Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
247 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 52 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 | #64116 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #63830 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#3713 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×#823 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #63578 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #4749 | — | 3 | 2 | no | maximal |
| 7 | #3828 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×C3×D12 | C22 | 1 | 2 | yes | — |
| 9 | C32×M16 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 11 | #3743 | — | 2 | 1 | no | — |
| 12 | C8×C3⋊D3 | — | 2 | 1 | no | — |
| 13 | C3×#208 | C22 | 1 | 2 | yes | — |
| 14 | #3716 | — | 2 | 2 | no | — |
| 15 | #3714 | — | 2 | 2 | no | — |
| 16 | #3713 | — | 2 | 1 | no | — |
| 17 | #3713 | — | 2 | 1 | no | — |
| 18 | #932 | — | 3 | 2 | no | — |
| 19 | C2×#213 | — | 3 | 2 | no | — |
| 20 | #908 | — | 3 | 2 | no | — |
| 21 | #907 | — | 3 | 2 | no | — |
| 22 | #907 | — | 6 | 1 | no | — |
| 23 | C3×#100 | — | 3 | 2 | no | — |
| 24 | #823 | D3 | 1 | 2 | yes | complemented |
| 25 | #822 | — | 3 | 2 | no | — |
| 26 | C22×C3×D3 | — | 2 | 2 | no | — |
| 27 | C2×C32×C4 | C23 | 1 | 1 | yes | characteristic |
| 28 | #638 | — | 2 | 1 | no | — |
| 29 | C4⋊(C3⋊D3) | D4 | 1 | 1 | yes | characteristic |
| 30 | C4×C3⋊D3 | — | 2 | 1 | no | — |
| 31 | #634 | D4 | 1 | 1 | yes | characteristic |
| 32 | C3×D12 | — | 2 | 2 | no | — |
| 33 | C3×D12 | — | 2 | 2 | no | — |
| 34 | C32×C8 | — | 2 | 1 | no | — |
| 35 | C3⋊(C3⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 36 | C3⋊(C3⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 37 | C3×C3⋊C8 | — | 2 | 2 | no | — |
| 38 | #472 | — | 9 | 1 | no | Sylow |
| 39 | C2×C3×D4 | — | 3 | 2 | no | — |
| 40 | #235 | — | 3 | 2 | no | — |
| 41 | #235 | — | 6 | 1 | no | — |
| 42 | C2×D12 | C2×D3 | 1 | 2 | yes | — |
| 43 | C3×M16 | — | 2 | 1 | no | — |
| 44 | C3×M16 | — | 3 | 2 | no | — |
| 45 | C3×M16 | C2×D3 | 1 | 2 | yes | — |
| 46 | #214 | — | 6 | 2 | no | — |
| 47 | #213 | — | 6 | 2 | no | — |
| 48 | #213 | — | 6 | 2 | no | — |
| 49 | #213 | — | 6 | 2 | no | — |
| 50 | #208 | C2×D3 | 1 | 2 | yes | — |
| 51 | C2×C3⋊C8 | — | 3 | 2 | no | — |
| 52 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 53 | D24 | — | 6 | 2 | no | — |
| 54 | #204 | — | 6 | 2 | no | — |
| 55 | ΓL(1,𝔽25) | — | 6 | 2 | no | — |
| 56 | ΓL(1,𝔽25) | — | 6 | 2 | no | — |
| 57 | D3×C8 | — | 6 | 2 | no | — |
| 58 | D3×C8 | — | 6 | 2 | no | — |
| 59 | C22×C32 | #41 | 1 | 1 | yes | characteristic |
| 60 | C2×C3⋊D3 | — | 2 | 1 | no | — |
| 61 | C2×C3×D3 | — | 4 | 2 | no | — |
| 62 | C2×C3×D3 | — | 8 | 2 | no | — |
| 63 | C32×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C32×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C3⋊Dic3 | — | 2 | 1 | no | — |
| 66 | AGL(1,ℤ/8) | — | 9 | 2 | no | — |
| 67 | C2×D8 | — | 9 | 1 | no | — |
| 68 | #131 | — | 9 | 1 | no | — |
| 69 | #108 | — | 9 | 1 | no | — |
| 70 | #100 | — | 3 | 2 | no | — |
| 71 | C23×C3 | — | 6 | 2 | no | — |
| 72 | C22×D3 | — | 2 | 2 | no | — |
| 73 | C3×D4 | — | 6 | 2 | no | — |
| 74 | C3×D4 | — | 6 | 2 | no | — |
| 75 | C2×C3×C4 | — | 2 | 1 | no | — |
| 76 | C2×C3×C4 | C22×D3 | 1 | 2 | yes | — |
| 77 | #67 | — | 6 | 2 | no | — |
| 78 | #67 | — | 6 | 2 | no | — |
| 79 | D12 | — | 2 | 2 | no | — |
| 80 | D12 | — | 2 | 2 | no | — |
| 81 | D12 | — | 3 | 2 | no | — |
| 82 | D12 | — | 6 | 1 | no | — |
| 83 | C4×D3 | — | 6 | 2 | no | — |
| 84 | C4×D3 | — | 6 | 2 | no | — |
| 85 | Dic6 | — | 3 | 2 | no | — |
| 86 | Dic6 | — | 6 | 1 | no | — |
| 87 | C3×C8 | — | 2 | 2 | no | — |
| 88 | C3×C8 | — | 2 | 2 | no | — |
| 89 | C3×C8 | — | 6 | 2 | no | — |
| 90 | C3⋊C8 | — | 2 | 2 | no | — |
| 91 | C3⋊C8 | — | 3 | 2 | no | — |
| 92 | C3⋊C8 | — | 3 | 2 | no | — |
| 93 | C3⋊C8 | — | 6 | 1 | no | — |
| 94 | C3⋊C8 | — | 6 | 1 | no | — |
| 95 | C2×C32 | — | 2 | 1 | no | — |
| 96 | C2×C32 | #121 | 1 | 1 | yes | characteristic |
| 97 | C3⋊D3 | — | 4 | 1 | no | — |
| 98 | C3×D3 | — | 8 | 2 | no | — |
| 99 | #41 | — | 9 | 1 | no | — |
| 100 | C2×D4 | — | 3 | 2 | no | — |
| 101 | SD16 | — | 18 | 2 | no | — |
| 102 | D8 | — | 18 | 2 | no | — |
| 103 | D8 | — | 18 | 1 | no | — |
| 104 | D8 | — | 18 | 1 | no | — |
| 105 | M16 | — | 3 | 2 | no | — |
| 106 | M16 | — | 18 | 1 | no | — |
| 107 | M16 | D32 | 1 | 1 | yes | characteristic |
| 108 | C2×C8 | — | 9 | 1 | no | — |
| 109 | C2×C8 | — | 18 | 1 | no | — |
| 110 | C22×C3 | — | 2 | 1 | no | — |
| 111 | C22×C3 | — | 12 | 2 | no | — |
| 112 | C22×C3 | — | 24 | 2 | no | — |
| 113 | C22×C3 | #235 | 1 | 2 | yes | — |
| 114 | C2×D3 | — | 4 | 2 | no | — |
| 115 | C2×D3 | — | 6 | 2 | no | — |
| 116 | C2×D3 | — | 6 | 2 | no | — |
| 117 | C2×D3 | — | 8 | 2 | no | — |
| 118 | C3×C4 | — | 2 | 1 | no | — |
| 119 | C3×C4 | — | 2 | 1 | no | — |
| 120 | C3×C4 | D3×D4 | 1 | 2 | yes | — |
| 121 | C3×C4 | D3×D4 | 1 | 2 | yes | — |
| 122 | Dic3 | — | 6 | 2 | no | — |
| 123 | Dic3 | — | 6 | 2 | no | — |
| 124 | C32 | #472 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 125 | C23 | — | 6 | 2 | no | — |
| 126 | Q8 | — | 9 | 1 | no | — |
| 127 | D4 | — | 6 | 2 | no | — |
| 128 | D4 | — | 6 | 2 | no | — |
| 129 | D4 | — | 9 | 1 | no | — |
| 130 | D4 | — | 18 | 1 | no | — |
| 131 | C2×C4 | — | 18 | 1 | no | — |
| 132 | C2×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 133 | C8 | — | 2 | 1 | no | — |
| 134 | C8 | — | 6 | 2 | no | — |
| 135 | C8 | — | 9 | 1 | no | — |
| 136 | C8 | — | 9 | 1 | no | — |
| 137 | C2×C3 | — | 2 | 2 | no | — |
| 138 | C2×C3 | — | 2 | 1 | no | — |
| 139 | C2×C3 | — | 4 | 1 | no | — |
| 140 | C2×C3 | — | 24 | 2 | no | — |
| 141 | C2×C3 | #884 | 1 | 2 | yes | — |
| 142 | D3 | — | 8 | 2 | no | — |
| 143 | D3 | — | 12 | 2 | no | — |
| 144 | D3 | — | 12 | 2 | no | — |
| 145 | C22 | — | 12 | 2 | no | — |
| 146 | C22 | — | 18 | 1 | no | — |
| 147 | C22 | — | 24 | 2 | no | — |
| 148 | C22 | #3798 | 1 | 1 | yes | characteristic |
| 149 | C4 | — | 18 | 1 | no | — |
| 150 | C4 | #3802 | 1 | 1 | yes | characteristic |
| 151 | C4 | #3811 | 1 | 1 | yes | characteristic |
| 152 | C3 | — | 2 | 1 | no | — |
| 153 | C3 | #4749 | 1 | 2 | yes | complemented |
| 154 | C2 | — | 2 | 1 | no | — |
| 155 | C2 | — | 24 | 2 | no | — |
| 156 | C2 | — | 36 | 1 | no | — |
| 157 | C2 | #63912 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #63912 |
| commutator, cosocle | C2×C32×C4 | C23 |
| Frattini | C2×C4 | C2×D32 |
| Fitting | C32×M16 | C22 |
| socle | C2×C32 | #121 |
| hypercenter | M16 | D32 |
| hypocenter | C32 | #472 |
| LCS | G ▷ C2×C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #64116 | C2 | semidirect |
| 2. #63830 | C2 | semidirect |
| 3. C2×#3713 | C2 | semidirect |
| 4. C3×#823 | C2 | semidirect |
| 5. #63578 | C2 | semidirect |
| 24. #823 | D3 | semidirect |
| 124. C32 | #472 | semidirect |
| 153. C3 | #4749 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64116 | 2 |
| #63830 | 2 |
| C2×#3713 | 2 |
| C3×#823 | 2 |
| #63578 | 2 |
| #4749 | 3 |