Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
182 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 25 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 | A4×D12 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×C4⋊S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3⋊(A4⋊C8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5415 | — | 3 | 1 | no | maximal |
| 5 | #5036 | — | 3 | 1 | no | maximal |
| 6 | C2×D3×A4 | — | 2 | 1 | no | — |
| 7 | C2×C3×S4 | — | 2 | 1 | no | — |
| 8 | C3×C4×A4 | C22 | 1 | 1 | yes | characteristic |
| 9 | #3713 | — | 4 | 1 | no | maximal |
| 10 | C22×D12 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | D4×A4 | — | 3 | 1 | no | — |
| 12 | C4⋊S4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C3×#121 | — | 3 | 1 | no | — |
| 14 | C2×#213 | — | 3 | 1 | no | — |
| 15 | C2×#213 | — | 3 | 1 | no | — |
| 16 | A4⋊C8 | — | 3 | 1 | no | — |
| 17 | A4⋊C8 | — | 6 | 1 | no | — |
| 18 | #830 | — | 3 | 1 | no | — |
| 19 | #809 | — | 3 | 1 | no | — |
| 20 | #809 | — | 3 | 1 | no | — |
| 21 | C2×C3×A4 | D4 | 1 | 1 | yes | characteristic |
| 22 | D3×A4 | — | 4 | 1 | no | — |
| 23 | C3×S4 | — | 4 | 1 | no | — |
| 24 | C3×D12 | — | 4 | 1 | no | — |
| 25 | C3×D12 | — | 4 | 1 | no | — |
| 26 | C3⋊(C3⋊C8) | — | 4 | 1 | no | — |
| 27 | #447 | — | 9 | 1 | no | Sylow |
| 28 | C23×D3 | — | 2 | 1 | no | — |
| 29 | C22×A4 | — | 6 | 1 | no | — |
| 30 | C2×S4 | — | 2 | 1 | no | — |
| 31 | C2×C3×D4 | — | 3 | 1 | no | — |
| 32 | C2×C3×D4 | — | 6 | 1 | no | — |
| 33 | C22×C3×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 34 | C2×D12 | — | 3 | 1 | no | — |
| 35 | C2×D12 | — | 3 | 1 | no | — |
| 36 | C2×D12 | — | 6 | 1 | no | — |
| 37 | C4×A4 | — | 2 | 1 | no | — |
| 38 | C4×A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 39 | C3×C4⋊C4 | — | 3 | 1 | no | — |
| 40 | C3×C22⋊C4 | — | 6 | 1 | no | — |
| 41 | #213 | — | 6 | 2 | no | — |
| 42 | #213 | — | 6 | 1 | no | — |
| 43 | #213 | — | 6 | 1 | no | — |
| 44 | #213 | — | 12 | 1 | no | — |
| 45 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 46 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 47 | C2×C3×D3 | — | 8 | 1 | no | — |
| 48 | C2×C3×D3 | — | 8 | 1 | no | — |
| 49 | C3×A4 | D8 | 1 | 1 | yes | characteristic, complemented |
| 50 | C32×C4 | — | 4 | 1 | no | — |
| 51 | C22×D4 | — | 3 | 1 | no | — |
| 52 | C2×D8 | — | 9 | 1 | no | — |
| 53 | C2×D8 | — | 9 | 1 | no | — |
| 54 | #121 | — | 3 | 1 | no | — |
| 55 | #102 | — | 9 | 1 | no | — |
| 56 | #102 | — | 9 | 1 | no | — |
| 57 | C22⋊C8 | — | 9 | 1 | no | — |
| 58 | C23×C3 | — | 6 | 1 | no | — |
| 59 | C23×C3 | #67 | 1 | 1 | yes | characteristic |
| 60 | C22×D3 | — | 4 | 1 | no | — |
| 61 | C22×D3 | — | 6 | 1 | no | — |
| 62 | C22×D3 | — | 6 | 1 | no | — |
| 63 | C22×D3 | — | 12 | 1 | no | — |
| 64 | C2×A4 | — | 2 | 1 | no | — |
| 65 | C2×A4 | — | 12 | 1 | no | — |
| 66 | C2×A4 | #67 | 1 | 1 | yes | characteristic |
| 67 | S4 | — | 4 | 1 | no | — |
| 68 | C3×D4 | — | 6 | 1 | no | — |
| 69 | C3×D4 | — | 6 | 1 | no | — |
| 70 | C3×D4 | — | 12 | 1 | no | — |
| 71 | C3×D4 | — | 12 | 1 | no | — |
| 72 | C3×D4 | — | 12 | 1 | no | — |
| 73 | C2×C3×C4 | — | 3 | 1 | no | — |
| 74 | C2×C3×C4 | — | 3 | 1 | no | — |
| 75 | C2×C3×C4 | — | 6 | 1 | no | — |
| 76 | D12 | — | 3 | 2 | no | — |
| 77 | D12 | — | 3 | 1 | no | — |
| 78 | D12 | — | 4 | 1 | no | — |
| 79 | D12 | — | 6 | 1 | no | — |
| 80 | D12 | S4 | 1 | 1 | yes | characteristic, complemented |
| 81 | C3⋊C8 | — | 6 | 1 | no | — |
| 82 | C3⋊C8 | — | 6 | 1 | no | — |
| 83 | C3⋊C8 | — | 12 | 1 | no | — |
| 84 | C3⋊C8 | — | 24 | 1 | no | — |
| 85 | C2×C32 | — | 4 | 1 | no | — |
| 86 | C3×D3 | — | 16 | 1 | no | — |
| 87 | C3×D3 | — | 16 | 1 | no | — |
| 88 | C24 | — | 6 | 1 | no | — |
| 89 | C2×D4 | — | 3 | 1 | no | — |
| 90 | C2×D4 | — | 6 | 1 | no | — |
| 91 | C2×D4 | — | 9 | 1 | no | — |
| 92 | C2×D4 | — | 9 | 1 | no | — |
| 93 | C2×D4 | — | 18 | 1 | no | — |
| 94 | C22×C4 | D32 | 1 | 1 | yes | characteristic |
| 95 | D8 | — | 18 | 1 | no | — |
| 96 | D8 | — | 18 | 2 | no | — |
| 97 | D8 | — | 18 | 1 | no | — |
| 98 | C2×C8 | — | 9 | 1 | no | — |
| 99 | C2×C8 | — | 9 | 1 | no | — |
| 100 | C4⋊C4 | — | 3 | 1 | no | — |
| 101 | C22⋊C4 | — | 6 | 1 | no | — |
| 102 | C22×C3 | — | 3 | 1 | no | — |
| 103 | C22×C3 | — | 3 | 1 | no | — |
| 104 | C22×C3 | — | 12 | 1 | no | — |
| 105 | C22×C3 | — | 12 | 1 | no | — |
| 106 | C22×C3 | — | 12 | 1 | no | — |
| 107 | C22×C3 | — | 24 | 1 | no | — |
| 108 | C22×C3 | #213 | 1 | 1 | yes | characteristic, complemented |
| 109 | C2×D3 | — | 2 | 1 | no | — |
| 110 | C2×D3 | — | 6 | 1 | no | — |
| 111 | C2×D3 | — | 8 | 1 | no | — |
| 112 | C2×D3 | — | 12 | 1 | no | — |
| 113 | C2×D3 | — | 12 | 1 | no | — |
| 114 | C2×D3 | — | 12 | 1 | no | — |
| 115 | C2×D3 | — | 12 | 1 | no | — |
| 116 | A4 | — | 2 | 1 | no | — |
| 117 | A4 | #213 | 1 | 1 | yes | characteristic, complemented |
| 118 | C3×C4 | — | 3 | 1 | no | — |
| 119 | C3×C4 | — | 4 | 1 | no | — |
| 120 | C3×C4 | — | 8 | 1 | no | — |
| 121 | C3×C4 | — | 12 | 1 | no | — |
| 122 | C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 123 | C32 | — | 4 | 1 | no | Sylow |
| 124 | C23 | — | 6 | 1 | no | — |
| 125 | C23 | — | 12 | 1 | no | — |
| 126 | C23 | — | 18 | 1 | no | — |
| 127 | C23 | — | 18 | 1 | no | — |
| 128 | C23 | — | 36 | 1 | no | — |
| 129 | C23 | #625 | 1 | 1 | yes | characteristic |
| 130 | D4 | — | 3 | 1 | no | — |
| 131 | D4 | — | 6 | 1 | no | — |
| 132 | D4 | — | 6 | 1 | no | — |
| 133 | D4 | — | 9 | 2 | no | — |
| 134 | D4 | — | 9 | 1 | no | — |
| 135 | D4 | — | 12 | 1 | no | — |
| 136 | D4 | — | 12 | 1 | no | — |
| 137 | D4 | — | 18 | 1 | no | — |
| 138 | C2×C4 | — | 3 | 1 | no | — |
| 139 | C2×C4 | — | 3 | 1 | no | — |
| 140 | C2×C4 | — | 6 | 1 | no | — |
| 141 | C8 | — | 18 | 1 | no | — |
| 142 | C8 | — | 18 | 1 | no | — |
| 143 | C2×C3 | — | 3 | 1 | no | — |
| 144 | C2×C3 | — | 3 | 1 | no | — |
| 145 | C2×C3 | — | 4 | 1 | no | — |
| 146 | C2×C3 | — | 8 | 1 | no | — |
| 147 | C2×C3 | — | 24 | 1 | no | — |
| 148 | C2×C3 | — | 48 | 1 | no | — |
| 149 | C2×C3 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 150 | D3 | — | 4 | 1 | no | — |
| 151 | D3 | — | 12 | 1 | no | — |
| 152 | D3 | — | 16 | 1 | no | — |
| 153 | C22 | — | 3 | 1 | no | — |
| 154 | C22 | — | 3 | 1 | no | — |
| 155 | C22 | — | 6 | 1 | no | — |
| 156 | C22 | — | 12 | 1 | no | — |
| 157 | C22 | — | 12 | 1 | no | — |
| 158 | C22 | — | 12 | 1 | no | — |
| 159 | C22 | — | 18 | 1 | no | — |
| 160 | C22 | — | 36 | 1 | no | — |
| 161 | C22 | — | 36 | 1 | no | — |
| 162 | C22 | — | 36 | 1 | no | — |
| 163 | C22 | — | 36 | 1 | no | — |
| 164 | C22 | #3713 | 1 | 1 | yes | characteristic, complemented |
| 165 | C4 | — | 3 | 1 | no | — |
| 166 | C4 | — | 12 | 1 | no | — |
| 167 | C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 168 | C3 | — | 4 | 1 | no | — |
| 169 | C3 | — | 8 | 1 | no | — |
| 170 | C3 | #5415 | 1 | 1 | yes | characteristic, complemented |
| 171 | C2 | — | 3 | 1 | no | — |
| 172 | C2 | — | 3 | 1 | no | — |
| 173 | C2 | — | 12 | 1 | no | — |
| 174 | C2 | — | 24 | 1 | no | — |
| 175 | C2 | — | 36 | 1 | no | — |
| 176 | C2 | #64209 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #64209 |
| commutator, cosocle | C3×C4×A4 | C22 |
| Frattini, hypercenter | C4 | D3×S4 |
| Fitting | C22×C3×C4 | C2×D3 |
| socle | C23×C3 | #67 |
| hypocenter | C3×A4 | D8 |
| LCS | G ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×D12 | C2 | semidirect |
| 2. C3×C4⋊S4 | C2 | semidirect |
| 3. C3⋊(A4⋊C8) | C2 | semidirect |
| 10. C22×D12 | D3 | semidirect |
| 12. C4⋊S4 | D3 | semidirect |
| 49. C3×A4 | D8 | semidirect |
| 80. D12 | S4 | semidirect |
| 108. C22×C3 | #213 | semidirect |
| 117. A4 | #213 | semidirect |
| 164. C22 | #3713 | semidirect |
| 170. C3 | #5415 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×D12 | 2 |
| C3×C4⋊S4 | 2 |
| C3⋊(A4⋊C8) | 2 |
| #5415 | 3 |
| #5036 | 3 |
| #3713 | 4 |