Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
626 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 626 conjugacy classes of subgroups — too many to draw. Only its 82 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 | C2×#6291 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×C3×#625 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #88813 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #63979 | — | 3 | 3 | no | maximal |
| 5 | C22×C3⋊(C3⋊D3) | C22 | 1 | 1 | yes | characteristic |
| 6 | C22×C32×D3 | C22 | 1 | 3 | yes | — |
| 7 | C2×C3×C3⋊Dic3 | C22 | 1 | 3 | yes | — |
| 8 | #6291 | — | 2 | 6 | no | — |
| 9 | C3×#625 | — | 2 | 6 | no | — |
| 10 | C2×#638 | — | 3 | 3 | no | — |
| 11 | C2×C3×#67 | — | 3 | 6 | no | — |
| 12 | C2×#626 | — | 3 | 6 | no | — |
| 13 | C2×#626 | — | 6 | 3 | no | — |
| 14 | C2×#625 | D3 | 1 | 3 | yes | complemented |
| 15 | Dic32 | — | 3 | 3 | no | — |
| 16 | C22×C33 | C23 | 1 | 1 | yes | characteristic |
| 17 | C2×C3⋊(C3⋊D3) | — | 2 | 3 | no | — |
| 18 | C2×C32×D3 | — | 2 | 3 | no | — |
| 19 | C2×C32×D3 | — | 2 | 6 | no | — |
| 20 | C3×C3⋊Dic3 | D4 | 1 | 6 | yes | complemented |
| 21 | #940 | — | 9 | 3 | no | — |
| 22 | C23×C32 | — | 3 | 3 | no | — |
| 23 | C22×C3⋊D3 | — | 3 | 3 | no | — |
| 24 | C22×C3⋊D3 | — | 6 | 3 | no | — |
| 25 | C22×C3⋊D3 | — | 12 | 1 | no | — |
| 26 | C22×C3×D3 | — | 2 | 3 | no | — |
| 27 | C22×C3×D3 | C2×D3 | 1 | 6 | yes | — |
| 28 | #638 | — | 6 | 6 | no | — |
| 29 | C2×C3⋊Dic3 | C2×D3 | 1 | 3 | yes | — |
| 30 | C3×#67 | — | 6 | 12 | no | — |
| 31 | C2×C3×Dic3 | — | 3 | 6 | no | — |
| 32 | C2×C3×Dic3 | — | 6 | 3 | no | — |
| 33 | #626 | — | 6 | 12 | no | — |
| 34 | #626 | — | 6 | 12 | no | — |
| 35 | #625 | — | 2 | 6 | no | — |
| 36 | C2×C33 | C2×D4 | 1 | 3 | yes | — |
| 37 | C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 38 | C32×D3 | — | 4 | 3 | no | — |
| 39 | C2×C3×D4 | — | 9 | 3 | no | — |
| 40 | C2×#67 | — | 3 | 6 | no | — |
| 41 | C2×#67 | — | 9 | 6 | no | — |
| 42 | C2×#67 | — | 18 | 3 | no | — |
| 43 | C2×D12 | — | 9 | 3 | no | — |
| 44 | C4×Dic3 | — | 9 | 3 | no | — |
| 45 | C22×C32 | — | 2 | 3 | no | — |
| 46 | C22×C32 | — | 4 | 1 | no | — |
| 47 | C22×C32 | — | 6 | 3 | no | — |
| 48 | C22×C32 | — | 6 | 6 | no | — |
| 49 | C22×C32 | C22×D3 | 1 | 3 | yes | — |
| 50 | C2×C3⋊D3 | — | 6 | 3 | no | — |
| 51 | C2×C3⋊D3 | — | 6 | 6 | no | — |
| 52 | C2×C3⋊D3 | — | 6 | 6 | no | — |
| 53 | C2×C3⋊D3 | — | 12 | 6 | no | — |
| 54 | C2×C3⋊D3 | — | 12 | 6 | no | — |
| 55 | C2×C3×D3 | — | 2 | 6 | no | — |
| 56 | C2×C3×D3 | — | 2 | 6 | no | — |
| 57 | C2×C3×D3 | — | 2 | 6 | no | — |
| 58 | C2×C3×D3 | — | 2 | 6 | no | — |
| 59 | C2×C3×D3 | — | 4 | 6 | no | — |
| 60 | C3⋊Dic3 | #67 | 1 | 6 | yes | complemented |
| 61 | C3×Dic3 | — | 3 | 12 | no | — |
| 62 | C3×Dic3 | — | 6 | 6 | no | — |
| 63 | #127 | — | 27 | 1 | no | Sylow |
| 64 | C33 | #127 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 65 | C23×C3 | — | 3 | 6 | no | — |
| 66 | C23×C3 | — | 6 | 3 | no | — |
| 67 | C22×D3 | — | 9 | 3 | no | — |
| 68 | C22×D3 | — | 18 | 3 | no | — |
| 69 | C22×D3 | — | 36 | 1 | no | — |
| 70 | C22×D3 | D32 | 1 | 3 | yes | — |
| 71 | C3×D4 | — | 18 | 6 | no | — |
| 72 | C2×C3×C4 | — | 9 | 3 | no | — |
| 73 | #67 | — | 6 | 12 | no | — |
| 74 | #67 | — | 18 | 12 | no | — |
| 75 | #67 | — | 18 | 12 | no | — |
| 76 | C2×Dic3 | — | 3 | 6 | no | — |
| 77 | C2×Dic3 | — | 6 | 3 | no | — |
| 78 | D12 | — | 18 | 6 | no | — |
| 79 | C2×C32 | — | 2 | 6 | no | — |
| 80 | C2×C32 | — | 2 | 3 | no | — |
| 81 | C2×C32 | — | 4 | 3 | no | — |
| 82 | C2×C32 | — | 12 | 3 | no | — |
| 83 | C2×C32 | D3×D4 | 1 | 6 | yes | — |
| 84 | C2×C32 | C2×#67 | 1 | 3 | yes | — |
| 85 | C3⋊D3 | — | 12 | 6 | no | — |
| 86 | C3⋊D3 | — | 12 | 4 | no | — |
| 87 | C3⋊D3 | — | 12 | 3 | no | — |
| 88 | C3×D3 | — | 4 | 6 | no | — |
| 89 | C3×D3 | — | 4 | 6 | no | — |
| 90 | C2×D4 | — | 9 | 3 | no | — |
| 91 | C2×D4 | — | 27 | 3 | no | — |
| 92 | C42 | — | 27 | 1 | no | — |
| 93 | C22×C3 | — | 2 | 3 | no | — |
| 94 | C22×C3 | — | 4 | 1 | no | — |
| 95 | C22×C3 | — | 6 | 6 | no | — |
| 96 | C22×C3 | — | 6 | 6 | no | — |
| 97 | C22×C3 | — | 6 | 6 | no | — |
| 98 | C22×C3 | — | 6 | 6 | no | — |
| 99 | C22×C3 | — | 12 | 6 | no | — |
| 100 | C22×C3 | C2×D32 | 1 | 3 | yes | — |
| 101 | C2×D3 | — | 2 | 6 | no | — |
| 102 | C2×D3 | — | 2 | 3 | no | — |
| 103 | C2×D3 | — | 18 | 6 | no | — |
| 104 | C2×D3 | — | 18 | 6 | no | — |
| 105 | C2×D3 | — | 18 | 3 | no | — |
| 106 | C2×D3 | — | 36 | 6 | no | — |
| 107 | C2×D3 | — | 36 | 6 | no | — |
| 108 | C3×C4 | — | 9 | 6 | no | — |
| 109 | Dic3 | — | 3 | 12 | no | — |
| 110 | Dic3 | — | 6 | 6 | no | — |
| 111 | C32 | — | 2 | 3 | no | — |
| 112 | C32 | — | 4 | 1 | no | — |
| 113 | C32 | #940 | 1 | 3 | yes | complemented |
| 114 | C23 | — | 3 | 3 | no | — |
| 115 | C23 | — | 27 | 1 | no | — |
| 116 | D4 | — | 18 | 6 | no | — |
| 117 | D4 | — | 54 | 6 | no | — |
| 118 | C2×C4 | — | 9 | 3 | no | — |
| 119 | C2×C3 | — | 2 | 3 | no | — |
| 120 | C2×C3 | — | 2 | 6 | no | — |
| 121 | C2×C3 | — | 4 | 3 | no | — |
| 122 | C2×C3 | — | 12 | 6 | no | — |
| 123 | C2×C3 | — | 12 | 6 | no | — |
| 124 | C2×C3 | D3×#67 | 1 | 6 | yes | — |
| 125 | C2×C3 | #3811 | 1 | 3 | yes | — |
| 126 | D3 | — | 4 | 3 | no | — |
| 127 | D3 | — | 36 | 6 | no | — |
| 128 | D3 | — | 36 | 4 | no | — |
| 129 | D3 | — | 36 | 3 | no | — |
| 130 | C22 | — | 6 | 6 | no | — |
| 131 | C22 | — | 6 | 3 | no | — |
| 132 | C22 | — | 54 | 3 | no | — |
| 133 | C22 | D33 | 1 | 1 | yes | characteristic |
| 134 | C4 | — | 9 | 6 | no | — |
| 135 | C3 | — | 2 | 3 | no | — |
| 136 | C3 | — | 4 | 1 | no | — |
| 137 | C3 | #63979 | 1 | 3 | yes | complemented |
| 138 | C2 | — | 12 | 3 | no | — |
| 139 | C2 | — | 108 | 1 | no | — |
| 140 | C2 | #88952 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D33 |
| commutator, Fitting, socle, cosocle | C22×C33 | C23 |
| hypocenter | C33 | #127 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6291 | C2 | semidirect |
| 2. C2×C3×#625 | C2 | semidirect |
| 3. #88813 | C2 | semidirect |
| 14. C2×#625 | D3 | semidirect |
| 20. C3×C3⋊Dic3 | D4 | semidirect |
| 60. C3⋊Dic3 | #67 | semidirect |
| 64. C33 | #127 | semidirect |
| 113. C32 | #940 | semidirect |
| 137. C3 | #63979 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6291 | 2 |
| C2×C3×#625 | 2 |
| #88813 | 2 |
| #63979 | 3 |