Subgroups — C₂²⋊C₄×3₋¹⁺² #88558

Order 432.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
3	C₃²×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic
4	C₉×C₂²⋊C₄	C₃	1	3	yes	maximal, complemented
5	C₂²×3₋¹⁺²	—	2	2	no	—
6	C₂²×3₋¹⁺²	C₄	1	2	yes	complemented
7	C₂²×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₄×3₋¹⁺²	—	2	2	no	—
9	C₂³×C₃²	C₂×C₃	1	1	yes	characteristic
10	C₂×C₃²×C₄	C₂×C₃	1	2	yes	—
11	C₂³×C₉	C₂×C₃	1	3	yes	—
12	C₂×C₄×C₉	C₂×C₃	1	6	yes	complemented
13	C₂×3₋¹⁺²	—	2	2	no	—
14	C₂×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
15	C₂×3₋¹⁺²	D₄	1	2	yes	—
16	C₃×C₂²⋊C₄	—	3	1	no	—
17	C₃×C₂²⋊C₄	C₃²	1	1	yes	characteristic
18	C₂²×C₃²	—	2	2	no	—
19	C₂²×C₃²	C₃×C₄	1	2	yes	—
20	C₂²×C₃²	C₂²×C₃	1	1	yes	characteristic
21	C₃²×C₄	—	2	2	no	—
22	C₂²×C₉	—	2	6	no	—
23	C₂²×C₉	C₃×C₄	1	6	yes	complemented
24	C₂²×C₉	C₂²×C₃	1	3	yes	—
25	C₄×C₉	—	2	6	no	—
26	3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
27	C₂³×C₃	—	3	1	no	—
28	C₂³×C₃	C₂×C₃²	1	1	yes	characteristic
29	C₂×C₃×C₄	—	3	2	no	—
30	C₂×C₃×C₄	C₂×C₃²	1	2	yes	—
31	C₂×C₃²	—	2	2	no	—
32	C₂×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
33	C₂×C₃²	C₃×D₄	1	2	yes	—
34	C₂×C₉	—	2	6	no	—
35	C₂×C₉	C₂×C₃×C₄	1	3	yes	—
36	C₂×C₉	C₃×D₄	1	6	yes	—
37	C₂²⋊C₄	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
38	C₂²×C₃	—	2	2	no	—
39	C₂²×C₃	—	3	2	no	—
40	C₂²×C₃	—	3	1	no	—
41	C₂²×C₃	—	6	2	no	—
42	C₂²×C₃	C₃²×C₄	1	2	yes	—
43	C₂²×C₃	C₂²×C₃²	1	1	yes	characteristic
44	C₃×C₄	—	2	2	no	—
45	C₃×C₄	—	6	2	no	—
46	C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
47	C₉	C₃×C₂²⋊C₄	1	3	yes	complemented
48	C₂³	C₂×3₋¹⁺²	1	1	yes	characteristic
49	C₂×C₄	C₂×3₋¹⁺²	1	2	yes	complemented
50	C₂×C₃	—	2	2	no	—
51	C₂×C₃	—	3	1	no	—
52	C₂×C₃	—	3	2	no	—
53	C₂×C₃	—	6	2	no	—
54	C₂×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
55	C₂×C₃	C₃²×D₄	1	2	yes	—
56	C₂²	—	2	2	no	—
57	C₂²	C₄×3₋¹⁺²	1	2	yes	complemented
58	C₂²	C₂²×3₋¹⁺²	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₃	—	3	1	no	—
61	C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
62	C₂	—	2	2	no	—
63	C₂	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
64	C₂	D₄×3₋¹⁺²	1	2	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²×C₃	C₂²×C₃²
commutator	C₂×C₃	C₂×C₃²×C₄
prime radical	C₂³×C₃²	C₂×C₃
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄×3₋¹⁺²	C₂	semidirect
4. C₉×C₂²⋊C₄	C₃	semidirect
6. C₂²×3₋¹⁺²	C₄	semidirect
12. C₂×C₄×C₉	C₂×C₃	semidirect
23. C₂²×C₉	C₃×C₄	semidirect
26. 3₋¹⁺²	C₂²⋊C₄	direct (N × H)
37. C₂²⋊C₄	3₋¹⁺²	direct (N × H)
47. C₉	C₃×C₂²⋊C₄	semidirect
49. C₂×C₄	C₂×3₋¹⁺²	semidirect
57. C₂²	C₄×3₋¹⁺²	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×3₋¹⁺²	2
C₂×C₄×3₋¹⁺²	2
C₃²×C₂²⋊C₄	3
C₉×C₂²⋊C₄	3

Subgroups — C22⋊C4×3−1+2 #88558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₄×3₋¹⁺² #88558