Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
191 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 30 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×Q8×A4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #5409 | C2 | 1 | 2 | yes | maximal |
| 3 | #1784 | — | 3 | 1 | no | maximal, Sylow |
| 4 | Q8×A4 | — | 2 | 2 | no | — |
| 5 | C2×C4×A4 | C4 | 1 | 2 | yes | — |
| 6 | C2×C4×A4 | C22 | 1 | 1 | yes | characteristic |
| 7 | A4⋊C8 | — | 2 | 2 | no | — |
| 8 | #836 | — | 4 | 1 | no | maximal |
| 9 | C23×Q8 | D3 | 1 | 1 | yes | characteristic |
| 10 | C2×#101 | — | 3 | 2 | no | — |
| 11 | C2×#101 | — | 3 | 1 | no | — |
| 12 | C2×#101 | — | 3 | 1 | no | — |
| 13 | #407 | — | 3 | 2 | no | — |
| 14 | C22×A4 | C2×C4 | 1 | 1 | yes | characteristic |
| 15 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 16 | C4×A4 | — | 2 | 2 | no | — |
| 17 | C4×A4 | D4 | 1 | 2 | yes | — |
| 18 | #208 | — | 4 | 2 | no | — |
| 19 | C22×Q8 | — | 2 | 2 | no | — |
| 20 | C22×Q8 | — | 3 | 1 | no | — |
| 21 | C22×Q8 | — | 3 | 1 | no | — |
| 22 | C22×Q8 | — | 3 | 2 | no | — |
| 23 | C22×Q8 | — | 6 | 2 | no | — |
| 24 | C23×C4 | Dic3 | 1 | 2 | yes | — |
| 25 | C23×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 26 | C2×M16 | — | 3 | 2 | no | — |
| 27 | C2×M16 | — | 3 | 2 | no | — |
| 28 | #101 | — | 6 | 4 | no | — |
| 29 | #101 | — | 6 | 2 | no | — |
| 30 | #101 | — | 6 | 1 | no | — |
| 31 | #101 | — | 6 | 1 | no | — |
| 32 | C22⋊C8 | — | 6 | 2 | no | — |
| 33 | C22⋊C8 | — | 6 | 2 | no | — |
| 34 | C2×A4 | — | 2 | 1 | no | — |
| 35 | C2×A4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 36 | C3×Q8 | — | 8 | 2 | no | — |
| 37 | C2×C3×C4 | — | 4 | 2 | no | — |
| 38 | C2×C3×C4 | — | 4 | 1 | no | — |
| 39 | C3⋊C8 | — | 8 | 2 | no | — |
| 40 | C24 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 41 | C2×Q8 | — | 3 | 4 | no | — |
| 42 | C2×Q8 | — | 3 | 2 | no | — |
| 43 | C2×Q8 | — | 3 | 1 | no | — |
| 44 | C2×Q8 | — | 6 | 1 | no | — |
| 45 | C2×Q8 | — | 6 | 2 | no | — |
| 46 | C2×Q8 | — | 6 | 2 | no | — |
| 47 | C2×Q8 | — | 6 | 2 | no | — |
| 48 | C2×Q8 | — | 6 | 4 | no | — |
| 49 | C2×Q8 | — | 6 | 4 | no | — |
| 50 | C2×Q8 | S4 | 1 | 1 | yes | characteristic |
| 51 | C22×C4 | — | 2 | 2 | no | — |
| 52 | C22×C4 | — | 3 | 1 | no | — |
| 53 | C22×C4 | — | 3 | 2 | no | — |
| 54 | C22×C4 | — | 3 | 2 | no | — |
| 55 | C22×C4 | — | 3 | 1 | no | — |
| 56 | C22×C4 | — | 3 | 2 | no | — |
| 57 | C22×C4 | — | 6 | 2 | no | — |
| 58 | C22×C4 | #67 | 1 | 2 | yes | — |
| 59 | M16 | — | 6 | 2 | no | — |
| 60 | M16 | — | 6 | 2 | no | — |
| 61 | M16 | — | 6 | 2 | no | — |
| 62 | M16 | — | 6 | 2 | no | — |
| 63 | C2×C8 | — | 6 | 2 | no | — |
| 64 | C2×C8 | — | 6 | 2 | no | — |
| 65 | C22×C3 | — | 4 | 1 | no | — |
| 66 | A4 | #101 | 1 | 1 | yes | characteristic, complemented |
| 67 | C3×C4 | — | 4 | 2 | no | — |
| 68 | C3×C4 | — | 8 | 2 | no | — |
| 69 | C23 | — | 2 | 1 | no | — |
| 70 | C23 | — | 3 | 1 | no | — |
| 71 | C23 | — | 3 | 1 | no | — |
| 72 | C23 | — | 6 | 1 | no | — |
| 73 | C23 | #217 | 1 | 1 | yes | characteristic |
| 74 | Q8 | — | 2 | 2 | no | — |
| 75 | Q8 | — | 6 | 4 | no | — |
| 76 | Q8 | — | 6 | 4 | no | — |
| 77 | Q8 | — | 6 | 2 | no | — |
| 78 | C2×C4 | — | 3 | 1 | no | — |
| 79 | C2×C4 | — | 3 | 2 | no | — |
| 80 | C2×C4 | — | 3 | 2 | no | — |
| 81 | C2×C4 | — | 3 | 2 | no | — |
| 82 | C2×C4 | — | 3 | 4 | no | — |
| 83 | C2×C4 | — | 3 | 2 | no | — |
| 84 | C2×C4 | — | 6 | 2 | no | — |
| 85 | C2×C4 | — | 6 | 2 | no | — |
| 86 | C2×C4 | — | 6 | 1 | no | — |
| 87 | C2×C4 | — | 6 | 2 | no | — |
| 88 | C2×C4 | SL(2,ℤ/4) | 1 | 2 | yes | — |
| 89 | C2×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 90 | C8 | — | 12 | 2 | no | — |
| 91 | C8 | — | 12 | 2 | no | — |
| 92 | C2×C3 | — | 4 | 1 | no | — |
| 93 | C2×C3 | — | 8 | 1 | no | — |
| 94 | C22 | — | 3 | 1 | no | — |
| 95 | C22 | — | 3 | 1 | no | — |
| 96 | C22 | — | 3 | 2 | no | — |
| 97 | C22 | — | 3 | 1 | no | — |
| 98 | C22 | — | 6 | 1 | no | — |
| 99 | C22 | — | 6 | 1 | no | — |
| 100 | C22 | — | 6 | 1 | no | — |
| 101 | C22 | #836 | 1 | 1 | yes | characteristic, complemented |
| 102 | C22 | C2×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 103 | C4 | — | 2 | 2 | no | — |
| 104 | C4 | — | 3 | 2 | no | — |
| 105 | C4 | — | 6 | 2 | no | — |
| 106 | C4 | GL(2,ℤ/4) | 1 | 2 | yes | — |
| 107 | C3 | — | 4 | 1 | no | Sylow |
| 108 | C2 | — | 2 | 1 | no | — |
| 109 | C2 | — | 3 | 1 | no | — |
| 110 | C2 | — | 3 | 1 | no | — |
| 111 | C2 | — | 6 | 1 | no | — |
| 112 | C2 | #5432 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 66. A4 | #101 | semidirect |
| 101. C22 | #836 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×A4 | 2 |
| #5409 | 2 |
| #1784 | 3 |
| #836 | 4 |