Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
788 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 788 conjugacy classes of subgroups — too many to draw. Only its 222 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 | #22268 | C2 | 1 | 3 | yes | maximal |
| 2 | #22246 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×#1705 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #9829 | C2 | 1 | 3 | yes | maximal |
| 5 | #9828 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #8817 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7845 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#498 | C22 | 1 | 3 | yes | — |
| 9 | C2×#492 | C22 | 1 | 3 | yes | — |
| 10 | C4×C4⋊Q8 | C22 | 1 | 3 | yes | — |
| 11 | C4×#127 | C22 | 1 | 1 | yes | characteristic |
| 12 | #1990 | C22 | 1 | 3 | yes | — |
| 13 | #1990 | C22 | 1 | 3 | yes | — |
| 14 | #1957 | C22 | 1 | 3 | yes | — |
| 15 | #1956 | C22 | 1 | 3 | yes | — |
| 16 | #1920 | — | 2 | 6 | no | — |
| 17 | #1919 | — | 2 | 6 | no | — |
| 18 | #1880 | — | 2 | 6 | no | — |
| 19 | #1877 | — | 2 | 6 | no | — |
| 20 | #1834 | C22 | 1 | 3 | yes | — |
| 21 | #1834 | C22 | 1 | 3 | yes | — |
| 22 | #1744 | C22 | 1 | 3 | yes | — |
| 23 | #1705 | C4 | 1 | 8 | yes | complemented |
| 24 | C2×#322 | C22 | 1 | 3 | yes | — |
| 25 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 26 | C22×SD16 | — | 2 | 6 | no | — |
| 27 | C2×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 28 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 29 | C2×C4×Q8 | — | 2 | 3 | no | — |
| 30 | C2×C4×D4 | — | 2 | 3 | no | — |
| 31 | #498 | — | 2 | 6 | no | — |
| 32 | #498 | C2×C4 | 1 | 12 | yes | complemented |
| 33 | #492 | — | 2 | 6 | no | — |
| 34 | #492 | C2×C4 | 1 | 12 | yes | — |
| 35 | C2×#106 | — | 2 | 6 | no | — |
| 36 | C2×#103 | — | 2 | 6 | no | — |
| 37 | C2×#102 | — | 2 | 6 | no | — |
| 38 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 39 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 40 | #391 | — | 2 | 3 | no | — |
| 41 | #390 | — | 2 | 3 | no | — |
| 42 | #389 | — | 2 | 3 | no | — |
| 43 | #389 | — | 2 | 3 | no | — |
| 44 | #384 | C23 | 1 | 1 | yes | characteristic |
| 45 | #384 | C23 | 1 | 3 | yes | — |
| 46 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 47 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 48 | C43 | C23 | 1 | 1 | yes | characteristic |
| 49 | #340 | — | 2 | 6 | no | — |
| 50 | #340 | — | 2 | 6 | no | — |
| 51 | #340 | — | 2 | 6 | no | — |
| 52 | #340 | — | 2 | 6 | no | — |
| 53 | #336 | D4 | 1 | 6 | yes | — |
| 54 | #336 | D4 | 1 | 6 | yes | — |
| 55 | #322 | — | 2 | 6 | no | — |
| 56 | C82 | C2×C4 | 1 | 4 | yes | complemented |
| 57 | C22×Q8 | — | 2 | 3 | no | — |
| 58 | C22×D4 | — | 2 | 3 | no | — |
| 59 | C23×C4 | — | 4 | 1 | no | — |
| 60 | C2×SD16 | — | 2 | 24 | no | — |
| 61 | C2×SD16 | — | 8 | 6 | no | — |
| 62 | C2×SD16 | — | 8 | 6 | no | — |
| 63 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 64 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 65 | C4⋊Q8 | — | 2 | 3 | no | — |
| 66 | C4⋊Q8 | — | 2 | 3 | no | — |
| 67 | C4⋊Q8 | — | 2 | 3 | no | — |
| 68 | C4⋊Q8 | C22×C4 | 1 | 6 | yes | — |
| 69 | #127 | — | 2 | 3 | no | — |
| 70 | #127 | C22×C4 | 1 | 2 | yes | — |
| 71 | C4×Q8 | — | 8 | 6 | no | — |
| 72 | C4×D4 | — | 8 | 6 | no | — |
| 73 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 74 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 75 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 76 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 77 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 78 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 79 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 80 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 81 | C2×C42 | — | 4 | 3 | no | — |
| 82 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 83 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 84 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 85 | #106 | — | 2 | 24 | no | — |
| 86 | #103 | — | 8 | 6 | no | — |
| 87 | #102 | — | 8 | 6 | no | — |
| 88 | C4×C8 | — | 2 | 6 | no | — |
| 89 | C4×C8 | C22×C4 | 1 | 6 | yes | — |
| 90 | C4×C8 | #41 | 1 | 6 | yes | — |
| 91 | #95 | — | 4 | 3 | no | — |
| 92 | #95 | — | 4 | 3 | no | — |
| 93 | C24 | — | 4 | 1 | no | — |
| 94 | C2×Q8 | — | 2 | 6 | no | — |
| 95 | C2×Q8 | — | 4 | 3 | no | — |
| 96 | C2×Q8 | — | 4 | 6 | no | — |
| 97 | C2×Q8 | — | 4 | 6 | no | — |
| 98 | C2×D4 | — | 2 | 6 | no | — |
| 99 | C2×D4 | — | 4 | 3 | no | — |
| 100 | C2×D4 | — | 4 | 6 | no | — |
| 101 | C2×D4 | — | 4 | 6 | no | — |
| 102 | C22×C4 | — | 4 | 3 | no | — |
| 103 | C22×C4 | — | 4 | 3 | no | — |
| 104 | C22×C4 | — | 4 | 1 | no | — |
| 105 | C22×C4 | — | 8 | 6 | no | — |
| 106 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 107 | C22×C4 | #127 | 1 | 3 | yes | — |
| 108 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 109 | SD16 | — | 8 | 24 | no | — |
| 110 | C2×C8 | — | 2 | 6 | no | — |
| 111 | C2×C8 | — | 2 | 6 | no | — |
| 112 | C2×C8 | — | 2 | 6 | no | — |
| 113 | C2×C8 | — | 2 | 6 | no | — |
| 114 | C2×C8 | — | 2 | 6 | no | — |
| 115 | C2×C8 | C4×D4 | 1 | 12 | yes | — |
| 116 | C4⋊C4 | — | 2 | 6 | no | — |
| 117 | C4⋊C4 | — | 2 | 6 | no | — |
| 118 | C4⋊C4 | — | 4 | 3 | no | — |
| 119 | C4⋊C4 | — | 4 | 3 | no | — |
| 120 | C4⋊C4 | — | 4 | 6 | no | — |
| 121 | C4⋊C4 | — | 4 | 6 | no | — |
| 122 | C4⋊C4 | — | 4 | 6 | no | — |
| 123 | C4⋊C4 | — | 4 | 6 | no | — |
| 124 | C4⋊C4 | — | 8 | 6 | no | — |
| 125 | C22⋊C4 | — | 8 | 6 | no | — |
| 126 | C42 | — | 2 | 6 | no | — |
| 127 | C42 | — | 2 | 6 | no | — |
| 128 | C42 | — | 8 | 6 | no | — |
| 129 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 130 | C42 | C2×#41 | 1 | 3 | yes | — |
| 131 | C23 | — | 4 | 2 | no | — |
| 132 | C23 | — | 8 | 3 | no | — |
| 133 | C23 | — | 8 | 3 | no | — |
| 134 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 135 | Q8 | — | 4 | 12 | no | — |
| 136 | Q8 | — | 8 | 6 | no | — |
| 137 | D4 | — | 4 | 12 | no | — |
| 138 | D4 | — | 8 | 6 | no | — |
| 139 | C2×C4 | — | 2 | 3 | no | — |
| 140 | C2×C4 | — | 2 | 3 | no | — |
| 141 | C2×C4 | — | 2 | 3 | no | — |
| 142 | C2×C4 | — | 2 | 3 | no | — |
| 143 | C2×C4 | — | 4 | 6 | no | — |
| 144 | C2×C4 | — | 8 | 3 | no | — |
| 145 | C2×C4 | — | 8 | 3 | no | — |
| 146 | C2×C4 | — | 8 | 3 | no | — |
| 147 | C2×C4 | — | 8 | 3 | no | — |
| 148 | C2×C4 | — | 8 | 3 | no | — |
| 149 | C2×C4 | — | 8 | 3 | no | — |
| 150 | C2×C4 | — | 16 | 4 | no | — |
| 151 | C2×C4 | C2×C4×D4 | 1 | 3 | yes | — |
| 152 | C2×C4 | #532 | 1 | 3 | yes | — |
| 153 | C2×C4 | #577 | 1 | 6 | yes | — |
| 154 | C2×C4 | #577 | 1 | 6 | yes | — |
| 155 | C8 | — | 2 | 12 | no | — |
| 156 | C8 | — | 4 | 6 | no | — |
| 157 | C22 | — | 8 | 6 | no | — |
| 158 | C22 | — | 16 | 1 | no | — |
| 159 | C22 | — | 16 | 3 | no | — |
| 160 | C22 | C4×#127 | 1 | 1 | yes | characteristic |
| 161 | C22 | #3154 | 1 | 3 | yes | — |
| 162 | C22 | #3154 | 1 | 3 | yes | — |
| 163 | C4 | — | 4 | 1 | no | — |
| 164 | C4 | — | 4 | 3 | no | — |
| 165 | C4 | — | 8 | 6 | no | — |
| 166 | C4 | — | 16 | 1 | no | — |
| 167 | C4 | — | 16 | 3 | no | — |
| 168 | C4 | #2949 | 1 | 6 | yes | — |
| 169 | C4 | #3174 | 1 | 6 | yes | — |
| 170 | C2 | — | 16 | 2 | no | — |
| 171 | C2 | #19733 | 1 | 1 | yes | characteristic |
| 172 | C2 | #19734 | 1 | 3 | yes | — |
| 173 | C2 | #21695 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ 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 |
|---|---|---|
| 2. #22246 | C2 | semidirect |
| 5. #9828 | C2 | semidirect |
| 6. #8817 | C2 | semidirect |
| 7. #7845 | C2 | semidirect |
| 23. #1705 | C4 | semidirect |
| 32. #498 | C2×C4 | semidirect |
| 56. C82 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22268 | 2 |
| #22246 | 2 |
| C2×#1705 | 2 |
| #9829 | 2 |
| #9828 | 2 |
| #8817 | 2 |
| #7845 | 2 |