Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
188 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 31 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 | #64230 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | D32⋊Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #63796 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #63791 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #63769 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #63730 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #63729 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×GO+(4,𝔽2) | — | 2 | 1 | no | — |
| 9 | #3802 | C22 | 1 | 1 | yes | characteristic |
| 10 | C4×D32 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3797 | C22 | 1 | 1 | yes | characteristic |
| 12 | C4⋊#157 | C22 | 1 | 1 | yes | characteristic |
| 13 | C4⋊#157 | C22 | 1 | 1 | yes | characteristic |
| 14 | #3773 | — | 2 | 1 | no | — |
| 15 | D32⋊C4 | — | 2 | 1 | no | — |
| 16 | D32⋊C4 | — | 2 | 1 | no | — |
| 17 | C8×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 18 | C3×#204 | — | 2 | 1 | no | — |
| 19 | C3×ΓL(1,𝔽25) | — | 2 | 1 | no | — |
| 20 | #3715 | — | 2 | 1 | no | — |
| 21 | #3711 | — | 2 | 1 | no | — |
| 22 | #3710 | C22 | 1 | 1 | yes | characteristic |
| 23 | D3×SD16 | — | 6 | 1 | no | — |
| 24 | D3×M16 | — | 6 | 1 | no | — |
| 25 | C2×D32 | — | 2 | 1 | no | — |
| 26 | C2×D32 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×#157 | — | 2 | 1 | no | — |
| 28 | C2×#157 | — | 2 | 1 | no | — |
| 29 | GO+(4,𝔽2) | — | 4 | 2 | no | — |
| 30 | C4×C3⋊D3 | C23 | 1 | 1 | yes | characteristic |
| 31 | C3×D12 | — | 2 | 1 | no | — |
| 32 | C3×C4×D3 | — | 2 | 1 | no | — |
| 33 | C3×Dic6 | — | 2 | 1 | no | — |
| 34 | #627 | — | 2 | 1 | no | — |
| 35 | #625 | — | 2 | 1 | no | — |
| 36 | #624 | — | 2 | 1 | no | — |
| 37 | #624 | D4 | 1 | 1 | yes | characteristic |
| 38 | D3×Dic3 | — | 2 | 1 | no | — |
| 39 | C32×C8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 40 | C3⋊(C3⋊C8) | D4 | 1 | 1 | yes | characteristic, complemented |
| 41 | C3×C3⋊C8 | — | 2 | 1 | no | — |
| 42 | #468 | — | 9 | 1 | no | maximal, Sylow |
| 43 | D3×Q8 | — | 6 | 1 | no | — |
| 44 | D3×D4 | — | 6 | 1 | no | — |
| 45 | C2×C4×D3 | — | 6 | 1 | no | — |
| 46 | C3×SD16 | — | 6 | 1 | no | — |
| 47 | C3×M16 | — | 6 | 1 | no | — |
| 48 | #215 | — | 6 | 1 | no | — |
| 49 | #214 | — | 6 | 1 | no | — |
| 50 | #208 | — | 6 | 1 | no | — |
| 51 | #204 | — | 2 | 1 | no | — |
| 52 | ΓL(1,𝔽25) | — | 2 | 1 | no | — |
| 53 | ΓL(1,𝔽25) | — | 6 | 1 | no | — |
| 54 | D3×C8 | — | 6 | 1 | no | — |
| 55 | D3×C8 | — | 6 | 1 | no | — |
| 56 | D3×C8 | — | 6 | 1 | no | — |
| 57 | C2×C3⋊D3 | C2×D4 | 1 | 1 | yes | characteristic |
| 58 | C2×C3×D3 | — | 2 | 1 | no | — |
| 59 | C2×C3×D3 | — | 4 | 1 | no | — |
| 60 | D32 | — | 2 | 1 | no | — |
| 61 | D32 | — | 2 | 1 | no | — |
| 62 | D32 | — | 4 | 1 | no | — |
| 63 | D32 | — | 4 | 1 | no | — |
| 64 | #157 | — | 4 | 1 | no | — |
| 65 | #157 | — | 4 | 1 | no | — |
| 66 | C32×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | C3⋊Dic3 | #41 | 1 | 1 | yes | characteristic |
| 68 | C3×Dic3 | — | 2 | 1 | no | — |
| 69 | C3×Dic3 | — | 4 | 1 | no | — |
| 70 | C2×SD16 | — | 9 | 1 | no | — |
| 71 | C2×M16 | — | 9 | 1 | no | — |
| 72 | C22⋊Q8 | — | 9 | 1 | no | — |
| 73 | #121 | — | 9 | 1 | no | — |
| 74 | #107 | — | 9 | 1 | no | — |
| 75 | #103 | — | 9 | 1 | no | — |
| 76 | #102 | — | 9 | 1 | no | — |
| 77 | C22×D3 | — | 6 | 1 | no | — |
| 78 | C22×D3 | — | 12 | 1 | no | — |
| 79 | C3×Q8 | — | 6 | 1 | no | — |
| 80 | C3×D4 | — | 6 | 1 | no | — |
| 81 | C2×C3×C4 | — | 6 | 1 | no | — |
| 82 | #67 | — | 12 | 1 | no | — |
| 83 | C2×Dic3 | — | 6 | 1 | no | — |
| 84 | D12 | — | 2 | 1 | no | — |
| 85 | C4×D3 | — | 2 | 1 | no | — |
| 86 | C4×D3 | — | 6 | 1 | no | — |
| 87 | C4×D3 | — | 6 | 1 | no | — |
| 88 | C4×D3 | — | 6 | 1 | no | — |
| 89 | C4×D3 | — | 6 | 1 | no | — |
| 90 | C4×D3 | — | 12 | 1 | no | — |
| 91 | Dic6 | — | 2 | 1 | no | — |
| 92 | Dic6 | — | 12 | 1 | no | — |
| 93 | C3×C8 | — | 2 | 1 | no | — |
| 94 | C3×C8 | — | 2 | 1 | no | — |
| 95 | C3×C8 | — | 6 | 1 | no | — |
| 96 | C3⋊C8 | — | 2 | 1 | no | — |
| 97 | C3⋊C8 | — | 6 | 1 | no | — |
| 98 | C3⋊C8 | — | 6 | 1 | no | — |
| 99 | C2×C32 | #121 | 1 | 1 | yes | characteristic |
| 100 | C3⋊D3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 101 | C3⋊D3 | #137 | 1 | 1 | yes | characteristic |
| 102 | C3×D3 | — | 4 | 1 | no | — |
| 103 | C3×D3 | — | 8 | 1 | no | — |
| 104 | C2×Q8 | — | 9 | 1 | no | — |
| 105 | C2×D4 | — | 9 | 1 | no | — |
| 106 | C2×D4 | — | 18 | 1 | no | — |
| 107 | C22×C4 | — | 9 | 1 | no | — |
| 108 | SD16 | — | 6 | 1 | no | — |
| 109 | SD16 | — | 18 | 1 | no | — |
| 110 | M16 | — | 6 | 1 | no | — |
| 111 | M16 | — | 18 | 1 | no | — |
| 112 | C2×C8 | — | 9 | 1 | no | — |
| 113 | C2×C8 | — | 9 | 1 | no | — |
| 114 | C4⋊C4 | — | 9 | 1 | no | — |
| 115 | C4⋊C4 | — | 9 | 1 | no | — |
| 116 | C4⋊C4 | — | 18 | 1 | no | — |
| 117 | C22⋊C4 | — | 18 | 1 | no | — |
| 118 | C22⋊C4 | — | 18 | 1 | no | — |
| 119 | C22×C3 | — | 6 | 1 | no | — |
| 120 | C22×C3 | — | 12 | 1 | no | — |
| 121 | C2×D3 | — | 2 | 1 | no | — |
| 122 | C2×D3 | — | 4 | 1 | no | — |
| 123 | C2×D3 | — | 6 | 1 | no | — |
| 124 | C2×D3 | — | 6 | 1 | no | — |
| 125 | C2×D3 | — | 12 | 1 | no | — |
| 126 | C2×D3 | — | 12 | 1 | no | — |
| 127 | C2×D3 | — | 24 | 1 | no | — |
| 128 | C2×D3 | — | 24 | 1 | no | — |
| 129 | C3×C4 | — | 2 | 1 | no | — |
| 130 | C3×C4 | — | 2 | 1 | no | — |
| 131 | C3×C4 | — | 6 | 1 | no | — |
| 132 | C3×C4 | — | 12 | 1 | no | — |
| 133 | Dic3 | — | 2 | 1 | no | — |
| 134 | Dic3 | — | 4 | 1 | no | — |
| 135 | Dic3 | — | 6 | 1 | no | — |
| 136 | Dic3 | — | 6 | 1 | no | — |
| 137 | C32 | #468 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 138 | C23 | — | 9 | 1 | no | — |
| 139 | C23 | — | 18 | 1 | no | — |
| 140 | Q8 | — | 6 | 1 | no | — |
| 141 | Q8 | — | 18 | 1 | no | — |
| 142 | D4 | — | 6 | 1 | no | — |
| 143 | D4 | — | 18 | 1 | no | — |
| 144 | D4 | — | 36 | 2 | no | — |
| 145 | C2×C4 | — | 6 | 1 | no | — |
| 146 | C2×C4 | — | 9 | 1 | no | — |
| 147 | C2×C4 | — | 9 | 1 | no | — |
| 148 | C2×C4 | — | 18 | 1 | no | — |
| 149 | C2×C4 | — | 18 | 1 | no | — |
| 150 | C2×C4 | — | 18 | 1 | no | — |
| 151 | C2×C4 | — | 18 | 1 | no | — |
| 152 | C8 | — | 6 | 1 | no | — |
| 153 | C8 | — | 9 | 1 | no | — |
| 154 | C8 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 155 | C2×C3 | — | 2 | 1 | no | — |
| 156 | C2×C3 | — | 2 | 1 | no | — |
| 157 | C2×C3 | — | 12 | 1 | no | — |
| 158 | C2×C3 | — | 24 | 1 | no | — |
| 159 | D3 | — | 4 | 1 | no | — |
| 160 | D3 | — | 6 | 1 | no | — |
| 161 | D3 | — | 6 | 1 | no | — |
| 162 | D3 | — | 6 | 1 | no | — |
| 163 | D3 | — | 6 | 1 | no | — |
| 164 | D3 | — | 8 | 1 | no | — |
| 165 | C22 | — | 6 | 1 | no | — |
| 166 | C22 | — | 9 | 1 | no | — |
| 167 | C22 | — | 12 | 1 | no | — |
| 168 | C22 | — | 18 | 1 | no | — |
| 169 | C22 | — | 18 | 1 | no | — |
| 170 | C22 | — | 36 | 1 | no | — |
| 171 | C22 | — | 36 | 1 | no | — |
| 172 | C4 | — | 6 | 1 | no | — |
| 173 | C4 | — | 9 | 1 | no | — |
| 174 | C4 | — | 12 | 1 | no | — |
| 175 | C4 | — | 36 | 1 | no | — |
| 176 | C4 | — | 36 | 1 | no | — |
| 177 | C4 | C2×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 178 | C3 | — | 2 | 1 | no | — |
| 179 | C3 | — | 2 | 1 | no | — |
| 180 | C2 | — | 9 | 1 | no | — |
| 181 | C2 | — | 9 | 1 | no | — |
| 182 | C2 | — | 12 | 1 | no | — |
| 183 | C2 | — | 24 | 1 | no | — |
| 184 | C2 | #64230 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. D32⋊Q8 | C2 | semidirect |
| 3. #63796 | C2 | semidirect |
| 4. #63791 | C2 | semidirect |
| 5. #63769 | C2 | semidirect |
| 6. #63730 | C2 | semidirect |
| 7. #63729 | C2 | semidirect |
| 39. C32×C8 | D4 | semidirect |
| 40. C3⋊(C3⋊C8) | D4 | semidirect |
| 137. C32 | #468 | semidirect |
| 154. C8 | GO+(4,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64230 | 2 |
| D32⋊Q8 | 2 |
| #63796 | 2 |
| #63791 | 2 |
| #63769 | 2 |
| #63730 | 2 |
| #63729 | 2 |
| #468 | 9 |