Subgroups — #64236

Order 288.

115 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#625	C₂	1	1	yes	maximal, characteristic, complemented
2	#3805	C₂	1	1	yes	maximal, characteristic, complemented
3	#3792	C₂	1	1	yes	maximal, characteristic, complemented
4	#3775	C₂	1	2	yes	maximal, complemented
5	#3774	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃×D₃	—	2	1	no	—
7	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
8	C₃×#67	—	2	1	no	—
9	#627	C₂²	1	1	yes	characteristic, complemented
10	#625	—	2	1	no	—
11	#625	C₂²	1	2	yes	—
12	#625	C₂²	1	1	yes	characteristic, complemented
13	D₃×Dic₃	—	2	1	no	—
14	#622	C₂²	1	2	yes	complemented
15	C₂×#67	—	6	1	no	—
16	#237	—	6	1	no	—
17	C₂²×C₃²	D₄	1	1	yes	characteristic
18	C₂×C₃×D₃	—	2	1	no	—
19	C₂×C₃×D₃	—	2	2	no	—
20	C₂×C₃×D₃	—	4	2	no	—
21	C₃⋊Dic₃	D₄	1	1	yes	characteristic
22	C₃⋊Dic₃	C₂³	1	1	yes	characteristic
23	C₃×Dic₃	—	2	1	no	—
24	AGL(1,ℤ/8)	—	9	1	no	maximal, Sylow
25	C₂³×C₃	—	6	1	no	—
26	C₂²×D₃	—	2	1	no	—
27	C₃×D₄	—	6	1	no	—
28	#67	—	2	1	no	—
29	#67	—	6	2	no	—
30	#67	—	6	1	no	—
31	#67	—	6	2	no	—
32	C₂×Dic₃	—	6	1	no	—
33	C₂×Dic₃	—	6	1	no	—
34	C₂×Dic₃	—	6	1	no	—
35	C₄×D₃	—	6	1	no	—
36	Dic₆	—	6	1	no	—
37	C₂×C₃²	—	2	1	no	—
38	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
39	C₃×D₃	—	4	1	no	—
40	C₃×D₃	—	4	2	no	—
41	#41	—	9	1	no	—
42	C₂×D₄	—	9	1	no	—
43	SD₁₆	—	9	2	no	—
44	D₈	—	9	2	no	—
45	M₁₆	—	9	1	no	—
46	C₂²×C₃	—	2	1	no	—
47	C₂²×C₃	—	2	1	no	—
48	C₂²×C₃	—	6	1	no	—
49	C₂²×C₃	—	6	2	no	—
50	C₂²×C₃	—	12	2	no	—
51	C₂×D₃	—	2	1	no	—
52	C₂×D₃	—	2	2	no	—
53	C₂×D₃	—	4	2	no	—
54	C₃×C₄	—	6	1	no	—
55	Dic₃	—	2	1	no	—
56	Dic₃	—	6	1	no	—
57	Dic₃	—	6	1	no	—
58	Dic₃	—	6	1	no	—
59	Dic₃	—	6	1	no	—
60	C₃²	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
61	C₂³	—	6	1	no	—
62	Q₈	—	9	1	no	—
63	D₄	—	6	1	no	—
64	D₄	—	9	1	no	—
65	D₄	—	9	2	no	—
66	D₄	—	18	1	no	—
67	C₂×C₄	—	9	1	no	—
68	C₂×C₄	—	18	1	no	—
69	C₈	—	9	2	no	—
70	C₂×C₃	—	2	2	no	—
71	C₂×C₃	—	2	1	no	—
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	4	1	no	—
74	C₂×C₃	—	12	2	no	—
75	C₂×C₃	—	12	1	no	—
76	D₃	—	4	2	no	—
77	D₃	—	4	1	no	—
78	C₂²	—	6	1	no	—
79	C₂²	—	6	2	no	—
80	C₂²	—	12	2	no	—
81	C₂²	GO⁺(4,𝔽₂)	1	1	yes	characteristic
82	C₄	—	6	1	no	—
83	C₄	—	9	1	no	—
84	C₄	—	9	1	no	—
85	C₃	—	2	1	no	—
86	C₃	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	—	12	2	no	—
89	C₂	—	12	1	no	—
90	C₂	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊Dic₃	C₂³
Fitting	C₂²×C₃²	D₄
socle	C₂×C₃²	C₂×D₄
hypercenter	C₂²	GO⁺(4,𝔽₂)
hypocenter	C₃²	AGL(1,ℤ/8)
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×#625	C₂	semidirect
2. #3805	C₂	semidirect
3. #3792	C₂	semidirect
4. #3775	C₂	semidirect
5. #3774	C₂	semidirect
9. #627	C₂²	semidirect
12. #625	C₂²	semidirect
14. #622	C₂²	semidirect
60. C₃²	AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×#625	2
#3805	2
#3792	2
#3775	2
#3774	2
AGL(1,ℤ/8)	9