Subgroups — #88938

Order 432.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6296	C₂	1	1	yes	maximal, characteristic
2	C₃×#627	C₂	1	1	yes	maximal, characteristic
3	C₃⋊#622	C₂	1	1	yes	maximal, characteristic
4	#3776	—	3	1	no	maximal
5	C₃×C₃⋊Dic₃	—	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	—	3	1	no	—
10	#627	—	6	1	no	—
11	#627	D₃	1	1	yes	characteristic
12	#622	—	3	1	no	—
13	C₂×C₃³	D₄	1	1	yes	characteristic
14	#216	—	9	1	no	maximal
15	C₃²×C₄	—	6	1	no	—
16	C₃⋊Dic₃	—	2	1	no	—
17	C₃⋊Dic₃	C₂×D₃	1	1	yes	characteristic
18	C₃×Dic₃	—	2	1	no	—
19	C₃×Dic₃	—	2	1	no	—
20	C₃×Dic₃	—	4	1	no	—
21	C₃×Dic₃	—	6	1	no	—
22	C₃×Dic₃	—	6	1	no	—
23	C₃×Dic₃	—	6	1	no	—
24	C₃×Dic₃	—	6	1	no	—
25	C₃×Dic₃	—	12	1	no	—
26	C₃³	Q₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₃×Q₈	—	9	1	no	—
28	Dic₆	—	6	1	no	—
29	Dic₆	—	9	1	no	—
30	Dic₆	—	18	1	no	—
31	C₃⋊C₈	—	9	1	no	—
32	C₂×C₃²	—	2	1	no	—
33	C₂×C₃²	—	2	1	no	—
34	C₂×C₃²	—	4	1	no	—
35	C₂×C₃²	—	4	1	no	—
36	C₂×C₃²	#67	1	1	yes	characteristic
37	Q₁₆	—	27	1	no	Sylow
38	C₃×C₄	—	6	1	no	—
39	C₃×C₄	—	6	1	no	—
40	C₃×C₄	—	9	1	no	—
41	C₃×C₄	—	12	1	no	—
42	C₃×C₄	—	18	1	no	—
43	Dic₃	—	2	1	no	—
44	Dic₃	—	6	1	no	—
45	Dic₃	—	6	1	no	—
46	Dic₃	—	6	1	no	—
47	Dic₃	—	6	1	no	—
48	Dic₃	—	12	1	no	—
49	C₃²	—	2	1	no	—
50	C₃²	—	2	1	no	—
51	C₃²	—	4	1	no	—
52	C₃²	—	4	1	no	—
53	C₃²	#216	1	1	yes	characteristic, complemented
54	Q₈	—	9	1	no	—
55	Q₈	—	27	1	no	—
56	C₈	—	27	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	4	1	no	—
60	C₂×C₃	—	4	1	no	—
61	C₂×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic
62	C₄	—	6	1	no	—
63	C₄	—	9	1	no	—
64	C₄	—	18	1	no	—
65	C₃	—	2	1	no	—
66	C₃	—	2	1	no	—
67	C₃	—	4	1	no	—
68	C₃	—	4	1	no	—
69	C₃	#3776	1	1	yes	characteristic, complemented
70	C₂	D₃²⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²⋊D₃
commutator, cosocle	C₃×C₃⋊Dic₃	C₂²
Fitting, socle, prime radical	C₂×C₃³	D₄
hypocenter	C₃³	Q₁₆
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
26. C₃³	Q₁₆	semidirect
53. C₃²	#216	semidirect
69. C₃	#3776	semidirect

Maximal subgroups

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

subgroup	index
#6296	2
C₃×#627	2
C₃⋊#622	2
#3776	3
#216	9