Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
342 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 58 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 | #64308 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #64156 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#942 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #63568 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #63559 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #5202 | — | 3 | 2 | no | maximal |
| 7 | C2×C32×Q8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×C4⋊(C3⋊D3) | C22 | 1 | 1 | yes | characteristic |
| 9 | C3×#235 | C22 | 1 | 2 | yes | — |
| 10 | #3811 | — | 2 | 2 | no | — |
| 11 | D3×D12 | — | 2 | 2 | no | — |
| 12 | #3799 | — | 2 | 2 | no | — |
| 13 | #3752 | — | 2 | 2 | no | — |
| 14 | C4×C3⋊Dic3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C3×#216 | — | 2 | 2 | no | — |
| 16 | C3×#215 | — | 2 | 2 | no | — |
| 17 | C3×#208 | C22 | 1 | 2 | yes | complemented |
| 18 | #1009 | — | 3 | 2 | no | — |
| 19 | C3×#137 | — | 3 | 2 | no | — |
| 20 | #947 | — | 3 | 2 | no | — |
| 21 | #947 | — | 6 | 1 | no | — |
| 22 | #942 | D3 | 1 | 2 | yes | complemented |
| 23 | #837 | — | 3 | 2 | no | — |
| 24 | #825 | — | 3 | 2 | no | — |
| 25 | #823 | — | 3 | 2 | no | — |
| 26 | C22×C3⋊D3 | D4 | 1 | 2 | yes | — |
| 27 | C2×D32 | — | 2 | 2 | no | — |
| 28 | C32×Q8 | — | 2 | 2 | no | — |
| 29 | C2×C32×C4 | — | 2 | 1 | no | — |
| 30 | C2×C32×C4 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×C3⋊Dic3 | — | 2 | 1 | no | — |
| 32 | C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 33 | C3×#67 | — | 2 | 2 | no | — |
| 34 | C3×D12 | D4 | 1 | 2 | yes | — |
| 35 | C3×C4×D3 | — | 2 | 2 | no | — |
| 36 | C3×Dic6 | D4 | 1 | 2 | yes | — |
| 37 | #626 | — | 2 | 4 | no | — |
| 38 | #624 | — | 2 | 2 | no | — |
| 39 | C3×C3⋊C8 | — | 2 | 2 | no | — |
| 40 | #455 | — | 9 | 1 | no | Sylow |
| 41 | C3×#41 | — | 3 | 2 | no | — |
| 42 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 43 | C2×C3×Q8 | C2×D3 | 1 | 2 | yes | complemented |
| 44 | #239 | — | 6 | 2 | no | — |
| 45 | D3×D4 | — | 6 | 2 | no | — |
| 46 | D3×D4 | — | 6 | 4 | no | — |
| 47 | #235 | — | 6 | 2 | no | — |
| 48 | #235 | C2×D3 | 1 | 2 | yes | — |
| 49 | C2×D12 | — | 3 | 2 | no | — |
| 50 | C2×D12 | — | 6 | 1 | no | — |
| 51 | C2×D12 | — | 6 | 2 | no | — |
| 52 | C3×Q16 | — | 6 | 2 | no | — |
| 53 | C3×SD16 | — | 6 | 2 | no | — |
| 54 | C3×M16 | — | 3 | 2 | no | — |
| 55 | #216 | — | 2 | 2 | no | — |
| 56 | #215 | — | 2 | 2 | no | — |
| 57 | (C2×D3)⋊C4 | — | 6 | 4 | no | — |
| 58 | (C2×D3)⋊C4 | — | 6 | 4 | no | — |
| 59 | C4×Dic3 | — | 3 | 2 | no | — |
| 60 | C4×Dic3 | — | 6 | 1 | no | — |
| 61 | #208 | C2×D3 | 1 | 2 | yes | complemented |
| 62 | C22×C32 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 64 | C2×C3⋊D3 | — | 4 | 2 | no | — |
| 65 | C2×C3×D3 | — | 2 | 2 | no | — |
| 66 | D32 | — | 8 | 2 | no | — |
| 67 | C32×C4 | — | 4 | 1 | no | — |
| 68 | C32×C4 | C2×D4 | 1 | 2 | yes | — |
| 69 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 70 | C3×Dic3 | — | 2 | 2 | no | — |
| 71 | H(5,𝔽2) | — | 9 | 1 | no | — |
| 72 | #137 | — | 3 | 2 | no | — |
| 73 | #124 | — | 9 | 1 | no | — |
| 74 | #104 | — | 9 | 2 | no | — |
| 75 | #100 | — | 9 | 1 | no | — |
| 76 | C22×D3 | — | 3 | 4 | no | — |
| 77 | C22×D3 | — | 6 | 2 | no | — |
| 78 | C22×D3 | — | 6 | 4 | no | — |
| 79 | C3×Q8 | — | 2 | 2 | no | — |
| 80 | C3×Q8 | — | 2 | 2 | no | — |
| 81 | C3×Q8 | — | 3 | 2 | no | — |
| 82 | C3×Q8 | — | 4 | 2 | no | — |
| 83 | C3×D4 | — | 3 | 2 | no | — |
| 84 | C3×D4 | — | 6 | 2 | no | — |
| 85 | C2×C3×C4 | — | 2 | 2 | no | — |
| 86 | C2×C3×C4 | — | 2 | 2 | no | — |
| 87 | C2×C3×C4 | — | 2 | 1 | no | — |
| 88 | C2×C3×C4 | — | 6 | 2 | no | — |
| 89 | C2×C3×C4 | C22×D3 | 1 | 2 | yes | — |
| 90 | #67 | — | 2 | 2 | no | — |
| 91 | #67 | — | 6 | 4 | no | — |
| 92 | C2×Dic3 | — | 6 | 2 | no | — |
| 93 | C2×Dic3 | — | 6 | 2 | no | — |
| 94 | D12 | — | 6 | 4 | no | — |
| 95 | D12 | — | 6 | 2 | no | — |
| 96 | D12 | — | 6 | 2 | no | — |
| 97 | D12 | — | 12 | 2 | no | — |
| 98 | D12 | #67 | 1 | 2 | yes | — |
| 99 | C4×D3 | — | 2 | 2 | no | — |
| 100 | C4×D3 | — | 6 | 4 | no | — |
| 101 | Dic6 | #67 | 1 | 2 | yes | — |
| 102 | C3×C8 | — | 6 | 2 | no | — |
| 103 | C3⋊C8 | — | 2 | 2 | no | — |
| 104 | C2×C32 | — | 2 | 1 | no | — |
| 105 | C2×C32 | #120 | 1 | 1 | yes | characteristic |
| 106 | C3⋊D3 | — | 4 | 2 | no | — |
| 107 | C3×D3 | — | 4 | 2 | no | — |
| 108 | #41 | — | 3 | 2 | no | — |
| 109 | #41 | — | 18 | 2 | no | — |
| 110 | C2×Q8 | D32 | 1 | 1 | yes | characteristic, complemented |
| 111 | C2×D4 | — | 9 | 1 | no | — |
| 112 | C2×D4 | — | 18 | 2 | no | — |
| 113 | C2×D4 | — | 18 | 2 | no | — |
| 114 | Q16 | — | 6 | 2 | no | — |
| 115 | SD16 | — | 6 | 2 | no | — |
| 116 | M16 | — | 3 | 2 | no | — |
| 117 | C22⋊C4 | — | 18 | 2 | no | — |
| 118 | C42 | — | 9 | 1 | no | — |
| 119 | C22×C3 | — | 2 | 1 | no | — |
| 120 | C22×C3 | — | 6 | 2 | no | — |
| 121 | C22×C3 | D3×D4 | 1 | 2 | yes | — |
| 122 | C2×D3 | — | 2 | 2 | no | — |
| 123 | C2×D3 | — | 6 | 4 | no | — |
| 124 | C2×D3 | — | 12 | 2 | no | — |
| 125 | C2×D3 | — | 12 | 4 | no | — |
| 126 | C2×D3 | — | 12 | 4 | no | — |
| 127 | C2×D3 | — | 24 | 4 | no | — |
| 128 | C3×C4 | — | 2 | 2 | no | — |
| 129 | C3×C4 | — | 4 | 2 | no | — |
| 130 | C3×C4 | — | 4 | 2 | no | — |
| 131 | C3×C4 | — | 6 | 2 | no | — |
| 132 | C3×C4 | D3×D4 | 1 | 2 | yes | — |
| 133 | C3×C4 | C2×#67 | 1 | 2 | yes | — |
| 134 | Dic3 | — | 2 | 2 | no | — |
| 135 | Dic3 | — | 12 | 2 | no | — |
| 136 | Dic3 | — | 12 | 2 | no | — |
| 137 | C32 | #455 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 138 | C23 | — | 9 | 2 | no | — |
| 139 | C23 | — | 18 | 2 | no | — |
| 140 | Q8 | — | 2 | 2 | no | — |
| 141 | Q8 | — | 3 | 2 | no | — |
| 142 | D4 | — | 3 | 2 | no | — |
| 143 | D4 | — | 6 | 2 | no | — |
| 144 | D4 | — | 18 | 4 | no | — |
| 145 | D4 | — | 18 | 2 | no | — |
| 146 | C2×C4 | — | 2 | 1 | no | — |
| 147 | C2×C4 | — | 6 | 2 | no | — |
| 148 | C2×C4 | — | 18 | 2 | no | — |
| 149 | C2×C4 | — | 18 | 1 | no | — |
| 150 | C2×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 151 | C8 | — | 6 | 2 | no | — |
| 152 | C2×C3 | — | 2 | 2 | no | — |
| 153 | C2×C3 | — | 2 | 2 | no | — |
| 154 | C2×C3 | — | 2 | 1 | no | — |
| 155 | C2×C3 | — | 12 | 2 | no | — |
| 156 | C2×C3 | #937 | 1 | 2 | yes | — |
| 157 | D3 | — | 4 | 2 | no | — |
| 158 | D3 | — | 12 | 4 | no | — |
| 159 | D3 | — | 24 | 2 | no | — |
| 160 | C22 | — | 6 | 2 | no | — |
| 161 | C22 | — | 18 | 2 | no | — |
| 162 | C22 | — | 36 | 2 | no | — |
| 163 | C22 | — | 72 | 2 | no | — |
| 164 | C22 | #3811 | 1 | 1 | yes | characteristic |
| 165 | C4 | — | 4 | 1 | no | — |
| 166 | C4 | — | 6 | 2 | no | — |
| 167 | C4 | — | 36 | 1 | no | — |
| 168 | C4 | D3×#67 | 1 | 2 | yes | — |
| 169 | C3 | — | 2 | 1 | no | — |
| 170 | C3 | #5202 | 1 | 2 | yes | complemented |
| 171 | C2 | — | 2 | 1 | no | — |
| 172 | C2 | — | 12 | 2 | no | — |
| 173 | C2 | — | 36 | 2 | no | — |
| 174 | C2 | #63983 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #64156 | C2 | semidirect |
| 3. C3×#942 | C2 | semidirect |
| 4. #63568 | C2 | semidirect |
| 5. #63559 | C2 | semidirect |
| 7. C2×C32×Q8 | C22 | semidirect |
| 14. C4×C3⋊Dic3 | C22 | semidirect |
| 17. C3×#208 | C22 | semidirect |
| 22. #942 | D3 | semidirect |
| 43. C2×C3×Q8 | C2×D3 | semidirect |
| 61. #208 | C2×D3 | semidirect |
| 110. C2×Q8 | D32 | semidirect |
| 137. C32 | #455 | semidirect |
| 170. C3 | #5202 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64308 | 2 |
| #64156 | 2 |
| C3×#942 | 2 |
| #63568 | 2 |
| #63559 | 2 |
| #5202 | 3 |