Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2076 conjugacy classes of subgroups; only the 338 normal subgroups are shown.
This group has 2076 conjugacy classes of subgroups — too many to draw. Only its 338 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×#3556 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #21322 | C2 | 1 | 3 | yes | maximal |
| 3 | C2×#1526 | C2 | 1 | 3 | yes | maximal |
| 4 | #8509 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #3556 | — | 2 | 12 | no | — |
| 6 | #3556 | C4 | 1 | 8 | yes | complemented |
| 7 | C2×#564 | C22 | 1 | 4 | yes | — |
| 8 | C2×#557 | — | 2 | 6 | no | — |
| 9 | C2×#541 | — | 2 | 6 | no | — |
| 10 | C2×#533 | C22 | 1 | 3 | yes | — |
| 11 | C2×C4×M16 | C22 | 1 | 3 | yes | — |
| 12 | C2×#384 | C22 | 1 | 1 | yes | characteristic |
| 13 | #1992 | C22 | 1 | 12 | yes | — |
| 14 | #1990 | — | 2 | 6 | no | — |
| 15 | #1736 | — | 2 | 6 | no | — |
| 16 | #1526 | — | 2 | 12 | no | — |
| 17 | C2×#330 | C22 | 1 | 12 | yes | — |
| 18 | C22×M16 | — | 2 | 3 | no | — |
| 19 | #564 | — | 2 | 24 | no | — |
| 20 | #564 | C2×C4 | 1 | 16 | yes | complemented |
| 21 | #557 | — | 2 | 48 | no | — |
| 22 | #557 | — | 2 | 48 | no | — |
| 23 | #541 | — | 2 | 48 | no | — |
| 24 | #541 | — | 2 | 48 | no | — |
| 25 | #533 | — | 2 | 12 | no | — |
| 26 | #533 | C2×C4 | 1 | 12 | yes | — |
| 27 | #533 | D4 | 1 | 12 | yes | — |
| 28 | C2×C4⋊Q8 | — | 2 | 6 | no | — |
| 29 | C2×C4⋊Q8 | D4 | 1 | 6 | yes | — |
| 30 | C2×#126 | — | 2 | 8 | no | — |
| 31 | C2×#125 | — | 2 | 6 | no | — |
| 32 | C2×#125 | — | 2 | 24 | no | — |
| 33 | C2×#125 | C23 | 1 | 6 | yes | — |
| 34 | C2×C22⋊Q8 | — | 2 | 6 | no | — |
| 35 | C2×C4×Q8 | — | 2 | 12 | no | — |
| 36 | C2×C4×Q8 | — | 2 | 6 | no | — |
| 37 | C2×#117 | D4 | 1 | 6 | yes | — |
| 38 | C22×C4⋊C4 | — | 2 | 3 | no | — |
| 39 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 40 | C4×M16 | — | 2 | 12 | no | — |
| 41 | C2×#97 | C23 | 1 | 6 | yes | — |
| 42 | C2×C4×C8 | — | 2 | 3 | no | — |
| 43 | #384 | — | 2 | 3 | no | — |
| 44 | #384 | C23 | 1 | 2 | yes | — |
| 45 | #340 | — | 2 | 24 | no | — |
| 46 | #330 | — | 2 | 24 | no | — |
| 47 | C22×Q8 | — | 2 | 6 | no | — |
| 48 | C23×C4 | — | 4 | 1 | no | — |
| 49 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 50 | C2×M16 | — | 2 | 12 | no | — |
| 51 | C2×M16 | — | 4 | 6 | no | — |
| 52 | C2×M16 | — | 4 | 6 | no | — |
| 53 | C22×C8 | — | 2 | 6 | no | — |
| 54 | C4⋊Q8 | — | 2 | 48 | no | — |
| 55 | C4⋊Q8 | — | 2 | 12 | no | — |
| 56 | C4⋊Q8 | C22⋊C4 | 1 | 24 | yes | — |
| 57 | #126 | — | 2 | 64 | no | — |
| 58 | #125 | — | 2 | 48 | no | — |
| 59 | #125 | — | 2 | 96 | no | — |
| 60 | #125 | — | 2 | 96 | no | — |
| 61 | #125 | — | 2 | 12 | no | — |
| 62 | #125 | C22×C4 | 1 | 12 | yes | — |
| 63 | #125 | C2×D4 | 1 | 12 | yes | — |
| 64 | C22⋊Q8 | — | 2 | 24 | no | — |
| 65 | C22⋊Q8 | — | 2 | 24 | no | — |
| 66 | C4×Q8 | — | 2 | 96 | no | — |
| 67 | C4×Q8 | — | 2 | 24 | no | — |
| 68 | C4×Q8 | — | 2 | 24 | no | — |
| 69 | #117 | — | 2 | 12 | no | — |
| 70 | #117 | C22⋊C4 | 1 | 24 | yes | — |
| 71 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 72 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 73 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 74 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 75 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 76 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 77 | C2×C4⋊C4 | C2×D4 | 1 | 12 | yes | — |
| 78 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 79 | C2×C42 | — | 2 | 8 | no | — |
| 80 | C2×C42 | — | 2 | 6 | no | — |
| 81 | C2×C42 | C22×C4 | 1 | 2 | yes | — |
| 82 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 83 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 84 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 85 | #97 | — | 2 | 12 | no | — |
| 86 | C4×C8 | — | 4 | 6 | no | — |
| 87 | C24 | #120 | 1 | 1 | yes | characteristic |
| 88 | C2×Q8 | — | 2 | 24 | no | — |
| 89 | C2×Q8 | — | 4 | 12 | no | — |
| 90 | C2×Q8 | — | 4 | 12 | no | — |
| 91 | C22×C4 | — | 2 | 6 | no | — |
| 92 | C22×C4 | — | 2 | 6 | no | — |
| 93 | C22×C4 | — | 2 | 12 | no | — |
| 94 | C22×C4 | — | 4 | 2 | no | — |
| 95 | C22×C4 | — | 8 | 6 | no | — |
| 96 | C22×C4 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 97 | C22×C4 | #120 | 1 | 6 | yes | — |
| 98 | C22×C4 | #120 | 1 | 3 | yes | — |
| 99 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 100 | M16 | — | 4 | 24 | no | — |
| 101 | C2×C8 | — | 2 | 12 | no | — |
| 102 | C2×C8 | — | 4 | 6 | no | — |
| 103 | C2×C8 | — | 4 | 6 | no | — |
| 104 | C4⋊C4 | — | 2 | 96 | no | — |
| 105 | C4⋊C4 | — | 2 | 24 | no | — |
| 106 | C4⋊C4 | — | 2 | 24 | no | — |
| 107 | C4⋊C4 | — | 2 | 12 | no | — |
| 108 | C4⋊C4 | — | 2 | 24 | no | — |
| 109 | C4⋊C4 | — | 4 | 12 | no | — |
| 110 | C4⋊C4 | — | 8 | 6 | no | — |
| 111 | C4⋊C4 | C2×C22⋊C4 | 1 | 24 | yes | — |
| 112 | C22⋊C4 | — | 2 | 24 | no | — |
| 113 | C42 | — | 2 | 32 | no | — |
| 114 | C42 | — | 2 | 24 | no | — |
| 115 | C42 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 116 | C42 | #120 | 1 | 6 | yes | — |
| 117 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 118 | C42 | C22×D4 | 1 | 3 | yes | — |
| 119 | C23 | — | 2 | 3 | no | — |
| 120 | C23 | — | 2 | 3 | no | — |
| 121 | C23 | #379 | 1 | 2 | yes | — |
| 122 | C23 | C2×#120 | 1 | 1 | yes | characteristic |
| 123 | Q8 | — | 4 | 48 | no | — |
| 124 | C2×C4 | — | 2 | 3 | no | — |
| 125 | C2×C4 | — | 2 | 3 | no | — |
| 126 | C2×C4 | — | 2 | 12 | no | — |
| 127 | C2×C4 | — | 2 | 12 | no | — |
| 128 | C2×C4 | — | 2 | 24 | no | — |
| 129 | C2×C4 | — | 4 | 12 | no | — |
| 130 | C2×C4 | — | 4 | 12 | no | — |
| 131 | C2×C4 | — | 8 | 6 | no | — |
| 132 | C2×C4 | — | 16 | 4 | no | — |
| 133 | C2×C4 | #379 | 1 | 6 | yes | — |
| 134 | C2×C4 | #454 | 1 | 6 | yes | — |
| 135 | C2×C4 | #456 | 1 | 6 | yes | — |
| 136 | C2×C4 | C22×C22⋊C4 | 1 | 3 | yes | — |
| 137 | C2×C4 | C2×#120 | 1 | 3 | yes | — |
| 138 | C8 | — | 4 | 12 | no | — |
| 139 | C22 | — | 2 | 6 | no | — |
| 140 | C22 | — | 4 | 1 | no | — |
| 141 | C22 | — | 4 | 3 | no | — |
| 142 | C22 | C2×#379 | 1 | 1 | yes | characteristic |
| 143 | C22 | C2×#454 | 1 | 3 | yes | — |
| 144 | C22 | C2×#456 | 1 | 3 | yes | — |
| 145 | C4 | — | 2 | 3 | no | — |
| 146 | C4 | — | 2 | 3 | no | — |
| 147 | C4 | — | 4 | 24 | no | — |
| 148 | C4 | — | 16 | 2 | no | — |
| 149 | C4 | #1793 | 1 | 12 | yes | — |
| 150 | C2 | — | 4 | 2 | no | — |
| 151 | C2 | #19083 | 1 | 1 | yes | characteristic |
| 152 | C2 | #19084 | 1 | 3 | yes | — |
| 153 | C2 | C2×#1793 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #8509 | C2 | semidirect |
| 6. #3556 | C4 | semidirect |
| 20. #564 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3556 | 2 |
| #21322 | 2 |
| C2×#1526 | 2 |
| #8509 | 2 |