Subgroups — #63569

Order 288.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#235	C₂	1	1	yes	maximal, characteristic
2	#3748	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#837	—	3	1	no	maximal
5	#805	—	3	1	no	maximal
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₃×#67	—	2	1	no	—
8	C₂×C₃×Dic₃	—	2	1	no	—
9	C₃×D₁₂	C₄	1	1	yes	characteristic, complemented
10	C₃×C₄×D₃	—	2	1	no	—
11	C₃×Dic₆	C₄	1	1	yes	characteristic, complemented
12	C₃⋊(C₃⋊C₈)	—	2	1	no	—
13	C₃×#41	—	3	1	no	—
14	#235	D₃	1	1	yes	characteristic
15	C₃×C₄²	—	3	1	no	—
16	C₄×Dic₃	D₃	1	1	yes	characteristic, complemented
17	#208	—	3	1	no	—
18	#208	—	3	1	no	—
19	#208	—	6	1	no	—
20	C₂²×C₃²	D₄	1	1	yes	characteristic
21	C₂×C₃×D₃	—	2	1	no	—
22	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₃²×C₄	D₄	1	1	yes	characteristic
24	C₃×Dic₃	—	2	2	no	—
25	C₃×Dic₃	—	2	1	no	—
26	#104	—	9	1	no	Sylow
27	C₃×Q₈	—	3	1	no	—
28	C₃×D₄	—	3	1	no	—
29	C₃×D₄	—	6	1	no	—
30	C₂×C₃×C₄	—	2	1	no	—
31	C₂×C₃×C₄	—	6	1	no	—
32	C₂×C₃×C₄	—	6	1	no	—
33	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
34	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
35	#67	—	2	1	no	—
36	C₂×Dic₃	—	2	1	no	—
37	D₁₂	Dic₃	1	1	yes	characteristic, complemented
38	C₄×D₃	—	2	1	no	—
39	Dic₆	Dic₃	1	1	yes	characteristic, complemented
40	C₃⋊C₈	—	6	2	no	—
41	C₃⋊C₈	—	6	1	no	—
42	C₃⋊C₈	—	6	1	no	—
43	C₂×C₃²	—	2	1	no	—
44	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
45	C₃×D₃	—	4	1	no	—
46	#41	—	3	1	no	—
47	M₁₆	—	9	1	no	—
48	C₄²	—	3	1	no	—
49	C₂²×C₃	—	2	1	no	—
50	C₂²×C₃	—	6	1	no	—
51	C₂²×C₃	#67	1	1	yes	characteristic
52	C₂²×C₃	#67	1	1	yes	characteristic
53	C₂×D₃	—	2	1	no	—
54	C₃×C₄	—	2	1	no	—
55	C₃×C₄	—	2	1	no	—
56	C₃×C₄	—	6	2	no	—
57	C₃×C₄	—	6	1	no	—
58	C₃×C₄	C₄×D₃	1	1	yes	characteristic
59	C₃×C₄	D₁₂	1	1	yes	characteristic
60	C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
61	C₃×C₄	#67	1	1	yes	characteristic
62	Dic₃	—	2	2	no	—
63	Dic₃	—	2	1	no	—
64	C₃²	#104	1	1	yes	characteristic, complemented, Sylow
65	Q₈	—	3	1	no	—
66	D₄	—	3	1	no	—
67	D₄	—	6	1	no	—
68	C₂×C₄	—	6	1	no	—
69	C₂×C₄	—	6	1	no	—
70	C₂×C₄	D₃²	1	1	yes	characteristic
71	C₈	—	18	1	no	—
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	—	4	1	no	—
76	C₂×C₃	—	12	1	no	—
77	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
78	C₂×C₃	#217	1	1	yes	characteristic
79	D₃	—	4	1	no	—
80	C₂²	—	6	1	no	—
81	C₂²	#625	1	1	yes	characteristic
82	C₄	—	6	2	no	—
83	C₄	—	6	1	no	—
84	C₄	D₃×Dic₃	1	1	yes	characteristic
85	C₄	#626	1	1	yes	characteristic
86	C₃	—	2	1	no	—
87	C₃	#805	1	1	yes	characteristic, complemented
88	C₃	#837	1	1	yes	characteristic, complemented
89	C₂	—	2	1	no	—
90	C₂	—	12	1	no	—
91	C₂	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#626
commutator	C₃²×C₄	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃²
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle	C₂×C₃²	C₂²⋊C₄
prime radical	C₃×#235	C₂
hypocenter	C₃²	#104
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. #3748	C₂	semidirect
3. C₃×C₄×Dic₃	C₂	semidirect
9. C₃×D₁₂	C₄	semidirect
11. C₃×Dic₆	C₄	semidirect
16. C₄×Dic₃	D₃	semidirect
37. D₁₂	Dic₃	semidirect
39. Dic₆	Dic₃	semidirect
64. C₃²	#104	semidirect
87. C₃	#805	semidirect
88. C₃	#837	semidirect

Maximal subgroups

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

subgroup	index
C₃×#235	2
#3748	2
C₃×C₄×Dic₃	2
#837	3
#805	3