Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
775 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 775 conjugacy classes of subgroups — too many to draw. Only its 150 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 | #60308 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #27693 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #12968 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #12807 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #12773 | C2 | 1 | 2 | yes | maximal |
| 6 | #3483 | C22 | 1 | 1 | yes | characteristic |
| 7 | #2991 | — | 2 | 3 | no | — |
| 8 | #2979 | — | 2 | 3 | no | — |
| 9 | #2979 | C22 | 1 | 3 | yes | — |
| 10 | C8×#41 | — | 2 | 3 | no | — |
| 11 | #2931 | — | 2 | 3 | no | — |
| 12 | #2931 | C22 | 1 | 3 | yes | — |
| 13 | #2916 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2126 | C22 | 1 | 6 | yes | — |
| 15 | #2102 | — | 2 | 3 | no | — |
| 16 | #2098 | — | 2 | 3 | no | — |
| 17 | #1945 | — | 2 | 3 | no | — |
| 18 | #1884 | C22 | 1 | 6 | yes | — |
| 19 | #1868 | C22 | 1 | 6 | yes | — |
| 20 | #1814 | — | 2 | 3 | no | — |
| 21 | #1812 | C22 | 1 | 3 | yes | — |
| 22 | #1391 | C22 | 1 | 6 | yes | — |
| 23 | #568 | C23 | 1 | 2 | yes | — |
| 24 | C2×#131 | — | 2 | 1 | no | — |
| 25 | C2×#131 | D4 | 1 | 3 | yes | — |
| 26 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 27 | Q82 | Q8 | 1 | 1 | yes | characteristic |
| 28 | #555 | — | 2 | 3 | no | — |
| 29 | #555 | Q8 | 1 | 3 | yes | — |
| 30 | #550 | — | 2 | 3 | no | — |
| 31 | #532 | — | 2 | 3 | no | — |
| 32 | #532 | C23 | 1 | 3 | yes | — |
| 33 | C4×#41 | — | 2 | 3 | no | — |
| 34 | C8×Q8 | — | 2 | 1 | no | — |
| 35 | C8×Q8 | — | 2 | 3 | no | — |
| 36 | #436 | — | 2 | 6 | no | — |
| 37 | #436 | — | 2 | 6 | no | — |
| 38 | #436 | — | 2 | 6 | no | — |
| 39 | C8×D4 | — | 2 | 3 | no | — |
| 40 | C8×D4 | — | 2 | 3 | no | — |
| 41 | C8×D4 | — | 2 | 6 | no | — |
| 42 | C8×D4 | — | 2 | 6 | no | — |
| 43 | #431 | — | 2 | 6 | no | — |
| 44 | #431 | — | 2 | 3 | no | — |
| 45 | #430 | — | 2 | 3 | no | — |
| 46 | C2×#108 | — | 2 | 3 | no | — |
| 47 | #427 | — | 2 | 3 | no | — |
| 48 | #424 | — | 2 | 6 | no | — |
| 49 | #423 | — | 2 | 3 | no | — |
| 50 | #423 | D4 | 1 | 3 | yes | — |
| 51 | #421 | — | 2 | 6 | no | — |
| 52 | #410 | C23 | 1 | 6 | yes | — |
| 53 | #408 | — | 2 | 3 | no | — |
| 54 | #408 | — | 2 | 3 | no | — |
| 55 | #408 | D4 | 1 | 3 | yes | — |
| 56 | #408 | C23 | 1 | 3 | yes | — |
| 57 | C2×C4×C8 | — | 2 | 3 | no | — |
| 58 | C2×C4×C8 | D4 | 1 | 3 | yes | — |
| 59 | #354 | — | 2 | 6 | no | — |
| 60 | #353 | — | 2 | 6 | no | — |
| 61 | #344 | — | 2 | 6 | no | — |
| 62 | #339 | — | 2 | 6 | no | — |
| 63 | H(3,ℤ/4) | — | 2 | 3 | no | — |
| 64 | C2×#41 | — | 2 | 3 | no | — |
| 65 | C2×#41 | — | 2 | 1 | no | — |
| 66 | C2×#41 | C2×D4 | 1 | 3 | yes | — |
| 67 | C2×#41 | C24 | 1 | 1 | yes | characteristic |
| 68 | #131 | — | 4 | 1 | no | — |
| 69 | #131 | — | 4 | 1 | no | — |
| 70 | #131 | — | 4 | 3 | no | — |
| 71 | #131 | — | 4 | 3 | no | — |
| 72 | #131 | — | 8 | 2 | no | — |
| 73 | #131 | — | 8 | 1 | no | — |
| 74 | #131 | — | 8 | 1 | no | — |
| 75 | C2×M16 | — | 2 | 3 | no | — |
| 76 | C2×M16 | — | 2 | 3 | no | — |
| 77 | C2×M16 | — | 2 | 6 | no | — |
| 78 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 79 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 80 | C22×C8 | — | 2 | 3 | no | — |
| 81 | C22×C8 | — | 2 | 3 | no | — |
| 82 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 83 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 84 | C4⋊Q8 | — | 2 | 3 | no | — |
| 85 | C4⋊Q8 | C2×Q8 | 1 | 3 | yes | — |
| 86 | #127 | — | 2 | 3 | no | — |
| 87 | #127 | #41 | 1 | 3 | yes | — |
| 88 | #124 | — | 2 | 3 | no | — |
| 89 | #124 | — | 2 | 3 | no | — |
| 90 | #124 | C2×Q8 | 1 | 3 | yes | — |
| 91 | #124 | #41 | 1 | 3 | yes | — |
| 92 | #121 | — | 2 | 6 | no | — |
| 93 | #121 | — | 2 | 6 | no | — |
| 94 | #121 | — | 2 | 6 | no | — |
| 95 | C4×Q8 | — | 2 | 1 | no | — |
| 96 | C4×Q8 | — | 2 | 3 | no | — |
| 97 | C4×D4 | — | 2 | 3 | no | — |
| 98 | C4×D4 | — | 2 | 6 | no | — |
| 99 | C4×D4 | — | 2 | 6 | no | — |
| 100 | C4×D4 | — | 2 | 3 | no | — |
| 101 | #117 | — | 2 | 3 | no | — |
| 102 | #117 | — | 2 | 6 | no | — |
| 103 | #117 | — | 2 | 6 | no | — |
| 104 | C2×C42 | — | 2 | 3 | no | — |
| 105 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 106 | #108 | — | 4 | 1 | no | — |
| 107 | #108 | — | 4 | 3 | no | — |
| 108 | #107 | — | 4 | 3 | no | — |
| 109 | #106 | — | 4 | 3 | no | — |
| 110 | C4⋊C8 | — | 2 | 3 | no | — |
| 111 | C4⋊C8 | — | 2 | 3 | no | — |
| 112 | C4⋊C8 | — | 2 | 6 | no | — |
| 113 | C4⋊C8 | — | 8 | 6 | no | — |
| 114 | #104 | — | 8 | 6 | no | — |
| 115 | #99 | — | 2 | 12 | no | — |
| 116 | C22⋊C8 | — | 2 | 6 | no | — |
| 117 | C22⋊C8 | — | 2 | 6 | no | — |
| 118 | C22⋊C8 | — | 2 | 6 | no | — |
| 119 | C4×C8 | — | 2 | 6 | no | — |
| 120 | C4×C8 | — | 2 | 3 | no | — |
| 121 | C4×C8 | — | 2 | 3 | no | — |
| 122 | #41 | — | 4 | 1 | no | — |
| 123 | #41 | — | 4 | 1 | no | — |
| 124 | #41 | — | 4 | 3 | no | — |
| 125 | #41 | — | 4 | 3 | no | — |
| 126 | #41 | — | 4 | 12 | no | — |
| 127 | #41 | — | 8 | 1 | no | — |
| 128 | #41 | — | 8 | 1 | no | — |
| 129 | C2×Q8 | — | 2 | 1 | no | — |
| 130 | C2×Q8 | — | 2 | 1 | no | — |
| 131 | C2×Q8 | C22⋊Q8 | 1 | 3 | yes | — |
| 132 | C2×Q8 | C22×Q8 | 1 | 1 | yes | characteristic |
| 133 | C2×D4 | — | 2 | 3 | no | — |
| 134 | C2×D4 | — | 2 | 6 | no | — |
| 135 | C2×D4 | — | 2 | 3 | no | — |
| 136 | C2×D4 | — | 2 | 3 | no | — |
| 137 | C2×D4 | C22⋊Q8 | 1 | 3 | yes | — |
| 138 | C2×D4 | C2×#41 | 1 | 3 | yes | — |
| 139 | C22×C4 | — | 2 | 6 | no | — |
| 140 | C22×C4 | — | 2 | 3 | no | — |
| 141 | C22×C4 | — | 2 | 3 | no | — |
| 142 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 143 | M16 | — | 4 | 3 | no | — |
| 144 | M16 | — | 4 | 3 | no | — |
| 145 | M16 | — | 4 | 6 | no | — |
| 146 | M16 | — | 4 | 2 | no | — |
| 147 | M16 | — | 8 | 3 | no | — |
| 148 | M16 | — | 8 | 3 | no | — |
| 149 | C2×C8 | — | 2 | 3 | no | — |
| 150 | C2×C8 | — | 2 | 3 | no | — |
| 151 | C2×C8 | — | 2 | 3 | no | — |
| 152 | C2×C8 | — | 2 | 3 | no | — |
| 153 | C2×C8 | — | 4 | 3 | no | — |
| 154 | C2×C8 | — | 4 | 3 | no | — |
| 155 | C2×C8 | — | 4 | 3 | no | — |
| 156 | C2×C8 | — | 4 | 6 | no | — |
| 157 | C2×C8 | — | 8 | 3 | no | — |
| 158 | C2×C8 | #120 | 1 | 3 | yes | — |
| 159 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 160 | C4⋊C4 | — | 2 | 3 | no | — |
| 161 | C4⋊C4 | — | 2 | 3 | no | — |
| 162 | C4⋊C4 | — | 2 | 6 | no | — |
| 163 | C4⋊C4 | — | 4 | 6 | no | — |
| 164 | C22⋊C4 | — | 2 | 6 | no | — |
| 165 | C22⋊C4 | — | 2 | 6 | no | — |
| 166 | C22⋊C4 | — | 2 | 6 | no | — |
| 167 | C22⋊C4 | — | 4 | 6 | no | — |
| 168 | C42 | — | 2 | 3 | no | — |
| 169 | C42 | — | 2 | 3 | no | — |
| 170 | C42 | — | 2 | 3 | no | — |
| 171 | C42 | — | 4 | 6 | no | — |
| 172 | C42 | C22⋊Q8 | 1 | 3 | yes | — |
| 173 | C42 | C22⋊Q8 | 1 | 3 | yes | — |
| 174 | C23 | — | 2 | 3 | no | — |
| 175 | C23 | #525 | 1 | 3 | yes | — |
| 176 | Q8 | — | 2 | 4 | no | — |
| 177 | Q8 | — | 4 | 1 | no | — |
| 178 | Q8 | — | 4 | 3 | no | — |
| 179 | Q8 | — | 8 | 1 | no | — |
| 180 | D4 | — | 4 | 3 | no | — |
| 181 | D4 | — | 4 | 3 | no | — |
| 182 | D4 | — | 4 | 12 | no | — |
| 183 | D4 | — | 4 | 6 | no | — |
| 184 | D4 | — | 8 | 3 | no | — |
| 185 | D4 | — | 8 | 3 | no | — |
| 186 | C2×C4 | — | 2 | 3 | no | — |
| 187 | C2×C4 | — | 2 | 3 | no | — |
| 188 | C2×C4 | — | 2 | 3 | no | — |
| 189 | C2×C4 | — | 2 | 3 | no | — |
| 190 | C2×C4 | — | 2 | 3 | no | — |
| 191 | C2×C4 | — | 4 | 6 | no | — |
| 192 | C2×C4 | — | 4 | 12 | no | — |
| 193 | C2×C4 | — | 4 | 3 | no | — |
| 194 | C2×C4 | — | 4 | 3 | no | — |
| 195 | C2×C4 | — | 8 | 6 | no | — |
| 196 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 197 | C2×C4 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 198 | C8 | — | 2 | 3 | no | — |
| 199 | C8 | — | 4 | 3 | no | — |
| 200 | C8 | — | 4 | 3 | no | — |
| 201 | C8 | — | 8 | 2 | no | — |
| 202 | C8 | #458 | 1 | 1 | yes | characteristic |
| 203 | C8 | #458 | 1 | 1 | yes | characteristic |
| 204 | C22 | — | 2 | 3 | no | — |
| 205 | C22 | — | 4 | 3 | no | — |
| 206 | C22 | — | 4 | 6 | no | — |
| 207 | C22 | — | 4 | 3 | no | — |
| 208 | C22 | #2848 | 1 | 1 | yes | characteristic |
| 209 | C4 | — | 2 | 6 | no | — |
| 210 | C4 | — | 2 | 3 | no | — |
| 211 | C4 | — | 4 | 3 | no | — |
| 212 | C4 | — | 4 | 3 | no | — |
| 213 | C4 | — | 16 | 6 | no | — |
| 214 | C4 | C2×#458 | 1 | 1 | yes | characteristic |
| 215 | C4 | #3027 | 1 | 1 | yes | characteristic |
| 216 | C2 | — | 2 | 1 | no | — |
| 217 | C2 | — | 4 | 3 | no | — |
| 218 | C2 | — | 8 | 3 | no | — |
| 219 | C2 | #32976 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3027 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #32976 |
| prime radical | #3483 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #27693 | C2 | semidirect |
| 3. #12968 | C2 | semidirect |
| 4. #12807 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60308 | 2 |
| #27693 | 2 |
| #12968 | 2 |
| #12807 | 2 |
| #12773 | 2 |