Subgroups — #88808

Order 432.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
3	#3710	—	3	1	no	maximal
4	D₃×C₃⋊C₈	—	3	2	no	maximal
5	C₂×C₃×C₃⋊D₃	C₄	1	1	yes	characteristic
6	C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₃×C₃⋊Dic₃	C₄	1	1	yes	characteristic
8	C₄×C₃⋊D₃	D₃	1	1	yes	characteristic
9	C₃×C₄×D₃	—	3	2	no	—
10	C₃×C₄×D₃	—	6	1	no	—
11	C₃⋊(C₃⋊C₈)	D₃	1	2	yes	complemented
12	C₃×C₃⋊C₈	—	3	2	no	—
13	C₃×C₃⋊C₈	—	3	2	no	—
14	C₃×C₃⋊C₈	—	6	2	no	—
15	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
16	C₃×C₃⋊D₃	C₈	1	2	yes	complemented
17	C₂×C₃⋊C₈	—	9	1	no	—
18	D₃×C₈	—	9	2	no	—
19	C₂×C₃⋊D₃	Dic₃	1	1	yes	characteristic
20	C₂×C₃×D₃	—	3	2	no	—
21	C₂×C₃×D₃	—	6	1	no	—
22	C₃²×C₄	—	2	2	no	—
23	C₃²×C₄	—	2	1	no	—
24	C₃²×C₄	—	4	1	no	—
25	C₃²×C₄	C₂×D₃	1	2	yes	—
26	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
27	C₃⋊Dic₃	Dic₃	1	1	yes	characteristic
28	C₃×Dic₃	—	3	2	no	—
29	C₃×Dic₃	—	6	1	no	—
30	C₃³	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃×C₄	—	9	1	no	—
32	C₄×D₃	—	3	2	no	—
33	C₄×D₃	—	6	1	no	—
34	C₃×C₈	—	9	2	no	—
35	C₃⋊C₈	—	3	2	no	—
36	C₃⋊C₈	—	3	2	no	—
37	C₃⋊C₈	—	6	2	no	—
38	C₂×C₃²	—	2	2	no	—
39	C₂×C₃²	—	2	1	no	—
40	C₂×C₃²	—	4	1	no	—
41	C₂×C₃²	C₄×D₃	1	2	yes	—
42	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
43	C₃⋊D₃	C₃⋊C₈	1	2	yes	complemented
44	C₃×D₃	—	3	4	no	—
45	C₃×D₃	—	6	2	no	—
46	C₂×C₈	—	27	1	no	Sylow
47	C₂²×C₃	—	9	1	no	—
48	C₂×D₃	—	3	2	no	—
49	C₂×D₃	—	6	1	no	—
50	C₃×C₄	—	2	2	no	—
51	C₃×C₄	—	2	1	no	—
52	C₃×C₄	—	4	1	no	—
53	C₃×C₄	—	9	1	no	—
54	C₃×C₄	D₃²	1	2	yes	—
55	C₃×C₄	D₃²	1	1	yes	characteristic
56	Dic₃	—	3	2	no	—
57	Dic₃	—	6	1	no	—
58	C₃²	—	2	2	no	—
59	C₃²	—	2	1	no	—
60	C₃²	—	4	1	no	—
61	C₃²	D₃×C₈	1	2	yes	complemented
62	C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic, complemented
63	C₂×C₄	—	9	1	no	—
64	C₈	—	9	2	no	—
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	—	4	1	no	—
68	C₂×C₃	—	9	2	no	—
69	C₂×C₃	D₃×Dic₃	1	2	yes	—
70	C₂×C₃	#624	1	1	yes	characteristic
71	D₃	—	3	4	no	—
72	D₃	—	6	2	no	—
73	C₂²	—	9	1	no	—
74	C₄	—	9	1	no	—
75	C₄	#1113	1	1	yes	characteristic
76	C₃	—	2	1	no	—
77	C₃	—	2	2	no	—
78	C₃	—	4	1	no	—
79	C₃	D₃×C₃⋊C₈	1	2	yes	complemented
80	C₃	#3710	1	1	yes	characteristic, complemented
81	C₂	—	9	2	no	—
82	C₂	#6294	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	#1113
commutator, hypocenter	C₃³	C₂×C₈
Fitting, cosocle	C₃³×C₄	C₂²
socle	C₂×C₃³	C₂×C₄
prime radical	C₂×C₃×C₃⋊D₃	C₄
LCS	G ▷ C₃³
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
11. C₃⋊(C₃⋊C₈)	D₃	semidirect
16. C₃×C₃⋊D₃	C₈	semidirect
30. C₃³	C₂×C₈	semidirect
43. C₃⋊D₃	C₃⋊C₈	semidirect
61. C₃²	D₃×C₈	semidirect
62. C₃²	C₂×C₃⋊C₈	semidirect
79. C₃	D₃×C₃⋊C₈	semidirect
80. C₃	#3710	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₃⋊D₃	2
C₃×C₃⋊(C₃⋊C₈)	2
#3710	3
D₃×C₃⋊C₈	3