Subgroups — C₂×#88529 #11721778

Order 864.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#272	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#88529	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×D₂₄	—	3	4	no	maximal
5	C₂²×#272	—	2	2	no	—
6	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₄⋊#272	—	2	2	no	—
8	C₄⋊#272	C₂²	1	4	yes	complemented
9	C₈×H(3,𝔽₃)	C₂²	1	2	yes	complemented
10	C₂×C₃×D₁₂	—	3	8	no	—
11	C₂×C₃²×C₈	D₃	1	4	yes	complemented
12	C₃×D₂₄	—	3	16	no	—
13	C₂²×H(3,𝔽₃)	D₄	1	1	yes	characteristic
14	C₂×#272	—	2	4	no	—
15	C₂×#272	—	4	4	no	—
16	C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
18	C₂×C₃×D₈	—	9	1	no	—
19	C₂×D₂₄	—	3	4	no	—
20	C₂²×C₃×D₃	—	6	8	no	—
21	C₂×C₃²×C₄	C₂×D₃	1	4	yes	—
22	C₃×D₁₂	—	3	16	no	—
23	C₃×D₁₂	—	6	8	no	—
24	C₃²×C₈	C₂×D₃	1	8	yes	complemented
25	C₂×H(3,𝔽₃)	D₈	1	2	yes	complemented
26	C₂×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
27	#272	—	4	4	no	—
28	C₂×C₃×D₄	—	9	2	no	—
29	C₂×D₁₂	—	3	8	no	—
30	C₃×D₈	—	9	4	no	—
31	C₂×C₃×C₈	—	3	4	no	—
32	C₂×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
33	D₂₄	—	3	16	no	—
34	C₂²×C₃²	D₁₂	1	4	yes	—
35	C₂×C₃×D₃	—	6	16	no	—
36	C₂×C₃×D₃	—	12	16	no	—
37	C₃²×C₄	D₁₂	1	4	yes	—
38	C₃²×C₄	C₂²×D₃	1	4	yes	—
39	C₂×D₈	—	9	1	no	Sylow
40	H(3,𝔽₃)	C₂×D₈	1	1	yes	characteristic, complemented, Sylow
41	C₂³×C₃	—	18	2	no	—
42	C₂²×D₃	—	6	8	no	—
43	C₃×D₄	—	9	4	no	—
44	C₃×D₄	—	18	2	no	—
45	C₂×C₃×C₄	—	3	4	no	—
46	C₂×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
47	D₁₂	—	3	16	no	—
48	D₁₂	—	6	8	no	—
49	C₃×C₈	—	3	8	no	—
50	C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
51	C₂×C₃²	D₂₄	1	8	yes	complemented
52	C₂×C₃²	C₂×D₁₂	1	4	yes	—
53	C₃×D₃	—	12	16	no	—
54	C₂×D₄	—	9	2	no	—
55	D₈	—	9	4	no	—
56	C₂×C₈	#272	1	1	yes	characteristic, complemented
57	C₂²×C₃	—	3	4	no	—
58	C₂²×C₃	—	18	4	no	—
59	C₂²×C₃	—	36	4	no	—
60	C₂²×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
61	C₂×D₃	—	6	16	no	—
62	C₂×D₃	—	12	16	no	—
63	C₃×C₄	—	3	4	no	—
64	C₃×C₄	—	3	4	no	—
65	C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
66	C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
67	C₃²	C₂×D₂₄	1	4	yes	complemented
68	C₂³	—	18	2	no	—
69	D₄	—	9	4	no	—
70	D₄	—	18	2	no	—
71	C₂×C₄	C₂×#272	1	1	yes	characteristic
72	C₈	C₂×#272	1	2	yes	complemented
73	C₂×C₃	—	3	4	no	—
74	C₂×C₃	—	3	8	no	—
75	C₂×C₃	—	36	4	no	—
76	C₂×C₃	#3745	1	2	yes	—
77	C₂×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
78	D₃	—	12	16	no	—
79	C₂²	—	18	4	no	—
80	C₂²	—	36	4	no	—
81	C₂²	C₄⋊#272	1	1	yes	characteristic
82	C₄	C₄⋊#272	1	1	yes	characteristic
83	C₄	C₂²×#272	1	1	yes	characteristic
84	C₃	—	3	4	no	—
85	C₃	C₂×#3745	1	1	yes	characteristic
86	C₂	—	36	4	no	—
87	C₂	#88529	1	2	yes	direct product factor
88	C₂	C₂×C₄⋊#272	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₄⋊(C₃⋊D₃)
commutator, cosocle	C₄×H(3,𝔽₃)	C₂³
Frattini	C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₈×H(3,𝔽₃)	C₂
hypercenter	C₂×C₃×C₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×D₈
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#272	C₂	semidirect
2. C₂×C₈×H(3,𝔽₃)	C₂	semidirect
3. #88529	C₂	direct (N × H)
8. C₄⋊#272	C₂²	semidirect
9. C₈×H(3,𝔽₃)	C₂²	semidirect
11. C₂×C₃²×C₈	D₃	semidirect
24. C₃²×C₈	C₂×D₃	semidirect
25. C₂×H(3,𝔽₃)	D₈	semidirect
40. H(3,𝔽₃)	C₂×D₈	semidirect
51. C₂×C₃²	D₂₄	semidirect
56. C₂×C₈	#272	semidirect
67. C₃²	C₂×D₂₄	semidirect
72. C₈	C₂×#272	semidirect
87. C₂	#88529	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⋊#272	2
C₂×C₈×H(3,𝔽₃)	2
#88529	2
C₂×C₃×D₂₄	3

Subgroups — C2×#88529 #11721778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#88529 #11721778