Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
450 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 89 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 | #64319 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D32×Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Q8×#157 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #64230 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C4×GO+(4,𝔽2) | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #64221 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | D32⋊Q8 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×GO+(4,𝔽2) | C22 | 1 | 3 | yes | — |
| 9 | Q8×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#239 | — | 2 | 1 | no | — |
| 11 | C3×D3×Q8 | — | 2 | 1 | no | — |
| 12 | #3802 | C22 | 1 | 3 | yes | — |
| 13 | C4×D32 | C22 | 1 | 3 | yes | — |
| 14 | C4×D32 | C22 | 1 | 3 | yes | — |
| 15 | #3797 | C22 | 1 | 3 | yes | — |
| 16 | #3795 | — | 2 | 3 | no | — |
| 17 | D3×Dic6 | — | 2 | 3 | no | — |
| 18 | C4⋊#157 | C22 | 1 | 3 | yes | — |
| 19 | C4×#157 | C22 | 1 | 3 | yes | — |
| 20 | #3773 | C22 | 1 | 3 | yes | — |
| 21 | D32⋊C4 | C22 | 1 | 6 | yes | — |
| 22 | D32⋊C4 | C22 | 1 | 3 | yes | — |
| 23 | D32⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 24 | D3×#41 | — | 6 | 1 | no | — |
| 25 | C2×D3×Q8 | — | 6 | 1 | no | — |
| 26 | C2×D32 | C23 | 1 | 3 | yes | — |
| 27 | C2×D32 | C23 | 1 | 1 | yes | characteristic |
| 28 | C2×#157 | C23 | 1 | 3 | yes | — |
| 29 | C2×#157 | C23 | 1 | 1 | yes | characteristic |
| 30 | GO+(4,𝔽2) | — | 2 | 6 | no | — |
| 31 | C32×Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 32 | C4×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 33 | #634 | D4 | 1 | 3 | yes | complemented |
| 34 | C3×D12 | — | 2 | 3 | no | — |
| 35 | C3×C4×D3 | — | 2 | 3 | no | — |
| 36 | C3×C4×D3 | — | 2 | 3 | no | — |
| 37 | C3×Dic6 | — | 2 | 3 | no | — |
| 38 | #627 | — | 2 | 3 | no | — |
| 39 | #625 | — | 2 | 3 | no | — |
| 40 | #624 | C23 | 1 | 3 | yes | — |
| 41 | #624 | C23 | 1 | 1 | yes | characteristic |
| 42 | D3×Dic3 | — | 2 | 3 | no | — |
| 43 | D3×Dic3 | — | 2 | 3 | no | — |
| 44 | #548 | — | 9 | 1 | no | maximal, Sylow |
| 45 | C3×#41 | — | 6 | 1 | no | — |
| 46 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 47 | #239 | — | 2 | 1 | no | — |
| 48 | D3×Q8 | — | 2 | 1 | no | — |
| 49 | D3×Q8 | — | 6 | 3 | no | — |
| 50 | D3×Q8 | — | 6 | 1 | no | — |
| 51 | D3×Q8 | — | 6 | 1 | no | — |
| 52 | D3×Q8 | — | 6 | 3 | no | — |
| 53 | #237 | — | 6 | 3 | no | — |
| 54 | D3×D4 | — | 6 | 3 | no | — |
| 55 | #235 | — | 6 | 3 | no | — |
| 56 | C2×C4×D3 | — | 6 | 3 | no | — |
| 57 | C2×C4×D3 | — | 6 | 3 | no | — |
| 58 | C2×Dic6 | — | 6 | 3 | no | — |
| 59 | C2×C3⋊D3 | C24 | 1 | 1 | yes | characteristic |
| 60 | C2×C3×D3 | — | 2 | 1 | no | — |
| 61 | C2×C3×D3 | — | 2 | 3 | no | — |
| 62 | D32 | — | 2 | 1 | no | — |
| 63 | D32 | — | 2 | 3 | no | — |
| 64 | D32 | — | 2 | 3 | no | — |
| 65 | D32 | #41 | 1 | 2 | yes | complemented |
| 66 | #157 | — | 2 | 3 | no | — |
| 67 | #157 | — | 2 | 1 | no | — |
| 68 | C32×C4 | C2×D4 | 1 | 3 | yes | — |
| 69 | C3⋊Dic3 | C2×D4 | 1 | 3 | yes | — |
| 70 | C3×Dic3 | — | 2 | 1 | no | — |
| 71 | C3×Dic3 | — | 2 | 3 | no | — |
| 72 | C2×#41 | — | 9 | 1 | no | — |
| 73 | C22×Q8 | — | 9 | 1 | no | — |
| 74 | #124 | — | 9 | 3 | no | — |
| 75 | C22⋊Q8 | — | 9 | 3 | no | — |
| 76 | #121 | — | 9 | 3 | no | — |
| 77 | C4×Q8 | — | 9 | 1 | no | — |
| 78 | C4×D4 | — | 9 | 3 | no | — |
| 79 | C22×D3 | — | 6 | 1 | no | — |
| 80 | C22×D3 | — | 6 | 3 | no | — |
| 81 | C3×Q8 | — | 2 | 1 | no | — |
| 82 | C3×Q8 | — | 2 | 1 | no | — |
| 83 | C3×Q8 | — | 6 | 3 | no | — |
| 84 | C3×D4 | — | 6 | 3 | no | — |
| 85 | C2×C3×C4 | — | 6 | 3 | no | — |
| 86 | C2×C3×C4 | — | 6 | 3 | no | — |
| 87 | #67 | — | 6 | 6 | no | — |
| 88 | C2×Dic3 | — | 6 | 3 | no | — |
| 89 | C2×Dic3 | — | 6 | 3 | no | — |
| 90 | D12 | — | 2 | 3 | no | — |
| 91 | C4×D3 | — | 2 | 3 | no | — |
| 92 | C4×D3 | — | 2 | 3 | no | — |
| 93 | C4×D3 | — | 6 | 1 | no | — |
| 94 | C4×D3 | — | 6 | 3 | no | — |
| 95 | C4×D3 | — | 6 | 3 | no | — |
| 96 | C4×D3 | — | 6 | 3 | no | — |
| 97 | C4×D3 | — | 6 | 3 | no | — |
| 98 | C4×D3 | — | 6 | 3 | no | — |
| 99 | Dic6 | — | 2 | 3 | no | — |
| 100 | Dic6 | — | 6 | 3 | no | — |
| 101 | Dic6 | — | 6 | 3 | no | — |
| 102 | Dic6 | — | 6 | 6 | no | — |
| 103 | C2×C32 | C22×D4 | 1 | 1 | yes | characteristic |
| 104 | C3⋊D3 | C2×#41 | 1 | 1 | yes | characteristic |
| 105 | C3⋊D3 | 2−1+4 | 1 | 1 | yes | characteristic |
| 106 | C3×D3 | — | 2 | 2 | no | — |
| 107 | C3×D3 | — | 4 | 3 | no | — |
| 108 | #41 | — | 6 | 1 | no | — |
| 109 | #41 | — | 18 | 3 | no | — |
| 110 | C2×Q8 | — | 6 | 1 | no | — |
| 111 | C2×Q8 | — | 9 | 3 | no | — |
| 112 | C2×Q8 | — | 9 | 1 | no | — |
| 113 | C2×Q8 | — | 18 | 3 | no | — |
| 114 | C2×D4 | — | 9 | 3 | no | — |
| 115 | C2×D4 | — | 9 | 3 | no | — |
| 116 | C22×C4 | — | 9 | 3 | no | — |
| 117 | C22×C4 | — | 9 | 3 | no | — |
| 118 | C4⋊C4 | — | 9 | 3 | no | — |
| 119 | C4⋊C4 | — | 9 | 3 | no | — |
| 120 | C22⋊C4 | — | 9 | 1 | no | — |
| 121 | C22⋊C4 | — | 9 | 3 | no | — |
| 122 | C22⋊C4 | — | 9 | 6 | no | — |
| 123 | C42 | — | 9 | 3 | no | — |
| 124 | C22×C3 | — | 6 | 1 | no | — |
| 125 | C22×C3 | — | 6 | 3 | no | — |
| 126 | C2×D3 | — | 2 | 1 | no | — |
| 127 | C2×D3 | — | 2 | 3 | no | — |
| 128 | C2×D3 | — | 6 | 2 | no | — |
| 129 | C2×D3 | — | 6 | 1 | no | — |
| 130 | C2×D3 | — | 6 | 1 | no | — |
| 131 | C2×D3 | — | 6 | 2 | no | — |
| 132 | C2×D3 | — | 12 | 3 | no | — |
| 133 | C2×D3 | — | 12 | 3 | no | — |
| 134 | C3×C4 | — | 2 | 3 | no | — |
| 135 | C3×C4 | — | 2 | 3 | no | — |
| 136 | C3×C4 | — | 6 | 3 | no | — |
| 137 | C3×C4 | — | 6 | 1 | no | — |
| 138 | Dic3 | — | 2 | 3 | no | — |
| 139 | Dic3 | — | 2 | 1 | no | — |
| 140 | Dic3 | — | 6 | 3 | no | — |
| 141 | Dic3 | — | 6 | 3 | no | — |
| 142 | C32 | #548 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 143 | C23 | — | 9 | 1 | no | — |
| 144 | C23 | — | 9 | 3 | no | — |
| 145 | Q8 | — | 6 | 3 | no | — |
| 146 | Q8 | — | 9 | 3 | no | — |
| 147 | Q8 | — | 18 | 3 | no | — |
| 148 | Q8 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 149 | D4 | — | 6 | 3 | no | — |
| 150 | D4 | — | 18 | 3 | no | — |
| 151 | D4 | — | 18 | 6 | no | — |
| 152 | C2×C4 | — | 6 | 3 | no | — |
| 153 | C2×C4 | — | 6 | 3 | no | — |
| 154 | C2×C4 | — | 9 | 3 | no | — |
| 155 | C2×C4 | — | 9 | 1 | no | — |
| 156 | C2×C4 | — | 9 | 3 | no | — |
| 157 | C2×C4 | — | 9 | 1 | no | — |
| 158 | C2×C4 | — | 9 | 3 | no | — |
| 159 | C2×C4 | — | 18 | 3 | no | — |
| 160 | C2×C4 | — | 18 | 3 | no | — |
| 161 | C2×C3 | — | 2 | 1 | no | — |
| 162 | C2×C3 | — | 2 | 1 | no | — |
| 163 | C2×C3 | — | 6 | 2 | no | — |
| 164 | C2×C3 | — | 12 | 3 | no | — |
| 165 | D3 | — | 2 | 2 | no | — |
| 166 | D3 | — | 4 | 3 | no | — |
| 167 | D3 | — | 6 | 1 | no | — |
| 168 | D3 | — | 6 | 1 | no | — |
| 169 | D3 | — | 6 | 1 | no | — |
| 170 | D3 | — | 6 | 1 | no | — |
| 171 | C22 | — | 6 | 1 | no | — |
| 172 | C22 | — | 6 | 3 | no | — |
| 173 | C22 | — | 9 | 2 | no | — |
| 174 | C22 | — | 9 | 1 | no | — |
| 175 | C22 | — | 18 | 1 | no | — |
| 176 | C22 | — | 18 | 3 | no | — |
| 177 | C22 | — | 18 | 3 | no | — |
| 178 | C4 | — | 6 | 1 | no | — |
| 179 | C4 | — | 6 | 3 | no | — |
| 180 | C4 | — | 9 | 3 | no | — |
| 181 | C4 | — | 18 | 3 | no | — |
| 182 | C4 | — | 18 | 1 | no | — |
| 183 | C4 | C2×GO+(4,𝔽2) | 1 | 3 | yes | — |
| 184 | C3 | — | 2 | 1 | no | — |
| 185 | C3 | — | 2 | 1 | no | — |
| 186 | C2 | — | 6 | 2 | no | — |
| 187 | C2 | — | 9 | 1 | no | — |
| 188 | C2 | — | 9 | 1 | no | — |
| 189 | C2 | — | 12 | 3 | no | — |
| 190 | C2 | C22×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊D3 | C24 |
| Fitting | C32×Q8 | D4 |
| socle | C2×C32 | C22×D4 |
| hypercenter | Q8 | GO+(4,𝔽2) |
| hypocenter | C32 | #548 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #64319 | C2 | semidirect |
| 2. D32×Q8 | C2 | semidirect |
| 3. Q8×#157 | C2 | semidirect |
| 4. #64230 | C2 | semidirect |
| 5. C4×GO+(4,𝔽2) | C2 | semidirect |
| 6. #64221 | C2 | semidirect |
| 7. D32⋊Q8 | C2 | semidirect |
| 31. C32×Q8 | D4 | semidirect |
| 33. #634 | D4 | semidirect |
| 65. D32 | #41 | semidirect |
| 142. C32 | #548 | semidirect |
| 148. Q8 | GO+(4,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64319 | 2 |
| D32×Q8 | 2 |
| Q8×#157 | 2 |
| #64230 | 2 |
| C4×GO+(4,𝔽2) | 2 |
| #64221 | 2 |
| D32⋊Q8 | 2 |
| #548 | 9 |