Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
720 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 720 conjugacy classes of subgroups — too many to draw. Only its 140 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 | #89042 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #89018 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C3×#3797 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #88965 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #88958 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #64316 | — | 3 | 3 | no | maximal |
| 7 | C4⋊(C3⋊(C3⋊D3)) | C22 | 1 | 1 | yes | characteristic |
| 8 | C3×C4×C3⋊D3 | C22 | 1 | 3 | yes | — |
| 9 | C32×Dic6 | C22 | 1 | 3 | yes | — |
| 10 | #6296 | C22 | 1 | 1 | yes | characteristic |
| 11 | #6295 | C22 | 1 | 3 | yes | — |
| 12 | #6292 | C22 | 1 | 6 | yes | — |
| 13 | #6289 | C22 | 1 | 6 | yes | — |
| 14 | C3×#627 | C22 | 1 | 6 | yes | — |
| 15 | C3×#624 | C22 | 1 | 6 | yes | — |
| 16 | #3832 | — | 3 | 3 | no | — |
| 17 | C3×D3×Q8 | — | 3 | 6 | no | — |
| 18 | #3799 | — | 3 | 6 | no | — |
| 19 | #3799 | — | 3 | 12 | no | — |
| 20 | #3799 | — | 6 | 3 | no | — |
| 21 | #3798 | — | 3 | 3 | no | — |
| 22 | #3798 | — | 3 | 6 | no | — |
| 23 | #3797 | — | 3 | 6 | no | — |
| 24 | #3797 | D3 | 1 | 3 | yes | complemented |
| 25 | C2×C3⋊(C3⋊D3) | C23 | 1 | 2 | yes | — |
| 26 | C2×C3×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 27 | C33×C4 | C23 | 1 | 1 | yes | characteristic |
| 28 | C3×C3⋊Dic3 | C23 | 1 | 3 | yes | — |
| 29 | C32×Dic3 | C23 | 1 | 6 | yes | — |
| 30 | #1007 | — | 9 | 3 | no | — |
| 31 | C32×Q8 | — | 3 | 3 | no | — |
| 32 | C4⋊(C3⋊D3) | — | 3 | 6 | no | — |
| 33 | C4⋊(C3⋊D3) | — | 3 | 3 | no | — |
| 34 | C4⋊(C3⋊D3) | — | 6 | 3 | no | — |
| 35 | C4⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 36 | C4×C3⋊D3 | — | 3 | 6 | no | — |
| 37 | C4×C3⋊D3 | C2×D3 | 1 | 3 | yes | — |
| 38 | C3×C4×D3 | — | 3 | 6 | no | — |
| 39 | C3×C4×D3 | — | 3 | 12 | no | — |
| 40 | C3×C4×D3 | — | 6 | 3 | no | — |
| 41 | C3×Dic6 | — | 2 | 3 | no | — |
| 42 | C3×Dic6 | — | 3 | 12 | no | — |
| 43 | C3×Dic6 | C2×D3 | 1 | 6 | yes | — |
| 44 | #627 | — | 3 | 3 | no | — |
| 45 | #627 | C2×D3 | 1 | 6 | yes | — |
| 46 | #626 | — | 3 | 6 | no | — |
| 47 | #626 | — | 3 | 12 | no | — |
| 48 | #626 | — | 6 | 6 | no | — |
| 49 | #625 | — | 3 | 3 | no | — |
| 50 | #624 | — | 3 | 12 | no | — |
| 51 | #624 | — | 6 | 6 | no | — |
| 52 | #624 | C2×D3 | 1 | 6 | yes | — |
| 53 | C2×C33 | C24 | 1 | 1 | yes | characteristic |
| 54 | C3⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 55 | C3×C3⋊D3 | — | 2 | 3 | no | — |
| 56 | C2×C3×Q8 | — | 9 | 3 | no | — |
| 57 | #239 | — | 9 | 6 | no | — |
| 58 | #239 | — | 9 | 6 | no | — |
| 59 | #239 | — | 18 | 3 | no | — |
| 60 | D3×Q8 | — | 3 | 6 | no | — |
| 61 | D3×Q8 | — | 9 | 6 | no | — |
| 62 | #235 | — | 9 | 12 | no | — |
| 63 | #235 | — | 9 | 3 | no | — |
| 64 | #235 | — | 9 | 3 | no | — |
| 65 | C2×C3⋊D3 | — | 3 | 6 | no | — |
| 66 | C2×C3⋊D3 | — | 6 | 6 | no | — |
| 67 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 68 | C2×C3⋊D3 | C22×D3 | 1 | 3 | yes | — |
| 69 | C2×C3×D3 | — | 3 | 6 | no | — |
| 70 | C2×C3×D3 | — | 6 | 3 | no | — |
| 71 | C32×C4 | — | 2 | 3 | no | — |
| 72 | C32×C4 | — | 3 | 6 | no | — |
| 73 | C32×C4 | — | 4 | 1 | no | — |
| 74 | C32×C4 | C22×D3 | 1 | 3 | yes | — |
| 75 | C3⋊Dic3 | C22×D3 | 1 | 3 | yes | — |
| 76 | C3×Dic3 | — | 2 | 6 | no | — |
| 77 | C3×Dic3 | — | 3 | 6 | no | — |
| 78 | C3×Dic3 | — | 6 | 3 | no | — |
| 79 | C3×Dic3 | C22×D3 | 1 | 12 | yes | — |
| 80 | 2−1+4 | — | 27 | 1 | no | Sylow |
| 81 | C33 | 2−1+4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 82 | C3×Q8 | — | 3 | 6 | no | — |
| 83 | C3×Q8 | — | 6 | 3 | no | — |
| 84 | C3×Q8 | — | 9 | 6 | no | — |
| 85 | C2×C3×C4 | — | 9 | 6 | no | — |
| 86 | C2×C3×C4 | — | 9 | 3 | no | — |
| 87 | #67 | — | 9 | 6 | no | — |
| 88 | #67 | — | 9 | 6 | no | — |
| 89 | D12 | — | 9 | 12 | no | — |
| 90 | D12 | — | 9 | 3 | no | — |
| 91 | D12 | — | 9 | 3 | no | — |
| 92 | D12 | — | 18 | 3 | no | — |
| 93 | D12 | — | 18 | 6 | no | — |
| 94 | D12 | — | 36 | 1 | no | — |
| 95 | C4×D3 | — | 3 | 6 | no | — |
| 96 | C4×D3 | — | 3 | 12 | no | — |
| 97 | C4×D3 | — | 6 | 3 | no | — |
| 98 | C4×D3 | — | 9 | 12 | no | — |
| 99 | C4×D3 | — | 9 | 6 | no | — |
| 100 | C4×D3 | — | 18 | 6 | no | — |
| 101 | Dic6 | — | 3 | 12 | no | — |
| 102 | Dic6 | — | 9 | 3 | no | — |
| 103 | Dic6 | D32 | 1 | 3 | yes | — |
| 104 | C2×C32 | — | 2 | 3 | no | — |
| 105 | C2×C32 | — | 4 | 1 | no | — |
| 106 | C2×C32 | C23×D3 | 1 | 3 | yes | — |
| 107 | C3⋊D3 | — | 2 | 3 | no | — |
| 108 | C3⋊D3 | — | 6 | 6 | no | — |
| 109 | C3⋊D3 | — | 12 | 6 | no | — |
| 110 | C3⋊D3 | — | 24 | 2 | no | — |
| 111 | C3×D3 | — | 6 | 6 | no | — |
| 112 | C3×D3 | — | 12 | 3 | no | — |
| 113 | #41 | — | 27 | 1 | no | — |
| 114 | #41 | — | 27 | 6 | no | — |
| 115 | #41 | — | 27 | 3 | no | — |
| 116 | C2×Q8 | — | 9 | 3 | no | — |
| 117 | C2×Q8 | — | 27 | 2 | no | — |
| 118 | C22×C3 | — | 9 | 3 | no | — |
| 119 | C2×D3 | — | 3 | 6 | no | — |
| 120 | C2×D3 | — | 6 | 3 | no | — |
| 121 | C2×D3 | — | 9 | 6 | no | — |
| 122 | C2×D3 | — | 18 | 6 | no | — |
| 123 | C2×D3 | — | 36 | 2 | no | — |
| 124 | C3×C4 | — | 2 | 3 | no | — |
| 125 | C3×C4 | — | 3 | 12 | no | — |
| 126 | C3×C4 | — | 4 | 1 | no | — |
| 127 | C3×C4 | — | 6 | 6 | no | — |
| 128 | C3×C4 | — | 9 | 3 | no | — |
| 129 | C3×C4 | C2×D32 | 1 | 3 | yes | — |
| 130 | Dic3 | — | 3 | 6 | no | — |
| 131 | Dic3 | — | 6 | 3 | no | — |
| 132 | Dic3 | C2×D32 | 1 | 6 | yes | — |
| 133 | C32 | — | 2 | 3 | no | — |
| 134 | C32 | — | 4 | 1 | no | — |
| 135 | C32 | #1007 | 1 | 3 | yes | complemented |
| 136 | Q8 | — | 3 | 3 | no | — |
| 137 | Q8 | — | 9 | 6 | no | — |
| 138 | Q8 | — | 27 | 1 | no | — |
| 139 | D4 | — | 27 | 6 | no | — |
| 140 | D4 | — | 27 | 3 | no | — |
| 141 | D4 | — | 27 | 1 | no | — |
| 142 | C2×C4 | — | 9 | 6 | no | — |
| 143 | C2×C4 | — | 9 | 3 | no | — |
| 144 | C2×C4 | — | 27 | 6 | no | — |
| 145 | C2×C3 | — | 2 | 3 | no | — |
| 146 | C2×C3 | — | 4 | 1 | no | — |
| 147 | C2×C3 | — | 18 | 3 | no | — |
| 148 | C2×C3 | C22×D32 | 1 | 3 | yes | — |
| 149 | D3 | — | 6 | 6 | no | — |
| 150 | D3 | — | 12 | 3 | no | — |
| 151 | D3 | — | 18 | 6 | no | — |
| 152 | D3 | — | 36 | 6 | no | — |
| 153 | D3 | — | 72 | 2 | no | — |
| 154 | C22 | — | 9 | 3 | no | — |
| 155 | C22 | — | 27 | 2 | no | — |
| 156 | C4 | — | 3 | 6 | no | — |
| 157 | C4 | — | 9 | 3 | no | — |
| 158 | C4 | D33 | 1 | 1 | yes | characteristic |
| 159 | C3 | — | 2 | 3 | no | — |
| 160 | C3 | — | 4 | 1 | no | — |
| 161 | C3 | #64316 | 1 | 3 | yes | complemented |
| 162 | C2 | — | 18 | 3 | no | — |
| 163 | C2 | — | 54 | 2 | no | — |
| 164 | C2 | C2×D33 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D33 |
| commutator, socle, cosocle | C2×C33 | C24 |
| Fitting | C33×C4 | C23 |
| hypercenter | C4 | D33 |
| hypocenter | C33 | 2−1+4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #89042 | C2 | semidirect |
| 2. #89018 | C2 | semidirect |
| 3. C3×#3797 | C2 | semidirect |
| 4. #88965 | C2 | semidirect |
| 5. #88958 | C2 | semidirect |
| 24. #3797 | D3 | semidirect |
| 81. C33 | 2−1+4 | semidirect |
| 135. C32 | #1007 | semidirect |
| 161. C3 | #64316 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #89042 | 2 |
| #89018 | 2 |
| C3×#3797 | 2 |
| #88965 | 2 |
| #88958 | 2 |
| #64316 | 3 |