Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
195 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 44 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 | #64121 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #63802 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#827 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #63551 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #63544 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #4915 | — | 3 | 2 | no | maximal |
| 7 | #3830 | — | 2 | 1 | no | — |
| 8 | #3797 | — | 2 | 1 | no | — |
| 9 | C32×D8 | C22 | 1 | 1 | yes | characteristic |
| 10 | #3754 | — | 2 | 1 | no | — |
| 11 | #3746 | C22 | 1 | 1 | yes | characteristic |
| 12 | C8×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 13 | C3×Dic12 | C22 | 1 | 2 | yes | — |
| 14 | #3718 | — | 2 | 1 | no | — |
| 15 | C3×C3⋊C16 | C22 | 1 | 2 | yes | — |
| 16 | D3×Q16 | — | 3 | 2 | no | — |
| 17 | #912 | — | 3 | 2 | no | — |
| 18 | #912 | — | 6 | 1 | no | — |
| 19 | C3×SD32 | — | 3 | 2 | no | — |
| 20 | #829 | — | 3 | 2 | no | — |
| 21 | #827 | D3 | 1 | 2 | yes | complemented |
| 22 | Dic24 | — | 3 | 2 | no | — |
| 23 | #798 | — | 3 | 2 | no | — |
| 24 | C32×D4 | — | 2 | 1 | no | — |
| 25 | #638 | — | 4 | 1 | no | — |
| 26 | C2×C3⋊Dic3 | — | 4 | 1 | no | — |
| 27 | C4×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 28 | #634 | — | 2 | 1 | no | — |
| 29 | C3×Dic6 | — | 2 | 2 | no | — |
| 30 | #627 | — | 4 | 1 | no | — |
| 31 | #624 | — | 4 | 1 | no | — |
| 32 | C32×C8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C3⋊(C3⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 34 | #510 | — | 9 | 1 | no | Sylow |
| 35 | D3×Q8 | — | 6 | 2 | no | — |
| 36 | #237 | — | 6 | 2 | no | — |
| 37 | #237 | — | 6 | 2 | no | — |
| 38 | C3×Q16 | — | 3 | 2 | no | — |
| 39 | C3×D8 | — | 2 | 1 | no | — |
| 40 | C3×D8 | C2×D3 | 1 | 2 | yes | — |
| 41 | #216 | — | 6 | 2 | no | — |
| 42 | #214 | — | 6 | 2 | no | — |
| 43 | #214 | — | 6 | 2 | no | — |
| 44 | Dic12 | — | 3 | 2 | no | — |
| 45 | Dic12 | — | 6 | 1 | no | — |
| 46 | Dic12 | C2×D3 | 1 | 2 | yes | — |
| 47 | D3×C8 | — | 3 | 2 | no | — |
| 48 | D3×C8 | — | 6 | 1 | no | — |
| 49 | C3×C16 | — | 3 | 2 | no | — |
| 50 | C3⋊C16 | C2×D3 | 1 | 2 | yes | — |
| 51 | C22×C32 | — | 4 | 1 | no | — |
| 52 | C2×C3⋊D3 | D8 | 1 | 1 | yes | characteristic |
| 53 | C32×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 55 | C3⋊Dic3 | D8 | 1 | 1 | yes | characteristic |
| 56 | C3×Dic3 | — | 4 | 2 | no | — |
| 57 | #135 | — | 9 | 1 | no | — |
| 58 | C2×Q16 | — | 9 | 1 | no | — |
| 59 | Q32 | — | 9 | 2 | no | — |
| 60 | SD32 | — | 3 | 2 | no | — |
| 61 | M32 | — | 9 | 1 | no | — |
| 62 | C3×Q8 | — | 6 | 2 | no | — |
| 63 | C3×D4 | — | 2 | 2 | no | — |
| 64 | C3×D4 | — | 2 | 2 | no | — |
| 65 | #67 | — | 12 | 2 | no | — |
| 66 | #67 | — | 12 | 2 | no | — |
| 67 | C2×Dic3 | — | 12 | 2 | no | — |
| 68 | C2×Dic3 | — | 12 | 2 | no | — |
| 69 | C4×D3 | — | 3 | 2 | no | — |
| 70 | C4×D3 | — | 6 | 1 | no | — |
| 71 | C4×D3 | — | 12 | 2 | no | — |
| 72 | Dic6 | — | 2 | 2 | no | — |
| 73 | Dic6 | — | 6 | 2 | no | — |
| 74 | Dic6 | — | 6 | 2 | no | — |
| 75 | Dic6 | — | 12 | 2 | no | — |
| 76 | C3×C8 | — | 2 | 1 | no | — |
| 77 | C3×C8 | C22×D3 | 1 | 2 | yes | — |
| 78 | C3⋊C8 | — | 3 | 2 | no | — |
| 79 | C3⋊C8 | — | 6 | 1 | no | — |
| 80 | C2×C32 | — | 8 | 1 | no | — |
| 81 | C2×C32 | C2×D8 | 1 | 1 | yes | characteristic |
| 82 | C3⋊D3 | — | 2 | 1 | no | — |
| 83 | #41 | — | 18 | 1 | no | — |
| 84 | C2×Q8 | — | 18 | 1 | no | — |
| 85 | Q16 | — | 3 | 2 | no | — |
| 86 | Q16 | — | 9 | 1 | no | — |
| 87 | Q16 | — | 18 | 1 | no | — |
| 88 | SD16 | — | 18 | 1 | no | — |
| 89 | D8 | D32 | 1 | 1 | yes | characteristic |
| 90 | C2×C8 | — | 9 | 1 | no | — |
| 91 | C16 | — | 3 | 2 | no | — |
| 92 | C22×C3 | — | 4 | 2 | no | — |
| 93 | C22×C3 | — | 4 | 2 | no | — |
| 94 | C2×D3 | — | 3 | 2 | no | — |
| 95 | C2×D3 | — | 6 | 1 | no | — |
| 96 | C3×C4 | — | 2 | 1 | no | — |
| 97 | C3×C4 | — | 12 | 2 | no | — |
| 98 | C3×C4 | D3×D4 | 1 | 2 | yes | — |
| 99 | Dic3 | — | 3 | 2 | no | — |
| 100 | Dic3 | — | 4 | 2 | no | — |
| 101 | Dic3 | — | 6 | 1 | no | — |
| 102 | Dic3 | — | 12 | 2 | no | — |
| 103 | Dic3 | — | 12 | 2 | no | — |
| 104 | C32 | #510 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 105 | Q8 | — | 6 | 2 | no | — |
| 106 | Q8 | — | 18 | 1 | no | — |
| 107 | Q8 | — | 36 | 1 | no | — |
| 108 | D4 | — | 2 | 1 | no | — |
| 109 | D4 | — | 36 | 1 | no | — |
| 110 | C2×C4 | — | 9 | 1 | no | — |
| 111 | C2×C4 | — | 36 | 1 | no | — |
| 112 | C2×C4 | — | 36 | 1 | no | — |
| 113 | C8 | — | 9 | 1 | no | — |
| 114 | C8 | C2×D32 | 1 | 1 | yes | characteristic |
| 115 | C2×C3 | — | 2 | 1 | no | — |
| 116 | C2×C3 | — | 8 | 2 | no | — |
| 117 | C2×C3 | — | 8 | 2 | no | — |
| 118 | C2×C3 | D3×D8 | 1 | 2 | yes | — |
| 119 | D3 | — | 6 | 2 | no | — |
| 120 | D3 | — | 12 | 1 | no | — |
| 121 | C22 | — | 4 | 1 | no | — |
| 122 | C22 | — | 9 | 1 | no | — |
| 123 | C4 | — | 9 | 1 | no | — |
| 124 | C4 | — | 12 | 2 | no | — |
| 125 | C4 | — | 36 | 1 | no | — |
| 126 | C4 | #3811 | 1 | 1 | yes | characteristic |
| 127 | C3 | — | 2 | 1 | no | — |
| 128 | C3 | #4915 | 1 | 2 | yes | complemented |
| 129 | C2 | — | 8 | 1 | no | — |
| 130 | C2 | — | 18 | 1 | no | — |
| 131 | C2 | #63926 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #63802 | C2 | semidirect |
| 3. C3×#827 | C2 | semidirect |
| 4. #63551 | C2 | semidirect |
| 5. #63544 | C2 | semidirect |
| 21. #827 | D3 | semidirect |
| 104. C32 | #510 | semidirect |
| 128. C3 | #4915 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64121 | 2 |
| #63802 | 2 |
| C3×#827 | 2 |
| #63551 | 2 |
| #63544 | 2 |
| #4915 | 3 |