Subgroups — C₂×#88864 #11723993

Order 864.

1848 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃³	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal, complemented
3	#88864	C₂	1	4	yes	maximal, direct product factor
4	C₂×#3757	—	3	13	no	maximal
5	C₂³×C₃³	—	2	4	no	—
6	C₂³×C₃³	C₄	1	4	yes	complemented
7	C₂³×C₃³	C₂²	1	2	yes	—
8	C₂³×C₃³	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
10	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	4	yes	complemented
11	C₂⁴×C₃²	D₃	1	13	yes	—
12	C₂²×C₃⋊Dic₃	—	3	26	no	—
13	#3757	—	3	52	no	—
14	C₂²×C₃³	—	2	8	no	—
15	C₂²×C₃³	—	2	4	no	—
16	C₂²×C₃³	C₂×C₄	1	2	yes	—
17	C₂²×C₃³	C₂×C₄	1	4	yes	complemented
18	C₂²×C₃³	D₄	1	4	yes	—
19	C₂²×C₃³	C₂³	1	1	yes	characteristic
20	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
21	C₂×#217	—	9	13	no	—
22	C₂³×C₃²	—	2	52	no	—
23	C₂³×C₃²	Dic₃	1	52	yes	complemented
24	C₂³×C₃²	C₂×D₃	1	26	yes	—
25	C₂³×C₃²	C₂×D₃	1	13	yes	—
26	C₂×C₃⋊Dic₃	—	3	52	no	—
27	C₂×C₃⋊Dic₃	—	6	52	no	—
28	C₂×C₃³	—	2	4	no	—
29	C₂×C₃³	C₂²⋊C₄	1	4	yes	complemented
30	C₂×C₃³	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₃³	C₂×D₄	1	2	yes	—
32	C₂⁴×C₃	C₃⋊D₃	1	13	yes	—
33	C₂²×Dic₃	—	9	26	no	—
34	#217	—	9	52	no	—
35	C₂²×C₃²	—	2	104	no	—
36	C₂²×C₃²	—	2	52	no	—
37	C₂²×C₃²	C₂×Dic₃	1	26	yes	—
38	C₂²×C₃²	C₂×Dic₃	1	52	yes	complemented
39	C₂²×C₃²	#67	1	52	yes	—
40	C₂²×C₃²	C₂²×D₃	1	13	yes	—
41	C₃⋊Dic₃	—	6	52	no	—
42	C₂×C₂²⋊C₄	—	27	1	no	Sylow
43	C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
44	C₂³×C₃	—	2	52	no	—
45	C₂³×C₃	C₃⋊Dic₃	1	52	yes	complemented
46	C₂³×C₃	C₂×C₃⋊D₃	1	26	yes	—
47	C₂³×C₃	C₂×C₃⋊D₃	1	13	yes	—
48	C₂×Dic₃	—	9	52	no	—
49	C₂×Dic₃	—	18	52	no	—
50	C₂×C₃²	—	2	52	no	—
51	C₂×C₃²	#217	1	52	yes	complemented
52	C₂×C₃²	C₂²×Dic₃	1	13	yes	—
53	C₂×C₃²	C₂×#67	1	26	yes	—
54	C₂⁴	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
55	C₂²×C₄	—	27	2	no	—
56	C₂²⋊C₄	—	27	4	no	—
57	C₂²×C₃	—	2	104	no	—
58	C₂²×C₃	—	2	52	no	—
59	C₂²×C₃	C₂×C₃⋊Dic₃	1	26	yes	—
60	C₂²×C₃	C₂×C₃⋊Dic₃	1	52	yes	complemented
61	C₂²×C₃	#638	1	52	yes	—
62	C₂²×C₃	C₂²×C₃⋊D₃	1	13	yes	—
63	Dic₃	—	18	52	no	—
64	C₃²	C₂×#217	1	13	yes	complemented
65	C₂³	—	2	4	no	—
66	C₂³	C₃⋊(C₃⋊Dic₃)	1	4	yes	complemented
67	C₂³	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
68	C₂³	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
69	C₂×C₄	—	27	4	no	—
70	C₂×C₄	—	54	4	no	—
71	C₂×C₃	—	2	52	no	—
72	C₂×C₃	#3757	1	52	yes	complemented
73	C₂×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
74	C₂×C₃	C₂×#638	1	26	yes	—
75	C₂²	—	2	8	no	—
76	C₂²	—	2	4	no	—
77	C₂²	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
78	C₂²	C₂×C₃⋊(C₃⋊Dic₃)	1	4	yes	complemented
79	C₂²	#6312	1	4	yes	—
80	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
81	C₄	—	54	4	no	—
82	C₃	C₂×#3757	1	13	yes	complemented
83	C₂	—	2	4	no	—
84	C₂	#88864	1	4	yes	direct product factor
85	C₂	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
86	C₂	C₂×#6312	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₂²×C₄
Frattini	C₂²	C₂²×C₃⋊(C₃⋊D₃)
Fitting, prime radical	C₂⁴×C₃³	C₂
socle	C₂³×C₃³	C₂²
cosocle	C₂²×C₃³	C₂³
hypercenter	C₂⁴	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
3. #88864	C₂	direct (N × H)
6. C₂³×C₃³	C₄	semidirect
10. C₂×C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
17. C₂²×C₃³	C₂×C₄	semidirect
23. C₂³×C₃²	Dic₃	semidirect
29. C₂×C₃³	C₂²⋊C₄	semidirect
38. C₂²×C₃²	C₂×Dic₃	semidirect
43. C₃³	C₂×C₂²⋊C₄	semidirect
45. C₂³×C₃	C₃⋊Dic₃	semidirect
51. C₂×C₃²	#217	semidirect
60. C₂²×C₃	C₂×C₃⋊Dic₃	semidirect
64. C₃²	C₂×#217	semidirect
66. C₂³	C₃⋊(C₃⋊Dic₃)	semidirect
72. C₂×C₃	#3757	semidirect
78. C₂²	C₂×C₃⋊(C₃⋊Dic₃)	semidirect
82. C₃	C₂×#3757	semidirect
84. C₂	#88864	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃³	2
C₂²×C₃⋊(C₃⋊Dic₃)	2
#88864	2
C₂×#3757	3

Subgroups — C2×#88864 #11723993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#88864 #11723993