Subgroups — #11724469

Order 864.

392 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊#226)	C₂	1	2	yes	maximal, complemented
3	#64265	—	3	12	no	maximal
4	#64160	—	3	1	no	maximal
5	#6308	—	4	1	no	maximal
6	C₃²×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₃²×#41	D₃	1	1	yes	characteristic
8	C₂×#634	—	3	1	no	—
9	C₃×#41⋊C₃	D₃	1	12	yes	—
10	C₃⋊#226	—	3	24	no	—
11	#3756	—	3	2	no	—
12	#3754	—	3	2	no	—
13	#3748	—	3	1	no	—
14	C₃³×C₄	—	4	1	no	—
15	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
16	#984	—	9	9	no	—
17	#951	—	9	4	no	—
18	C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
19	C₃²×D₄	—	3	1	no	—
20	C₂×C₃²×C₄	—	3	1	no	—
21	C₂×C₃⋊Dic₃	—	6	1	no	—
22	#634	—	3	2	no	—
23	#634	—	6	1	no	—
24	#634	—	12	12	no	—
25	C₃×SL(2,𝔽₃)	C₂×D₃	1	12	yes	—
26	C₃⋊(C₃⋊C₈)	—	3	2	no	—
27	C₂×C₃³	—	4	1	no	—
28	C₃×#41	C₃⋊D₃	1	4	yes	—
29	C₂×Dic₆	—	9	4	no	—
30	#41⋊C₃	C₃⋊D₃	1	9	yes	—
31	#226	—	9	18	no	—
32	#216	—	9	8	no	—
33	#214	—	9	8	no	—
34	#208	—	9	4	no	—
35	C₂²×C₃²	—	3	1	no	—
36	C₃²×C₄	—	3	1	no	—
37	C₃²×C₄	—	4	12	no	—
38	C₃²×C₄	S₄	1	1	yes	characteristic
39	C₃⋊Dic₃	—	6	2	no	—
40	C₃⋊Dic₃	—	12	24	no	—
41	#137	—	27	1	no	Sylow
42	C₃³	—	4	1	no	Sylow
43	C₃×Q₈	C₂×C₃⋊D₃	1	4	yes	—
44	C₃×D₄	—	3	4	no	—
45	C₂×C₃×C₄	—	3	4	no	—
46	C₂×Dic₃	—	18	4	no	—
47	Dic₆	—	9	8	no	—
48	Dic₆	—	18	4	no	—
49	Dic₆	—	36	9	no	—
50	SL(2,𝔽₃)	C₂×C₃⋊D₃	1	9	yes	—
51	C₃⋊C₈	—	9	8	no	—
52	C₂×C₃²	—	4	12	no	—
53	C₂×C₃²	—	6	1	no	—
54	C₂×C₃²	C₂×S₄	1	1	yes	characteristic
55	#41	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
56	C₂×Q₈	—	27	1	no	—
57	Q₁₆	—	27	2	no	—
58	SD₁₆	—	27	2	no	—
59	M₁₆	—	27	1	no	—
60	C₂²×C₃	—	3	4	no	—
61	C₃×C₄	—	3	4	no	—
62	C₃×C₄	—	4	9	no	—
63	C₃×C₄	C₃⋊S₄	1	4	yes	—
64	Dic₃	—	18	8	no	—
65	Dic₃	—	36	18	no	—
66	C₃²	—	4	12	no	—
67	C₃²	#984	1	1	yes	characteristic, complemented
68	Q₈	—	27	2	no	—
69	Q₈	—	54	1	no	—
70	Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
71	D₄	—	3	1	no	—
72	C₂×C₄	—	3	1	no	—
73	C₂×C₄	—	54	1	no	—
74	C₈	—	27	2	no	—
75	C₂×C₃	—	4	9	no	—
76	C₂×C₃	—	6	4	no	—
77	C₂×C₃	C₂×C₃⋊S₄	1	4	yes	—
78	C₂²	—	3	1	no	—
79	C₄	—	3	1	no	—
80	C₄	—	54	2	no	—
81	C₄	C₃⋊(C₃⋊S₄)	1	1	yes	characteristic
82	C₃	—	4	9	no	—
83	C₃	#64265	1	4	yes	complemented
84	C₂	—	6	1	no	—
85	C₂	C₂×C₃⋊(C₃⋊S₄)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊(C₃⋊S₄)
commutator, cosocle, hypocenter	C₃²×SL(2,𝔽₃)	C₂²
Fitting	C₃²×#41	D₃
socle	C₂×C₃²	C₂×S₄
prime radical	C₃²×#41⋊C₃	C₂
hypercenter	C₄	C₃⋊(C₃⋊S₄)
LCS	G ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊#226)	C₂	semidirect
67. C₃²	#984	semidirect
83. C₃	#64265	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#41⋊C₃	2
C₃⋊(C₃⋊#226)	2
#64265	3
#64160	3
#6308	4