Subgroups — #63863

Order 288.

167 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	C₂×C₃×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₃)⋊Dic₃	C₂	1	4	yes	maximal, complemented
5	#947	—	3	1	no	maximal
6	#936	—	3	1	no	maximal
7	C₂²×C₃×D₃	C₂²	1	2	yes	—
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
10	C₂×C₃×Dic₃	C₂²	1	2	yes	—
11	C₃×D₁₂	—	2	2	no	—
12	C₃×Dic₆	—	2	2	no	—
13	C₂×C₃×Q₈	—	3	1	no	—
14	C₂×C₃×D₄	—	3	1	no	—
15	C₂×D₁₂	D₃	1	1	yes	characteristic
16	C₂×Dic₆	D₃	1	1	yes	characteristic, complemented
17	#217	—	3	4	no	—
18	(C₂×D₃)⋊C₄	—	3	4	no	—
19	C₄×Dic₃	—	3	1	no	—
20	C₄×Dic₃	—	3	1	no	—
21	C₄×Dic₃	—	6	1	no	—
22	C₂²×C₃²	C₂³	1	1	yes	characteristic
23	C₂×C₃×D₃	—	2	2	no	—
24	C₂×C₃×D₃	—	2	4	no	—
25	C₃²×C₄	D₄	1	2	yes	—
26	C₃⋊Dic₃	—	2	2	no	—
27	C₃×Dic₃	—	2	2	no	—
28	#124	—	9	1	no	Sylow
29	C₂³×C₃	—	3	2	no	—
30	C₂²×D₃	C₂×D₃	1	2	yes	—
31	C₃×Q₈	—	6	2	no	—
32	C₃×D₄	—	6	2	no	—
33	C₂×C₃×C₄	—	2	1	no	—
34	C₂×C₃×C₄	—	3	2	no	—
35	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂×Dic₃	—	3	2	no	—
38	C₂×Dic₃	—	3	2	no	—
39	C₂×Dic₃	—	6	2	no	—
40	C₂×Dic₃	C₂×D₃	1	2	yes	—
41	D₁₂	—	2	2	no	—
42	Dic₆	—	2	2	no	—
43	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃²	#41	1	2	yes	—
45	C₃×D₃	—	4	2	no	—
46	C₂×Q₈	—	3	1	no	—
47	C₂×D₄	—	3	1	no	—
48	C₂²⋊C₄	—	9	4	no	—
49	C₄²	—	9	1	no	—
50	C₂²×C₃	—	2	1	no	—
51	C₂²×C₃	—	6	4	no	—
52	C₂²×C₃	—	6	2	no	—
53	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
54	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
55	C₂×D₃	—	2	4	no	—
56	C₂×D₃	—	2	2	no	—
57	C₃×C₄	—	2	2	no	—
58	C₃×C₄	—	6	2	no	—
59	C₃×C₄	#67	1	2	yes	—
60	C₃×C₄	#67	1	2	yes	—
61	Dic₃	—	2	2	no	—
62	Dic₃	—	6	4	no	—
63	Dic₃	—	6	2	no	—
64	Dic₃	—	6	2	no	—
65	C₃²	#124	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	3	2	no	—
67	Q₈	—	6	2	no	—
68	D₄	—	6	2	no	—
69	C₂×C₄	—	3	2	no	—
70	C₂×C₄	—	9	2	no	—
71	C₂×C₄	D₃²	1	1	yes	characteristic
72	C₂×C₃	—	2	2	no	—
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	12	2	no	—
75	C₂×C₃	#237	1	2	yes	—
76	C₂×C₃	#239	1	2	yes	—
77	C₂×C₃	C₂×#67	1	1	yes	characteristic
78	C₂×C₃	C₂×#67	1	1	yes	characteristic
79	D₃	—	4	2	no	—
80	C₂²	—	6	4	no	—
81	C₂²	—	6	2	no	—
82	C₂²	C₂×D₃²	1	1	yes	characteristic
83	C₄	—	6	2	no	—
84	C₄	—	18	2	no	—
85	C₄	#625	1	2	yes	—
86	C₃	—	2	1	no	—
87	C₃	#936	1	1	yes	characteristic, complemented
88	C₃	#947	1	1	yes	characteristic, complemented
89	C₂	—	12	2	no	—
90	C₂	#3796	1	2	yes	—
91	C₂	C₂×#625	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃²
commutator, socle, cosocle	C₂²×C₃²	C₂³
Fitting	C₂×C₃²×C₄	C₂²
prime radical	C₂×C₃×D₁₂	C₂
hypercenter	C₂×C₄	D₃²
hypocenter	C₃²	#124
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×Dic₆	C₂	semidirect
3. C₄×C₃⋊Dic₃	C₂	semidirect
4. (C₂×D₃)⋊Dic₃	C₂	semidirect
16. C₂×Dic₆	D₃	semidirect
65. C₃²	#124	semidirect
87. C₃	#936	semidirect
88. C₃	#947	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
C₂×C₃×Dic₆	2
C₄×C₃⋊Dic₃	2
(C₂×D₃)⋊Dic₃	2
#947	3
#936	3