Subgroups — #88790

Order 432.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3753	—	3	1	no	maximal
5	#3714	—	3	4	no	maximal
6	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
7	C₂×C₃²×D₃	—	2	1	no	—
8	C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₃²×D₄	—	3	1	no	—
10	C₄⋊(C₃⋊D₃)	—	3	1	no	—
11	C₄⋊(C₃⋊D₃)	—	3	4	no	—
12	C₄⋊(C₃⋊D₃)	—	6	4	no	—
13	C₃×D₁₂	D₃	1	4	yes	complemented
14	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic, complemented
15	C₃×C₃⋊C₈	—	3	4	no	—
16	C₂×C₃³	D₄	1	1	yes	characteristic
17	C₃⋊(C₃⋊D₃)	—	4	1	no	—
18	C₃²×D₃	—	4	1	no	—
19	#213	—	9	4	no	—
20	D₂₄	—	9	1	no	—
21	C₂²×C₃²	—	6	1	no	—
22	C₂×C₃⋊D₃	—	6	8	no	—
23	C₂×C₃⋊D₃	—	6	1	no	—
24	C₂×C₃⋊D₃	—	6	4	no	—
25	C₂×C₃×D₃	—	2	4	no	—
26	C₃²×C₄	—	2	4	no	—
27	C₃²×C₄	C₂×D₃	1	4	yes	—
28	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
29	C₃³	D₈	1	1	yes	characteristic, complemented, Sylow
30	C₃×D₄	—	3	4	no	—
31	D₁₂	—	9	4	no	—
32	D₁₂	—	9	1	no	—
33	D₁₂	—	18	4	no	—
34	D₁₂	C₃⋊D₃	1	1	yes	characteristic, complemented
35	C₃×C₈	—	9	1	no	—
36	C₃⋊C₈	—	3	4	no	—
37	C₂×C₃²	—	2	4	no	—
38	C₂×C₃²	—	12	1	no	—
39	C₂×C₃²	D₁₂	1	1	yes	characteristic
40	C₂×C₃²	#67	1	4	yes	—
41	C₃⋊D₃	—	12	8	no	—
42	C₃⋊D₃	—	12	1	no	—
43	C₃⋊D₃	—	12	4	no	—
44	C₃×D₃	—	4	4	no	—
45	D₈	—	27	1	no	Sylow
46	C₂²×C₃	—	6	4	no	—
47	C₂×D₃	—	2	1	no	—
48	C₂×D₃	—	18	8	no	—
49	C₂×D₃	—	18	4	no	—
50	C₂×D₃	—	18	1	no	—
51	C₃×C₄	—	2	4	no	—
52	C₃×C₄	D₃²	1	4	yes	—
53	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
54	C₃²	—	2	4	no	—
55	C₃²	D₂₄	1	1	yes	characteristic, complemented
56	C₃²	#213	1	4	yes	complemented
57	D₄	—	3	1	no	—
58	D₄	—	27	1	no	—
59	C₈	—	9	1	no	—
60	C₂×C₃	—	2	4	no	—
61	C₂×C₃	—	12	4	no	—
62	C₂×C₃	#626	1	4	yes	—
63	C₂×C₃	#638	1	1	yes	characteristic
64	D₃	—	4	1	no	—
65	D₃	—	36	8	no	—
66	D₃	—	36	4	no	—
67	D₃	—	36	1	no	—
68	C₂²	—	6	1	no	—
69	C₂²	—	54	1	no	—
70	C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
71	C₃	—	2	4	no	—
72	C₃	#3714	1	4	yes	complemented
73	C₃	#3753	1	1	yes	characteristic, complemented
74	C₂	—	12	1	no	—
75	C₂	—	108	1	no	—
76	C₂	#6291	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6291
commutator, Fitting, cosocle	C₃³×C₄	C₂²
Frattini, hypercenter	C₄	D₃×C₃⋊D₃
socle	C₂×C₃³	D₄
hypocenter	C₃³	D₈
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
2. C₃²×D₁₂	C₂	semidirect
3. C₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
13. C₃×D₁₂	D₃	semidirect
14. C₃⋊(C₃⋊C₈)	D₃	semidirect
29. C₃³	D₈	semidirect
34. D₁₂	C₃⋊D₃	semidirect
55. C₃²	D₂₄	semidirect
56. C₃²	#213	semidirect
72. C₃	#3714	semidirect
73. C₃	#3753	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₃⋊(C₃⋊D₃))	2
C₃²×D₁₂	2
C₃×C₃⋊(C₃⋊C₈)	2
#3753	3
#3714	3