Subgroups — #10589134

Order 576.

105 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#63639	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃×#812	C₂	1	1	yes	maximal, characteristic
4	#4511	—	3	1	no	maximal
5	#4463	—	3	1	no	maximal
6	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
8	C₂×C₃×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
10	C₃×C₃⋊C₁₆	—	2	1	no	—
11	C₃×M₃₂	—	3	1	no	—
12	C₃×C₄×C₈	—	3	1	no	—
13	C₈×Dic₃	D₃	1	1	yes	characteristic
14	#812	—	3	1	no	—
15	#812	—	3	1	no	—
16	#812	—	6	1	no	—
17	#812	D₃	1	1	yes	characteristic
18	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×Dic₃	—	2	1	no	—
20	C₃²×C₈	D₄	1	1	yes	characteristic
21	C₃²×C₈	Q₈	1	1	yes	characteristic
22	C₃×C₃⋊C₈	C₈	1	2	yes	—
23	#364	—	9	1	no	Sylow
24	C₂×C₃×C₈	—	2	1	no	—
25	C₂×C₃×C₈	—	3	1	no	—
26	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
27	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
28	C₃×C₄²	—	3	1	no	—
29	C₄×Dic₃	Dic₃	1	1	yes	characteristic
30	C₂×C₃⋊C₈	Dic₃	1	1	yes	characteristic
31	C₃×C₁₆	—	6	1	no	—
32	C₃⋊C₁₆	—	2	1	no	—
33	C₃⋊C₁₆	—	6	2	no	—
34	C₃⋊C₁₆	—	6	1	no	—
35	C₃⋊C₁₆	—	6	1	no	—
36	C₂²×C₃²	M₁₆	1	1	yes	characteristic
37	C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
38	C₃²×C₄	C₂×C₈	1	1	yes	characteristic
39	C₃×Dic₃	—	2	2	no	—
40	M₃₂	—	3	1	no	—
41	M₃₂	—	9	1	no	—
42	C₄×C₈	—	3	1	no	—
43	C₂×C₃×C₄	—	2	1	no	—
44	C₂×C₃×C₄	—	6	1	no	—
45	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
47	C₂×Dic₃	—	2	1	no	—
48	C₃×C₈	—	2	1	no	—
49	C₃×C₈	—	2	1	no	—
50	C₃×C₈	—	3	2	no	—
51	C₃×C₈	Dic₆	1	1	yes	characteristic
52	C₃×C₈	Dic₆	1	1	yes	characteristic
53	C₃×C₈	D₁₂	1	1	yes	characteristic
54	C₃×C₈	#67	1	1	yes	characteristic
55	C₃⋊C₈	C₃⋊C₈	1	2	yes	—
56	C₂×C₃²	—	2	1	no	—
57	C₂×C₃²	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₈	—	3	1	no	—
59	C₂×C₈	D₃²	1	1	yes	characteristic
60	C₄²	—	3	1	no	—
61	C₁₆	—	6	1	no	—
62	C₁₆	—	18	1	no	—
63	C₂²×C₃	—	2	1	no	—
64	C₂²×C₃	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
65	C₂²×C₃	#208	1	1	yes	characteristic
66	C₃×C₄	—	2	1	no	—
67	C₃×C₄	—	2	1	no	—
68	C₃×C₄	—	6	2	no	—
69	C₃×C₄	D₃×C₈	1	1	yes	characteristic
70	C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
71	C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
72	C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
73	Dic₃	—	2	2	no	—
74	C₃²	#364	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄	—	6	1	no	—
76	C₂×C₄	D₃×Dic₃	1	1	yes	characteristic
77	C₈	—	3	2	no	—
78	C₈	#626	1	1	yes	characteristic
79	C₈	#627	1	1	yes	characteristic
80	C₂×C₃	—	2	1	no	—
81	C₂×C₃	—	2	1	no	—
82	C₂×C₃	—	2	1	no	—
83	C₂×C₃	—	4	1	no	—
84	C₂×C₃	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
85	C₂×C₃	Dic₃⋊C₈	1	1	yes	characteristic
86	C₂²	#3711	1	1	yes	characteristic
87	C₄	—	6	2	no	—
88	C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
89	C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
90	C₃	—	2	1	no	—
91	C₃	#4463	1	1	yes	characteristic, complemented
92	C₃	#4511	1	1	yes	characteristic, complemented
93	C₂	—	2	1	no	—
94	C₂	Dic₃⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#626
commutator	C₃²×C₄	C₂×C₈
Frattini, hypercenter	C₂×C₈	D₃²
Fitting, cosocle	C₂×C₃²×C₈	C₂²
socle	C₂×C₃²	C₄⋊C₈
prime radical	C₂²×C₃²	M₁₆
hypocenter	C₃²	#364
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
74. C₃²	#364	semidirect
91. C₃	#4463	semidirect
92. C₃	#4511	semidirect

Maximal subgroups

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

subgroup	index
#63639	2
C₃×C₈×Dic₃	2
C₃×#812	2
#4511	3
#4463	3