Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
236 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 43 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 | #89040 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#3797 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C4×C3⋊#157 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #88934 | C2 | 1 | 4 | yes | maximal |
| 5 | #64222 | — | 3 | 1 | no | maximal |
| 6 | C2×C3⋊#157 | C22 | 1 | 2 | yes | — |
| 7 | C3×C4×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3×#634 | — | 2 | 1 | no | — |
| 9 | C32×Dic6 | — | 2 | 1 | no | — |
| 10 | #6296 | — | 2 | 1 | no | — |
| 11 | #6294 | C22 | 1 | 2 | yes | — |
| 12 | C3×#627 | — | 2 | 1 | no | — |
| 13 | C3×#624 | C22 | 1 | 2 | yes | — |
| 14 | C3×D3×Q8 | — | 6 | 1 | no | — |
| 15 | #3797 | — | 3 | 1 | no | — |
| 16 | #3797 | D3 | 1 | 1 | yes | characteristic |
| 17 | D3×Dic6 | — | 6 | 1 | no | — |
| 18 | C4×#157 | — | 3 | 1 | no | — |
| 19 | #3773 | — | 3 | 4 | no | — |
| 20 | C2×C3×C3⋊D3 | C23 | 1 | 1 | yes | characteristic |
| 21 | C3⋊#157 | Q8 | 1 | 4 | yes | complemented |
| 22 | C33×C4 | D4 | 1 | 1 | yes | characteristic |
| 23 | C3×C3⋊Dic3 | — | 2 | 2 | no | — |
| 24 | C3×C3⋊Dic3 | D4 | 1 | 1 | yes | characteristic |
| 25 | C32×Dic3 | — | 2 | 2 | no | — |
| 26 | #944 | — | 9 | 1 | no | maximal |
| 27 | C2×#157 | — | 3 | 2 | no | — |
| 28 | C32×Q8 | — | 6 | 1 | no | — |
| 29 | C4×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 30 | #634 | — | 2 | 1 | no | — |
| 31 | C3×C4×D3 | — | 6 | 2 | no | — |
| 32 | C3×C4×D3 | — | 6 | 1 | no | — |
| 33 | C3×C4×D3 | — | 6 | 1 | no | — |
| 34 | C3×Dic6 | — | 2 | 1 | no | — |
| 35 | C3×Dic6 | — | 2 | 1 | no | — |
| 36 | C3×Dic6 | — | 4 | 1 | no | — |
| 37 | C3×Dic6 | — | 6 | 2 | no | — |
| 38 | C3×Dic6 | — | 6 | 1 | no | — |
| 39 | C3×Dic6 | — | 6 | 1 | no | — |
| 40 | C3×Dic6 | — | 12 | 1 | no | — |
| 41 | #627 | — | 2 | 1 | no | — |
| 42 | #627 | — | 6 | 2 | no | — |
| 43 | #627 | — | 6 | 1 | no | — |
| 44 | #624 | — | 3 | 2 | no | — |
| 45 | #624 | C2×D3 | 1 | 2 | yes | — |
| 46 | D3×Dic3 | — | 6 | 2 | no | — |
| 47 | C2×C33 | C2×D4 | 1 | 1 | yes | characteristic |
| 48 | C3×C3⋊D3 | C2×Q8 | 1 | 2 | yes | — |
| 49 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 50 | D3×Q8 | — | 6 | 1 | no | — |
| 51 | D3×Q8 | — | 18 | 1 | no | — |
| 52 | C2×Dic6 | — | 9 | 1 | no | — |
| 53 | Dic3⋊C4 | — | 9 | 4 | no | — |
| 54 | C4×Dic3 | — | 9 | 1 | no | — |
| 55 | C2×C3⋊D3 | C22×D3 | 1 | 1 | yes | characteristic |
| 56 | C2×C3×D3 | — | 6 | 1 | no | — |
| 57 | C2×C3×D3 | — | 6 | 1 | no | — |
| 58 | #157 | — | 3 | 4 | no | — |
| 59 | C32×C4 | — | 2 | 1 | no | — |
| 60 | C32×C4 | — | 2 | 1 | no | — |
| 61 | C32×C4 | — | 4 | 1 | no | — |
| 62 | C32×C4 | — | 4 | 1 | no | — |
| 63 | C32×C4 | — | 6 | 2 | no | — |
| 64 | C32×C4 | #67 | 1 | 1 | yes | characteristic |
| 65 | C3⋊Dic3 | — | 2 | 2 | no | — |
| 66 | C3⋊Dic3 | #67 | 1 | 1 | yes | characteristic |
| 67 | C3×Dic3 | — | 2 | 2 | no | — |
| 68 | C3×Dic3 | — | 2 | 2 | no | — |
| 69 | C3×Dic3 | — | 4 | 2 | no | — |
| 70 | C3×Dic3 | — | 6 | 1 | no | — |
| 71 | C3×Dic3 | — | 6 | 1 | no | — |
| 72 | C3×Dic3 | — | 6 | 2 | no | — |
| 73 | C3×Dic3 | — | 6 | 2 | no | — |
| 74 | C3×Dic3 | — | 12 | 2 | no | — |
| 75 | C4⋊Q8 | — | 27 | 1 | no | Sylow |
| 76 | C33 | C4⋊Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 77 | C3×Q8 | — | 6 | 1 | no | — |
| 78 | C3×Q8 | — | 6 | 1 | no | — |
| 79 | C3×Q8 | — | 12 | 1 | no | — |
| 80 | C3×Q8 | — | 18 | 1 | no | — |
| 81 | C3×Q8 | — | 18 | 1 | no | — |
| 82 | C2×C3×C4 | — | 9 | 2 | no | — |
| 83 | C2×C3×C4 | — | 9 | 1 | no | — |
| 84 | C2×Dic3 | — | 9 | 2 | no | — |
| 85 | C2×Dic3 | — | 9 | 2 | no | — |
| 86 | C4×D3 | — | 6 | 2 | no | — |
| 87 | C4×D3 | — | 6 | 1 | no | — |
| 88 | C4×D3 | — | 6 | 1 | no | — |
| 89 | C4×D3 | — | 18 | 2 | no | — |
| 90 | Dic6 | — | 2 | 1 | no | — |
| 91 | Dic6 | — | 6 | 2 | no | — |
| 92 | Dic6 | — | 6 | 1 | no | — |
| 93 | Dic6 | — | 6 | 1 | no | — |
| 94 | Dic6 | — | 12 | 1 | no | — |
| 95 | Dic6 | — | 18 | 1 | no | — |
| 96 | Dic6 | — | 18 | 2 | no | — |
| 97 | C2×C32 | — | 2 | 1 | no | — |
| 98 | C2×C32 | — | 2 | 1 | no | — |
| 99 | C2×C32 | — | 4 | 1 | no | — |
| 100 | C2×C32 | — | 4 | 1 | no | — |
| 101 | C2×C32 | C2×#67 | 1 | 1 | yes | characteristic |
| 102 | C3⋊D3 | D3×Q8 | 1 | 2 | yes | — |
| 103 | C3×D3 | — | 6 | 2 | no | — |
| 104 | C3×D3 | — | 6 | 2 | no | — |
| 105 | C2×Q8 | — | 9 | 1 | no | — |
| 106 | C2×Q8 | — | 27 | 1 | no | — |
| 107 | C4⋊C4 | — | 27 | 4 | no | — |
| 108 | C42 | — | 27 | 1 | no | — |
| 109 | C22×C3 | — | 9 | 1 | no | — |
| 110 | C2×D3 | — | 6 | 1 | no | — |
| 111 | C2×D3 | — | 6 | 1 | no | — |
| 112 | C3×C4 | — | 2 | 1 | no | — |
| 113 | C3×C4 | — | 2 | 1 | no | — |
| 114 | C3×C4 | — | 4 | 1 | no | — |
| 115 | C3×C4 | — | 4 | 1 | no | — |
| 116 | C3×C4 | — | 6 | 2 | no | — |
| 117 | C3×C4 | — | 6 | 2 | no | — |
| 118 | C3×C4 | — | 9 | 1 | no | — |
| 119 | C3×C4 | — | 12 | 2 | no | — |
| 120 | C3×C4 | — | 18 | 2 | no | — |
| 121 | C3×C4 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 122 | Dic3 | — | 2 | 2 | no | — |
| 123 | Dic3 | — | 6 | 1 | no | — |
| 124 | Dic3 | — | 6 | 1 | no | — |
| 125 | Dic3 | — | 6 | 2 | no | — |
| 126 | Dic3 | — | 6 | 2 | no | — |
| 127 | Dic3 | — | 9 | 4 | no | — |
| 128 | Dic3 | — | 12 | 2 | no | — |
| 129 | C32 | — | 2 | 1 | no | — |
| 130 | C32 | — | 2 | 1 | no | — |
| 131 | C32 | — | 4 | 1 | no | — |
| 132 | C32 | — | 4 | 1 | no | — |
| 133 | C32 | #944 | 1 | 1 | yes | characteristic, complemented |
| 134 | Q8 | — | 6 | 1 | no | — |
| 135 | Q8 | — | 18 | 1 | no | — |
| 136 | Q8 | — | 18 | 1 | no | — |
| 137 | Q8 | — | 54 | 1 | no | — |
| 138 | C2×C4 | — | 9 | 2 | no | — |
| 139 | C2×C4 | — | 9 | 1 | no | — |
| 140 | C2×C4 | — | 27 | 2 | no | — |
| 141 | C2×C4 | — | 27 | 2 | no | — |
| 142 | C2×C3 | — | 2 | 1 | no | — |
| 143 | C2×C3 | — | 2 | 1 | no | — |
| 144 | C2×C3 | — | 4 | 1 | no | — |
| 145 | C2×C3 | — | 4 | 1 | no | — |
| 146 | C2×C3 | — | 9 | 2 | no | — |
| 147 | C2×C3 | C2×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 148 | D3 | — | 6 | 2 | no | — |
| 149 | D3 | — | 6 | 2 | no | — |
| 150 | C22 | — | 9 | 1 | no | — |
| 151 | C4 | — | 6 | 2 | no | — |
| 152 | C4 | — | 9 | 1 | no | — |
| 153 | C4 | — | 18 | 2 | no | — |
| 154 | C4 | — | 27 | 4 | no | — |
| 155 | C4 | D32⋊D3 | 1 | 1 | yes | characteristic |
| 156 | C3 | — | 2 | 1 | no | — |
| 157 | C3 | — | 2 | 1 | no | — |
| 158 | C3 | — | 4 | 1 | no | — |
| 159 | C3 | — | 4 | 1 | no | — |
| 160 | C3 | #64222 | 1 | 1 | yes | characteristic, complemented |
| 161 | C2 | — | 9 | 2 | no | — |
| 162 | C2 | C2×D32⋊D3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32⋊D3 |
| commutator, prime radical, cosocle | C2×C3×C3⋊D3 | C23 |
| Fitting | C33×C4 | D4 |
| socle | C2×C33 | C2×D4 |
| hypercenter | C4 | D32⋊D3 |
| hypocenter | C33 | C4⋊Q8 |
| LCS | G ▷ C2×C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 21. C3⋊#157 | Q8 | semidirect |
| 76. C33 | C4⋊Q8 | semidirect |
| 133. C32 | #944 | semidirect |
| 160. C3 | #64222 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #89040 | 2 |
| C3×#3797 | 2 |
| C4×C3⋊#157 | 2 |
| #88934 | 2 |
| #64222 | 3 |
| #944 | 9 |