Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
382 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 72 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×#3832 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #63818 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C3×#836 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #63559 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #4751 | — | 3 | 2 | no | maximal |
| 6 | C2×C32×Q8 | C22 | 1 | 1 | yes | characteristic |
| 7 | #3832 | — | 2 | 2 | no | — |
| 8 | #3832 | — | 2 | 2 | no | — |
| 9 | C2×C4⋊(C3⋊D3) | C4 | 1 | 2 | yes | — |
| 10 | C2×C4⋊(C3⋊D3) | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×C3⋊D3 | C4 | 1 | 2 | yes | — |
| 12 | C2×C4×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 13 | C3×#208 | C22 | 1 | 4 | yes | — |
| 14 | #3712 | — | 2 | 2 | no | — |
| 15 | #3710 | — | 2 | 2 | no | — |
| 16 | C2×#239 | — | 3 | 2 | no | — |
| 17 | C2×#239 | — | 6 | 1 | no | — |
| 18 | D3×M16 | — | 3 | 4 | no | — |
| 19 | C3×#101 | — | 3 | 2 | no | — |
| 20 | #836 | D3 | 1 | 2 | yes | complemented |
| 21 | #823 | — | 3 | 4 | no | — |
| 22 | C22×C3⋊D3 | C2×C4 | 1 | 2 | yes | — |
| 23 | C22×C3⋊D3 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C32×Q8 | — | 2 | 2 | no | — |
| 25 | C2×C32×C4 | C2×C4 | 1 | 2 | yes | — |
| 26 | C2×C32×C4 | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×C3⋊Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C4⋊(C3⋊D3) | — | 2 | 4 | no | — |
| 29 | C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 30 | C4×C3⋊D3 | — | 2 | 2 | no | — |
| 31 | C4×C3⋊D3 | — | 2 | 2 | no | — |
| 32 | C4×C3⋊D3 | D4 | 1 | 2 | yes | — |
| 33 | C4×C3⋊D3 | D4 | 1 | 2 | yes | — |
| 34 | C3×C3⋊C8 | — | 2 | 4 | no | — |
| 35 | #413 | — | 9 | 1 | no | Sylow |
| 36 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 37 | C2×C3×Q8 | C2×D3 | 1 | 2 | yes | — |
| 38 | #239 | — | 6 | 4 | no | — |
| 39 | #239 | — | 6 | 4 | no | — |
| 40 | #239 | — | 6 | 4 | no | — |
| 41 | #239 | — | 6 | 4 | no | — |
| 42 | C2×D12 | — | 3 | 4 | no | — |
| 43 | C2×D12 | — | 3 | 2 | no | — |
| 44 | C2×D12 | — | 6 | 2 | no | — |
| 45 | C2×D12 | — | 6 | 1 | no | — |
| 46 | C2×C4×D3 | — | 3 | 2 | no | — |
| 47 | C2×C4×D3 | — | 3 | 4 | no | — |
| 48 | C2×C4×D3 | — | 6 | 2 | no | — |
| 49 | C2×C4×D3 | — | 6 | 1 | no | — |
| 50 | C3×M16 | — | 3 | 4 | no | — |
| 51 | #208 | C2×D3 | 1 | 4 | yes | — |
| 52 | ΓL(1,𝔽25) | — | 6 | 4 | no | — |
| 53 | D3×C8 | — | 6 | 4 | no | — |
| 54 | C22×C32 | C22×C4 | 1 | 1 | yes | characteristic |
| 55 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 56 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 57 | C2×C3⋊D3 | — | 4 | 2 | no | — |
| 58 | C2×C3⋊D3 | C22⋊C4 | 1 | 2 | yes | — |
| 59 | C32×C4 | — | 2 | 2 | no | — |
| 60 | C32×C4 | C2×D4 | 1 | 2 | yes | — |
| 61 | C3⋊Dic3 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C3⋊Dic3 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C2×#41 | — | 9 | 1 | no | — |
| 64 | C2×M16 | — | 9 | 2 | no | — |
| 65 | #101 | — | 3 | 2 | no | — |
| 66 | #100 | — | 9 | 2 | no | — |
| 67 | C22×D3 | — | 3 | 2 | no | — |
| 68 | C22×D3 | — | 3 | 4 | no | — |
| 69 | C22×D3 | — | 6 | 1 | no | — |
| 70 | C22×D3 | — | 6 | 2 | no | — |
| 71 | C3×Q8 | — | 2 | 4 | no | — |
| 72 | C3×Q8 | — | 2 | 4 | no | — |
| 73 | C2×C3×C4 | — | 2 | 2 | no | — |
| 74 | C2×C3×C4 | — | 2 | 1 | no | — |
| 75 | C2×C3×C4 | C4×D3 | 1 | 4 | yes | — |
| 76 | C2×C3×C4 | C22×D3 | 1 | 2 | yes | — |
| 77 | C2×Dic3 | — | 3 | 2 | no | — |
| 78 | C2×Dic3 | — | 6 | 1 | no | — |
| 79 | D12 | — | 6 | 8 | no | — |
| 80 | D12 | — | 6 | 8 | no | — |
| 81 | D12 | — | 6 | 4 | no | — |
| 82 | D12 | — | 6 | 4 | no | — |
| 83 | C4×D3 | — | 3 | 4 | no | — |
| 84 | C4×D3 | — | 3 | 4 | no | — |
| 85 | C4×D3 | — | 6 | 2 | no | — |
| 86 | C4×D3 | — | 6 | 2 | no | — |
| 87 | C4×D3 | — | 6 | 4 | no | — |
| 88 | C4×D3 | — | 6 | 4 | no | — |
| 89 | C4×D3 | — | 6 | 4 | no | — |
| 90 | C4×D3 | — | 6 | 4 | no | — |
| 91 | C3×C8 | — | 6 | 4 | no | — |
| 92 | C3⋊C8 | — | 2 | 4 | no | — |
| 93 | C2×C32 | — | 2 | 1 | no | — |
| 94 | C2×C32 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 95 | C3⋊D3 | — | 2 | 2 | no | — |
| 96 | C3⋊D3 | — | 4 | 2 | no | — |
| 97 | #41 | — | 18 | 2 | no | — |
| 98 | #41 | — | 18 | 2 | no | — |
| 99 | C2×Q8 | D32 | 1 | 1 | yes | characteristic |
| 100 | C2×D4 | — | 9 | 2 | no | — |
| 101 | C2×D4 | — | 9 | 1 | no | — |
| 102 | C22×C4 | — | 9 | 1 | no | — |
| 103 | C22×C4 | — | 9 | 2 | no | — |
| 104 | M16 | — | 3 | 4 | no | — |
| 105 | M16 | — | 18 | 2 | no | — |
| 106 | C2×C8 | — | 18 | 2 | no | — |
| 107 | C22×C3 | — | 2 | 1 | no | — |
| 108 | C22×C3 | C2×C4×D3 | 1 | 2 | yes | — |
| 109 | C2×D3 | — | 3 | 4 | no | — |
| 110 | C2×D3 | — | 6 | 2 | no | — |
| 111 | C2×D3 | — | 6 | 4 | no | — |
| 112 | C2×D3 | — | 6 | 4 | no | — |
| 113 | C2×D3 | — | 12 | 2 | no | — |
| 114 | C2×D3 | — | 12 | 2 | no | — |
| 115 | C2×D3 | — | 12 | 4 | no | — |
| 116 | C2×D3 | — | 12 | 4 | no | — |
| 117 | C3×C4 | — | 2 | 4 | no | — |
| 118 | C3×C4 | — | 2 | 4 | no | — |
| 119 | C3×C4 | — | 2 | 2 | no | — |
| 120 | C3×C4 | D3×D4 | 1 | 4 | yes | — |
| 121 | Dic3 | — | 3 | 2 | no | — |
| 122 | Dic3 | — | 3 | 2 | no | — |
| 123 | Dic3 | — | 6 | 1 | no | — |
| 124 | Dic3 | — | 6 | 1 | no | — |
| 125 | C32 | #413 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 126 | C23 | — | 9 | 1 | no | — |
| 127 | C23 | — | 9 | 2 | no | — |
| 128 | Q8 | — | 2 | 2 | no | — |
| 129 | D4 | — | 18 | 4 | no | — |
| 130 | D4 | — | 18 | 2 | no | — |
| 131 | C2×C4 | — | 9 | 1 | no | — |
| 132 | C2×C4 | — | 9 | 2 | no | — |
| 133 | C2×C4 | — | 9 | 2 | no | — |
| 134 | C2×C4 | — | 18 | 2 | no | — |
| 135 | C2×C4 | — | 18 | 2 | no | — |
| 136 | C2×C4 | #624 | 1 | 2 | yes | — |
| 137 | C2×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 138 | C8 | — | 6 | 4 | no | — |
| 139 | C2×C3 | — | 2 | 2 | no | — |
| 140 | C2×C3 | — | 2 | 2 | no | — |
| 141 | C2×C3 | — | 2 | 1 | no | — |
| 142 | C2×C3 | D3×C22⋊C4 | 1 | 2 | yes | — |
| 143 | D3 | — | 6 | 4 | no | — |
| 144 | D3 | — | 12 | 2 | no | — |
| 145 | D3 | — | 12 | 4 | no | — |
| 146 | D3 | — | 12 | 4 | no | — |
| 147 | C22 | — | 9 | 2 | no | — |
| 148 | C22 | — | 18 | 2 | no | — |
| 149 | C22 | — | 36 | 1 | no | — |
| 150 | C22 | — | 36 | 2 | no | — |
| 151 | C22 | C2×#624 | 1 | 1 | yes | characteristic |
| 152 | C4 | — | 2 | 2 | no | — |
| 153 | C4 | — | 9 | 1 | no | — |
| 154 | C4 | — | 9 | 1 | no | — |
| 155 | C4 | #3811 | 1 | 2 | yes | — |
| 156 | C3 | — | 2 | 1 | no | — |
| 157 | C3 | #4751 | 1 | 2 | yes | complemented |
| 158 | C2 | — | 2 | 1 | no | — |
| 159 | C2 | — | 18 | 2 | no | — |
| 160 | C2 | — | 36 | 2 | no | — |
| 161 | C2 | #63974 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #63818 | C2 | semidirect |
| 3. C3×#836 | C2 | semidirect |
| 4. #63559 | C2 | semidirect |
| 20. #836 | D3 | semidirect |
| 125. C32 | #413 | semidirect |
| 157. C3 | #4751 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3832 | 2 |
| #63818 | 2 |
| C3×#836 | 2 |
| #63559 | 2 |
| #4751 | 3 |