Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
615 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 615 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 | #33266 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C42×D8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13130 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #13109 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #13012 | C2 | 1 | 3 | yes | maximal |
| 6 | #12935 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #12894 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C4×D8 | C22 | 1 | 3 | yes | — |
| 10 | #2860 | — | 2 | 1 | no | — |
| 11 | C42×D4 | — | 2 | 1 | no | — |
| 12 | C2×#366 | C22 | 1 | 3 | yes | — |
| 13 | C2×#357 | C22 | 1 | 3 | yes | — |
| 14 | C2×#357 | C22 | 1 | 3 | yes | — |
| 15 | #1970 | C22 | 1 | 3 | yes | — |
| 16 | #1942 | C22 | 1 | 3 | yes | — |
| 17 | #1934 | — | 2 | 3 | no | — |
| 18 | C4×#107 | C22 | 1 | 3 | yes | — |
| 19 | C4×#102 | — | 2 | 3 | no | — |
| 20 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1380 | C22 | 1 | 3 | yes | — |
| 22 | #1380 | C22 | 1 | 3 | yes | — |
| 23 | #1374 | C22 | 1 | 3 | yes | — |
| 24 | #1374 | C22 | 1 | 3 | yes | — |
| 25 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 26 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×#126 | — | 4 | 1 | no | — |
| 28 | C2×C4×D4 | — | 2 | 3 | no | — |
| 29 | C22×C42 | — | 4 | 1 | no | — |
| 30 | C2×D16 | — | 2 | 6 | no | — |
| 31 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 32 | C22×C16 | C23 | 1 | 3 | yes | — |
| 33 | C4×D8 | — | 2 | 6 | no | — |
| 34 | C4×D8 | — | 2 | 6 | no | — |
| 35 | C2×#107 | C23 | 1 | 3 | yes | — |
| 36 | C2×#107 | C23 | 1 | 3 | yes | — |
| 37 | C2×#102 | — | 2 | 3 | no | — |
| 38 | C2×#102 | — | 2 | 3 | no | — |
| 39 | C2×C4×C8 | D4 | 1 | 3 | yes | — |
| 40 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 41 | #390 | — | 2 | 3 | no | — |
| 42 | #389 | — | 2 | 3 | no | — |
| 43 | #388 | — | 4 | 3 | no | — |
| 44 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 45 | C4×C22⋊C4 | — | 4 | 3 | no | — |
| 46 | C43 | D4 | 1 | 1 | yes | characteristic |
| 47 | #366 | — | 2 | 6 | no | — |
| 48 | #357 | — | 2 | 6 | no | — |
| 49 | #357 | — | 2 | 6 | no | — |
| 50 | #340 | — | 2 | 3 | no | — |
| 51 | #340 | — | 2 | 3 | no | — |
| 52 | C22×D4 | — | 2 | 1 | no | — |
| 53 | C22×D4 | — | 2 | 1 | no | — |
| 54 | C23×C4 | — | 4 | 3 | no | — |
| 55 | C2×D8 | — | 2 | 3 | no | — |
| 56 | C2×D8 | — | 2 | 3 | no | — |
| 57 | C2×D8 | — | 2 | 3 | no | — |
| 58 | C2×D8 | #41 | 1 | 6 | yes | — |
| 59 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 60 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 61 | #126 | — | 8 | 4 | no | — |
| 62 | C4×D4 | — | 2 | 12 | no | — |
| 63 | C4×D4 | — | 4 | 6 | no | — |
| 64 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 65 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 66 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 67 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 68 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 69 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 70 | C2×C42 | — | 4 | 3 | no | — |
| 71 | C2×C42 | — | 8 | 4 | no | — |
| 72 | C2×C42 | D8 | 1 | 3 | yes | — |
| 73 | C2×C42 | D8 | 1 | 1 | yes | characteristic |
| 74 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 75 | D16 | — | 4 | 4 | no | — |
| 76 | C2×C16 | — | 2 | 3 | no | — |
| 77 | C2×C16 | — | 2 | 3 | no | — |
| 78 | C2×C16 | — | 2 | 3 | no | — |
| 79 | C2×C16 | — | 2 | 3 | no | — |
| 80 | #107 | — | 2 | 3 | no | — |
| 81 | #107 | — | 2 | 3 | no | — |
| 82 | #107 | — | 2 | 3 | no | — |
| 83 | #107 | #41 | 1 | 6 | yes | — |
| 84 | #102 | — | 2 | 12 | no | — |
| 85 | #102 | — | 4 | 6 | no | — |
| 86 | C4×C8 | — | 2 | 3 | no | — |
| 87 | C4×C8 | — | 2 | 3 | no | — |
| 88 | C4×C8 | — | 4 | 3 | no | — |
| 89 | #95 | — | 4 | 3 | no | — |
| 90 | #95 | — | 4 | 3 | no | — |
| 91 | C24 | — | 4 | 1 | no | — |
| 92 | C24 | — | 4 | 1 | no | — |
| 93 | C2×D4 | — | 2 | 6 | no | — |
| 94 | C2×D4 | — | 4 | 3 | no | — |
| 95 | C2×D4 | — | 4 | 3 | no | — |
| 96 | C2×D4 | — | 4 | 3 | no | — |
| 97 | C22×C4 | — | 4 | 12 | no | — |
| 98 | C22×C4 | — | 4 | 3 | no | — |
| 99 | C22×C4 | — | 4 | 3 | no | — |
| 100 | C22×C4 | — | 8 | 6 | no | — |
| 101 | C22×C4 | — | 8 | 6 | no | — |
| 102 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 103 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 104 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 105 | D8 | — | 2 | 6 | no | — |
| 106 | D8 | — | 4 | 1 | no | — |
| 107 | D8 | — | 4 | 3 | no | — |
| 108 | D8 | #126 | 1 | 4 | yes | complemented |
| 109 | C2×C8 | — | 2 | 3 | no | — |
| 110 | C2×C8 | — | 2 | 3 | no | — |
| 111 | C2×C8 | — | 2 | 3 | no | — |
| 112 | C2×C8 | C2×#41 | 1 | 3 | yes | — |
| 113 | C2×C8 | C2×#41 | 1 | 3 | yes | — |
| 114 | C4⋊C4 | — | 2 | 6 | no | — |
| 115 | C4⋊C4 | — | 4 | 3 | no | — |
| 116 | C4⋊C4 | — | 4 | 3 | no | — |
| 117 | C4⋊C4 | — | 4 | 3 | no | — |
| 118 | C4⋊C4 | — | 8 | 6 | no | — |
| 119 | C22⋊C4 | — | 4 | 12 | no | — |
| 120 | C22⋊C4 | — | 8 | 6 | no | — |
| 121 | C22⋊C4 | — | 8 | 6 | no | — |
| 122 | C42 | — | 2 | 3 | no | — |
| 123 | C42 | — | 2 | 3 | no | — |
| 124 | C42 | — | 4 | 3 | no | — |
| 125 | C42 | — | 8 | 6 | no | — |
| 126 | C42 | D16 | 1 | 4 | yes | complemented |
| 127 | C16 | — | 4 | 1 | no | — |
| 128 | C16 | — | 4 | 3 | no | — |
| 129 | C23 | — | 4 | 6 | no | — |
| 130 | C23 | — | 8 | 1 | no | — |
| 131 | C23 | — | 8 | 3 | no | — |
| 132 | C23 | — | 8 | 3 | no | — |
| 133 | C23 | — | 8 | 1 | no | — |
| 134 | C23 | — | 8 | 3 | no | — |
| 135 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 136 | D4 | — | 2 | 4 | no | — |
| 137 | D4 | — | 4 | 6 | no | — |
| 138 | D4 | — | 8 | 1 | no | — |
| 139 | D4 | — | 8 | 3 | no | — |
| 140 | C2×C4 | — | 2 | 3 | no | — |
| 141 | C2×C4 | — | 2 | 3 | no | — |
| 142 | C2×C4 | — | 2 | 3 | no | — |
| 143 | C2×C4 | — | 2 | 3 | no | — |
| 144 | C2×C4 | — | 2 | 3 | no | — |
| 145 | C2×C4 | — | 4 | 6 | no | — |
| 146 | C2×C4 | — | 8 | 24 | no | — |
| 147 | C2×C4 | — | 8 | 3 | no | — |
| 148 | C2×C4 | — | 8 | 3 | no | — |
| 149 | C2×C4 | — | 8 | 3 | no | — |
| 150 | C2×C4 | — | 8 | 3 | no | — |
| 151 | C2×C4 | — | 8 | 3 | no | — |
| 152 | C2×C4 | C2×D16 | 1 | 6 | yes | — |
| 153 | C2×C4 | #525 | 1 | 3 | yes | — |
| 154 | C2×C4 | #575 | 1 | 3 | yes | — |
| 155 | C8 | — | 4 | 3 | no | — |
| 156 | C8 | C2×#126 | 1 | 1 | yes | characteristic |
| 157 | C8 | #529 | 1 | 3 | yes | — |
| 158 | C22 | — | 4 | 4 | no | — |
| 159 | C22 | — | 8 | 6 | no | — |
| 160 | C22 | — | 8 | 6 | no | — |
| 161 | C22 | — | 16 | 1 | no | — |
| 162 | C22 | — | 16 | 3 | no | — |
| 163 | C22 | — | 16 | 3 | no | — |
| 164 | C22 | #3100 | 1 | 3 | yes | — |
| 165 | C22 | C22×D16 | 1 | 1 | yes | characteristic |
| 166 | C22 | #3414 | 1 | 3 | yes | — |
| 167 | C4 | — | 2 | 6 | no | — |
| 168 | C4 | — | 4 | 3 | no | — |
| 169 | C4 | — | 8 | 6 | no | — |
| 170 | C4 | — | 16 | 3 | no | — |
| 171 | C4 | #2852 | 1 | 1 | yes | characteristic |
| 172 | C4 | #3111 | 1 | 3 | yes | — |
| 173 | C2 | — | 8 | 4 | no | — |
| 174 | C2 | — | 32 | 1 | no | — |
| 175 | C2 | #33078 | 1 | 1 | yes | characteristic |
| 176 | C2 | #33660 | 1 | 3 | yes | — |
| 177 | C2 | #33671 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33266 | C2 | semidirect |
| 2. C42×D8 | C2 | semidirect |
| 3. #13130 | C2 | semidirect |
| 4. #13109 | C2 | semidirect |
| 6. #12935 | C2 | semidirect |
| 7. #12894 | C2 | semidirect |
| 108. D8 | #126 | semidirect |
| 126. C42 | D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33266 | 2 |
| C42×D8 | 2 |
| #13130 | 2 |
| #13109 | 2 |
| #13012 | 2 |
| #12935 | 2 |
| #12894 | 2 |