Subgroups — #63558

Order 288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₁₂	C₂	1	1	yes	maximal, characteristic
2	#3748	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	#833	—	3	1	no	maximal
5	#823	—	3	1	no	maximal
6	C₂²×C₃×D₃	C₄	1	2	yes	—
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₃×D₁₂	—	2	1	no	—
9	C₃×D₁₂	—	2	1	no	—
10	C₃⋊(C₃⋊C₈)	—	2	1	no	—
11	C₃×C₃⋊C₈	—	2	1	no	—
12	C₂×C₃×D₄	—	3	1	no	—
13	C₂×D₁₂	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, complemented
19	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×D₃	—	2	2	no	—
21	C₂×C₃×D₃	—	4	2	no	—
22	C₃²×C₄	D₄	1	1	yes	characteristic
23	C₃²×C₄	D₄	1	1	yes	characteristic
24	#100	—	9	1	no	Sylow
25	C₂³×C₃	—	3	2	no	—
26	C₂²×D₃	Dic₃	1	2	yes	—
27	C₃×D₄	—	6	1	no	—
28	C₃×D₄	—	6	1	no	—
29	C₂×C₃×C₄	—	2	1	no	—
30	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
32	D₁₂	—	2	1	no	—
33	D₁₂	—	2	1	no	—
34	C₃×C₈	—	6	1	no	—
35	C₃⋊C₈	—	2	1	no	—
36	C₃⋊C₈	—	6	2	no	—
37	C₃⋊C₈	—	6	1	no	—
38	C₃⋊C₈	—	6	1	no	—
39	C₂×C₃²	—	2	1	no	—
40	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
41	C₃×D₃	—	4	2	no	—
42	C₂×D₄	—	3	1	no	—
43	M₁₆	—	3	1	no	—
44	M₁₆	—	9	1	no	—
45	C₂²×C₃	—	2	1	no	—
46	C₂²×C₃	—	6	2	no	—
47	C₂²×C₃	—	12	2	no	—
48	C₂²×C₃	C₄×D₃	1	1	yes	characteristic
49	C₂²×C₃	C₂×Dic₃	1	1	yes	characteristic
50	C₂×D₃	—	2	2	no	—
51	C₂×D₃	—	4	2	no	—
52	C₃×C₄	—	2	1	no	—
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	D₁₂	1	1	yes	characteristic
55	C₃×C₄	#67	1	1	yes	characteristic
56	C₃×C₄	#67	1	1	yes	characteristic
57	C₃×C₄	#67	1	1	yes	characteristic
58	C₃²	#100	1	1	yes	characteristic, complemented, Sylow
59	C₂³	—	3	2	no	—
60	D₄	—	6	1	no	—
61	D₄	—	6	1	no	—
62	C₂×C₄	D₃²	1	1	yes	characteristic
63	C₈	—	6	1	no	—
64	C₈	—	18	1	no	—
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	—	2	1	no	—
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	—	4	1	no	—
69	C₂×C₃	—	12	2	no	—
70	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
71	C₂×C₃	#217	1	1	yes	characteristic
72	D₃	—	4	2	no	—
73	C₂²	—	6	2	no	—
74	C₂²	—	12	2	no	—
75	C₂²	D₃×Dic₃	1	1	yes	characteristic
76	C₄	#625	1	1	yes	characteristic
77	C₄	#626	1	1	yes	characteristic
78	C₃	—	2	1	no	—
79	C₃	#823	1	1	yes	characteristic, complemented
80	C₃	#833	1	1	yes	characteristic, complemented
81	C₂	—	2	1	no	—
82	C₂	—	12	2	no	—
83	C₂	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×D₃)⋊Dic₃
commutator	C₂²×C₃²	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃²
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle	C₂×C₃²	C₂²⋊C₄
prime radical	C₂×C₃×D₁₂	C₂
hypocenter	C₃²	#100
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
2. #3748	C₂	semidirect
3. C₃×#208	C₂	semidirect
18. #208	D₃	semidirect
58. C₃²	#100	semidirect
79. C₃	#823	semidirect
80. C₃	#833	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₁₂	2
#3748	2
C₃×#208	2
#833	3
#823	3