Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
414 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 85 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 | #64320 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #64316 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #64289 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #64227 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #64226 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #64224 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #64223 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #3832 | C22 | 1 | 1 | yes | characteristic |
| 9 | C3×#239 | — | 2 | 1 | no | — |
| 10 | C3×D3×Q8 | — | 2 | 1 | no | — |
| 11 | #3802 | C22 | 1 | 3 | yes | — |
| 12 | D3×D12 | — | 2 | 3 | no | — |
| 13 | #3799 | — | 2 | 3 | no | — |
| 14 | #3798 | C22 | 1 | 3 | yes | — |
| 15 | #3798 | C22 | 1 | 3 | yes | complemented |
| 16 | #3797 | C22 | 1 | 3 | yes | complemented |
| 17 | #3788 | C22 | 1 | 3 | yes | complemented |
| 18 | #3787 | C22 | 1 | 3 | yes | complemented |
| 19 | #3776 | C22 | 1 | 3 | yes | complemented |
| 20 | #3775 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | #3775 | C22 | 1 | 3 | yes | complemented |
| 22 | #3775 | C22 | 1 | 6 | yes | complemented |
| 23 | #3774 | C22 | 1 | 3 | yes | complemented |
| 24 | #1009 | — | 6 | 1 | no | — |
| 25 | #1007 | — | 6 | 1 | no | — |
| 26 | C2×D32 | — | 2 | 3 | no | — |
| 27 | C32×Q8 | D4 | 1 | 1 | yes | characteristic |
| 28 | C4⋊(C3⋊D3) | D4 | 1 | 3 | yes | — |
| 29 | C4×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 30 | C3×D12 | — | 2 | 3 | no | — |
| 31 | C3×C4×D3 | — | 2 | 3 | no | — |
| 32 | C3×C4×D3 | — | 2 | 3 | no | — |
| 33 | C3×Dic6 | — | 2 | 3 | no | — |
| 34 | #627 | C23 | 1 | 3 | yes | — |
| 35 | #627 | C23 | 1 | 1 | yes | characteristic |
| 36 | #626 | — | 2 | 3 | no | — |
| 37 | #626 | — | 2 | 3 | no | — |
| 38 | #625 | C23 | 1 | 1 | yes | characteristic |
| 39 | #625 | C23 | 1 | 3 | yes | — |
| 40 | #624 | — | 2 | 3 | no | — |
| 41 | #622 | C23 | 1 | 3 | yes | — |
| 42 | #622 | C23 | 1 | 1 | yes | characteristic |
| 43 | #577 | — | 9 | 1 | no | maximal, Sylow |
| 44 | C3×#41 | — | 6 | 1 | no | — |
| 45 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 46 | #239 | — | 2 | 1 | no | — |
| 47 | #239 | — | 6 | 1 | no | — |
| 48 | #239 | — | 6 | 1 | no | — |
| 49 | #239 | — | 6 | 3 | no | — |
| 50 | D3×Q8 | — | 2 | 1 | no | — |
| 51 | D3×Q8 | — | 6 | 3 | no | — |
| 52 | D3×D4 | — | 6 | 3 | no | — |
| 53 | D3×D4 | — | 6 | 3 | no | — |
| 54 | #235 | — | 6 | 3 | no | — |
| 55 | #235 | — | 6 | 3 | no | — |
| 56 | #235 | — | 6 | 3 | no | — |
| 57 | C2×D12 | — | 6 | 3 | no | — |
| 58 | C2×C3⋊D3 | C2×D4 | 1 | 3 | yes | — |
| 59 | C2×C3×D3 | — | 2 | 1 | no | — |
| 60 | C2×C3×D3 | — | 2 | 3 | no | — |
| 61 | D32 | — | 8 | 3 | no | — |
| 62 | C32×C4 | C2×D4 | 1 | 3 | yes | — |
| 63 | C3⋊Dic3 | C24 | 1 | 1 | yes | characteristic |
| 64 | C3×Dic3 | — | 2 | 3 | no | — |
| 65 | C3×Dic3 | — | 2 | 1 | no | — |
| 66 | 2−1+4 | — | 9 | 1 | no | — |
| 67 | H(5,𝔽2) | — | 9 | 1 | no | — |
| 68 | #137 | — | 9 | 3 | no | — |
| 69 | AGL(1,ℤ/8) | — | 9 | 3 | no | — |
| 70 | #135 | — | 9 | 3 | no | — |
| 71 | C2×SD16 | — | 9 | 3 | no | — |
| 72 | #131 | — | 9 | 1 | no | — |
| 73 | C22×D3 | — | 6 | 6 | no | — |
| 74 | C3×Q8 | — | 2 | 1 | no | — |
| 75 | C3×Q8 | — | 2 | 1 | no | — |
| 76 | C3×Q8 | — | 6 | 3 | no | — |
| 77 | C3×D4 | — | 6 | 3 | no | — |
| 78 | C2×C3×C4 | — | 6 | 3 | no | — |
| 79 | C2×C3×C4 | — | 6 | 3 | no | — |
| 80 | #67 | — | 6 | 1 | no | — |
| 81 | #67 | — | 6 | 3 | no | — |
| 82 | #67 | — | 6 | 3 | no | — |
| 83 | #67 | — | 6 | 3 | no | — |
| 84 | D12 | — | 2 | 3 | no | — |
| 85 | D12 | — | 6 | 3 | no | — |
| 86 | D12 | — | 6 | 3 | no | — |
| 87 | D12 | — | 6 | 3 | no | — |
| 88 | D12 | — | 6 | 3 | no | — |
| 89 | C4×D3 | — | 2 | 3 | no | — |
| 90 | C4×D3 | — | 2 | 3 | no | — |
| 91 | C4×D3 | — | 6 | 3 | no | — |
| 92 | C4×D3 | — | 6 | 3 | no | — |
| 93 | C4×D3 | — | 6 | 6 | no | — |
| 94 | Dic6 | — | 2 | 3 | no | — |
| 95 | Dic6 | — | 6 | 1 | no | — |
| 96 | Dic6 | — | 6 | 3 | no | — |
| 97 | C2×C32 | C22×D4 | 1 | 1 | yes | characteristic |
| 98 | C3⋊D3 | — | 2 | 3 | no | — |
| 99 | C3×D3 | — | 4 | 3 | no | — |
| 100 | C3×D3 | — | 4 | 1 | no | — |
| 101 | #41 | — | 6 | 1 | no | — |
| 102 | #41 | — | 9 | 3 | no | — |
| 103 | #41 | — | 9 | 3 | no | — |
| 104 | #41 | — | 9 | 1 | no | — |
| 105 | #41 | — | 18 | 3 | no | — |
| 106 | C2×Q8 | — | 6 | 1 | no | — |
| 107 | C2×Q8 | — | 9 | 3 | no | — |
| 108 | C2×D4 | — | 9 | 3 | no | — |
| 109 | C2×D4 | — | 18 | 3 | no | — |
| 110 | Q16 | — | 9 | 3 | no | — |
| 111 | SD16 | — | 9 | 3 | no | — |
| 112 | SD16 | — | 9 | 6 | no | — |
| 113 | SD16 | — | 9 | 1 | no | — |
| 114 | D8 | — | 9 | 3 | no | — |
| 115 | M16 | — | 9 | 3 | no | — |
| 116 | C2×C8 | — | 9 | 3 | no | — |
| 117 | C22×C3 | — | 6 | 1 | no | — |
| 118 | C22×C3 | — | 6 | 3 | no | — |
| 119 | C2×D3 | — | 2 | 1 | no | — |
| 120 | C2×D3 | — | 2 | 3 | no | — |
| 121 | C2×D3 | — | 6 | 3 | no | — |
| 122 | C2×D3 | — | 6 | 3 | no | — |
| 123 | C2×D3 | — | 24 | 6 | no | — |
| 124 | C3×C4 | — | 2 | 3 | no | — |
| 125 | C3×C4 | — | 2 | 3 | no | — |
| 126 | C3×C4 | — | 6 | 3 | no | — |
| 127 | C3×C4 | — | 6 | 1 | no | — |
| 128 | Dic3 | — | 2 | 3 | no | — |
| 129 | Dic3 | — | 2 | 1 | no | — |
| 130 | Dic3 | — | 6 | 1 | no | — |
| 131 | Dic3 | — | 6 | 1 | no | — |
| 132 | C32 | #577 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 133 | C23 | — | 18 | 3 | no | — |
| 134 | Q8 | — | 6 | 3 | no | — |
| 135 | Q8 | — | 9 | 3 | no | — |
| 136 | Q8 | — | 9 | 1 | no | — |
| 137 | Q8 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 138 | D4 | — | 6 | 3 | no | — |
| 139 | D4 | — | 9 | 1 | no | — |
| 140 | D4 | — | 9 | 3 | no | — |
| 141 | D4 | — | 9 | 3 | no | — |
| 142 | D4 | — | 18 | 3 | no | — |
| 143 | D4 | — | 18 | 3 | no | — |
| 144 | C2×C4 | — | 6 | 3 | no | — |
| 145 | C2×C4 | — | 6 | 3 | no | — |
| 146 | C2×C4 | — | 9 | 3 | no | — |
| 147 | C2×C4 | — | 18 | 3 | no | — |
| 148 | C8 | — | 9 | 1 | no | — |
| 149 | C8 | — | 9 | 3 | no | — |
| 150 | C2×C3 | — | 2 | 1 | no | — |
| 151 | C2×C3 | — | 2 | 1 | no | — |
| 152 | C2×C3 | — | 12 | 3 | no | — |
| 153 | C2×C3 | — | 12 | 1 | no | — |
| 154 | D3 | — | 4 | 3 | no | — |
| 155 | D3 | — | 4 | 1 | no | — |
| 156 | D3 | — | 12 | 3 | no | — |
| 157 | D3 | — | 12 | 3 | no | — |
| 158 | C22 | — | 6 | 1 | no | — |
| 159 | C22 | — | 6 | 3 | no | — |
| 160 | C22 | — | 9 | 3 | no | — |
| 161 | C22 | — | 72 | 3 | no | — |
| 162 | C4 | — | 6 | 1 | no | — |
| 163 | C4 | — | 6 | 3 | no | — |
| 164 | C4 | — | 9 | 1 | no | — |
| 165 | C4 | C2×GO+(4,𝔽2) | 1 | 3 | yes | — |
| 166 | C3 | — | 2 | 1 | no | — |
| 167 | C3 | — | 2 | 1 | no | — |
| 168 | C2 | — | 12 | 3 | no | — |
| 169 | C2 | — | 12 | 1 | no | — |
| 170 | C2 | — | 18 | 3 | no | — |
| 171 | C2 | C22×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C3⋊Dic3 | C24 |
| Fitting | C32×Q8 | D4 |
| socle | C2×C32 | C22×D4 |
| hypercenter | Q8 | GO+(4,𝔽2) |
| hypocenter | C32 | #577 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #64320 | C2 | semidirect |
| 2. #64316 | C2 | semidirect |
| 3. #64289 | C2 | semidirect |
| 4. #64227 | C2 | semidirect |
| 5. #64226 | C2 | semidirect |
| 6. #64224 | C2 | semidirect |
| 7. #64223 | C2 | semidirect |
| 15. #3798 | C22 | semidirect |
| 16. #3797 | C22 | semidirect |
| 17. #3788 | C22 | semidirect |
| 18. #3787 | C22 | semidirect |
| 19. #3776 | C22 | semidirect |
| 20. #3775 | C22 | semidirect |
| 21. #3775 | C22 | semidirect |
| 22. #3775 | C22 | semidirect |
| 23. #3774 | C22 | semidirect |
| 132. C32 | #577 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64320 | 2 |
| #64316 | 2 |
| #64289 | 2 |
| #64227 | 2 |
| #64226 | 2 |
| #64224 | 2 |
| #64223 | 2 |
| #577 | 9 |