Subgroups — C₂×D₄×3₋¹⁺² #88758

Order 432.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×C₃²
commutator, Frattini, cosocle	C₂×C₃	C₂³×C₃²
prime radical	C₂×C₃²×D₄	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
1. C₂³×3₋¹⁺²	C₂	semidirect
2. D₄×3₋¹⁺²	C₂	direct (N × H)
3. C₂×C₄×3₋¹⁺²	C₂	semidirect
5. C₂×D₄×C₉	C₃	semidirect
8. C₂²×3₋¹⁺²	C₂²	semidirect
9. C₄×3₋¹⁺²	C₂²	semidirect
13. C₂³×C₉	C₂×C₃	semidirect
14. D₄×C₉	C₂×C₃	semidirect
15. C₂×C₄×C₉	C₂×C₃	semidirect
17. C₂×3₋¹⁺²	D₄	direct (N × H)
27. C₂²×C₉	C₂²×C₃	semidirect
28. C₄×C₉	C₂²×C₃	semidirect
29. 3₋¹⁺²	C₂×D₄	direct (N × H)
40. C₂×C₉	C₃×D₄	semidirect
42. C₂×D₄	3₋¹⁺²	direct (N × H)
52. C₉	C₂×C₃×D₄	semidirect
53. C₂³	C₂×3₋¹⁺²	semidirect
54. D₄	C₂×3₋¹⁺²	direct (N × H)
55. C₂×C₄	C₂×3₋¹⁺²	semidirect
64. C₂²	C₂²×3₋¹⁺²	semidirect
65. C₄	C₂²×3₋¹⁺²	semidirect
69. C₂	D₄×3₋¹⁺²	direct (N × H)

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
D₄×3₋¹⁺²	2
C₂×C₄×3₋¹⁺²	2
C₂×C₃²×D₄	3
C₂×D₄×C₉	3

Subgroups — C2×D4×3−1+2 #88758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×3₋¹⁺² #88758