Subgroups — #11720314

Order 864.

148 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	C₄⋊((C₃⋊Dic₃)⋊C₃)	C₂	1	2	yes	maximal
3	#63573	—	3	2	no	maximal
4	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₂×(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
6	#6180	C₄	1	2	yes	complemented
7	C₄⋊(C₃⋊Dic₃)	D₃	1	2	yes	—
8	C₃×C₄⋊Dic₃	—	3	2	no	—
9	C₂×C₃×C₃⋊C₈	—	3	2	no	—
10	C₂×C₃×C₃⋊C₈	—	6	1	no	—
11	C₂²×H(3,𝔽₃)	D₄	1	1	yes	characteristic
12	C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
13	C₄×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
14	(C₃⋊Dic₃)⋊C₃	—	4	2	no	—
15	#817	—	9	1	no	—
16	#808	—	9	2	no	—
17	C₂×C₃²×C₄	—	2	1	no	—
18	C₂×C₃²×C₄	C₂×D₃	1	2	yes	—
19	C₂×C₃⋊Dic₃	—	2	2	no	—
20	C₂×C₃×Dic₃	—	6	2	no	—
21	C₃×C₃⋊C₈	—	3	4	no	—
22	C₃×C₃⋊C₈	—	6	2	no	—
23	C₂×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
24	C₂×H(3,𝔽₃)	SD₁₆	1	2	yes	—
25	C₂×C₃×C₈	—	9	1	no	—
26	C₃×C₄⋊C₄	—	9	2	no	—
27	C₄⋊Dic₃	—	3	2	no	—
28	C₄⋊Dic₃	—	18	2	no	—
29	C₂×C₃⋊C₈	—	3	2	no	—
30	C₂×C₃⋊C₈	—	6	1	no	—
31	C₂²×C₃²	—	2	1	no	—
32	C₂²×C₃²	#67	1	2	yes	—
33	C₃²×C₄	—	2	1	no	—
34	C₃²×C₄	—	2	1	no	—
35	C₃²×C₄	Dic₆	1	2	yes	—
36	C₃²×C₄	C₄×D₃	1	2	yes	—
37	C₃⋊Dic₃	—	4	2	no	—
38	C₃×Dic₃	—	12	2	no	—
39	#106	—	27	1	no	Sylow
40	H(3,𝔽₃)	#106	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃×C₄	—	6	2	no	—
42	C₂×C₃×C₄	—	12	1	no	—
43	C₂×C₃×C₄	—	18	2	no	—
44	C₂×C₃×C₄	D₃²	1	1	yes	characteristic
45	C₂×Dic₃	—	6	2	no	—
46	C₂×Dic₃	—	18	2	no	—
47	C₃×C₈	—	9	2	no	—
48	C₃⋊C₈	—	3	4	no	—
49	C₃⋊C₈	—	6	2	no	—
50	C₂×C₃²	—	2	2	no	—
51	C₂×C₃²	—	2	1	no	—
52	C₂×C₃²	Dic₃⋊C₄	1	2	yes	—
53	C₂×C₃²	#214	1	2	yes	—
54	C₂×C₃²	#215	1	2	yes	—
55	C₂×C₈	—	9	1	no	—
56	C₄⋊C₄	—	9	2	no	—
57	C₂²×C₃	—	6	2	no	—
58	C₂²×C₃	—	12	1	no	—
59	C₂²×C₃	#625	1	1	yes	characteristic
60	C₃×C₄	—	6	2	no	—
61	C₃×C₄	—	6	2	no	—
62	C₃×C₄	—	12	1	no	—
63	C₃×C₄	—	12	1	no	—
64	C₃×C₄	—	36	2	no	—
65	C₃×C₄	#624	1	1	yes	characteristic
66	C₃×C₄	#627	1	1	yes	characteristic
67	Dic₃	—	12	2	no	—
68	Dic₃	—	36	2	no	—
69	C₃²	—	2	1	no	—
70	C₃²	#808	1	2	yes	complemented
71	C₂×C₄	—	18	2	no	—
72	C₂×C₄	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
73	C₈	—	9	2	no	—
74	C₂×C₃	—	6	2	no	—
75	C₂×C₃	—	6	2	no	—
76	C₂×C₃	—	6	2	no	—
77	C₂×C₃	—	12	2	no	—
78	C₂×C₃	—	12	1	no	—
79	C₂×C₃	#3715	1	2	yes	—
80	C₂×C₃	#3724	1	1	yes	characteristic
81	C₂²	#6198	1	1	yes	characteristic
82	C₄	—	36	2	no	—
83	C₄	#6196	1	1	yes	characteristic
84	C₄	#6199	1	1	yes	characteristic
85	C₃	—	6	2	no	—
86	C₃	—	12	1	no	—
87	C₃	#63574	1	1	yes	characteristic
88	C₂	#88431	1	2	yes	—
89	C₂	#88449	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#6198
commutator	C₄×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₄	D₃²
Fitting, cosocle	C₂×C₄×H(3,𝔽₃)	C₂²
socle	C₂²×C₃	#625
prime radical	C₂²×H(3,𝔽₃)	D₄
hypercenter	C₂×C₄	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	#106
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
6. #6180	C₄	semidirect
40. H(3,𝔽₃)	#106	semidirect
70. C₃²	#808	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
C₄⋊((C₃⋊Dic₃)⋊C₃)	2
#63573	3