Subgroups — D₄×#272 #88743

Order 432.

297 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#272	C₂	1	2	yes	maximal, complemented
2	D₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#6235	C₂	1	2	yes	maximal, complemented
4	C₄⋊#272	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#272	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₃×D₄	—	3	4	no	maximal
7	C₂²×H(3,𝔽₃)	C₂²	1	2	yes	complemented
8	C₂×#272	—	2	4	no	—
9	C₂×#272	C₂²	1	1	yes	characteristic
10	C₂×#272	C₂²	1	2	yes	complemented
11	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
12	#1084	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₃×D₃	—	3	8	no	—
14	C₃²×D₄	D₃	1	4	yes	complemented
15	C₃×#67	—	3	8	no	—
16	C₃×D₁₂	—	3	4	no	—
17	C₃×C₄×D₃	—	3	4	no	—
18	C₂×H(3,𝔽₃)	—	2	2	no	—
19	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
20	#272	—	2	2	no	—
21	#272	D₄	1	2	yes	direct product factor
22	C₂×C₃×D₄	—	9	1	no	—
23	D₃×D₄	—	3	4	no	—
24	C₂²×C₃²	C₂×D₃	1	8	yes	complemented
25	C₂×C₃×D₃	—	3	4	no	—
26	C₂×C₃×D₃	—	3	8	no	—
27	C₂×C₃×D₃	—	6	16	no	—
28	C₃²×C₄	C₂×D₃	1	4	yes	complemented
29	C₃×Dic₃	—	3	4	no	—
30	H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
31	C₂³×C₃	—	9	2	no	—
32	C₂²×D₃	—	3	8	no	—
33	C₃×D₄	—	3	4	no	—
34	C₃×D₄	—	9	2	no	—
35	C₃×D₄	—	9	1	no	—
36	C₃×D₄	C₃⋊D₃	1	1	yes	characteristic
37	C₂×C₃×C₄	—	9	1	no	—
38	#67	—	3	8	no	—
39	D₁₂	—	3	4	no	—
40	C₄×D₃	—	3	4	no	—
41	C₂×C₃²	—	2	8	no	—
42	C₂×C₃²	C₂²×D₃	1	4	yes	—
43	C₃×D₃	—	3	8	no	—
44	C₃×D₃	—	6	8	no	—
45	C₂×D₄	—	9	1	no	Sylow
46	C₂²×C₃	—	3	8	no	—
47	C₂²×C₃	—	9	1	no	—
48	C₂²×C₃	—	9	2	no	—
49	C₂²×C₃	—	18	4	no	—
50	C₂²×C₃	C₂×C₃⋊D₃	1	2	yes	—
51	C₂×D₃	—	3	4	no	—
52	C₂×D₃	—	3	8	no	—
53	C₂×D₃	—	6	16	no	—
54	C₃×C₄	—	3	4	no	—
55	C₃×C₄	—	9	1	no	—
56	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
57	Dic₃	—	3	4	no	—
58	C₃²	D₃×D₄	1	4	yes	complemented
59	C₂³	—	9	2	no	—
60	D₄	—	9	2	no	—
61	D₄	—	9	1	no	—
62	D₄	#272	1	1	yes	characteristic, direct product factor
63	C₂×C₄	—	9	1	no	—
64	C₂×C₃	—	2	2	no	—
65	C₂×C₃	—	3	4	no	—
66	C₂×C₃	—	6	8	no	—
67	C₂×C₃	—	9	2	no	—
68	C₂×C₃	—	18	2	no	—
69	C₂×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
70	D₃	—	3	8	no	—
71	D₃	—	6	8	no	—
72	C₂²	—	9	1	no	—
73	C₂²	—	9	2	no	—
74	C₂²	—	18	4	no	—
75	C₂²	C₂×#272	1	2	yes	complemented
76	C₄	—	9	1	no	—
77	C₄	C₂×#272	1	1	yes	characteristic, complemented
78	C₃	—	3	4	no	—
79	C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
80	C₂	—	2	2	no	—
81	C₂	—	9	2	no	—
82	C₂	—	18	2	no	—
83	C₂	C₂²×#272	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂²×C₃⋊D₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³
Fitting	D₄×H(3,𝔽₃)	C₂
hypercenter	C₃×D₄	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#272	C₂	semidirect
2. D₄×H(3,𝔽₃)	C₂	semidirect
3. #6235	C₂	semidirect
4. C₄⋊#272	C₂	semidirect
5. C₄×#272	C₂	semidirect
7. C₂²×H(3,𝔽₃)	C₂²	semidirect
10. C₂×#272	C₂²	semidirect
11. C₄×H(3,𝔽₃)	C₂²	semidirect
12. #1084	C₂²	semidirect
14. C₃²×D₄	D₃	semidirect
21. #272	D₄	direct (N × H)
24. C₂²×C₃²	C₂×D₃	semidirect
28. C₃²×C₄	C₂×D₃	semidirect
30. H(3,𝔽₃)	C₂×D₄	semidirect
58. C₃²	D₃×D₄	semidirect
62. D₄	#272	direct (N × H)
75. C₂²	C₂×#272	semidirect
77. C₄	C₂×#272	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#272	2
D₄×H(3,𝔽₃)	2
#6235	2
C₄⋊#272	2
C₄×#272	2
C₃×D₃×D₄	3

Subgroups — D4×#272 #88743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#272 #88743