Subgroups — #11720315

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	#63575	—	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	#818	—	9	1	no	—
16	#807	—	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)	D₈	1	1	yes	characteristic
25	C₂×H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic
26	C₂×C₃×C₈	—	9	1	no	—
27	C₃×C₄⋊C₄	—	9	2	no	—
28	C₄⋊Dic₃	—	3	2	no	—
29	C₄⋊Dic₃	—	18	2	no	—
30	C₂×C₃⋊C₈	—	3	2	no	—
31	C₂×C₃⋊C₈	—	6	1	no	—
32	C₂²×C₃²	—	2	1	no	—
33	C₂²×C₃²	#67	1	2	yes	—
34	C₃²×C₄	—	2	1	no	—
35	C₃²×C₄	—	2	1	no	—
36	C₃²×C₄	Dic₆	1	2	yes	—
37	C₃²×C₄	C₄×D₃	1	2	yes	—
38	C₃⋊Dic₃	—	4	2	no	—
39	C₃×Dic₃	—	12	2	no	—
40	#107	—	27	1	no	Sylow
41	H(3,𝔽₃)	#107	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₃×C₄	—	6	1	no	—
43	C₂×C₃×C₄	—	6	2	no	—
44	C₂×C₃×C₄	—	18	2	no	—
45	C₂×C₃×C₄	D₃²	1	1	yes	characteristic
46	C₂×Dic₃	—	6	2	no	—
47	C₂×Dic₃	—	18	2	no	—
48	C₃×C₈	—	9	2	no	—
49	C₃⋊C₈	—	3	4	no	—
50	C₃⋊C₈	—	6	2	no	—
51	C₂×C₃²	—	2	1	no	—
52	C₂×C₃²	—	2	1	no	—
53	C₂×C₃²	—	2	1	no	—
54	C₂×C₃²	Dic₃⋊C₄	1	2	yes	—
55	C₂×C₃²	#213	1	2	yes	—
56	C₂×C₃²	#216	1	2	yes	—
57	C₂×C₈	—	9	1	no	—
58	C₄⋊C₄	—	9	2	no	—
59	C₂²×C₃	—	6	2	no	—
60	C₂²×C₃	—	6	1	no	—
61	C₂²×C₃	#625	1	1	yes	characteristic
62	C₃×C₄	—	6	1	no	—
63	C₃×C₄	—	6	2	no	—
64	C₃×C₄	—	6	2	no	—
65	C₃×C₄	—	6	1	no	—
66	C₃×C₄	—	36	2	no	—
67	C₃×C₄	#624	1	1	yes	characteristic
68	C₃×C₄	#627	1	1	yes	characteristic
69	Dic₃	—	12	2	no	—
70	Dic₃	—	36	2	no	—
71	C₃²	—	2	1	no	—
72	C₃²	#807	1	2	yes	complemented
73	C₂×C₄	—	18	2	no	—
74	C₂×C₄	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
75	C₈	—	9	2	no	—
76	C₂×C₃	—	6	2	no	—
77	C₂×C₃	—	6	1	no	—
78	C₂×C₃	—	6	1	no	—
79	C₂×C₃	—	6	2	no	—
80	C₂×C₃	—	6	1	no	—
81	C₂×C₃	—	6	2	no	—
82	C₂×C₃	#3713	1	1	yes	characteristic
83	C₂×C₃	#3718	1	1	yes	characteristic
84	C₂×C₃	#3724	1	1	yes	characteristic
85	C₂²	#6198	1	1	yes	characteristic
86	C₄	—	36	2	no	—
87	C₄	#6196	1	1	yes	characteristic
88	C₄	#6199	1	1	yes	characteristic
89	C₃	—	6	1	no	—
90	C₃	—	6	2	no	—
91	C₃	#63576	1	1	yes	characteristic
92	C₂	#88432	1	1	yes	characteristic
93	C₂	#88433	1	1	yes	characteristic
94	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)	#107
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
41. H(3,𝔽₃)	#107	semidirect
72. C₃²	#807	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
#63575	3