Subgroups — #88936

Order 432.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6295	C₂	1	1	yes	maximal, characteristic
2	C₃×#627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊#622	C₂	1	1	yes	maximal, characteristic, complemented
4	#3775	—	3	1	no	maximal
5	C₂×C₃×C₃⋊D₃	—	2	1	no	—
6	C₃×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
7	C₃²×Dic₃	—	2	1	no	—
8	C₃×Dic₆	—	6	1	no	—
9	#627	D₃	1	1	yes	characteristic, complemented
10	#626	—	6	1	no	—
11	#625	—	3	1	no	—
12	#622	—	3	1	no	—
13	C₂×C₃³	D₄	1	1	yes	characteristic
14	C₃×C₃⋊D₃	—	4	1	no	—
15	#215	—	9	1	no	maximal
16	C₂×C₃⋊D₃	—	2	1	no	—
17	C₂×C₃×D₃	—	6	1	no	—
18	C₂×C₃×D₃	—	6	1	no	—
19	C₂×C₃×D₃	—	12	1	no	—
20	C₃²×C₄	—	6	1	no	—
21	C₃⋊Dic₃	C₂×D₃	1	1	yes	characteristic
22	C₃×Dic₃	—	2	1	no	—
23	C₃×Dic₃	—	2	1	no	—
24	C₃×Dic₃	—	4	1	no	—
25	C₃×Dic₃	—	6	1	no	—
26	C₃×Dic₃	—	6	1	no	—
27	C₃³	SD₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	9	1	no	—
29	#67	—	18	1	no	—
30	D₁₂	—	9	1	no	—
31	Dic₆	—	6	1	no	—
32	C₃⋊C₈	—	9	1	no	—
33	C₂×C₃²	—	2	1	no	—
34	C₂×C₃²	—	2	1	no	—
35	C₂×C₃²	—	4	1	no	—
36	C₂×C₃²	—	4	1	no	—
37	C₂×C₃²	#67	1	1	yes	characteristic
38	C₃⋊D₃	—	4	1	no	—
39	C₃×D₃	—	12	1	no	—
40	C₃×D₃	—	12	2	no	—
41	C₃×D₃	—	12	1	no	—
42	SD₁₆	—	27	1	no	Sylow
43	C₂²×C₃	—	18	1	no	—
44	C₂×D₃	—	6	1	no	—
45	C₂×D₃	—	6	1	no	—
46	C₂×D₃	—	12	1	no	—
47	C₃×C₄	—	6	1	no	—
48	C₃×C₄	—	6	1	no	—
49	C₃×C₄	—	9	1	no	—
50	C₃×C₄	—	12	1	no	—
51	Dic₃	—	2	1	no	—
52	Dic₃	—	6	1	no	—
53	Dic₃	—	6	1	no	—
54	C₃²	—	2	1	no	—
55	C₃²	—	2	1	no	—
56	C₃²	—	4	1	no	—
57	C₃²	—	4	1	no	—
58	C₃²	#215	1	1	yes	characteristic, complemented
59	Q₈	—	9	1	no	—
60	D₄	—	27	1	no	—
61	C₈	—	27	1	no	—
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	—	2	1	no	—
64	C₂×C₃	—	4	1	no	—
65	C₂×C₃	—	4	1	no	—
66	C₂×C₃	—	36	1	no	—
67	C₂×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic
68	D₃	—	12	1	no	—
69	D₃	—	12	2	no	—
70	D₃	—	12	1	no	—
71	C₂²	—	18	1	no	—
72	C₄	—	6	1	no	—
73	C₄	—	9	1	no	—
74	C₃	—	2	1	no	—
75	C₃	—	2	1	no	—
76	C₃	—	4	1	no	—
77	C₃	—	4	1	no	—
78	C₃	#3775	1	1	yes	characteristic, complemented
79	C₂	—	36	1	no	—
80	C₂	D₃²⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²⋊D₃
commutator, cosocle	C₃×C₃⋊Dic₃	C₂²
Fitting, socle	C₂×C₃³	D₄
prime radical	#6295	C₂
hypocenter	C₃³	SD₁₆
LCS	G ▷ C₃×C₃⋊Dic₃ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₃×#627	C₂	semidirect
3. C₃⋊#622	C₂	semidirect
9. #627	D₃	semidirect
27. C₃³	SD₁₆	semidirect
58. C₃²	#215	semidirect
78. C₃	#3775	semidirect

Maximal subgroups

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

subgroup	index
#6295	2
C₃×#627	2
C₃⋊#622	2
#3775	3
#215	9