Subgroups — #11720316

Order 864.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6180	C₂	1	1	yes	maximal, characteristic
2	#3748⋊C₃	C₂	1	2	yes	maximal
3	#63577	—	3	2	no	maximal
4	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	#6180	C₄	1	2	yes	—
6	(C₃⋊(C₃⋊C₈))⋊C₃	—	2	2	no	—
7	#3748	D₃	1	2	yes	—
8	C₃×#208	—	3	2	no	—
9	C₂×C₃×C₃⋊C₈	—	3	2	no	—
10	C₂×C₃×C₃⋊C₈	—	6	1	no	—
11	C₂²×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
12	C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
13	C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
14	#822	—	9	2	no	—
15	#819	—	9	1	no	—
16	C₂×C₃²×C₄	—	2	1	no	—
17	C₂×C₃²×C₄	C₂×D₃	1	2	yes	—
18	C₃⋊(C₃⋊C₈)	—	2	2	no	—
19	C₃×C₃⋊C₈	—	3	4	no	—
20	C₃×C₃⋊C₈	—	6	2	no	—
21	C₃×C₃⋊C₈	—	6	2	no	—
22	C₂×H(3,𝔽₃)	—	2	1	no	—
23	C₂×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
24	C₃×M₁₆	—	9	2	no	—
25	C₂×C₃×C₈	—	9	1	no	—
26	#208	—	3	2	no	—
27	#208	—	18	2	no	—
28	C₂×C₃⋊C₈	—	3	2	no	—
29	C₂×C₃⋊C₈	—	6	1	no	—
30	C₂²×C₃²	—	2	1	no	—
31	C₂²×C₃²	Dic₆	1	2	yes	—
32	C₃²×C₄	—	2	1	no	—
33	C₃²×C₄	—	2	1	no	—
34	C₃²×C₄	C₄×D₃	1	2	yes	—
35	C₃²×C₄	#67	1	2	yes	—
36	#108	—	27	1	no	Sylow
37	H(3,𝔽₃)	#108	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₃×C₄	—	6	2	no	—
39	C₂×C₃×C₄	—	12	1	no	—
40	C₂×C₃×C₄	D₃²	1	1	yes	characteristic
41	C₃×C₈	—	9	2	no	—
42	C₃×C₈	—	18	2	no	—
43	C₃⋊C₈	—	3	4	no	—
44	C₃⋊C₈	—	6	2	no	—
45	C₃⋊C₈	—	6	2	no	—
46	C₃⋊C₈	—	18	2	no	—
47	C₂×C₃²	—	2	2	no	—
48	C₂×C₃²	—	2	2	no	—
49	C₂×C₃²	—	2	1	no	—
50	C₂×C₃²	Dic₃⋊C₄	1	2	yes	—
51	M₁₆	—	9	2	no	—
52	C₂×C₈	—	9	1	no	—
53	C₂²×C₃	—	6	2	no	—
54	C₂²×C₃	—	12	1	no	—
55	C₂²×C₃	#627	1	1	yes	characteristic
56	C₃×C₄	—	6	2	no	—
57	C₃×C₄	—	6	2	no	—
58	C₃×C₄	—	12	1	no	—
59	C₃×C₄	—	12	1	no	—
60	C₃×C₄	#624	1	1	yes	characteristic
61	C₃×C₄	#625	1	1	yes	characteristic
62	C₃²	—	2	1	no	—
63	C₃²	#822	1	2	yes	complemented
64	C₂×C₄	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
65	C₈	—	9	2	no	—
66	C₈	—	18	2	no	—
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	—	6	2	no	—
69	C₂×C₃	—	12	2	no	—
70	C₂×C₃	—	12	2	no	—
71	C₂×C₃	—	12	1	no	—
72	C₂×C₃	#3724	1	1	yes	characteristic
73	C₂²	#6196	1	1	yes	characteristic
74	C₄	#6198	1	1	yes	characteristic
75	C₄	#6199	1	1	yes	characteristic
76	C₃	—	6	2	no	—
77	C₃	—	12	1	no	—
78	C₃	#63578	1	1	yes	characteristic
79	C₂	—	2	1	no	—
80	C₂	#88449	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#6198
commutator	C₄×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₄	D₃²
Fitting, cosocle	C₂×C₄×H(3,𝔽₃)	C₂²
socle	C₂×C₃	#3724
prime radical	C₂²×H(3,𝔽₃)	Q₈
hypercenter	C₂×C₄	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	#108
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
37. H(3,𝔽₃)	#108	semidirect
63. C₃²	#822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6180	2
#3748⋊C₃	2
#63577	3