Subgroups — C₃×#3713 #88771

Order 432.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₁₂	C₂	1	2	yes	maximal, complemented
2	C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#213	—	3	2	no	maximal
4	#3713	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃²×D₃	—	2	2	no	—
6	C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₃²×D₄	—	3	2	no	—
8	C₃×D₁₂	—	2	2	no	—
9	C₃×D₁₂	D₃	1	2	yes	complemented
10	C₃×D₁₂	C₂×C₃	1	2	yes	complemented
11	C₃⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
12	C₃×C₃⋊C₈	—	3	2	no	—
13	C₃×C₃⋊C₈	—	6	1	no	—
14	C₂×C₃³	D₄	1	1	yes	characteristic
15	C₃²×D₃	—	4	2	no	—
16	C₃×D₈	—	9	1	no	—
17	#213	—	3	2	no	—
18	C₂²×C₃²	—	6	2	no	—
19	C₂×C₃×D₃	—	2	4	no	—
20	C₂×C₃×D₃	—	2	2	no	—
21	C₂×C₃×D₃	—	2	2	no	—
22	C₃²×C₄	—	2	1	no	—
23	C₃²×C₄	—	2	2	no	—
24	C₃²×C₄	—	4	1	no	—
25	C₃²×C₄	C₂×D₃	1	2	yes	—
26	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
27	C₃³	D₈	1	1	yes	characteristic, complemented, Sylow
28	C₃×D₄	—	3	2	no	—
29	C₃×D₄	—	3	2	no	—
30	C₃×D₄	—	6	2	no	—
31	D₁₂	C₃×D₃	1	2	yes	complemented
32	C₃×C₈	—	9	1	no	—
33	C₃⋊C₈	—	3	2	no	—
34	C₃⋊C₈	—	6	1	no	—
35	C₂×C₃²	—	2	1	no	—
36	C₂×C₃²	—	2	2	no	—
37	C₂×C₃²	—	4	1	no	—
38	C₂×C₃²	—	12	2	no	—
39	C₂×C₃²	#67	1	2	yes	—
40	C₂×C₃²	C₃×D₄	1	1	yes	characteristic
41	C₃×D₃	—	4	4	no	—
42	C₃×D₃	—	4	2	no	—
43	C₃×D₃	—	4	2	no	—
44	D₈	—	9	1	no	Sylow
45	C₂²×C₃	—	6	4	no	—
46	C₂²×C₃	—	6	2	no	—
47	C₂²×C₃	—	6	2	no	—
48	C₂×D₃	—	2	2	no	—
49	C₃×C₄	—	2	1	no	—
50	C₃×C₄	—	2	2	no	—
51	C₃×C₄	—	4	1	no	—
52	C₃×C₄	D₃²	1	1	yes	characteristic
53	C₃×C₄	C₂×C₃×D₃	1	2	yes	—
54	C₃²	—	2	1	no	—
55	C₃²	—	2	2	no	—
56	C₃²	—	4	1	no	—
57	C₃²	#213	1	2	yes	complemented
58	C₃²	C₃×D₈	1	1	yes	characteristic, complemented
59	D₄	—	3	2	no	—
60	C₈	—	9	1	no	—
61	C₂×C₃	—	2	2	no	—
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	—	4	1	no	—
64	C₂×C₃	—	12	4	no	—
65	C₂×C₃	—	12	2	no	—
66	C₂×C₃	—	12	2	no	—
67	C₂×C₃	#625	1	1	yes	characteristic
68	C₂×C₃	C₃×#67	1	2	yes	—
69	D₃	—	4	2	no	—
70	C₂²	—	6	2	no	—
71	C₄	C₃×D₃²	1	1	yes	characteristic
72	C₃	—	2	2	no	—
73	C₃	—	2	1	no	—
74	C₃	—	4	1	no	—
75	C₃	#3713	1	1	yes	characteristic, direct product factor
76	C₃	C₃×#213	1	2	yes	complemented
77	C₂	—	12	2	no	—
78	C₂	C₃×#625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#625
commutator, cosocle	C₃²×C₄	C₂²×C₃
Frattini	C₄	C₃×D₃²
Fitting	C₃³×C₄	C₂²
socle	C₂×C₃³	D₄
hypercenter	C₃×C₄	D₃²
hypocenter	C₃²	C₃×D₈
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
1. C₃²×D₁₂	C₂	semidirect
2. C₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
4. #3713	C₃	direct (N × H)
9. C₃×D₁₂	D₃	semidirect
10. C₃×D₁₂	C₂×C₃	semidirect
11. C₃⋊(C₃⋊C₈)	C₂×C₃	semidirect
27. C₃³	D₈	semidirect
31. D₁₂	C₃×D₃	semidirect
57. C₃²	#213	semidirect
58. C₃²	C₃×D₈	semidirect
75. C₃	#3713	direct (N × H)
76. C₃	C₃×#213	semidirect

Maximal subgroups

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

subgroup	index
C₃²×D₁₂	2
C₃×C₃⋊(C₃⋊C₈)	2
C₃×#213	3
#3713	3

Subgroups — C3×#3713 #88771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3713 #88771