Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
769 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 769 conjugacy classes of subgroups — too many to draw. Only its 128 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 | #53303 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #29907 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #7828 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #7822 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #7718 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #2854 | C22 | 1 | 3 | yes | — |
| 7 | #2800 | — | 2 | 3 | no | — |
| 8 | #2727 | C22 | 1 | 3 | yes | — |
| 9 | #2705 | — | 2 | 3 | no | — |
| 10 | #2685 | — | 2 | 3 | no | — |
| 11 | #2679 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2678 | — | 2 | 3 | no | — |
| 13 | C2×#384 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | #2100 | C22 | 1 | 6 | yes | complemented |
| 15 | #2094 | C22 | 1 | 6 | yes | — |
| 16 | #2084 | — | 2 | 3 | no | — |
| 17 | #2071 | — | 2 | 3 | no | — |
| 18 | #1874 | — | 2 | 3 | no | — |
| 19 | #1853 | — | 2 | 3 | no | — |
| 20 | #1817 | C22 | 1 | 3 | yes | complemented |
| 21 | #1271 | C22 | 1 | 6 | yes | complemented |
| 22 | #1270 | C22 | 1 | 6 | yes | — |
| 23 | C2×#126 | — | 2 | 3 | no | — |
| 24 | C2×#123 | — | 2 | 3 | no | — |
| 25 | C2×#121 | D4 | 1 | 3 | yes | — |
| 26 | C2×#120 | — | 2 | 1 | no | — |
| 27 | C2×#120 | — | 2 | 3 | no | — |
| 28 | C22×C4⋊C4 | — | 2 | 3 | no | — |
| 29 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 30 | C2×#106 | — | 2 | 6 | no | — |
| 31 | C2×#102 | — | 2 | 6 | no | — |
| 32 | C2×C22⋊C8 | — | 2 | 3 | no | — |
| 33 | #399 | — | 2 | 3 | no | — |
| 34 | #397 | — | 2 | 3 | no | — |
| 35 | #396 | — | 2 | 3 | no | — |
| 36 | #396 | — | 2 | 1 | no | — |
| 37 | #396 | — | 2 | 3 | no | — |
| 38 | #394 | — | 2 | 3 | no | — |
| 39 | #394 | — | 2 | 3 | no | — |
| 40 | #394 | — | 2 | 3 | no | — |
| 41 | #394 | — | 2 | 3 | no | — |
| 42 | #392 | — | 2 | 3 | no | — |
| 43 | #390 | — | 2 | 3 | no | — |
| 44 | #390 | — | 2 | 3 | no | — |
| 45 | #390 | D4 | 1 | 3 | yes | — |
| 46 | #390 | C23 | 1 | 3 | yes | — |
| 47 | #389 | — | 2 | 3 | no | — |
| 48 | #389 | D4 | 1 | 3 | yes | — |
| 49 | #389 | C23 | 1 | 3 | yes | — |
| 50 | #388 | — | 2 | 3 | no | — |
| 51 | #388 | — | 2 | 3 | no | — |
| 52 | #388 | — | 2 | 3 | no | — |
| 53 | #388 | — | 2 | 3 | no | — |
| 54 | #387 | — | 2 | 3 | no | — |
| 55 | #386 | — | 2 | 3 | no | — |
| 56 | #384 | — | 2 | 3 | no | — |
| 57 | #384 | C23 | 1 | 2 | yes | — |
| 58 | #380 | D4 | 1 | 3 | yes | — |
| 59 | #379 | — | 2 | 3 | no | — |
| 60 | #340 | — | 2 | 6 | no | — |
| 61 | #340 | — | 2 | 6 | no | — |
| 62 | #336 | — | 2 | 3 | no | — |
| 63 | #336 | C23 | 1 | 6 | yes | — |
| 64 | C25 | — | 2 | 1 | no | — |
| 65 | C22×D4 | — | 2 | 3 | no | — |
| 66 | C22×D4 | — | 2 | 3 | no | — |
| 67 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 68 | C23×C4 | — | 4 | 1 | no | — |
| 69 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 70 | C22×C8 | — | 2 | 6 | no | — |
| 71 | #126 | — | 8 | 6 | no | — |
| 72 | #123 | — | 8 | 6 | no | — |
| 73 | #121 | — | 4 | 6 | no | — |
| 74 | #120 | — | 8 | 6 | no | — |
| 75 | #120 | — | 8 | 2 | no | — |
| 76 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 77 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 78 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 79 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 80 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 81 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 82 | C2×C4⋊C4 | — | 8 | 6 | no | — |
| 83 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 84 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 85 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 86 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 87 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 88 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 89 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 90 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 91 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 92 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 93 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 94 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 95 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 96 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 97 | C2×C42 | — | 2 | 3 | no | — |
| 98 | C2×C42 | — | 4 | 2 | no | — |
| 99 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 100 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 101 | #106 | — | 8 | 6 | no | — |
| 102 | #102 | — | 8 | 6 | no | — |
| 103 | C22⋊C8 | — | 4 | 6 | no | — |
| 104 | #95 | — | 2 | 1 | no | — |
| 105 | #95 | — | 2 | 3 | no | — |
| 106 | #95 | — | 2 | 3 | no | — |
| 107 | #95 | — | 2 | 3 | no | — |
| 108 | #95 | — | 2 | 3 | no | — |
| 109 | C24 | — | 2 | 3 | no | — |
| 110 | C24 | — | 8 | 3 | no | — |
| 111 | C24 | — | 8 | 1 | no | — |
| 112 | C24 | — | 8 | 3 | no | — |
| 113 | C24 | #120 | 1 | 1 | yes | characteristic |
| 114 | C2×D4 | — | 2 | 6 | no | — |
| 115 | C2×D4 | — | 4 | 3 | no | — |
| 116 | C2×D4 | — | 4 | 3 | no | — |
| 117 | C2×D4 | — | 4 | 6 | no | — |
| 118 | C2×D4 | — | 8 | 3 | no | — |
| 119 | C2×D4 | — | 8 | 3 | no | — |
| 120 | C2×D4 | — | 8 | 3 | no | — |
| 121 | C2×D4 | — | 8 | 3 | no | — |
| 122 | C2×D4 | — | 8 | 3 | no | — |
| 123 | C22×C4 | — | 2 | 6 | no | — |
| 124 | C22×C4 | — | 2 | 6 | no | — |
| 125 | C22×C4 | — | 2 | 3 | no | — |
| 126 | C22×C4 | — | 2 | 3 | no | — |
| 127 | C22×C4 | — | 2 | 3 | no | — |
| 128 | C22×C4 | — | 4 | 3 | no | — |
| 129 | C22×C4 | — | 4 | 2 | no | — |
| 130 | C22×C4 | — | 8 | 6 | no | — |
| 131 | C22×C4 | #120 | 1 | 3 | yes | — |
| 132 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 133 | C2×C8 | — | 4 | 6 | no | — |
| 134 | C2×C8 | — | 4 | 6 | no | — |
| 135 | C2×C8 | — | 4 | 6 | no | — |
| 136 | C4⋊C4 | — | 2 | 6 | no | — |
| 137 | C4⋊C4 | — | 4 | 3 | no | — |
| 138 | C4⋊C4 | — | 4 | 3 | no | — |
| 139 | C4⋊C4 | — | 4 | 6 | no | — |
| 140 | C4⋊C4 | — | 4 | 6 | no | — |
| 141 | C4⋊C4 | — | 4 | 6 | no | — |
| 142 | C4⋊C4 | — | 8 | 3 | no | — |
| 143 | C4⋊C4 | — | 8 | 6 | no | — |
| 144 | C4⋊C4 | — | 8 | 3 | no | — |
| 145 | C4⋊C4 | — | 8 | 3 | no | — |
| 146 | C22⋊C4 | — | 4 | 6 | no | — |
| 147 | C22⋊C4 | — | 4 | 6 | no | — |
| 148 | C22⋊C4 | — | 8 | 3 | no | — |
| 149 | C22⋊C4 | — | 8 | 3 | no | — |
| 150 | C22⋊C4 | — | 8 | 3 | no | — |
| 151 | C22⋊C4 | — | 8 | 3 | no | — |
| 152 | C22⋊C4 | — | 8 | 3 | no | — |
| 153 | C22⋊C4 | — | 8 | 3 | no | — |
| 154 | C22⋊C4 | — | 8 | 3 | no | — |
| 155 | C22⋊C4 | — | 8 | 3 | no | — |
| 156 | C22⋊C4 | — | 8 | 3 | no | — |
| 157 | C42 | — | 4 | 3 | no | — |
| 158 | C42 | — | 4 | 1 | no | — |
| 159 | C42 | — | 8 | 3 | no | — |
| 160 | C23 | — | 2 | 3 | no | — |
| 161 | C23 | — | 2 | 3 | no | — |
| 162 | C23 | — | 4 | 3 | no | — |
| 163 | C23 | — | 4 | 3 | no | — |
| 164 | C23 | — | 4 | 3 | no | — |
| 165 | C23 | — | 4 | 3 | no | — |
| 166 | C23 | — | 4 | 3 | no | — |
| 167 | C23 | — | 4 | 3 | no | — |
| 168 | C23 | — | 4 | 3 | no | — |
| 169 | C23 | — | 16 | 6 | no | — |
| 170 | C23 | — | 16 | 6 | no | — |
| 171 | C23 | — | 16 | 2 | no | — |
| 172 | C23 | #458 | 1 | 2 | yes | — |
| 173 | C23 | C2×#120 | 1 | 1 | yes | characteristic |
| 174 | D4 | — | 8 | 6 | no | — |
| 175 | D4 | — | 16 | 6 | no | — |
| 176 | D4 | — | 16 | 6 | no | — |
| 177 | C2×C4 | — | 2 | 3 | no | — |
| 178 | C2×C4 | — | 2 | 3 | no | — |
| 179 | C2×C4 | — | 2 | 3 | no | — |
| 180 | C2×C4 | — | 2 | 3 | no | — |
| 181 | C2×C4 | — | 2 | 3 | no | — |
| 182 | C2×C4 | — | 4 | 12 | no | — |
| 183 | C2×C4 | — | 4 | 3 | no | — |
| 184 | C2×C4 | — | 4 | 3 | no | — |
| 185 | C2×C4 | — | 4 | 3 | no | — |
| 186 | C2×C4 | — | 4 | 3 | no | — |
| 187 | C2×C4 | — | 4 | 3 | no | — |
| 188 | C2×C4 | — | 4 | 3 | no | — |
| 189 | C2×C4 | — | 4 | 3 | no | — |
| 190 | C2×C4 | — | 4 | 3 | no | — |
| 191 | C2×C4 | — | 4 | 3 | no | — |
| 192 | C2×C4 | — | 8 | 6 | no | — |
| 193 | C2×C4 | — | 16 | 4 | no | — |
| 194 | C2×C4 | #577 | 1 | 6 | yes | — |
| 195 | C8 | — | 8 | 6 | no | — |
| 196 | C22 | — | 2 | 6 | no | — |
| 197 | C22 | — | 4 | 3 | no | — |
| 198 | C22 | — | 4 | 1 | no | — |
| 199 | C22 | — | 8 | 3 | no | — |
| 200 | C22 | — | 8 | 3 | no | — |
| 201 | C22 | — | 8 | 3 | no | — |
| 202 | C22 | — | 8 | 3 | no | — |
| 203 | C22 | — | 8 | 3 | no | — |
| 204 | C22 | — | 8 | 3 | no | — |
| 205 | C22 | — | 8 | 3 | no | — |
| 206 | C22 | — | 16 | 8 | no | — |
| 207 | C22 | #3009 | 1 | 3 | yes | — |
| 208 | C22 | #3021 | 1 | 3 | yes | — |
| 209 | C22 | C2×#458 | 1 | 1 | yes | characteristic |
| 210 | C4 | — | 2 | 6 | no | — |
| 211 | C4 | — | 4 | 3 | no | — |
| 212 | C4 | — | 8 | 3 | no | — |
| 213 | C4 | — | 8 | 3 | no | — |
| 214 | C4 | — | 8 | 3 | no | — |
| 215 | C4 | — | 16 | 2 | no | — |
| 216 | C2 | — | 4 | 2 | no | — |
| 217 | C2 | — | 16 | 3 | no | — |
| 218 | C2 | #18233 | 1 | 3 | yes | — |
| 219 | C2 | #18955 | 1 | 3 | yes | — |
| 220 | C2 | #19094 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #53303 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #29907 | C2 | semidirect |
| 3. #7828 | C2 | semidirect |
| 4. #7822 | C2 | semidirect |
| 5. #7718 | C2 | semidirect |
| 13. C2×#384 | C22 | semidirect |
| 14. #2100 | C22 | semidirect |
| 20. #1817 | C22 | semidirect |
| 21. #1271 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #53303 | 2 |
| #29907 | 2 |
| #7828 | 2 |
| #7822 | 2 |
| #7718 | 2 |