Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
526 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 124 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 | #64149 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#935 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #63953 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #63952 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #63897 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #63838 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #5664 | — | 3 | 2 | no | maximal |
| 8 | C2×C32×D4 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#638 | C22 | 1 | 4 | yes | — |
| 10 | C2×C4⋊(C3⋊D3) | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×C3×Dic3 | C22 | 1 | 2 | yes | — |
| 12 | C2×#627 | C22 | 1 | 2 | yes | — |
| 13 | C2×#624 | C22 | 1 | 2 | yes | — |
| 14 | C4×C3⋊Dic3 | C22 | 1 | 1 | yes | characteristic |
| 15 | C3×#217 | C22 | 1 | 4 | yes | — |
| 16 | C3×#217 | C22 | 1 | 2 | yes | — |
| 17 | C3×Dic3⋊C4 | C22 | 1 | 4 | yes | — |
| 18 | #3724 | C22 | 1 | 2 | yes | — |
| 19 | #3722 | C22 | 1 | 4 | yes | — |
| 20 | #3722 | C22 | 1 | 2 | yes | — |
| 21 | Dic32 | C22 | 1 | 4 | yes | — |
| 22 | C3×#123 | — | 3 | 2 | no | — |
| 23 | #940 | — | 3 | 2 | no | — |
| 24 | #940 | — | 6 | 1 | no | — |
| 25 | #935 | D3 | 1 | 2 | yes | complemented |
| 26 | C4×#67 | — | 3 | 4 | no | — |
| 27 | #894 | — | 3 | 4 | no | — |
| 28 | #887 | — | 3 | 4 | no | — |
| 29 | #885 | — | 3 | 4 | no | — |
| 30 | #885 | — | 3 | 4 | no | — |
| 31 | #67⋊C4 | — | 3 | 4 | no | — |
| 32 | C23×C32 | C23 | 1 | 2 | yes | — |
| 33 | C22×C3⋊D3 | C23 | 1 | 2 | yes | — |
| 34 | C32×D4 | — | 4 | 1 | no | — |
| 35 | C2×C32×C4 | C23 | 1 | 1 | yes | characteristic |
| 36 | #638 | — | 2 | 8 | no | — |
| 37 | C2×C3⋊Dic3 | C23 | 1 | 2 | yes | — |
| 38 | C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 39 | C2×C3×Dic3 | — | 4 | 2 | no | — |
| 40 | C2×C3×Dic3 | C23 | 1 | 4 | yes | — |
| 41 | C2×C3×Dic3 | C23 | 1 | 4 | yes | — |
| 42 | #627 | — | 2 | 4 | no | — |
| 43 | #624 | — | 4 | 2 | no | — |
| 44 | #559 | — | 9 | 1 | no | Sylow |
| 45 | C2×C3×D4 | — | 2 | 1 | no | — |
| 46 | C2×C3×D4 | C2×D3 | 1 | 2 | yes | — |
| 47 | C22×C3×C4 | — | 3 | 2 | no | — |
| 48 | C2×#67 | — | 3 | 4 | no | — |
| 49 | C2×#67 | — | 3 | 4 | no | — |
| 50 | C2×#67 | — | 6 | 4 | no | — |
| 51 | C22×Dic3 | C2×D3 | 1 | 2 | yes | — |
| 52 | C2×D12 | — | 3 | 2 | no | — |
| 53 | C2×D12 | — | 6 | 1 | no | — |
| 54 | C2×C4×D3 | — | 3 | 4 | no | — |
| 55 | C2×Dic6 | — | 3 | 4 | no | — |
| 56 | C3×C4⋊C4 | — | 3 | 4 | no | — |
| 57 | C3×C22⋊C4 | — | 3 | 4 | no | — |
| 58 | C3×C22⋊C4 | — | 3 | 2 | no | — |
| 59 | #217 | C2×D3 | 1 | 2 | yes | — |
| 60 | #217 | C2×D3 | 1 | 4 | yes | — |
| 61 | (C2×D3)⋊C4 | — | 3 | 4 | no | — |
| 62 | (C2×D3)⋊C4 | — | 3 | 4 | no | — |
| 63 | (C2×D3)⋊C4 | — | 3 | 4 | no | — |
| 64 | Dic3⋊C4 | — | 3 | 4 | no | — |
| 65 | Dic3⋊C4 | C2×D3 | 1 | 4 | yes | — |
| 66 | C4×Dic3 | — | 3 | 2 | no | — |
| 67 | C4×Dic3 | — | 3 | 4 | no | — |
| 68 | C4×Dic3 | — | 3 | 4 | no | — |
| 69 | C4×Dic3 | — | 6 | 1 | no | — |
| 70 | C22×C32 | — | 2 | 4 | no | — |
| 71 | C22×C32 | — | 2 | 2 | no | — |
| 72 | C22×C32 | C24 | 1 | 1 | yes | characteristic |
| 73 | C2×C3⋊D3 | — | 2 | 4 | no | — |
| 74 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 75 | C32×C4 | — | 2 | 1 | no | — |
| 76 | C3⋊Dic3 | #41 | 1 | 4 | yes | — |
| 77 | C3×Dic3 | — | 2 | 4 | no | — |
| 78 | C3×Dic3 | — | 2 | 4 | no | — |
| 79 | #127 | — | 9 | 1 | no | — |
| 80 | #124 | — | 9 | 4 | no | — |
| 81 | #123 | — | 3 | 2 | no | — |
| 82 | #123 | — | 9 | 2 | no | — |
| 83 | C4×Q8 | — | 9 | 2 | no | — |
| 84 | C4×D4 | — | 9 | 4 | no | — |
| 85 | C23×C3 | — | 2 | 2 | no | — |
| 86 | C23×C3 | C22×D3 | 1 | 2 | yes | — |
| 87 | C23×C3 | C22×D3 | 1 | 2 | yes | — |
| 88 | C22×D3 | — | 3 | 4 | no | — |
| 89 | C22×D3 | — | 6 | 2 | no | — |
| 90 | C3×D4 | — | 4 | 2 | no | — |
| 91 | C3×D4 | — | 4 | 2 | no | — |
| 92 | C2×C3×C4 | — | 2 | 1 | no | — |
| 93 | C2×C3×C4 | — | 3 | 4 | no | — |
| 94 | C2×C3×C4 | — | 3 | 4 | no | — |
| 95 | C2×C3×C4 | — | 12 | 2 | no | — |
| 96 | C2×C3×C4 | C22×D3 | 1 | 2 | yes | — |
| 97 | #67 | — | 6 | 8 | no | — |
| 98 | #67 | — | 6 | 8 | no | — |
| 99 | #67 | — | 12 | 8 | no | — |
| 100 | C2×Dic3 | — | 3 | 4 | no | — |
| 101 | C2×Dic3 | — | 4 | 2 | no | — |
| 102 | C2×Dic3 | — | 6 | 2 | no | — |
| 103 | C2×Dic3 | C22×D3 | 1 | 4 | yes | — |
| 104 | C2×Dic3 | C22×D3 | 1 | 4 | yes | — |
| 105 | D12 | — | 12 | 2 | no | — |
| 106 | D12 | — | 12 | 2 | no | — |
| 107 | C4×D3 | — | 12 | 4 | no | — |
| 108 | Dic6 | — | 6 | 8 | no | — |
| 109 | C2×C32 | — | 4 | 2 | no | — |
| 110 | C2×C32 | C2×#41 | 1 | 2 | yes | — |
| 111 | C2×C32 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 112 | C3⋊D3 | — | 4 | 2 | no | — |
| 113 | C2×Q8 | — | 9 | 2 | no | — |
| 114 | C2×D4 | — | 9 | 4 | no | — |
| 115 | C2×D4 | — | 9 | 1 | no | — |
| 116 | C2×D4 | D32 | 1 | 1 | yes | characteristic |
| 117 | C22×C4 | — | 3 | 2 | no | — |
| 118 | C22×C4 | — | 9 | 2 | no | — |
| 119 | C4⋊C4 | — | 3 | 4 | no | — |
| 120 | C4⋊C4 | — | 9 | 2 | no | — |
| 121 | C22⋊C4 | — | 3 | 4 | no | — |
| 122 | C22⋊C4 | — | 3 | 2 | no | — |
| 123 | C22⋊C4 | — | 9 | 4 | no | — |
| 124 | C22⋊C4 | — | 9 | 2 | no | — |
| 125 | C42 | — | 9 | 4 | no | — |
| 126 | C42 | — | 9 | 1 | no | — |
| 127 | C22×C3 | — | 2 | 2 | no | — |
| 128 | C22×C3 | — | 2 | 2 | no | — |
| 129 | C22×C3 | — | 2 | 4 | no | — |
| 130 | C22×C3 | — | 2 | 4 | no | — |
| 131 | C22×C3 | — | 2 | 1 | no | — |
| 132 | C22×C3 | — | 4 | 4 | no | — |
| 133 | C22×C3 | — | 4 | 2 | no | — |
| 134 | C22×C3 | C23×D3 | 1 | 2 | yes | — |
| 135 | C2×D3 | — | 6 | 4 | no | — |
| 136 | C2×D3 | — | 6 | 4 | no | — |
| 137 | C2×D3 | — | 6 | 4 | no | — |
| 138 | C2×D3 | — | 12 | 2 | no | — |
| 139 | C2×D3 | — | 12 | 4 | no | — |
| 140 | C3×C4 | — | 2 | 2 | no | — |
| 141 | C3×C4 | — | 2 | 2 | no | — |
| 142 | C3×C4 | — | 6 | 4 | no | — |
| 143 | C3×C4 | — | 6 | 4 | no | — |
| 144 | Dic3 | — | 2 | 4 | no | — |
| 145 | Dic3 | — | 2 | 4 | no | — |
| 146 | Dic3 | — | 3 | 8 | no | — |
| 147 | Dic3 | — | 6 | 4 | no | — |
| 148 | C32 | #559 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 149 | C23 | — | 9 | 2 | no | — |
| 150 | C23 | C2×D32 | 1 | 2 | yes | — |
| 151 | Q8 | — | 18 | 4 | no | — |
| 152 | D4 | — | 4 | 1 | no | — |
| 153 | D4 | — | 18 | 8 | no | — |
| 154 | D4 | — | 36 | 1 | no | — |
| 155 | C2×C4 | — | 3 | 4 | no | — |
| 156 | C2×C4 | — | 3 | 4 | no | — |
| 157 | C2×C4 | — | 9 | 2 | no | — |
| 158 | C2×C4 | — | 12 | 2 | no | — |
| 159 | C2×C4 | — | 36 | 2 | no | — |
| 160 | C2×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 161 | C2×C3 | — | 2 | 2 | no | — |
| 162 | C2×C3 | — | 2 | 1 | no | — |
| 163 | C2×C3 | — | 4 | 2 | no | — |
| 164 | C2×C3 | — | 4 | 2 | no | — |
| 165 | C2×C3 | — | 8 | 2 | no | — |
| 166 | C2×C3 | #1004 | 1 | 2 | yes | — |
| 167 | C2×C3 | D3×#41 | 1 | 4 | yes | — |
| 168 | D3 | — | 12 | 4 | no | — |
| 169 | D3 | — | 24 | 2 | no | — |
| 170 | C22 | — | 2 | 4 | no | — |
| 171 | C22 | — | 2 | 2 | no | — |
| 172 | C22 | — | 18 | 4 | no | — |
| 173 | C22 | — | 18 | 2 | no | — |
| 174 | C22 | C22×D32 | 1 | 1 | yes | characteristic |
| 175 | C4 | — | 2 | 1 | no | — |
| 176 | C4 | — | 6 | 4 | no | — |
| 177 | C4 | — | 6 | 4 | no | — |
| 178 | C4 | — | 9 | 4 | no | — |
| 179 | C3 | — | 2 | 1 | no | — |
| 180 | C3 | #5664 | 1 | 2 | yes | complemented |
| 181 | C2 | — | 4 | 2 | no | — |
| 182 | C2 | — | 36 | 2 | no | — |
| 183 | C2 | #64312 | 1 | 2 | yes | — |
| 184 | C2 | #64330 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C2×C32×D4 | C22 |
| prime radical | #64149 | C2 |
| hypercenter | C2×D4 | D32 |
| hypocenter | C32 | #559 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#935 | C2 | semidirect |
| 3. #63953 | C2 | semidirect |
| 4. #63952 | C2 | semidirect |
| 5. #63897 | C2 | semidirect |
| 6. #63838 | C2 | semidirect |
| 25. #935 | D3 | semidirect |
| 148. C32 | #559 | semidirect |
| 180. C3 | #5664 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64149 | 2 |
| C3×#935 | 2 |
| #63953 | 2 |
| #63952 | 2 |
| #63897 | 2 |
| #63838 | 2 |
| #5664 | 3 |