C4⋊C4 | — | 2 | 8 | no | — |
| 166 | C4⋊C4 | — | 2 | 8 | no | — |
| 167 | C4⋊C4 | — | 4 | 8 | no | — |
| 168 | C4⋊C4 | — | 4 | 2 | no | — |
| 169 | C4⋊C4 | — | 4 | 4 | no | — |
| 170 | C4⋊C4 | — | 8 | 8 | no | — |
| 171 | C4⋊C4 | — | 8 | 8 | no | — |
| 172 | C4⋊C4 | — | 8 | 8 | no | — |
| 173 | C4⋊C4 | — | 8 | 8 | no | — |
| 174 | C4⋊C4 | — | 8 | 8 | no | — |
| 175 | C4⋊C4 | — | 8 | 8 | no | — |
| 176 | C4⋊C4 | C22×Q8 | 1 | 16 | yes | — |
| 177 | C42 | — | 2 | 1 | no | — |
| 178 | C42 | — | 2 | 4 | no | — |
| 179 | C42 | — | 2 | 1 | no | — |
| 180 | C42 | — | 2 | 4 | no | — |
| 181 | C42 | — | 8 | 8 | no | — |
| 182 | C42 | — | 8 | 8 | no | — |
| 183 | C42 | C22×Q8 | 1 | 4 | yes | — |
| 184 | C42 | 2−1+4 | 1 | 4 | yes | — |
| 185 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 186 | Q8 | — | 4 | 8 | no | — |
| 187 | C2×C4 | — | 2 | 8 | no | — |
| 188 | C2×C4 | — | 2 | 8 | no | — |
| 189 | C2×C4 | — | 2 | 4 | no | — |
| 190 | C2×C4 | — | 2 | 4 | no | — |
| 191 | C2×C4 | — | 2 | 4 | no | — |
| 192 | C2×C4 | — | 2 | 16 | no | — |
| 193 | C2×C4 | — | 2 | 2 | no | — |
| 194 | C2×C4 | — | 2 | 4 | no | — |
| 195 | C2×C4 | — | 2 | 4 | no | — |
| 196 | C2×C4 | — | 4 | 8 | no | — |
| 197 | C2×C4 | — | 4 | 8 | no | — |
| 198 | C2×C4 | — | 4 | 8 | no | — |
| 199 | C2×C4 | — | 4 | 8 | no | — |
| 200 | C2×C4 | — | 4 | 8 | no | — |
| 201 | C2×C4 | D4×Q8 | 1 | 8 | yes | — |
| 202 | C2×C4 | C23×Q8 | 1 | 2 | yes | — |
| 203 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 204 | C2×C4 | C2×2−1+4 | 1 | 2 | yes | — |
| 205 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 206 | C8 | — | 2 | 4 | no | — |
| 207 | C8 | — | 4 | 1 | no | — |
| 208 | C8 | — | 4 | 4 | no | — |
| 209 | C8 | Q82 | 1 | 4 | yes | — |
| 210 | C22 | C2×D4×Q8 | 1 | 2 | yes | — |
| 211 | C22 | #3496 | 1 | 1 | yes | characteristic |
| 212 | C22 | #3584 | 1 | 4 | yes | — |
| 213 | C4 | — | 2 | 8 | no | — |
| 214 | C4 | — | 4 | 4 | no | — |
| 215 | C4 | — | 4 | 4 | no | — |
| 216 | C4 | — | 4 | 2 | no | — |
| 217 | C4 | — | 4 | 16 | no | — |
| 218 | C4 | — | 8 | 8 | no | — |
| 219 | C4 | C2×Q82 | 1 | 1 | yes | characteristic |
| 220 | C4 | #3502 | 1 | 2 | yes | — |
| 221 | C4 | #3550 | 1 | 1 | yes | characteristic |
| 222 | C2 | #59832 | 1 | 1 | yes | characteristic |
| 223 | C2 | #61375 | 1 | 4 | yes | — |
| 224 | C2 | #62274 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 48. #499 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #45427 | 2 |
| #32691 | 2 |
| #29294 | 2 |
| #28811 | 2 |
| #28678 | 2 |
| #26459 | 2 |
| #23159 | 2 |
| #21722 | 2 |
| #21715 | 2 |