Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
551 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 114 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 | #88748 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | Q8×(C3⋊D3)⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #88652 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #88651 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #88650 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #64315 | — | 3 | 2 | no | maximal |
| 7 | Q8×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 8 | C4⋊#272 | C22 | 1 | 3 | yes | — |
| 9 | C4×#272 | C22 | 1 | 3 | yes | — |
| 10 | C4×(C3⋊D3)⋊C3 | C22 | 1 | 6 | yes | — |
| 11 | #634⋊C3 | C22 | 1 | 6 | yes | — |
| 12 | #6200 | C22 | 1 | 6 | yes | — |
| 13 | #6199 | C22 | 1 | 6 | yes | — |
| 14 | #6198 | C22 | 1 | 1 | yes | characteristic |
| 15 | #6196 | C22 | 1 | 3 | yes | — |
| 16 | Q8×C3⋊D3 | D3 | 1 | 2 | yes | complemented |
| 17 | C3×#239 | — | 3 | 2 | no | — |
| 18 | C3×#239 | — | 6 | 1 | no | — |
| 19 | C3×D3×Q8 | — | 3 | 2 | no | — |
| 20 | #3798 | — | 3 | 6 | no | — |
| 21 | #3796 | — | 3 | 6 | no | — |
| 22 | #3795 | — | 3 | 6 | no | — |
| 23 | D3×Dic6 | — | 3 | 6 | no | — |
| 24 | C2×#272 | C23 | 1 | 3 | yes | — |
| 25 | C2×(C3⋊D3)⋊C3 | C23 | 1 | 2 | yes | — |
| 26 | C4×H(3,𝔽3) | C23 | 1 | 3 | yes | — |
| 27 | #1084 | C23 | 1 | 1 | yes | characteristic |
| 28 | (C3⋊Dic3)⋊C3 | C23 | 1 | 6 | yes | — |
| 29 | #1010 | — | 9 | 1 | no | — |
| 30 | #1007 | — | 9 | 2 | no | — |
| 31 | C32×Q8 | — | 2 | 1 | no | — |
| 32 | C32×Q8 | C2×D3 | 1 | 2 | yes | — |
| 33 | C4×C3⋊D3 | C2×D3 | 1 | 6 | yes | — |
| 34 | #634 | C2×D3 | 1 | 6 | yes | — |
| 35 | C3×D12 | — | 3 | 6 | no | — |
| 36 | C3×D12 | — | 6 | 3 | no | — |
| 37 | C3×C4×D3 | — | 3 | 6 | no | — |
| 38 | C3×C4×D3 | — | 3 | 6 | no | — |
| 39 | C3×C4×D3 | — | 6 | 3 | no | — |
| 40 | C3×Dic6 | — | 3 | 6 | no | — |
| 41 | #627 | — | 3 | 6 | no | — |
| 42 | #626 | — | 3 | 2 | no | — |
| 43 | #626 | — | 3 | 6 | no | — |
| 44 | #625 | — | 3 | 6 | no | — |
| 45 | D3×Dic3 | — | 3 | 12 | no | — |
| 46 | C2×H(3,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 47 | #272 | — | 2 | 3 | no | — |
| 48 | (C3⋊D3)⋊C3 | — | 2 | 2 | no | — |
| 49 | C3×#41 | — | 9 | 1 | no | — |
| 50 | C2×C3×Q8 | — | 9 | 2 | no | — |
| 51 | #239 | — | 3 | 2 | no | — |
| 52 | #239 | — | 6 | 1 | no | — |
| 53 | #239 | — | 9 | 6 | no | — |
| 54 | D3×Q8 | — | 3 | 2 | no | — |
| 55 | D3×Q8 | — | 9 | 2 | no | — |
| 56 | D3×Q8 | — | 9 | 6 | no | — |
| 57 | #237 | — | 9 | 6 | no | — |
| 58 | #235 | — | 9 | 3 | no | — |
| 59 | #235 | — | 9 | 6 | no | — |
| 60 | #235 | — | 9 | 6 | no | — |
| 61 | C2×Dic6 | — | 9 | 3 | no | — |
| 62 | C2×C3⋊D3 | C22×D3 | 1 | 2 | yes | — |
| 63 | C2×C3×D3 | — | 3 | 2 | no | — |
| 64 | C2×C3×D3 | — | 3 | 6 | no | — |
| 65 | C2×C3×D3 | — | 6 | 3 | no | — |
| 66 | C32×C4 | — | 2 | 3 | no | — |
| 67 | C32×C4 | C22×D3 | 1 | 6 | yes | — |
| 68 | C3⋊Dic3 | C22×D3 | 1 | 6 | yes | — |
| 69 | C3×Dic3 | — | 3 | 6 | no | — |
| 70 | C3×Dic3 | — | 3 | 2 | no | — |
| 71 | C3×Dic3 | — | 6 | 1 | no | — |
| 72 | 2−1+4 | — | 27 | 1 | no | Sylow |
| 73 | H(3,𝔽3) | 2−1+4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 74 | C3×Q8 | — | 3 | 2 | no | — |
| 75 | C3×Q8 | — | 6 | 1 | no | — |
| 76 | C3×Q8 | — | 9 | 6 | no | — |
| 77 | C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 78 | C3×D4 | — | 9 | 3 | no | — |
| 79 | C2×C3×C4 | — | 9 | 6 | no | — |
| 80 | C2×C3×C4 | — | 9 | 3 | no | — |
| 81 | #67 | — | 9 | 6 | no | — |
| 82 | #67 | — | 9 | 2 | no | — |
| 83 | #67 | — | 9 | 6 | no | — |
| 84 | C2×Dic3 | — | 9 | 6 | no | — |
| 85 | D12 | — | 3 | 6 | no | — |
| 86 | D12 | — | 6 | 3 | no | — |
| 87 | D12 | — | 9 | 1 | no | — |
| 88 | D12 | — | 9 | 6 | no | — |
| 89 | C4×D3 | — | 3 | 6 | no | — |
| 90 | C4×D3 | — | 3 | 6 | no | — |
| 91 | C4×D3 | — | 6 | 3 | no | — |
| 92 | C4×D3 | — | 9 | 6 | no | — |
| 93 | C4×D3 | — | 9 | 12 | no | — |
| 94 | Dic6 | — | 3 | 6 | no | — |
| 95 | Dic6 | — | 9 | 6 | no | — |
| 96 | Dic6 | — | 9 | 3 | no | — |
| 97 | Dic6 | — | 9 | 6 | no | — |
| 98 | C2×C32 | — | 2 | 1 | no | — |
| 99 | C2×C32 | C23×D3 | 1 | 2 | yes | — |
| 100 | C3⋊D3 | — | 2 | 2 | no | — |
| 101 | C3×D3 | — | 6 | 2 | no | — |
| 102 | C3×D3 | — | 6 | 6 | no | — |
| 103 | C3×D3 | — | 12 | 3 | no | — |
| 104 | #41 | — | 9 | 1 | no | — |
| 105 | #41 | — | 27 | 3 | no | — |
| 106 | #41 | — | 27 | 6 | no | — |
| 107 | C2×Q8 | — | 9 | 2 | no | — |
| 108 | C2×Q8 | — | 27 | 3 | no | — |
| 109 | C22×C3 | — | 9 | 2 | no | — |
| 110 | C22×C3 | — | 9 | 3 | no | — |
| 111 | C2×D3 | — | 3 | 6 | no | — |
| 112 | C2×D3 | — | 3 | 2 | no | — |
| 113 | C2×D3 | — | 6 | 3 | no | — |
| 114 | C2×D3 | — | 9 | 2 | no | — |
| 115 | C3×C4 | — | 3 | 6 | no | — |
| 116 | C3×C4 | — | 6 | 3 | no | — |
| 117 | C3×C4 | — | 9 | 6 | no | — |
| 118 | C3×C4 | — | 9 | 1 | no | — |
| 119 | C3×C4 | C2×D32 | 1 | 3 | yes | — |
| 120 | Dic3 | — | 3 | 2 | no | — |
| 121 | Dic3 | — | 3 | 6 | no | — |
| 122 | Dic3 | — | 6 | 1 | no | — |
| 123 | Dic3 | — | 9 | 6 | no | — |
| 124 | C32 | — | 2 | 1 | no | — |
| 125 | C32 | #1007 | 1 | 2 | yes | complemented |
| 126 | Q8 | — | 9 | 6 | no | — |
| 127 | Q8 | — | 27 | 3 | no | — |
| 128 | Q8 | ((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 129 | D4 | — | 9 | 3 | no | — |
| 130 | D4 | — | 27 | 1 | no | — |
| 131 | D4 | — | 27 | 6 | no | — |
| 132 | C2×C4 | — | 9 | 6 | no | — |
| 133 | C2×C4 | — | 9 | 3 | no | — |
| 134 | C2×C4 | — | 27 | 6 | no | — |
| 135 | C2×C3 | — | 3 | 2 | no | — |
| 136 | C2×C3 | — | 6 | 1 | no | — |
| 137 | C2×C3 | — | 18 | 2 | no | — |
| 138 | C2×C3 | — | 18 | 3 | no | — |
| 139 | C2×C3 | C22×D32 | 1 | 1 | yes | characteristic |
| 140 | D3 | — | 6 | 6 | no | — |
| 141 | D3 | — | 6 | 2 | no | — |
| 142 | D3 | — | 12 | 3 | no | — |
| 143 | D3 | — | 18 | 2 | no | — |
| 144 | C22 | — | 9 | 2 | no | — |
| 145 | C22 | — | 9 | 3 | no | — |
| 146 | C4 | — | 9 | 1 | no | — |
| 147 | C4 | — | 9 | 6 | no | — |
| 148 | C4 | C2×((C3⋊D3)⋊C3)⋊C2 | 1 | 3 | yes | — |
| 149 | C3 | — | 3 | 2 | no | — |
| 150 | C3 | — | 6 | 1 | no | — |
| 151 | C3 | #64316 | 1 | 1 | yes | characteristic |
| 152 | C2 | — | 18 | 3 | no | — |
| 153 | C2 | — | 18 | 2 | no | — |
| 154 | C2 | C22×((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #88748 | C2 | semidirect |
| 2. Q8×(C3⋊D3)⋊C3 | C2 | semidirect |
| 3. #88652 | C2 | semidirect |
| 4. #88651 | C2 | semidirect |
| 5. #88650 | C2 | semidirect |
| 16. Q8×C3⋊D3 | D3 | semidirect |
| 73. H(3,𝔽3) | 2−1+4 | semidirect |
| 125. C32 | #1007 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #88748 | 2 |
| Q8×(C3⋊D3)⋊C3 | 2 |
| #88652 | 2 |
| #88651 | 2 |
| #88650 | 2 |
| #64315 | 3 |