Subgroups — #63560

Order 288.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Dic₆	C₂	1	1	yes	maximal, characteristic
2	#3748	C₂	1	1	yes	maximal, characteristic
3	C₃×#208	C₂	1	1	yes	maximal, characteristic
4	#836	—	3	1	no	maximal
5	#824	—	3	1	no	maximal
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃×Dic₃	C₄	1	2	yes	—
8	C₃×Dic₆	—	2	1	no	—
9	C₃×Dic₆	—	2	1	no	—
10	C₃⋊(C₃⋊C₈)	—	2	1	no	—
11	C₃×C₃⋊C₈	—	2	1	no	—
12	C₂×C₃×Q₈	—	3	1	no	—
13	C₂×Dic₆	D₃	1	1	yes	characteristic
14	C₃×M₁₆	—	3	1	no	—
15	#208	—	3	1	no	—
16	#208	—	3	1	no	—
17	#208	—	6	1	no	—
18	#208	D₃	1	1	yes	characteristic
19	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
20	C₃²×C₄	D₄	1	1	yes	characteristic
21	C₃²×C₄	D₄	1	1	yes	characteristic
22	C₃×Dic₃	—	2	2	no	—
23	#101	—	9	1	no	Sylow
24	C₃×Q₈	—	6	1	no	—
25	C₃×Q₈	—	6	1	no	—
26	C₂×C₃×C₄	—	2	1	no	—
27	C₂×C₃×C₄	—	3	2	no	—
28	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
30	C₂×Dic₃	Dic₃	1	2	yes	—
31	Dic₆	—	2	1	no	—
32	Dic₆	—	2	1	no	—
33	C₃×C₈	—	6	1	no	—
34	C₃⋊C₈	—	2	1	no	—
35	C₃⋊C₈	—	6	2	no	—
36	C₃⋊C₈	—	6	1	no	—
37	C₃⋊C₈	—	6	1	no	—
38	C₂×C₃²	—	2	1	no	—
39	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×Q₈	—	3	1	no	—
41	M₁₆	—	3	1	no	—
42	M₁₆	—	9	1	no	—
43	C₂²×C₃	—	2	1	no	—
44	C₂²×C₃	C₄×D₃	1	1	yes	characteristic
45	C₂²×C₃	C₂×Dic₃	1	1	yes	characteristic
46	C₃×C₄	—	2	1	no	—
47	C₃×C₄	—	2	1	no	—
48	C₃×C₄	—	6	2	no	—
49	C₃×C₄	D₁₂	1	1	yes	characteristic
50	C₃×C₄	#67	1	1	yes	characteristic
51	C₃×C₄	#67	1	1	yes	characteristic
52	C₃×C₄	#67	1	1	yes	characteristic
53	Dic₃	—	2	2	no	—
54	C₃²	#101	1	1	yes	characteristic, complemented, Sylow
55	Q₈	—	6	1	no	—
56	Q₈	—	6	1	no	—
57	C₂×C₄	—	3	2	no	—
58	C₂×C₄	D₃²	1	1	yes	characteristic
59	C₈	—	6	1	no	—
60	C₈	—	18	1	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	—	2	1	no	—
64	C₂×C₃	—	4	1	no	—
65	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
66	C₂×C₃	#217	1	1	yes	characteristic
67	C₂²	D₃×Dic₃	1	1	yes	characteristic
68	C₄	—	6	2	no	—
69	C₄	#625	1	1	yes	characteristic
70	C₄	#626	1	1	yes	characteristic
71	C₃	—	2	1	no	—
72	C₃	#824	1	1	yes	characteristic, complemented
73	C₃	#836	1	1	yes	characteristic, complemented
74	C₂	—	2	1	no	—
75	C₂	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×D₃)⋊Dic₃
commutator, prime radical	C₂²×C₃²	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃²
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle	C₂×C₃²	C₂²⋊C₄
hypocenter	C₃²	#101
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
54. C₃²	#101	semidirect
72. C₃	#824	semidirect
73. C₃	#836	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Dic₆	2
#3748	2
C₃×#208	2
#836	3
#824	3