Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
201 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 44 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 | #33825 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13712 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13711 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13517 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #7420 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #7420 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7363 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3176 | — | 2 | 1 | no | — |
| 9 | #2228 | — | 2 | 1 | no | — |
| 10 | #2228 | — | 2 | 1 | no | — |
| 11 | #2218 | — | 2 | 1 | no | — |
| 12 | #2215 | C22 | 1 | 1 | yes | characteristic |
| 13 | #2215 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2206 | — | 2 | 1 | no | — |
| 15 | #2206 | — | 2 | 1 | no | — |
| 16 | C2×#366 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2151 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1952 | — | 2 | 1 | no | — |
| 19 | #1424 | — | 2 | 1 | no | — |
| 20 | #1423 | — | 2 | 1 | no | — |
| 21 | #1415 | — | 2 | 1 | no | — |
| 22 | #1415 | — | 2 | 1 | no | — |
| 23 | C22⋊C32 | C22 | 1 | 1 | yes | characteristic |
| 24 | C22⋊C32 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1387 | — | 2 | 1 | no | — |
| 26 | #1381 | C22 | 1 | 1 | yes | characteristic |
| 27 | #540 | — | 4 | 1 | no | — |
| 28 | C2×D16 | — | 2 | 1 | no | — |
| 29 | C2×D16 | D4 | 1 | 1 | yes | characteristic |
| 30 | C2×D16 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×M32 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 33 | #499 | — | 2 | 1 | no | — |
| 34 | #493 | — | 2 | 1 | no | — |
| 35 | #490 | — | 4 | 1 | no | — |
| 36 | #469 | — | 4 | 1 | no | — |
| 37 | #466 | — | 2 | 1 | no | — |
| 38 | #466 | — | 2 | 1 | no | — |
| 39 | #442 | — | 4 | 1 | no | — |
| 40 | C4×D8 | — | 2 | 1 | no | — |
| 41 | C4×D8 | — | 2 | 1 | no | — |
| 42 | C2×#108 | — | 2 | 1 | no | — |
| 43 | #428 | — | 4 | 1 | no | — |
| 44 | C2×#107 | — | 2 | 1 | no | — |
| 45 | #405 | D4 | 1 | 1 | yes | characteristic |
| 46 | C2×C32 | — | 2 | 1 | no | — |
| 47 | C2×C32 | — | 2 | 1 | no | — |
| 48 | #367 | — | 2 | 1 | no | — |
| 49 | #366 | — | 2 | 1 | no | — |
| 50 | #366 | — | 2 | 1 | no | — |
| 51 | #366 | D4 | 1 | 1 | yes | characteristic |
| 52 | #366 | D4 | 1 | 1 | yes | characteristic |
| 53 | C4⋊C16 | — | 2 | 1 | no | — |
| 54 | C4⋊C16 | — | 2 | 1 | no | — |
| 55 | #357 | — | 2 | 1 | no | — |
| 56 | #357 | — | 2 | 1 | no | — |
| 57 | #357 | — | 2 | 1 | no | — |
| 58 | #357 | — | 2 | 1 | no | — |
| 59 | #343 | — | 4 | 1 | no | — |
| 60 | C2×D8 | — | 2 | 1 | no | — |
| 61 | C2×D8 | — | 2 | 1 | no | — |
| 62 | C2×D8 | — | 4 | 1 | no | — |
| 63 | C2×M16 | — | 4 | 1 | no | — |
| 64 | C2×M16 | D8 | 1 | 1 | yes | characteristic |
| 65 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | #127 | — | 4 | 1 | no | — |
| 67 | #125 | — | 4 | 1 | no | — |
| 68 | #123 | — | 8 | 1 | no | — |
| 69 | #123 | — | 8 | 1 | no | — |
| 70 | #121 | — | 4 | 1 | no | — |
| 71 | #121 | — | 4 | 1 | no | — |
| 72 | #121 | — | 8 | 1 | no | — |
| 73 | #121 | — | 8 | 1 | no | — |
| 74 | C4×D4 | — | 4 | 1 | no | — |
| 75 | C4×D4 | — | 4 | 1 | no | — |
| 76 | #117 | D8 | 1 | 1 | yes | characteristic |
| 77 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 78 | D16 | — | 4 | 1 | no | — |
| 79 | D16 | — | 4 | 1 | no | — |
| 80 | D16 | — | 8 | 1 | no | — |
| 81 | M32 | — | 4 | 1 | no | — |
| 82 | C2×C16 | — | 2 | 1 | no | — |
| 83 | C2×C16 | — | 4 | 1 | no | — |
| 84 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | #108 | — | 2 | 2 | no | — |
| 87 | #108 | — | 4 | 1 | no | — |
| 88 | #107 | — | 2 | 1 | no | — |
| 89 | #107 | — | 2 | 1 | no | — |
| 90 | #107 | — | 4 | 1 | no | — |
| 91 | #107 | — | 4 | 1 | no | — |
| 92 | #107 | — | 8 | 1 | no | — |
| 93 | #106 | — | 4 | 1 | no | — |
| 94 | #106 | — | 8 | 1 | no | — |
| 95 | #102 | — | 4 | 1 | no | — |
| 96 | #102 | — | 4 | 1 | no | — |
| 97 | #102 | — | 4 | 1 | no | — |
| 98 | #102 | — | 4 | 1 | no | — |
| 99 | #97 | — | 2 | 1 | no | — |
| 100 | C4×C8 | — | 2 | 1 | no | — |
| 101 | C32 | — | 4 | 1 | no | — |
| 102 | C32 | — | 4 | 1 | no | — |
| 103 | C2×D4 | — | 4 | 1 | no | — |
| 104 | C2×D4 | — | 4 | 1 | no | — |
| 105 | C2×D4 | — | 8 | 1 | no | — |
| 106 | C2×D4 | — | 8 | 1 | no | — |
| 107 | C2×D4 | — | 8 | 1 | no | — |
| 108 | C2×D4 | — | 8 | 1 | no | — |
| 109 | C22×C4 | — | 8 | 1 | no | — |
| 110 | C22×C4 | — | 8 | 1 | no | — |
| 111 | C22×C4 | — | 8 | 1 | no | — |
| 112 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 113 | D8 | — | 4 | 1 | no | — |
| 114 | D8 | — | 4 | 1 | no | — |
| 115 | D8 | — | 4 | 1 | no | — |
| 116 | D8 | — | 4 | 1 | no | — |
| 117 | D8 | — | 8 | 2 | no | — |
| 118 | M16 | — | 4 | 1 | no | — |
| 119 | M16 | — | 4 | 2 | no | — |
| 120 | M16 | — | 8 | 1 | no | — |
| 121 | C2×C8 | — | 2 | 1 | no | — |
| 122 | C2×C8 | — | 2 | 1 | no | — |
| 123 | C2×C8 | — | 2 | 1 | no | — |
| 124 | C2×C8 | — | 8 | 1 | no | — |
| 125 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 126 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 127 | C4⋊C4 | — | 4 | 1 | no | — |
| 128 | C4⋊C4 | — | 4 | 1 | no | — |
| 129 | C4⋊C4 | — | 8 | 1 | no | — |
| 130 | C4⋊C4 | — | 8 | 1 | no | — |
| 131 | C4⋊C4 | — | 8 | 1 | no | — |
| 132 | C4⋊C4 | — | 8 | 1 | no | — |
| 133 | C4⋊C4 | D16 | 1 | 1 | yes | characteristic |
| 134 | C4⋊C4 | D16 | 1 | 1 | yes | characteristic |
| 135 | C22⋊C4 | — | 2 | 1 | no | — |
| 136 | C22⋊C4 | — | 8 | 1 | no | — |
| 137 | C22⋊C4 | — | 8 | 1 | no | — |
| 138 | C22⋊C4 | — | 8 | 1 | no | — |
| 139 | C22⋊C4 | — | 8 | 1 | no | — |
| 140 | C42 | — | 2 | 1 | no | — |
| 141 | C16 | — | 2 | 1 | no | — |
| 142 | C16 | — | 2 | 1 | no | — |
| 143 | C16 | — | 4 | 1 | no | — |
| 144 | C23 | — | 8 | 1 | no | — |
| 145 | C23 | — | 8 | 1 | no | — |
| 146 | C23 | #509 | 1 | 1 | yes | characteristic |
| 147 | D4 | — | 8 | 1 | no | — |
| 148 | D4 | — | 8 | 1 | no | — |
| 149 | D4 | — | 8 | 1 | no | — |
| 150 | D4 | — | 8 | 1 | no | — |
| 151 | D4 | — | 32 | 1 | no | — |
| 152 | D4 | — | 32 | 1 | no | — |
| 153 | D4 | — | 32 | 1 | no | — |
| 154 | D4 | — | 32 | 1 | no | — |
| 155 | C2×C4 | — | 2 | 1 | no | — |
| 156 | C2×C4 | — | 2 | 1 | no | — |
| 157 | C2×C4 | — | 2 | 1 | no | — |
| 158 | C2×C4 | — | 2 | 1 | no | — |
| 159 | C2×C4 | — | 8 | 1 | no | — |
| 160 | C2×C4 | — | 8 | 1 | no | — |
| 161 | C2×C4 | — | 16 | 1 | no | — |
| 162 | C2×C4 | — | 32 | 1 | no | — |
| 163 | C2×C4 | — | 32 | 1 | no | — |
| 164 | C2×C4 | — | 32 | 1 | no | — |
| 165 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 166 | C2×C4 | C2×D16 | 1 | 1 | yes | characteristic |
| 167 | C8 | — | 2 | 1 | no | — |
| 168 | C8 | — | 4 | 1 | no | — |
| 169 | C8 | — | 8 | 2 | no | — |
| 170 | C8 | #456 | 1 | 1 | yes | characteristic |
| 171 | C8 | #458 | 1 | 1 | yes | characteristic |
| 172 | C22 | — | 2 | 1 | no | — |
| 173 | C22 | — | 2 | 1 | no | — |
| 174 | C22 | — | 2 | 1 | no | — |
| 175 | C22 | — | 16 | 1 | no | — |
| 176 | C22 | — | 16 | 1 | no | — |
| 177 | C22 | — | 16 | 1 | no | — |
| 178 | C22 | — | 16 | 1 | no | — |
| 179 | C22 | — | 16 | 1 | no | — |
| 180 | C22 | — | 16 | 1 | no | — |
| 181 | C22 | #2184 | 1 | 1 | yes | characteristic |
| 182 | C4 | — | 2 | 1 | no | — |
| 183 | C4 | — | 4 | 1 | no | — |
| 184 | C4 | — | 4 | 1 | no | — |
| 185 | C4 | — | 16 | 1 | no | — |
| 186 | C4 | — | 16 | 1 | no | — |
| 187 | C4 | — | 32 | 1 | no | — |
| 188 | C4 | #1610 | 1 | 1 | yes | characteristic |
| 189 | C4 | #2191 | 1 | 1 | yes | characteristic |
| 190 | C2 | — | 4 | 1 | no | — |
| 191 | C2 | — | 32 | 1 | no | — |
| 192 | C2 | — | 32 | 1 | no | — |
| 193 | C2 | #12288 | 1 | 1 | yes | characteristic |
| 194 | C2 | #13669 | 1 | 1 | yes | characteristic |
| 195 | C2 | #13671 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2184 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | #33825 | C2 |
| LCS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2151 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13712 | C2 | semidirect |
| 3. #13711 | C2 | semidirect |
| 4. #13517 | C2 | semidirect |
| 5. #7420 | C2 | semidirect |
| 6. #7420 | C2 | semidirect |
| 7. #7363 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33825 | 2 |
| #13712 | 2 |
| #13711 | 2 |
| #13517 | 2 |
| #7420 | 2 |
| #7363 | 2 |