Subgroups — #88545

Order 432.

121 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	C₄⋊#272	C₂	1	1	yes	maximal, characteristic, complemented
3	#6180	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#213	—	3	1	no	maximal
5	C₃×#213	—	3	3	no	maximal
6	C₂²×H(3,𝔽₃)	—	2	1	no	—
7	C₂×#272	—	2	1	no	—
8	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	C₃²×D₄	D₃	1	1	yes	characteristic, complemented
10	C₃²×D₄	D₃	1	3	yes	complemented
11	C₃×D₁₂	—	3	1	no	—
12	C₃×D₁₂	—	3	3	no	—
13	C₃×C₃⋊C₈	—	3	1	no	—
14	C₃×C₃⋊C₈	—	3	3	no	—
15	C₂×H(3,𝔽₃)	—	4	1	no	—
16	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	#272	—	4	1	no	—
18	C₃×D₈	—	9	1	no	—
19	#213	—	3	1	no	—
20	#213	—	3	3	no	—
21	C₂²×C₃²	—	2	1	no	—
22	C₂²×C₃²	—	2	3	no	—
23	C₂×C₃×D₃	—	6	1	no	—
24	C₂×C₃×D₃	—	6	3	no	—
25	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₃²×C₄	C₂×D₃	1	3	yes	—
27	H(3,𝔽₃)	D₈	1	1	yes	characteristic, complemented, Sylow
28	C₃×D₄	—	3	1	no	—
29	C₃×D₄	—	6	3	no	—
30	C₃×D₄	—	9	1	no	—
31	C₃×D₄	C₃⋊D₃	1	1	yes	characteristic
32	D₁₂	—	3	1	no	—
33	D₁₂	—	3	3	no	—
34	C₃×C₈	—	9	1	no	—
35	C₃⋊C₈	—	3	1	no	—
36	C₃⋊C₈	—	3	3	no	—
37	C₂×C₃²	—	4	1	no	—
38	C₂×C₃²	—	4	3	no	—
39	C₂×C₃²	#67	1	1	yes	characteristic
40	C₂×C₃²	#67	1	3	yes	—
41	C₃×D₃	—	12	1	no	—
42	C₃×D₃	—	12	3	no	—
43	D₈	—	9	1	no	Sylow
44	C₂²×C₃	—	2	1	no	—
45	C₂²×C₃	—	6	1	no	—
46	C₂²×C₃	—	6	3	no	—
47	C₂²×C₃	—	18	1	no	—
48	C₂×D₃	—	6	1	no	—
49	C₂×D₃	—	6	3	no	—
50	C₃×C₄	—	3	1	no	—
51	C₃×C₄	—	6	3	no	—
52	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₃²	#213	1	1	yes	characteristic, complemented
54	C₃²	#213	1	3	yes	complemented
55	D₄	—	9	1	no	—
56	D₄	#272	1	1	yes	characteristic, complemented
57	C₈	—	9	1	no	—
58	C₂×C₃	—	3	1	no	—
59	C₂×C₃	—	4	1	no	—
60	C₂×C₃	—	6	3	no	—
61	C₂×C₃	—	12	1	no	—
62	C₂×C₃	—	12	3	no	—
63	C₂×C₃	—	36	1	no	—
64	C₂×C₃	#638	1	1	yes	characteristic
65	D₃	—	12	1	no	—
66	D₃	—	12	3	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	18	1	no	—
69	C₄	C₂×#272	1	1	yes	characteristic
70	C₃	—	3	1	no	—
71	C₃	—	6	3	no	—
72	C₃	#3753	1	1	yes	characteristic
73	C₂	—	4	1	no	—
74	C₂	—	36	1	no	—
75	C₂	#6235	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#638
commutator, cosocle	C₄×H(3,𝔽₃)	C₂²
Frattini	C₃×C₄	C₂×C₃⋊D₃
Fitting	D₄×H(3,𝔽₃)	C₂
hypercenter	C₃×D₄	C₃⋊D₃
hypocenter	H(3,𝔽₃)	D₈
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ 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. C₄⋊#272	C₂	semidirect
3. #6180	C₂	semidirect
9. C₃²×D₄	D₃	semidirect
10. C₃²×D₄	D₃	semidirect
27. H(3,𝔽₃)	D₈	semidirect
53. C₃²	#213	semidirect
54. C₃²	#213	semidirect
56. 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
C₄⋊#272	2
#6180	2
C₃×#213	3