Subgroups — C₂×#88852 #11723944

Order 864.

1064 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	2	yes	maximal, complemented
2	C₂×C₃³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#88852	C₂	1	4	yes	maximal, direct product factor
4	C₂×#3745	—	3	13	no	maximal
5	C₂²×C₃⋊(C₃⋊D₃)	—	2	2	no	—
6	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	2	no	—
8	C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	4	yes	complemented
9	C₃³×C₈	C₂²	1	2	yes	complemented
10	C₂×C₄⋊(C₃⋊D₃)	—	3	26	no	—
11	C₂×C₃²×C₈	D₃	1	13	yes	complemented
12	#3745	—	3	52	no	—
13	C₂²×C₃³	D₄	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊D₃)	—	2	4	no	—
15	C₂×C₃⋊(C₃⋊D₃)	—	4	4	no	—
16	C₃³×C₄	D₄	1	1	yes	characteristic
17	C₃³×C₄	C₂³	1	1	yes	characteristic
18	C₂×D₂₄	—	9	13	no	—
19	C₂²×C₃⋊D₃	—	6	26	no	—
20	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
21	C₄⋊(C₃⋊D₃)	—	3	52	no	—
22	C₄⋊(C₃⋊D₃)	—	6	26	no	—
23	C₃²×C₈	C₂×D₃	1	26	yes	complemented
24	C₂×C₃³	D₈	1	2	yes	complemented
25	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
26	C₃⋊(C₃⋊D₃)	—	4	4	no	—
27	C₂×D₁₂	—	9	26	no	—
28	C₂×C₃×C₈	C₃⋊D₃	1	13	yes	complemented
29	D₂₄	—	9	52	no	—
30	C₂²×C₃²	D₁₂	1	13	yes	—
31	C₂×C₃⋊D₃	—	6	52	no	—
32	C₂×C₃⋊D₃	—	12	52	no	—
33	C₃²×C₄	D₁₂	1	13	yes	—
34	C₃²×C₄	C₂²×D₃	1	13	yes	—
35	C₂×D₈	—	27	1	no	Sylow
36	C₃³	C₂×D₈	1	1	yes	characteristic, complemented, Sylow
37	C₂²×D₃	—	18	26	no	—
38	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
39	D₁₂	—	9	52	no	—
40	D₁₂	—	18	26	no	—
41	C₃×C₈	C₂×C₃⋊D₃	1	26	yes	complemented
42	C₂×C₃²	D₂₄	1	26	yes	complemented
43	C₂×C₃²	C₂×D₁₂	1	13	yes	—
44	C₃⋊D₃	—	12	52	no	—
45	C₂×D₄	—	27	2	no	—
46	D₈	—	27	4	no	—
47	C₂×C₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
48	C₂²×C₃	C₄⋊(C₃⋊D₃)	1	13	yes	—
49	C₂×D₃	—	18	52	no	—
50	C₂×D₃	—	36	52	no	—
51	C₃×C₄	C₄⋊(C₃⋊D₃)	1	13	yes	—
52	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
53	C₃²	C₂×D₂₄	1	13	yes	complemented
54	C₂³	—	54	2	no	—
55	D₄	—	27	4	no	—
56	D₄	—	54	2	no	—
57	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
58	C₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
59	C₂×C₃	#3745	1	26	yes	complemented
60	C₂×C₃	C₂×C₄⋊(C₃⋊D₃)	1	13	yes	—
61	D₃	—	36	52	no	—
62	C₂²	—	54	4	no	—
63	C₂²	—	108	4	no	—
64	C₂²	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
65	C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
66	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
67	C₃	C₂×#3745	1	13	yes	complemented
68	C₂	—	108	4	no	—
69	C₂	#88852	1	2	yes	direct product factor
70	C₂	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊(C₃⋊(C₃⋊D₃))
commutator, cosocle	C₃³×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×C₈	C₂
socle	C₂²×C₃³	D₄
hypercenter	C₂×C₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×D₈
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
2. C₂×C₃³×C₈	C₂	semidirect
3. #88852	C₂	direct (N × H)
8. C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	semidirect
9. C₃³×C₈	C₂²	semidirect
11. C₂×C₃²×C₈	D₃	semidirect
23. C₃²×C₈	C₂×D₃	semidirect
24. C₂×C₃³	D₈	semidirect
28. C₂×C₃×C₈	C₃⋊D₃	semidirect
36. C₃³	C₂×D₈	semidirect
41. C₃×C₈	C₂×C₃⋊D₃	semidirect
42. C₂×C₃²	D₂₄	semidirect
47. C₂×C₈	C₃⋊(C₃⋊D₃)	semidirect
53. C₃²	C₂×D₂₄	semidirect
58. C₈	C₂×C₃⋊(C₃⋊D₃)	semidirect
59. C₂×C₃	#3745	semidirect
67. C₃	C₂×#3745	semidirect
69. C₂	#88852	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊D₃))	2
C₂×C₃³×C₈	2
#88852	2
C₂×#3745	3

Subgroups — C2×#88852 #11723944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#88852 #11723944