Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
791 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 791 conjugacy classes of subgroups — too many to draw. Only its 168 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 | C2×#2206 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #21697 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #13448 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #12899 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #12813 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | C22×D16 | C22 | 1 | 4 | yes | — |
| 7 | C2×#493 | C22 | 1 | 2 | yes | — |
| 8 | C2×#442 | — | 2 | 2 | no | — |
| 9 | C2×C4×D8 | C22 | 1 | 4 | yes | — |
| 10 | #2328 | — | 2 | 2 | no | — |
| 11 | #2206 | — | 2 | 16 | no | — |
| 12 | C2×C4⋊C16 | C22 | 1 | 4 | yes | — |
| 13 | C2×C4⋊C16 | C22 | 1 | 2 | yes | — |
| 14 | C2×#357 | C22 | 1 | 8 | yes | — |
| 15 | #1972 | — | 2 | 2 | no | — |
| 16 | #1970 | C22 | 1 | 4 | yes | — |
| 17 | #1961 | — | 2 | 2 | no | — |
| 18 | #1846 | C22 | 1 | 2 | yes | — |
| 19 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1380 | C22 | 1 | 4 | yes | — |
| 21 | C22×D8 | — | 2 | 2 | no | — |
| 22 | C22×D8 | C23 | 1 | 4 | yes | — |
| 23 | C2×#127 | — | 2 | 2 | no | — |
| 24 | C2×C4×D4 | — | 2 | 4 | no | — |
| 25 | C2×D16 | — | 2 | 16 | no | — |
| 26 | C2×D16 | — | 4 | 4 | no | — |
| 27 | C22×C16 | C23 | 1 | 4 | yes | — |
| 28 | #493 | — | 2 | 4 | no | — |
| 29 | #493 | D4 | 1 | 8 | yes | complemented |
| 30 | #442 | — | 4 | 8 | no | — |
| 31 | C4×D8 | — | 2 | 8 | no | — |
| 32 | C4×D8 | — | 2 | 8 | no | — |
| 33 | C2×#107 | C23 | 1 | 4 | yes | — |
| 34 | C2×#106 | — | 2 | 2 | no | — |
| 35 | C2×#102 | — | 2 | 4 | no | — |
| 36 | C2×#102 | — | 2 | 4 | no | — |
| 37 | C2×#97 | D4 | 1 | 2 | yes | — |
| 38 | C2×#97 | D4 | 1 | 1 | yes | characteristic |
| 39 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 41 | #390 | — | 2 | 4 | no | — |
| 42 | #386 | — | 4 | 8 | no | — |
| 43 | #382 | — | 2 | 2 | no | — |
| 44 | #376 | D4 | 1 | 1 | yes | characteristic |
| 45 | C4⋊C16 | — | 2 | 8 | no | — |
| 46 | C4⋊C16 | — | 4 | 2 | no | — |
| 47 | #357 | — | 2 | 16 | no | — |
| 48 | C22×D4 | — | 2 | 4 | no | — |
| 49 | C22×D4 | — | 4 | 4 | no | — |
| 50 | C23×C4 | — | 4 | 4 | no | — |
| 51 | C2×D8 | — | 2 | 8 | no | — |
| 52 | C2×D8 | — | 2 | 4 | no | — |
| 53 | C2×D8 | — | 2 | 4 | no | — |
| 54 | C2×D8 | — | 2 | 4 | no | — |
| 55 | C2×D8 | — | 2 | 4 | no | — |
| 56 | C2×D8 | — | 2 | 4 | no | — |
| 57 | C2×D8 | — | 4 | 4 | no | — |
| 58 | C2×D8 | — | 4 | 4 | no | — |
| 59 | C2×D8 | C2×D4 | 1 | 8 | yes | — |
| 60 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 61 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 62 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 63 | #127 | — | 2 | 8 | no | — |
| 64 | #127 | — | 4 | 4 | no | — |
| 65 | C4×D4 | — | 4 | 8 | no | — |
| 66 | C4×D4 | — | 4 | 8 | no | — |
| 67 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 68 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 69 | C2×C22⋊C4 | — | 4 | 8 | no | — |
| 70 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 72 | D16 | — | 4 | 16 | no | — |
| 73 | C2×C16 | — | 2 | 8 | no | — |
| 74 | C2×C16 | — | 2 | 4 | no | — |
| 75 | #107 | — | 2 | 4 | no | — |
| 76 | #107 | — | 2 | 4 | no | — |
| 77 | #106 | — | 4 | 4 | no | — |
| 78 | #102 | — | 4 | 8 | no | — |
| 79 | #102 | — | 4 | 8 | no | — |
| 80 | #97 | — | 2 | 4 | no | — |
| 81 | #97 | — | 4 | 1 | no | — |
| 82 | C4×C8 | — | 2 | 1 | no | — |
| 83 | C4×C8 | — | 2 | 1 | no | — |
| 84 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 85 | C4×C8 | #41 | 1 | 4 | yes | — |
| 86 | #95 | — | 4 | 4 | no | — |
| 87 | #95 | D8 | 1 | 4 | yes | — |
| 88 | C24 | — | 4 | 4 | no | — |
| 89 | C2×D4 | — | 2 | 8 | no | — |
| 90 | C2×D4 | — | 4 | 4 | no | — |
| 91 | C2×D4 | — | 4 | 4 | no | — |
| 92 | C2×D4 | — | 4 | 4 | no | — |
| 93 | C2×D4 | — | 4 | 4 | no | — |
| 94 | C2×D4 | — | 4 | 4 | no | — |
| 95 | C2×D4 | — | 4 | 16 | no | — |
| 96 | C2×D4 | — | 16 | 8 | no | — |
| 97 | C22×C4 | — | 4 | 4 | no | — |
| 98 | C22×C4 | — | 8 | 8 | no | — |
| 99 | C22×C4 | — | 16 | 8 | no | — |
| 100 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 101 | C22×C4 | C2×D8 | 1 | 4 | yes | — |
| 102 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 103 | D8 | — | 2 | 8 | no | — |
| 104 | D8 | — | 2 | 8 | no | — |
| 105 | D8 | — | 4 | 16 | no | — |
| 106 | D8 | — | 4 | 4 | no | — |
| 107 | D8 | — | 4 | 4 | no | — |
| 108 | C2×C8 | — | 2 | 2 | no | — |
| 109 | C2×C8 | — | 2 | 2 | no | — |
| 110 | C2×C8 | — | 2 | 1 | no | — |
| 111 | C2×C8 | — | 2 | 2 | no | — |
| 112 | C2×C8 | #121 | 1 | 4 | yes | — |
| 113 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 114 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 115 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 116 | C2×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 117 | C4⋊C4 | — | 4 | 4 | no | — |
| 118 | C4⋊C4 | — | 4 | 4 | no | — |
| 119 | C22⋊C4 | — | 8 | 8 | no | — |
| 120 | C22⋊C4 | — | 16 | 8 | no | — |
| 121 | C42 | — | 2 | 1 | no | — |
| 122 | C42 | — | 2 | 1 | no | — |
| 123 | C42 | #121 | 1 | 4 | yes | — |
| 124 | C42 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 125 | C16 | — | 4 | 4 | no | — |
| 126 | C23 | — | 4 | 8 | no | — |
| 127 | C23 | — | 8 | 8 | no | — |
| 128 | C23 | — | 8 | 8 | no | — |
| 129 | C23 | — | 8 | 4 | no | — |
| 130 | C23 | — | 8 | 4 | no | — |
| 131 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 132 | D4 | — | 4 | 8 | no | — |
| 133 | D4 | — | 4 | 8 | no | — |
| 134 | D4 | — | 8 | 4 | no | — |
| 135 | D4 | — | 8 | 4 | no | — |
| 136 | D4 | — | 16 | 16 | no | — |
| 137 | C2×C4 | — | 2 | 8 | no | — |
| 138 | C2×C4 | — | 2 | 4 | no | — |
| 139 | C2×C4 | — | 2 | 2 | no | — |
| 140 | C2×C4 | — | 8 | 8 | no | — |
| 141 | C2×C4 | — | 8 | 4 | no | — |
| 142 | C2×C4 | — | 16 | 16 | no | — |
| 143 | C2×C4 | #459 | 1 | 8 | yes | — |
| 144 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 145 | C2×C4 | #537 | 1 | 1 | yes | characteristic |
| 146 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 147 | C2×C4 | #575 | 1 | 1 | yes | characteristic |
| 148 | C8 | — | 2 | 4 | no | — |
| 149 | C8 | — | 4 | 1 | no | — |
| 150 | C8 | D42 | 1 | 1 | yes | characteristic |
| 151 | C8 | #547 | 1 | 2 | yes | — |
| 152 | C8 | #555 | 1 | 1 | yes | characteristic |
| 153 | C22 | — | 8 | 16 | no | — |
| 154 | C22 | — | 8 | 8 | no | — |
| 155 | C22 | — | 16 | 8 | no | — |
| 156 | C22 | — | 16 | 4 | no | — |
| 157 | C22 | — | 16 | 4 | no | — |
| 158 | C22 | C2×#459 | 1 | 2 | yes | — |
| 159 | C22 | #3245 | 1 | 1 | yes | characteristic |
| 160 | C22 | #3415 | 1 | 4 | yes | — |
| 161 | C4 | — | 2 | 8 | no | — |
| 162 | C4 | — | 4 | 2 | no | — |
| 163 | C4 | — | 16 | 4 | no | — |
| 164 | C4 | #2537 | 1 | 1 | yes | characteristic |
| 165 | C4 | #3036 | 1 | 2 | yes | — |
| 166 | C4 | #3253 | 1 | 1 | yes | characteristic |
| 167 | C2 | — | 16 | 8 | no | — |
| 168 | C2 | #28251 | 1 | 1 | yes | characteristic |
| 169 | C2 | #33585 | 1 | 4 | yes | — |
| 170 | C2 | #33825 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2206 | C2 | semidirect |
| 2. #21697 | C2 | semidirect |
| 3. #13448 | C2 | semidirect |
| 4. #12899 | C2 | semidirect |
| 5. #12813 | C2 | semidirect |
| 29. #493 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2206 | 2 |
| #21697 | 2 |
| #13448 | 2 |
| #12899 | 2 |
| #12813 | 2 |