Subgroups — #63577

Order 288.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3748	C₂	1	1	yes	maximal, characteristic
2	C₃×#208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#822	—	3	1	no	maximal
5	#819	—	3	1	no	maximal
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊C₈)	—	2	1	no	—
8	C₃×C₃⋊C₈	—	2	1	no	—
9	C₃×C₃⋊C₈	C₄	1	2	yes	—
10	C₃×M₁₆	—	3	1	no	—
11	C₂×C₃×C₈	—	3	1	no	—
12	#208	—	3	1	no	—
13	#208	—	3	1	no	—
14	#208	—	6	1	no	—
15	#208	D₃	1	1	yes	characteristic
16	C₂×C₃⋊C₈	D₃	1	1	yes	characteristic
17	C₂²×C₃²	Q₈	1	1	yes	characteristic
18	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₃²×C₄	D₄	1	1	yes	characteristic
20	#108	—	9	1	no	Sylow
21	C₂×C₃×C₄	—	2	1	no	—
22	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
23	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
24	C₃×C₈	—	3	2	no	—
25	C₃×C₈	—	6	1	no	—
26	C₃⋊C₈	—	2	1	no	—
27	C₃⋊C₈	—	6	2	no	—
28	C₃⋊C₈	—	6	1	no	—
29	C₃⋊C₈	—	6	1	no	—
30	C₃⋊C₈	Dic₃	1	2	yes	—
31	C₂×C₃²	—	2	1	no	—
32	C₂×C₃²	C₄⋊C₄	1	1	yes	characteristic
33	M₁₆	—	3	1	no	—
34	M₁₆	—	9	1	no	—
35	C₂×C₈	—	3	1	no	—
36	C₂²×C₃	—	2	1	no	—
37	C₂²×C₃	Dic₆	1	1	yes	characteristic
38	C₂²×C₃	Dic₆	1	1	yes	characteristic
39	C₃×C₄	—	2	1	no	—
40	C₃×C₄	—	2	1	no	—
41	C₃×C₄	C₄×D₃	1	1	yes	characteristic
42	C₃×C₄	D₁₂	1	1	yes	characteristic
43	C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
44	C₃×C₄	#67	1	1	yes	characteristic
45	C₃²	#108	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄	D₃²	1	1	yes	characteristic
47	C₈	—	3	2	no	—
48	C₈	—	6	1	no	—
49	C₈	—	18	1	no	—
50	C₂×C₃	—	2	1	no	—
51	C₂×C₃	—	2	1	no	—
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	—	4	1	no	—
54	C₂×C₃	Dic₃⋊C₄	1	1	yes	characteristic
55	C₂×C₃	C₄⋊Dic₃	1	1	yes	characteristic
56	C₂²	#627	1	1	yes	characteristic
57	C₄	D₃×Dic₃	1	1	yes	characteristic
58	C₄	#626	1	1	yes	characteristic
59	C₃	—	2	1	no	—
60	C₃	#819	1	1	yes	characteristic, complemented
61	C₃	#822	1	1	yes	characteristic, complemented
62	C₂	—	2	1	no	—
63	C₂	Dic₃⋊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	C₄⋊C₄
prime radical	C₂²×C₃²	Q₈
hypocenter	C₃²	#108
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
45. C₃²	#108	semidirect
60. C₃	#819	semidirect
61. C₃	#822	semidirect

Maximal subgroups

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

subgroup	index
#3748	2
C₃×#208	2
C₂×C₃×C₃⋊C₈	2
#822	3
#819	3