Subgroups — #88744

Order 432.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	#6235	C₂	1	2	yes	maximal, complemented
3	C₂×#1084	C₂	1	2	yes	maximal, complemented
4	C₄×#272	C₂	1	1	yes	maximal, characteristic, complemented
5	#6229	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#237	—	3	4	no	maximal
7	C₂²×H(3,𝔽₃)	C₂²	1	2	yes	—
8	C₂×#272	C₂²	1	1	yes	characteristic
9	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	#1084	C₂²	1	2	yes	—
11	#1084	C₂²	1	1	yes	characteristic
12	C₃²×D₄	D₃	1	4	yes	complemented
13	C₃×#67	—	3	8	no	—
14	C₂×C₃×Dic₃	—	3	8	no	—
15	C₃×C₄×D₃	—	3	4	no	—
16	C₃×Dic₆	—	3	4	no	—
17	C₂×H(3,𝔽₃)	—	2	2	no	—
18	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
19	#272	—	2	1	no	—
20	C₃×#41	—	9	1	no	—
21	#237	—	3	4	no	—
22	C₂²×C₃²	C₂×D₃	1	8	yes	—
23	C₂×C₃×D₃	—	3	4	no	—
24	C₃²×C₄	C₂×D₃	1	4	yes	—
25	C₃×Dic₃	—	3	8	no	—
26	C₃×Dic₃	—	3	4	no	—
27	H(3,𝔽₃)	#41	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	9	1	no	—
29	C₃×D₄	—	3	4	no	—
30	C₃×D₄	—	9	2	no	—
31	C₃×D₄	C₃⋊D₃	1	1	yes	characteristic
32	C₂×C₃×C₄	—	9	2	no	—
33	C₂×C₃×C₄	—	9	1	no	—
34	#67	—	3	8	no	—
35	C₂×Dic₃	—	3	8	no	—
36	C₄×D₃	—	3	4	no	—
37	Dic₆	—	3	4	no	—
38	C₂×C₃²	—	2	8	no	—
39	C₂×C₃²	C₂²×D₃	1	4	yes	—
40	C₃×D₃	—	6	4	no	—
41	#41	—	9	1	no	Sylow
42	C₂²×C₃	—	3	8	no	—
43	C₂²×C₃	—	9	1	no	—
44	C₂²×C₃	C₂×C₃⋊D₃	1	2	yes	—
45	C₂×D₃	—	3	4	no	—
46	C₃×C₄	—	3	4	no	—
47	C₃×C₄	—	9	1	no	—
48	C₃×C₄	—	9	2	no	—
49	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
50	Dic₃	—	3	8	no	—
51	Dic₃	—	3	4	no	—
52	C₃²	#237	1	4	yes	complemented
53	Q₈	—	9	1	no	—
54	D₄	—	9	2	no	—
55	D₄	#272	1	1	yes	characteristic, complemented
56	C₂×C₄	—	9	2	no	—
57	C₂×C₄	—	9	1	no	—
58	C₂×C₃	—	2	2	no	—
59	C₂×C₃	—	3	4	no	—
60	C₂×C₃	—	6	8	no	—
61	C₂×C₃	—	18	1	no	—
62	C₂×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	D₃	—	6	4	no	—
64	C₂²	—	9	1	no	—
65	C₂²	C₂×#272	1	2	yes	—
66	C₄	—	9	1	no	—
67	C₄	—	9	2	no	—
68	C₄	C₂×#272	1	1	yes	characteristic
69	C₃	—	3	4	no	—
70	C₃	#3830	1	1	yes	characteristic
71	C₂	—	2	2	no	—
72	C₂	—	18	1	no	—
73	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)	#41
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. D₄×H(3,𝔽₃)	C₂	semidirect
2. #6235	C₂	semidirect
3. C₂×#1084	C₂	semidirect
4. C₄×#272	C₂	semidirect
5. #6229	C₂	semidirect
12. C₃²×D₄	D₃	semidirect
27. H(3,𝔽₃)	#41	semidirect
52. C₃²	#237	semidirect
55. D₄	#272	semidirect

Maximal subgroups

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

subgroup	index
D₄×H(3,𝔽₃)	2
#6235	2
C₂×#1084	2
C₄×#272	2
#6229	2
C₃×#237	3