Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1297 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1297 conjugacy classes of subgroups — too many to draw. Only its 234 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 | #33301 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33227 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #30259 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #30258 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×#1721 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | C2×#1717 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #7843 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#493 | — | 2 | 4 | no | — |
| 9 | C2×#493 | C22 | 1 | 4 | yes | — |
| 10 | C2×#493 | C22 | 1 | 4 | yes | — |
| 11 | C2×#492 | C22 | 1 | 4 | yes | — |
| 12 | C2×#487 | C22 | 1 | 2 | yes | — |
| 13 | C2×#486 | C22 | 1 | 4 | yes | — |
| 14 | C2×#486 | C22 | 1 | 4 | yes | — |
| 15 | C2×#486 | C22 | 1 | 2 | yes | — |
| 16 | #2867 | C22 | 1 | 2 | yes | — |
| 17 | #2862 | C22 | 1 | 1 | yes | characteristic |
| 18 | #2858 | C22 | 1 | 1 | yes | characteristic |
| 19 | #2014 | — | 2 | 8 | no | — |
| 20 | #2011 | — | 2 | 8 | no | — |
| 21 | #1958 | — | 2 | 4 | no | — |
| 22 | #1744 | C22 | 1 | 2 | yes | — |
| 23 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1721 | — | 2 | 16 | no | — |
| 25 | #1717 | — | 2 | 16 | no | — |
| 26 | C2×#322 | C22 | 1 | 4 | yes | — |
| 27 | C22×SD16 | — | 2 | 8 | no | — |
| 28 | C22×D8 | — | 2 | 8 | no | — |
| 29 | C22×D8 | — | 2 | 8 | no | — |
| 30 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 32 | C2×#127 | — | 2 | 4 | no | — |
| 33 | C2×#127 | — | 2 | 4 | no | — |
| 34 | C2×#127 | — | 2 | 4 | no | — |
| 35 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×#127 | C23 | 1 | 4 | yes | — |
| 37 | C2×#124 | — | 2 | 2 | no | — |
| 38 | #493 | — | 2 | 8 | no | — |
| 39 | #493 | — | 2 | 8 | no | — |
| 40 | #493 | — | 2 | 8 | no | — |
| 41 | #493 | — | 2 | 8 | no | — |
| 42 | #493 | — | 4 | 16 | no | — |
| 43 | #492 | — | 2 | 8 | no | — |
| 44 | #492 | — | 2 | 8 | no | — |
| 45 | #487 | — | 2 | 8 | no | — |
| 46 | #486 | — | 2 | 8 | no | — |
| 47 | #486 | — | 2 | 8 | no | — |
| 48 | #486 | — | 2 | 8 | no | — |
| 49 | #486 | — | 2 | 8 | no | — |
| 50 | #486 | — | 2 | 8 | no | — |
| 51 | C2×#103 | — | 2 | 4 | no | — |
| 52 | C2×#102 | — | 2 | 4 | no | — |
| 53 | C2×#102 | — | 2 | 8 | no | — |
| 54 | C2×#102 | — | 2 | 8 | no | — |
| 55 | C2×C4×C8 | D4 | 1 | 4 | yes | — |
| 56 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 57 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 58 | #391 | — | 2 | 4 | no | — |
| 59 | #390 | — | 2 | 4 | no | — |
| 60 | #382 | — | 2 | 2 | no | — |
| 61 | C43 | C23 | 1 | 1 | yes | characteristic |
| 62 | #336 | D4 | 1 | 8 | yes | — |
| 63 | #322 | D4 | 1 | 16 | yes | complemented |
| 64 | C22×Q8 | — | 2 | 1 | no | — |
| 65 | C22×Q8 | — | 2 | 2 | no | — |
| 66 | C22×D4 | — | 2 | 1 | no | — |
| 67 | C22×D4 | — | 2 | 4 | no | — |
| 68 | C22×D4 | — | 2 | 4 | no | — |
| 69 | C22×D4 | — | 2 | 4 | no | — |
| 70 | C22×D4 | — | 2 | 2 | no | — |
| 71 | C22×D4 | — | 4 | 8 | no | — |
| 72 | C2×SD16 | — | 4 | 16 | no | — |
| 73 | C2×SD16 | — | 8 | 16 | no | — |
| 74 | C2×D8 | — | 4 | 32 | no | — |
| 75 | C2×D8 | — | 4 | 16 | no | — |
| 76 | C2×D8 | — | 4 | 16 | no | — |
| 77 | C2×D8 | — | 8 | 16 | no | — |
| 78 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 79 | C22×C8 | C2×D4 | 1 | 8 | yes | — |
| 80 | C4⋊Q8 | — | 2 | 4 | no | — |
| 81 | C4⋊Q8 | — | 2 | 4 | no | — |
| 82 | C4⋊Q8 | — | 2 | 4 | no | — |
| 83 | #127 | — | 2 | 4 | no | — |
| 84 | #127 | — | 2 | 4 | no | — |
| 85 | #127 | — | 2 | 4 | no | — |
| 86 | #127 | — | 2 | 4 | no | — |
| 87 | #127 | — | 2 | 4 | no | — |
| 88 | #127 | — | 4 | 16 | no | — |
| 89 | #127 | — | 8 | 8 | no | — |
| 90 | #127 | — | 8 | 8 | no | — |
| 91 | #124 | — | 4 | 8 | no | — |
| 92 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 93 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 94 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 95 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 96 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 97 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 98 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 99 | #103 | — | 4 | 8 | no | — |
| 100 | #102 | — | 4 | 16 | no | — |
| 101 | #102 | — | 4 | 16 | no | — |
| 102 | #102 | — | 4 | 8 | no | — |
| 103 | C4×C8 | — | 2 | 8 | no | — |
| 104 | C4×C8 | C2×D4 | 1 | 8 | yes | — |
| 105 | C4×C8 | C2×D4 | 1 | 8 | yes | — |
| 106 | C4×C8 | C2×D4 | 1 | 8 | yes | — |
| 107 | #95 | — | 4 | 4 | no | — |
| 108 | C24 | — | 4 | 1 | no | — |
| 109 | C24 | — | 4 | 4 | no | — |
| 110 | C2×Q8 | — | 4 | 2 | no | — |
| 111 | C2×Q8 | — | 4 | 4 | no | — |
| 112 | C2×Q8 | — | 4 | 4 | no | — |
| 113 | C2×Q8 | — | 4 | 8 | no | — |
| 114 | C2×D4 | — | 4 | 2 | no | — |
| 115 | C2×D4 | — | 4 | 4 | no | — |
| 116 | C2×D4 | — | 4 | 4 | no | — |
| 117 | C2×D4 | — | 4 | 4 | no | — |
| 118 | C2×D4 | — | 4 | 4 | no | — |
| 119 | C2×D4 | — | 4 | 4 | no | — |
| 120 | C2×D4 | — | 4 | 4 | no | — |
| 121 | C2×D4 | — | 4 | 4 | no | — |
| 122 | C2×D4 | — | 4 | 4 | no | — |
| 123 | C2×D4 | — | 4 | 16 | no | — |
| 124 | C2×D4 | — | 4 | 16 | no | — |
| 125 | C2×D4 | — | 4 | 16 | no | — |
| 126 | C2×D4 | — | 4 | 8 | no | — |
| 127 | C2×D4 | — | 8 | 16 | no | — |
| 128 | C2×D4 | — | 8 | 16 | no | — |
| 129 | C2×D4 | — | 8 | 16 | no | — |
| 130 | C22×C4 | — | 4 | 2 | no | — |
| 131 | C22×C4 | — | 4 | 1 | no | — |
| 132 | C22×C4 | #127 | 1 | 4 | yes | — |
| 133 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 134 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 135 | SD16 | — | 8 | 32 | no | — |
| 136 | D8 | — | 8 | 32 | no | — |
| 137 | D8 | — | 8 | 32 | no | — |
| 138 | C2×C8 | — | 2 | 8 | no | — |
| 139 | C2×C8 | — | 2 | 16 | no | — |
| 140 | C2×C8 | #127 | 1 | 16 | yes | — |
| 141 | C2×C8 | #127 | 1 | 8 | yes | — |
| 142 | C4⋊C4 | — | 4 | 2 | no | — |
| 143 | C4⋊C4 | — | 4 | 4 | no | — |
| 144 | C4⋊C4 | — | 4 | 4 | no | — |
| 145 | C4⋊C4 | — | 4 | 2 | no | — |
| 146 | C22⋊C4 | — | 8 | 8 | no | — |
| 147 | C42 | — | 2 | 8 | no | — |
| 148 | C42 | AGL(1,ℤ/8) | 1 | 8 | yes | — |
| 149 | C42 | C22×D4 | 1 | 4 | yes | — |
| 150 | C23 | — | 8 | 1 | no | — |
| 151 | C23 | — | 8 | 4 | no | — |
| 152 | C23 | — | 8 | 2 | no | — |
| 153 | C23 | — | 8 | 8 | no | — |
| 154 | C23 | — | 8 | 8 | no | — |
| 155 | C23 | — | 8 | 8 | no | — |
| 156 | C23 | — | 8 | 4 | no | — |
| 157 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 158 | Q8 | — | 8 | 4 | no | — |
| 159 | Q8 | — | 8 | 8 | no | — |
| 160 | D4 | — | 8 | 8 | no | — |
| 161 | D4 | — | 8 | 8 | no | — |
| 162 | D4 | — | 8 | 4 | no | — |
| 163 | D4 | — | 8 | 8 | no | — |
| 164 | D4 | — | 8 | 8 | no | — |
| 165 | D4 | — | 8 | 8 | no | — |
| 166 | D4 | — | 8 | 8 | no | — |
| 167 | D4 | — | 8 | 8 | no | — |
| 168 | D4 | — | 16 | 32 | no | — |
| 169 | C2×C4 | — | 2 | 8 | no | — |
| 170 | C2×C4 | — | 8 | 1 | no | — |
| 171 | C2×C4 | — | 8 | 2 | no | — |
| 172 | C2×C4 | — | 8 | 2 | no | — |
| 173 | C2×C4 | — | 8 | 4 | no | — |
| 174 | C2×C4 | #496 | 1 | 8 | yes | — |
| 175 | C2×C4 | C2×#127 | 1 | 2 | yes | — |
| 176 | C2×C4 | C2×#127 | 1 | 4 | yes | — |
| 177 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 178 | C2×C4 | #576 | 1 | 8 | yes | — |
| 179 | C8 | — | 2 | 8 | no | — |
| 180 | C8 | — | 2 | 16 | no | — |
| 181 | C22 | — | 16 | 4 | no | — |
| 182 | C22 | — | 16 | 1 | no | — |
| 183 | C22 | — | 16 | 2 | no | — |
| 184 | C22 | — | 16 | 4 | no | — |
| 185 | C22 | — | 16 | 4 | no | — |
| 186 | C22 | — | 16 | 16 | no | — |
| 187 | C22 | — | 16 | 4 | no | — |
| 188 | C22 | #2867 | 1 | 1 | yes | characteristic |
| 189 | C22 | C2×#496 | 1 | 2 | yes | — |
| 190 | C22 | #3155 | 1 | 4 | yes | — |
| 191 | C4 | — | 2 | 8 | no | — |
| 192 | C4 | — | 16 | 1 | no | — |
| 193 | C4 | — | 16 | 2 | no | — |
| 194 | C4 | #3151 | 1 | 4 | yes | — |
| 195 | C4 | #3155 | 1 | 8 | yes | — |
| 196 | C2 | — | 32 | 4 | no | — |
| 197 | C2 | — | 32 | 1 | no | — |
| 198 | C2 | #19698 | 1 | 4 | yes | — |
| 199 | C2 | #33304 | 1 | 1 | yes | characteristic |
| 200 | C2 | #33308 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33301 | C2 | semidirect |
| 2. #33227 | C2 | semidirect |
| 3. #30259 | C2 | semidirect |
| 4. #30258 | C2 | semidirect |
| 5. C2×#1721 | C2 | semidirect |
| 6. C2×#1717 | C2 | semidirect |
| 7. #7843 | C2 | semidirect |
| 63. #322 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33301 | 2 |
| #33227 | 2 |
| #30259 | 2 |
| #30258 | 2 |
| C2×#1721 | 2 |
| C2×#1717 | 2 |
| #7843 | 2 |