Subgroups — #11724242

Order 864.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#622	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#3771	C₂	1	2	yes	maximal, complemented
3	#64218	—	3	1	no	maximal
4	C₂×C₃×C₃⋊Dic₃	C₄	1	1	yes	characteristic
5	C₃×#622	C₄	1	1	yes	characteristic
6	C₃×#622	C₂²	1	1	yes	characteristic
7	C₂×#622	D₃	1	1	yes	characteristic
8	#3771	—	3	2	no	—
9	C₂²×C₃³	C₈	1	1	yes	characteristic
10	C₃×C₃⋊Dic₃	C₈	1	1	yes	characteristic
11	C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
12	#812	—	9	1	no	maximal
13	C₂×C₃⋊Dic₃	Dic₃	1	1	yes	characteristic
14	C₂×C₃×Dic₃	—	12	1	no	—
15	#622	Dic₃	1	1	yes	characteristic
16	#622	C₂×D₃	1	1	yes	characteristic
17	C₂×C₃³	—	2	1	no	—
18	C₂×C₃³	C₂×C₈	1	1	yes	characteristic
19	C₂×C₃×C₈	—	9	1	no	—
20	C₃⋊C₁₆	—	9	2	no	—
21	C₂²×C₃²	—	4	1	no	—
22	C₂²×C₃²	—	8	1	no	—
23	C₂²×C₃²	C₃⋊C₈	1	1	yes	characteristic
24	C₃⋊Dic₃	C₃⋊C₈	1	1	yes	characteristic
25	C₃⋊Dic₃	C₂×Dic₃	1	1	yes	characteristic
26	C₃×Dic₃	—	12	1	no	—
27	C₃×Dic₃	—	12	1	no	—
28	M₃₂	—	27	1	no	Sylow
29	C₃³	M₃₂	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₃×C₄	—	9	1	no	—
31	C₂×Dic₃	—	12	1	no	—
32	C₃×C₈	—	9	1	no	—
33	C₃×C₈	—	9	1	no	—
34	C₂×C₃²	—	2	1	no	—
35	C₂×C₃²	—	4	2	no	—
36	C₂×C₃²	—	4	1	no	—
37	C₂×C₃²	—	8	2	no	—
38	C₂×C₃²	—	8	1	no	—
39	C₂×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
40	C₂×C₈	—	9	1	no	—
41	C₁₆	—	27	2	no	—
42	C₂²×C₃	—	4	1	no	—
43	C₂²×C₃	—	8	1	no	—
44	C₂²×C₃	AGL(1,𝔽₉)	1	1	yes	characteristic
45	C₃×C₄	—	9	1	no	—
46	C₃×C₄	—	9	1	no	—
47	Dic₃	—	12	1	no	—
48	Dic₃	—	12	1	no	—
49	C₃²	—	4	1	no	—
50	C₃²	—	8	1	no	—
51	C₃²	#812	1	1	yes	characteristic, complemented
52	C₂×C₄	—	9	1	no	—
53	C₈	—	9	1	no	—
54	C₈	—	9	1	no	—
55	C₂×C₃	—	2	1	no	—
56	C₂×C₃	—	4	2	no	—
57	C₂×C₃	—	4	1	no	—
58	C₂×C₃	—	8	2	no	—
59	C₂×C₃	—	8	1	no	—
60	C₂×C₃	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
61	C₂²	C₃⋊AGL(1,𝔽₉)	1	1	yes	characteristic
62	C₄	—	9	1	no	—
63	C₄	—	9	1	no	—
64	C₃	—	4	1	no	—
65	C₃	—	8	1	no	—
66	C₃	#64218	1	1	yes	characteristic, complemented
67	C₂	—	2	1	no	—
68	C₂	C₂×C₃⋊AGL(1,𝔽₉)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊AGL(1,𝔽₉)
commutator, socle	C₂×C₃³	C₂×C₈
Fitting, prime radical	C₂²×C₃³	C₈
cosocle	C₃×#622	C₂²
hypercenter	C₂²	C₃⋊AGL(1,𝔽₉)
hypocenter	C₃³	M₃₂
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₃⋊#3771	C₂	semidirect
29. C₃³	M₃₂	semidirect
51. C₃²	#812	semidirect
66. C₃	#64218	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#622	2
C₃⋊#3771	2
#64218	3
#812	9