Subgroups — #88934

Order 432.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊#157	C₂	1	1	yes	maximal, characteristic
2	#6294	C₂	1	1	yes	maximal, characteristic
3	C₃×#624	C₂	1	1	yes	maximal, characteristic
4	#3773	—	3	1	no	maximal
5	C₂×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₃⋊#157	C₄	1	2	yes	complemented
7	C₃×C₃⋊Dic₃	—	2	1	no	—
8	C₃²×Dic₃	—	2	1	no	—
9	C₂×#157	—	3	1	no	—
10	C₃×C₄×D₃	—	6	1	no	—
11	#624	—	3	1	no	—
12	#624	D₃	1	1	yes	characteristic
13	D₃×Dic₃	—	6	1	no	—
14	C₂×C₃³	D₄	1	1	yes	characteristic
15	C₃×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
16	C₃×C₃⋊D₃	Q₈	1	1	yes	characteristic
17	Dic₃⋊C₄	—	9	1	no	maximal
18	C₂×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
19	C₂×C₃×D₃	—	6	1	no	—
20	C₂×C₃×D₃	—	6	1	no	—
21	#157	—	3	2	no	—
22	C₃²×C₄	—	6	1	no	—
23	C₃⋊Dic₃	—	2	1	no	—
24	C₃×Dic₃	—	2	1	no	—
25	C₃×Dic₃	—	2	1	no	—
26	C₃×Dic₃	—	4	1	no	—
27	C₃×Dic₃	—	6	1	no	—
28	C₃×Dic₃	—	6	1	no	—
29	C₃×Dic₃	—	12	1	no	—
30	C₃³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃×C₄	—	9	1	no	—
32	C₂×Dic₃	—	9	1	no	—
33	C₂×Dic₃	—	9	1	no	—
34	C₄×D₃	—	6	1	no	—
35	C₄×D₃	—	18	1	no	—
36	C₂×C₃²	—	2	1	no	—
37	C₂×C₃²	—	2	1	no	—
38	C₂×C₃²	—	4	1	no	—
39	C₂×C₃²	—	4	1	no	—
40	C₂×C₃²	#67	1	1	yes	characteristic
41	C₃⋊D₃	Dic₆	1	1	yes	characteristic
42	C₃⋊D₃	C₄×D₃	1	1	yes	characteristic
43	C₃×D₃	—	6	1	no	—
44	C₃×D₃	—	6	1	no	—
45	C₃×D₃	—	6	1	no	—
46	C₃×D₃	—	6	1	no	—
47	C₄⋊C₄	—	27	1	no	Sylow
48	C₂²×C₃	—	9	1	no	—
49	C₂×D₃	—	6	1	no	—
50	C₂×D₃	—	6	1	no	—
51	C₃×C₄	—	6	1	no	—
52	C₃×C₄	—	6	1	no	—
53	C₃×C₄	—	12	1	no	—
54	C₃×C₄	—	18	1	no	—
55	Dic₃	—	2	1	no	—
56	Dic₃	—	6	1	no	—
57	Dic₃	—	6	1	no	—
58	Dic₃	—	9	2	no	—
59	Dic₃	—	12	1	no	—
60	C₃²	—	2	1	no	—
61	C₃²	—	2	1	no	—
62	C₃²	—	4	1	no	—
63	C₃²	—	4	1	no	—
64	C₃²	Dic₃⋊C₄	1	1	yes	characteristic, complemented
65	C₂×C₄	—	9	1	no	—
66	C₂×C₄	—	27	1	no	—
67	C₂×C₄	—	27	1	no	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	2	1	no	—
70	C₂×C₃	—	4	1	no	—
71	C₂×C₃	—	4	1	no	—
72	C₂×C₃	—	9	1	no	—
73	C₂×C₃	—	9	1	no	—
74	C₂×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic
75	D₃	—	6	1	no	—
76	D₃	—	6	1	no	—
77	D₃	—	6	1	no	—
78	D₃	—	6	1	no	—
79	C₂²	—	9	1	no	—
80	C₄	—	6	1	no	—
81	C₄	—	18	1	no	—
82	C₄	—	27	2	no	—
83	C₃	—	2	1	no	—
84	C₃	—	2	1	no	—
85	C₃	—	4	1	no	—
86	C₃	—	4	1	no	—
87	C₃	#3773	1	1	yes	characteristic, complemented
88	C₂	—	9	1	no	—
89	C₂	—	9	1	no	—
90	C₂	D₃²⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²⋊D₃
commutator	C₃×C₃⋊D₃	C₂×C₄
Fitting, socle	C₂×C₃³	D₄
prime radical, cosocle	C₂×C₃×C₃⋊D₃	C₂²
hypocenter	C₃³	C₄⋊C₄
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
6. C₃⋊#157	C₄	semidirect
30. C₃³	C₄⋊C₄	semidirect
64. C₃²	Dic₃⋊C₄	semidirect
87. C₃	#3773	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊#157	2
#6294	2
C₃×#624	2
#3773	3
Dic₃⋊C₄	9