Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
299 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #21660 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13602 | C2 | 1 | 2 | yes | maximal |
| 3 | #12809 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #12807 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #7497 | C2 | 1 | 2 | yes | maximal |
| 6 | #3131 | C22 | 1 | 1 | yes | characteristic |
| 7 | #2960 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 9 | #2132 | C22 | 1 | 2 | yes | — |
| 10 | #2126 | C22 | 1 | 2 | yes | — |
| 11 | #1994 | C4 | 1 | 1 | yes | characteristic, complemented |
| 12 | #1962 | C22 | 1 | 1 | yes | characteristic |
| 13 | #1938 | — | 2 | 1 | no | — |
| 14 | #1938 | — | 2 | 1 | no | — |
| 15 | #1837 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | #1813 | — | 2 | 1 | no | — |
| 17 | #1812 | — | 2 | 1 | no | — |
| 18 | #1765 | — | 2 | 1 | no | — |
| 19 | #1765 | — | 2 | 1 | no | — |
| 20 | C4×#97 | C4 | 1 | 1 | yes | characteristic, complemented |
| 21 | #1414 | — | 2 | 4 | no | — |
| 22 | #1392 | — | 2 | 1 | no | — |
| 23 | #1391 | — | 2 | 1 | no | — |
| 24 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 25 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 26 | #533 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | #532 | C23 | 1 | 1 | yes | characteristic |
| 28 | #488 | D4 | 1 | 2 | yes | — |
| 29 | #488 | Q8 | 1 | 2 | yes | — |
| 30 | #487 | — | 2 | 1 | no | — |
| 31 | #486 | — | 2 | 1 | no | — |
| 32 | #446 | — | 4 | 1 | no | — |
| 33 | C8×Q8 | — | 4 | 1 | no | — |
| 34 | #436 | — | 4 | 1 | no | — |
| 35 | C8×D4 | — | 4 | 1 | no | — |
| 36 | C2×#108 | — | 4 | 1 | no | — |
| 37 | #427 | — | 4 | 1 | no | — |
| 38 | #424 | — | 2 | 1 | no | — |
| 39 | #423 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | #423 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C2×#104 | — | 2 | 1 | no | — |
| 42 | C2×#104 | — | 2 | 1 | no | — |
| 43 | #416 | — | 2 | 1 | no | — |
| 44 | #416 | — | 2 | 1 | no | — |
| 45 | #413 | — | 2 | 2 | no | — |
| 46 | #410 | — | 2 | 2 | no | — |
| 47 | #408 | D4 | 1 | 1 | yes | characteristic |
| 48 | #408 | D4 | 1 | 1 | yes | characteristic |
| 49 | #405 | — | 2 | 1 | no | — |
| 50 | C2×#97 | — | 2 | 1 | no | — |
| 51 | C2×#97 | — | 2 | 1 | no | — |
| 52 | C2×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C2×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C43 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | #356 | — | 2 | 2 | no | — |
| 56 | #355 | — | 2 | 2 | no | — |
| 57 | #354 | — | 2 | 2 | no | — |
| 58 | #353 | — | 2 | 2 | no | — |
| 59 | #341 | — | 2 | 1 | no | — |
| 60 | #341 | — | 2 | 1 | no | — |
| 61 | #339 | — | 2 | 1 | no | — |
| 62 | #339 | — | 2 | 1 | no | — |
| 63 | #338 | — | 4 | 1 | no | — |
| 64 | H(3,ℤ/4) | — | 4 | 1 | no | — |
| 65 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | #131 | — | 4 | 1 | no | — |
| 68 | #131 | — | 4 | 1 | no | — |
| 69 | #131 | — | 4 | 1 | no | — |
| 70 | #131 | — | 4 | 1 | no | — |
| 71 | C2×M16 | — | 2 | 1 | no | — |
| 72 | C2×M16 | — | 2 | 1 | no | — |
| 73 | C2×M16 | — | 2 | 1 | no | — |
| 74 | C2×M16 | — | 4 | 2 | no | — |
| 75 | C2×M16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 76 | C2×M16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 77 | C22×C8 | — | 2 | 1 | no | — |
| 78 | C22×C8 | — | 2 | 1 | no | — |
| 79 | C22×C8 | — | 2 | 1 | no | — |
| 80 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 81 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C4⋊Q8 | — | 2 | 1 | no | — |
| 83 | C4⋊Q8 | #41 | 1 | 1 | yes | characteristic |
| 84 | #127 | #41 | 1 | 1 | yes | characteristic |
| 85 | #125 | C4⋊C4 | 1 | 2 | yes | — |
| 86 | #124 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | #124 | C2×Q8 | 1 | 1 | yes | characteristic |
| 88 | C22⋊Q8 | — | 4 | 1 | no | — |
| 89 | #121 | — | 4 | 1 | no | — |
| 90 | C4×Q8 | — | 4 | 1 | no | — |
| 91 | C4×D4 | — | 4 | 1 | no | — |
| 92 | #117 | — | 2 | 1 | no | — |
| 93 | #117 | — | 4 | 2 | no | — |
| 94 | C2×C42 | — | 2 | 1 | no | — |
| 95 | C2×C42 | — | 4 | 1 | no | — |
| 96 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 97 | #108 | — | 8 | 1 | no | — |
| 98 | #108 | — | 8 | 1 | no | — |
| 99 | #107 | — | 8 | 1 | no | — |
| 100 | #106 | — | 8 | 1 | no | — |
| 101 | C4⋊C8 | — | 4 | 1 | no | — |
| 102 | C4⋊C8 | — | 4 | 1 | no | — |
| 103 | C4⋊C8 | — | 4 | 2 | no | — |
| 104 | #104 | — | 4 | 2 | no | — |
| 105 | #104 | — | 4 | 2 | no | — |
| 106 | #103 | — | 2 | 2 | no | — |
| 107 | #103 | — | 2 | 2 | no | — |
| 108 | #102 | — | 2 | 2 | no | — |
| 109 | #102 | — | 2 | 2 | no | — |
| 110 | #101 | — | 4 | 2 | no | — |
| 111 | #100 | — | 4 | 2 | no | — |
| 112 | #99 | — | 4 | 2 | no | — |
| 113 | #99 | — | 4 | 2 | no | — |
| 114 | C22⋊C8 | — | 4 | 1 | no | — |
| 115 | C22⋊C8 | — | 4 | 1 | no | — |
| 116 | #97 | — | 4 | 2 | no | — |
| 117 | #97 | — | 4 | 1 | no | — |
| 118 | #97 | — | 4 | 1 | no | — |
| 119 | #97 | — | 4 | 1 | no | — |
| 120 | C4×C8 | — | 2 | 1 | no | — |
| 121 | C4×C8 | — | 2 | 1 | no | — |
| 122 | C4×C8 | — | 2 | 1 | no | — |
| 123 | C4×C8 | — | 4 | 1 | no | — |
| 124 | C4×C8 | C4⋊C4 | 1 | 2 | yes | — |
| 125 | #41 | — | 4 | 1 | no | — |
| 126 | #41 | — | 4 | 1 | no | — |
| 127 | #41 | — | 4 | 1 | no | — |
| 128 | #41 | — | 4 | 1 | no | — |
| 129 | C2×Q8 | — | 4 | 1 | no | — |
| 130 | C2×Q8 | #121 | 1 | 1 | yes | characteristic |
| 131 | C2×Q8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 132 | C2×D4 | — | 2 | 1 | no | — |
| 133 | C2×D4 | — | 2 | 1 | no | — |
| 134 | C2×D4 | — | 4 | 1 | no | — |
| 135 | C2×D4 | #121 | 1 | 1 | yes | characteristic |
| 136 | C2×D4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 137 | C22×C4 | — | 2 | 1 | no | — |
| 138 | C22×C4 | — | 2 | 1 | no | — |
| 139 | C22×C4 | — | 2 | 1 | no | — |
| 140 | C22×C4 | — | 4 | 1 | no | — |
| 141 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 142 | M16 | — | 4 | 1 | no | — |
| 143 | M16 | — | 4 | 1 | no | — |
| 144 | M16 | — | 4 | 1 | no | — |
| 145 | M16 | — | 4 | 1 | no | — |
| 146 | M16 | — | 8 | 1 | no | — |
| 147 | M16 | — | 8 | 1 | no | — |
| 148 | M16 | — | 8 | 2 | no | — |
| 149 | M16 | — | 8 | 2 | no | — |
| 150 | C2×C8 | — | 2 | 1 | no | — |
| 151 | C2×C8 | — | 2 | 1 | no | — |
| 152 | C2×C8 | — | 2 | 2 | no | — |
| 153 | C2×C8 | — | 2 | 1 | no | — |
| 154 | C2×C8 | — | 2 | 1 | no | — |
| 155 | C2×C8 | — | 2 | 1 | no | — |
| 156 | C2×C8 | — | 4 | 1 | no | — |
| 157 | C2×C8 | — | 4 | 1 | no | — |
| 158 | C2×C8 | — | 4 | 1 | no | — |
| 159 | C2×C8 | — | 4 | 1 | no | — |
| 160 | C2×C8 | — | 4 | 1 | no | — |
| 161 | C2×C8 | — | 4 | 1 | no | — |
| 162 | C2×C8 | — | 4 | 1 | no | — |
| 163 | C2×C8 | — | 4 | 1 | no | — |
| 164 | C2×C8 | — | 8 | 2 | no | — |
| 165 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 166 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 167 | C4⋊C4 | — | 2 | 2 | no | — |
| 168 | C4⋊C4 | — | 4 | 1 | no | — |
| 169 | C4⋊C4 | — | 4 | 2 | no | — |
| 170 | C4⋊C4 | — | 4 | 1 | no | — |
| 171 | C4⋊C4 | — | 8 | 2 | no | — |
| 172 | C22⋊C4 | — | 4 | 1 | no | — |
| 173 | C22⋊C4 | — | 4 | 1 | no | — |
| 174 | C22⋊C4 | — | 4 | 1 | no | — |
| 175 | C22⋊C4 | — | 8 | 2 | no | — |
| 176 | C42 | — | 2 | 1 | no | — |
| 177 | C42 | — | 2 | 2 | no | — |
| 178 | C42 | — | 4 | 1 | no | — |
| 179 | C42 | — | 4 | 1 | no | — |
| 180 | C42 | — | 8 | 2 | no | — |
| 181 | C42 | — | 8 | 2 | no | — |
| 182 | C42 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 183 | C42 | #117 | 1 | 1 | yes | characteristic |
| 184 | C23 | — | 2 | 1 | no | — |
| 185 | C23 | — | 2 | 1 | no | — |
| 186 | C23 | #410 | 1 | 1 | yes | characteristic |
| 187 | Q8 | — | 4 | 1 | no | — |
| 188 | Q8 | — | 4 | 1 | no | — |
| 189 | Q8 | — | 8 | 2 | no | — |
| 190 | D4 | — | 4 | 1 | no | — |
| 191 | D4 | — | 4 | 1 | no | — |
| 192 | D4 | — | 8 | 2 | no | — |
| 193 | D4 | — | 8 | 1 | no | — |
| 194 | D4 | — | 8 | 1 | no | — |
| 195 | C2×C4 | — | 2 | 1 | no | — |
| 196 | C2×C4 | — | 2 | 1 | no | — |
| 197 | C2×C4 | — | 2 | 1 | no | — |
| 198 | C2×C4 | — | 2 | 1 | no | — |
| 199 | C2×C4 | — | 2 | 1 | no | — |
| 200 | C2×C4 | — | 2 | 1 | no | — |
| 201 | C2×C4 | — | 4 | 2 | no | — |
| 202 | C2×C4 | — | 4 | 1 | no | — |
| 203 | C2×C4 | — | 4 | 1 | no | — |
| 204 | C2×C4 | — | 4 | 1 | no | — |
| 205 | C2×C4 | — | 4 | 1 | no | — |
| 206 | C2×C4 | — | 4 | 2 | no | — |
| 207 | C2×C4 | — | 4 | 2 | no | — |
| 208 | C2×C4 | — | 8 | 2 | no | — |
| 209 | C2×C4 | — | 16 | 2 | no | — |
| 210 | C2×C4 | #380 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | C2×#99 | 1 | 1 | yes | characteristic |
| 212 | C8 | — | 2 | 1 | no | — |
| 213 | C8 | — | 4 | 2 | no | — |
| 214 | C8 | — | 4 | 1 | no | — |
| 215 | C8 | — | 4 | 1 | no | — |
| 216 | C8 | — | 4 | 1 | no | — |
| 217 | C8 | — | 8 | 1 | no | — |
| 218 | C8 | — | 16 | 2 | no | — |
| 219 | C8 | #354 | 1 | 1 | yes | characteristic |
| 220 | C8 | #354 | 1 | 1 | yes | characteristic |
| 221 | C22 | — | 2 | 1 | no | — |
| 222 | C22 | — | 4 | 1 | no | — |
| 223 | C22 | — | 4 | 1 | no | — |
| 224 | C22 | — | 4 | 1 | no | — |
| 225 | C22 | — | 8 | 1 | no | — |
| 226 | C22 | — | 8 | 1 | no | — |
| 227 | C22 | #1799 | 1 | 1 | yes | characteristic |
| 228 | C4 | — | 2 | 2 | no | — |
| 229 | C4 | — | 2 | 1 | no | — |
| 230 | C4 | — | 4 | 1 | no | — |
| 231 | C4 | — | 4 | 1 | no | — |
| 232 | C4 | — | 4 | 1 | no | — |
| 233 | C4 | — | 8 | 2 | no | — |
| 234 | C4 | — | 8 | 2 | no | — |
| 235 | C4 | — | 32 | 2 | no | — |
| 236 | C4 | C2×#354 | 1 | 1 | yes | characteristic |
| 237 | C4 | #2126 | 1 | 1 | yes | characteristic |
| 238 | C2 | — | 2 | 1 | no | — |
| 239 | C2 | — | 4 | 1 | no | — |
| 240 | C2 | — | 8 | 1 | no | — |
| 241 | C2 | — | 8 | 1 | no | — |
| 242 | C2 | #12798 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2126 |
| commutator | C2×C42 | C22×C4 |
| Frattini, prime radical, cosocle | #532 | C23 |
| socle | C2 | #12798 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. #2960 | C4 | semidirect |
| 11. #1994 | C4 | semidirect |
| 15. #1837 | C4 | semidirect |
| 20. C4×#97 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21660 | 2 |
| #13602 | 2 |
| #12809 | 2 |
| #12807 | 2 |
| #7497 | 2 |