Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
621 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 621 conjugacy classes of subgroups — too many to draw. Only its 58 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 | #10589567 | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 2 | C2×C3×H(3,𝔽4) | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C23×C22⋊C9 | — | 4 | 2 | no | maximal |
| 4 | C22×Q8⋊C9 | — | 4 | 3 | no | maximal |
| 5 | C3×H(3,𝔽4) | — | 3 | 3 | no | — |
| 6 | C3×H(3,𝔽4) | — | 3 | 2 | no | — |
| 7 | C3×H(3,𝔽4) | C2×C3 | 1 | 1 | yes | characteristic |
| 8 | C2×C3×#124 | — | 3 | 3 | no | — |
| 9 | C2×C3×#120 | — | 3 | 2 | no | — |
| 10 | C22×C22⋊C9 | — | 4 | 12 | no | — |
| 11 | C22×C22⋊C9 | — | 4 | 2 | no | — |
| 12 | C2×Q8⋊C9 | — | 4 | 6 | no | — |
| 13 | C2×Q8⋊C9 | — | 4 | 3 | no | — |
| 14 | C2×H(3,𝔽4) | C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C25×C3 | A4 | 1 | 2 | yes | — |
| 16 | C22×C3×Q8 | A4 | 1 | 3 | yes | — |
| 17 | C22×C3×D4 | — | 3 | 3 | no | — |
| 18 | C3×#124 | — | 3 | 3 | no | — |
| 19 | C3×#124 | — | 3 | 12 | no | — |
| 20 | C3×#124 | — | 3 | 3 | no | — |
| 21 | C3×#124 | — | 3 | 3 | no | — |
| 22 | C3×#124 | — | 3 | 3 | no | — |
| 23 | C3×#120 | — | 3 | 6 | no | — |
| 24 | C3×#120 | — | 3 | 6 | no | — |
| 25 | C3×#120 | — | 3 | 2 | no | — |
| 26 | C3×#120 | — | 3 | 2 | no | — |
| 27 | C2×C3×C22⋊C4 | — | 3 | 6 | no | — |
| 28 | C2×C3×C42 | — | 3 | 1 | no | — |
| 29 | C23×C9 | — | 16 | 1 | no | — |
| 30 | C2×C22⋊C9 | — | 4 | 6 | no | — |
| 31 | C2×C22⋊C9 | — | 4 | 8 | no | — |
| 32 | Q8⋊C9 | — | 4 | 3 | no | — |
| 33 | H(3,𝔽4) | — | 3 | 3 | no | — |
| 34 | H(3,𝔽4) | — | 3 | 2 | no | — |
| 35 | H(3,𝔽4) | C2×C9 | 1 | 1 | yes | characteristic, complemented |
| 36 | C2×#124 | — | 3 | 3 | no | — |
| 37 | C2×#120 | — | 3 | 2 | no | — |
| 38 | C24×C3 | — | 3 | 2 | no | — |
| 39 | C24×C3 | — | 3 | 2 | no | — |
| 40 | C24×C3 | — | 4 | 12 | no | — |
| 41 | C24×C3 | C2×A4 | 1 | 2 | yes | — |
| 42 | C2×C3×Q8 | — | 3 | 3 | no | — |
| 43 | C2×C3×Q8 | — | 4 | 6 | no | — |
| 44 | C2×C3×Q8 | C2×A4 | 1 | 3 | yes | — |
| 45 | C2×C3×D4 | — | 3 | 6 | no | — |
| 46 | C2×C3×D4 | — | 3 | 3 | no | — |
| 47 | C2×C3×D4 | — | 3 | 3 | no | — |
| 48 | C2×C3×D4 | — | 12 | 6 | no | — |
| 49 | C22×C3×C4 | — | 3 | 3 | no | — |
| 50 | C3×C22⋊C4 | — | 3 | 6 | no | — |
| 51 | C3×C22⋊C4 | — | 3 | 12 | no | — |
| 52 | C3×C22⋊C4 | — | 3 | 6 | no | — |
| 53 | C3×C42 | — | 3 | 3 | no | — |
| 54 | C3×C42 | — | 3 | 1 | no | — |
| 55 | C22×C9 | — | 16 | 6 | no | — |
| 56 | C22×C9 | — | 16 | 1 | no | — |
| 57 | C22⋊C9 | — | 4 | 2 | no | — |
| 58 | C25 | C22⋊C9 | 1 | 2 | yes | complemented |
| 59 | C22×Q8 | C22⋊C9 | 1 | 3 | yes | complemented |
| 60 | C22×D4 | — | 3 | 3 | no | — |
| 61 | #124 | — | 3 | 3 | no | — |
| 62 | #124 | — | 3 | 12 | no | — |
| 63 | #124 | — | 3 | 3 | no | — |
| 64 | #124 | — | 3 | 3 | no | — |
| 65 | #124 | — | 3 | 3 | no | — |
| 66 | #120 | — | 3 | 6 | no | — |
| 67 | #120 | — | 3 | 6 | no | — |
| 68 | #120 | — | 3 | 2 | no | — |
| 69 | #120 | — | 3 | 2 | no | — |
| 70 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 71 | C2×C42 | — | 3 | 1 | no | — |
| 72 | C23×C3 | — | 3 | 2 | no | — |
| 73 | C23×C3 | — | 3 | 2 | no | — |
| 74 | C23×C3 | — | 4 | 6 | no | — |
| 75 | C23×C3 | — | 4 | 8 | no | — |
| 76 | C23×C3 | — | 6 | 12 | no | — |
| 77 | C23×C3 | — | 12 | 6 | no | — |
| 78 | C23×C3 | — | 12 | 8 | no | — |
| 79 | C23×C3 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 80 | C3×Q8 | — | 4 | 3 | no | — |
| 81 | C3×Q8 | — | 12 | 3 | no | — |
| 82 | C3×D4 | — | 12 | 3 | no | — |
| 83 | C3×D4 | — | 12 | 3 | no | — |
| 84 | C3×D4 | — | 12 | 6 | no | — |
| 85 | C2×C3×C4 | — | 3 | 3 | no | — |
| 86 | C2×C3×C4 | — | 3 | 3 | no | — |
| 87 | C2×C3×C4 | — | 6 | 6 | no | — |
| 88 | C2×C9 | — | 16 | 3 | no | — |
| 89 | C2×C9 | — | 16 | 4 | no | — |
| 90 | C24 | — | 3 | 2 | no | — |
| 91 | C24 | — | 3 | 2 | no | — |
| 92 | C24 | — | 4 | 12 | no | — |
| 93 | C24 | C2×C22⋊C9 | 1 | 2 | yes | complemented |
| 94 | C2×Q8 | — | 3 | 3 | no | — |
| 95 | C2×Q8 | — | 4 | 6 | no | — |
| 96 | C2×Q8 | C2×C22⋊C9 | 1 | 3 | yes | complemented |
| 97 | C2×D4 | — | 3 | 3 | no | — |
| 98 | C2×D4 | — | 3 | 3 | no | — |
| 99 | C2×D4 | — | 3 | 6 | no | — |
| 100 | C2×D4 | — | 12 | 6 | no | — |
| 101 | C22×C4 | — | 3 | 3 | no | — |
| 102 | C22⋊C4 | — | 3 | 6 | no | — |
| 103 | C22⋊C4 | — | 3 | 12 | no | — |
| 104 | C22⋊C4 | — | 3 | 6 | no | — |
| 105 | C42 | — | 3 | 1 | no | — |
| 106 | C42 | — | 3 | 3 | no | — |
| 107 | C22×C3 | — | 4 | 2 | no | — |
| 108 | C22×C3 | — | 6 | 6 | no | — |
| 109 | C22×C3 | — | 6 | 6 | no | — |
| 110 | C22×C3 | — | 12 | 2 | no | — |
| 111 | C22×C3 | — | 12 | 8 | no | — |
| 112 | C22×C3 | — | 12 | 8 | no | — |
| 113 | C22×C3 | #997 | 1 | 6 | yes | — |
| 114 | C22×C3 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 115 | C3×C4 | — | 6 | 3 | no | — |
| 116 | C3×C4 | — | 6 | 3 | no | — |
| 117 | C9 | — | 16 | 1 | no | Sylow |
| 118 | C23 | — | 3 | 2 | no | — |
| 119 | C23 | — | 3 | 2 | no | — |
| 120 | C23 | — | 4 | 6 | no | — |
| 121 | C23 | — | 4 | 8 | no | — |
| 122 | C23 | — | 6 | 12 | no | — |
| 123 | C23 | — | 12 | 6 | no | — |
| 124 | C23 | — | 12 | 8 | no | — |
| 125 | C23 | C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
| 126 | Q8 | — | 4 | 3 | no | — |
| 127 | Q8 | — | 12 | 3 | no | — |
| 128 | D4 | — | 12 | 3 | no | — |
| 129 | D4 | — | 12 | 3 | no | — |
| 130 | D4 | — | 12 | 6 | no | — |
| 131 | C2×C4 | — | 3 | 3 | no | — |
| 132 | C2×C4 | — | 3 | 3 | no | — |
| 133 | C2×C4 | — | 6 | 6 | no | — |
| 134 | C2×C3 | — | 12 | 2 | no | — |
| 135 | C2×C3 | — | 12 | 2 | no | — |
| 136 | C2×C3 | #5465 | 1 | 4 | yes | — |
| 137 | C2×C3 | C2×#997 | 1 | 3 | yes | — |
| 138 | C22 | — | 4 | 2 | no | — |
| 139 | C22 | — | 6 | 6 | no | — |
| 140 | C22 | — | 6 | 6 | no | — |
| 141 | C22 | — | 12 | 2 | no | — |
| 142 | C22 | — | 12 | 8 | no | — |
| 143 | C22 | — | 12 | 8 | no | — |
| 144 | C22 | #63703 | 1 | 6 | yes | — |
| 145 | C22 | C2×C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
| 146 | C4 | — | 6 | 3 | no | — |
| 147 | C4 | — | 6 | 3 | no | — |
| 148 | C3 | C2×#5465 | 1 | 1 | yes | characteristic |
| 149 | C2 | — | 12 | 2 | no | — |
| 150 | C2 | — | 12 | 2 | no | — |
| 151 | C2 | #10589567 | 1 | 4 | yes | direct product factor |
| 152 | C2 | C2×#63703 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C3 | C22⋊A4 |
| commutator, hypocenter | H(3,𝔽4) | C2×C9 |
| Frattini | C22×C3 | C2×C22⋊A4 |
| Fitting, prime radical | C2×C3×H(3,𝔽4) | C3 |
| cosocle | C3×H(3,𝔽4) | C2×C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10589567 | C2 | direct (N × H) |
| 14. C2×H(3,𝔽4) | C9 | semidirect |
| 35. H(3,𝔽4) | C2×C9 | semidirect |
| 58. C25 | C22⋊C9 | semidirect |
| 59. C22×Q8 | C22⋊C9 | semidirect |
| 93. C24 | C2×C22⋊C9 | semidirect |
| 96. C2×Q8 | C2×C22⋊C9 | semidirect |
| 151. C2 | #10589567 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10589567 | 2 |
| C2×C3×H(3,𝔽4) | 3 |
| C23×C22⋊C9 | 4 |
| C22×Q8⋊C9 | 4 |