Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
172 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 18 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 | #5950 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | B(4,𝔽2)⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #3027 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | #997 | C22 | 1 | 1 | yes | characteristic |
| 5 | D4×A4 | — | 4 | 1 | no | maximal |
| 6 | #585 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | #535 | — | 3 | 1 | no | — |
| 8 | B(4,𝔽2) | — | 3 | 2 | no | — |
| 9 | B(4,𝔽2) | C2×C3 | 1 | 2 | yes | complemented |
| 10 | #410 | — | 3 | 1 | no | — |
| 11 | C22×A4 | — | 4 | 2 | no | — |
| 12 | #41⋊C3 | — | 8 | 1 | no | — |
| 13 | C4×A4 | — | 4 | 1 | no | — |
| 14 | C4×A4 | — | 8 | 1 | no | — |
| 15 | 2−1+4 | — | 6 | 1 | no | — |
| 16 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 17 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 18 | H(5,𝔽2) | C22×C3 | 1 | 1 | yes | characteristic |
| 19 | C2×#41 | — | 3 | 1 | no | — |
| 20 | C2×#41 | — | 6 | 1 | no | — |
| 21 | C2×#41 | — | 6 | 1 | no | — |
| 22 | C22×D4 | A4 | 1 | 1 | yes | characteristic, complemented |
| 23 | #127 | — | 6 | 1 | no | — |
| 24 | #124 | — | 6 | 1 | no | — |
| 25 | #121 | — | 6 | 1 | no | — |
| 26 | #121 | — | 6 | 1 | no | — |
| 27 | #120 | — | 3 | 2 | no | — |
| 28 | #120 | — | 3 | 2 | no | — |
| 29 | #117 | — | 3 | 1 | no | — |
| 30 | #99 | — | 3 | 2 | no | — |
| 31 | #99 | — | 3 | 2 | no | — |
| 32 | C2×A4 | — | 4 | 1 | no | — |
| 33 | C2×A4 | — | 8 | 1 | no | — |
| 34 | C2×A4 | — | 8 | 2 | no | — |
| 35 | C3×D4 | — | 16 | 1 | no | — |
| 36 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 37 | C24 | C2×A4 | 1 | 2 | yes | — |
| 38 | #41 | — | 2 | 1 | no | — |
| 39 | #41 | — | 6 | 2 | no | — |
| 40 | #41 | — | 6 | 2 | no | — |
| 41 | #41 | — | 6 | 1 | no | — |
| 42 | #41 | — | 6 | 2 | no | — |
| 43 | #41 | — | 6 | 1 | no | — |
| 44 | #41 | — | 6 | 2 | no | — |
| 45 | #41 | — | 6 | 2 | no | — |
| 46 | #41 | — | 6 | 1 | no | — |
| 47 | C2×Q8 | — | 3 | 1 | no | — |
| 48 | C2×Q8 | — | 6 | 1 | no | — |
| 49 | C2×Q8 | — | 6 | 1 | no | — |
| 50 | C2×D4 | — | 3 | 1 | no | — |
| 51 | C2×D4 | — | 3 | 1 | no | — |
| 52 | C2×D4 | — | 3 | 1 | no | — |
| 53 | C2×D4 | — | 6 | 1 | no | — |
| 54 | C2×D4 | — | 6 | 1 | no | — |
| 55 | C2×D4 | — | 6 | 1 | no | — |
| 56 | C2×D4 | — | 6 | 2 | no | — |
| 57 | C2×D4 | — | 6 | 1 | no | — |
| 58 | C2×D4 | — | 6 | 1 | no | — |
| 59 | C2×D4 | — | 6 | 1 | no | — |
| 60 | C2×D4 | — | 6 | 2 | no | — |
| 61 | C2×D4 | — | 6 | 2 | no | — |
| 62 | C22×C4 | — | 2 | 1 | no | — |
| 63 | C22×C4 | — | 3 | 1 | no | — |
| 64 | C22×C4 | — | 3 | 1 | no | — |
| 65 | C22×C4 | — | 6 | 1 | no | — |
| 66 | C22×C4 | C2×A4 | 1 | 1 | yes | characteristic |
| 67 | C4⋊C4 | — | 6 | 1 | no | — |
| 68 | C22⋊C4 | — | 3 | 2 | no | — |
| 69 | C22⋊C4 | — | 6 | 2 | no | — |
| 70 | C22⋊C4 | — | 6 | 2 | no | — |
| 71 | C42 | — | 6 | 1 | no | — |
| 72 | C22×C3 | — | 16 | 2 | no | — |
| 73 | A4 | — | 4 | 1 | no | — |
| 74 | A4 | — | 8 | 1 | no | — |
| 75 | C3×C4 | — | 16 | 1 | no | — |
| 76 | C23 | — | 2 | 1 | no | — |
| 77 | C23 | — | 3 | 1 | no | — |
| 78 | C23 | — | 3 | 1 | no | — |
| 79 | C23 | — | 6 | 2 | no | — |
| 80 | C23 | — | 6 | 1 | no | — |
| 81 | C23 | — | 8 | 2 | no | — |
| 82 | C23 | C22×A4 | 1 | 1 | yes | characteristic |
| 83 | Q8 | — | 2 | 1 | no | — |
| 84 | Q8 | — | 6 | 2 | no | — |
| 85 | Q8 | — | 6 | 1 | no | — |
| 86 | D4 | — | 4 | 1 | no | — |
| 87 | D4 | — | 6 | 2 | no | — |
| 88 | D4 | — | 6 | 2 | no | — |
| 89 | D4 | — | 6 | 2 | no | — |
| 90 | D4 | — | 6 | 1 | no | — |
| 91 | D4 | — | 6 | 1 | no | — |
| 92 | D4 | — | 6 | 2 | no | — |
| 93 | D4 | — | 12 | 1 | no | — |
| 94 | D4 | — | 24 | 2 | no | — |
| 95 | D4 | — | 24 | 2 | no | — |
| 96 | C2×C4 | — | 3 | 1 | no | — |
| 97 | C2×C4 | — | 3 | 1 | no | — |
| 98 | C2×C4 | — | 3 | 1 | no | — |
| 99 | C2×C4 | — | 3 | 1 | no | — |
| 100 | C2×C4 | — | 6 | 1 | no | — |
| 101 | C2×C4 | — | 6 | 1 | no | — |
| 102 | C2×C4 | — | 6 | 2 | no | — |
| 103 | C2×C4 | — | 6 | 1 | no | — |
| 104 | C2×C4 | — | 6 | 1 | no | — |
| 105 | C2×C4 | — | 6 | 1 | no | — |
| 106 | C2×C4 | — | 6 | 1 | no | — |
| 107 | C2×C4 | — | 6 | 2 | no | — |
| 108 | C2×C3 | — | 16 | 1 | no | — |
| 109 | C2×C3 | — | 32 | 2 | no | — |
| 110 | C22 | — | 3 | 1 | no | — |
| 111 | C22 | — | 4 | 1 | no | — |
| 112 | C22 | — | 4 | 2 | no | — |
| 113 | C22 | — | 6 | 1 | no | — |
| 114 | C22 | — | 6 | 1 | no | — |
| 115 | C22 | — | 8 | 1 | no | — |
| 116 | C22 | — | 12 | 2 | no | — |
| 117 | C22 | — | 12 | 2 | no | — |
| 118 | C22 | — | 12 | 1 | no | — |
| 119 | C22 | — | 12 | 1 | no | — |
| 120 | C22 | — | 24 | 1 | no | — |
| 121 | C4 | — | 3 | 1 | no | — |
| 122 | C4 | — | 6 | 1 | no | — |
| 123 | C4 | — | 6 | 1 | no | — |
| 124 | C4 | — | 12 | 2 | no | — |
| 125 | C4 | #120⋊C3 | 1 | 1 | yes | characteristic |
| 126 | C3 | — | 16 | 1 | no | Sylow |
| 127 | C2 | — | 6 | 1 | no | — |
| 128 | C2 | — | 8 | 2 | no | — |
| 129 | C2 | — | 12 | 1 | no | — |
| 130 | C2 | — | 12 | 1 | no | — |
| 131 | C2 | C2×#120⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | #3027 | C3 |
| hypercenter | C4 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5950 | C2 | semidirect |
| 2. B(4,𝔽2)⋊C3 | C2 | semidirect |
| 3. #3027 | C3 | semidirect |
| 6. #585 | C2×C3 | semidirect |
| 9. B(4,𝔽2) | C2×C3 | semidirect |
| 22. C22×D4 | A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5950 | 2 |
| B(4,𝔽2)⋊C3 | 2 |
| #3027 | 3 |
| D4×A4 | 4 |