Subgroups — #949

Order 96.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#215	C₂	1	2	yes	maximal, complemented
4	#213	C₂	1	2	yes	maximal, complemented
5	#208	C₂	1	1	yes	maximal, characteristic, complemented
6	AGL(1,ℤ/8)	—	3	1	no	maximal, Sylow
7	C₂²×D₃	—	2	1	no	—
8	C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
9	C₃×D₄	—	2	1	no	—
10	C₃×D₄	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₃×C₄	—	2	1	no	—
12	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
13	D₁₂	—	2	1	no	—
14	D₁₂	C₂²	1	2	yes	—
15	C₃⋊C₈	C₂²	1	2	yes	complemented
16	#41	D₃	1	1	yes	characteristic, complemented
17	C₂×D₄	—	3	1	no	—
18	SD₁₆	—	3	2	no	—
19	D₈	—	3	2	no	—
20	M₁₆	—	3	1	no	—
21	C₂²×C₃	—	2	1	no	—
22	C₂²×C₃	D₄	1	1	yes	characteristic
23	C₂×D₃	—	2	2	no	—
24	C₂×D₃	—	4	2	no	—
25	C₃×C₄	—	2	1	no	—
26	C₃×C₄	D₄	1	1	yes	characteristic
27	C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂³	—	6	1	no	—
29	Q₈	C₂×D₃	1	1	yes	characteristic, complemented
30	D₄	—	2	1	no	—
31	D₄	—	3	2	no	—
32	D₄	—	6	1	no	—
33	D₄	C₂×D₃	1	1	yes	characteristic, complemented
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	C₂×D₃	1	1	yes	characteristic
36	C₈	—	3	2	no	—
37	C₂×C₃	—	2	1	no	—
38	C₂×C₃	—	4	1	no	—
39	C₂×C₃	C₂×D₄	1	1	yes	characteristic
40	D₃	—	4	2	no	—
41	C₂²	—	2	1	no	—
42	C₂²	—	6	2	no	—
43	C₂²	—	12	2	no	—
44	C₂²	#67	1	1	yes	characteristic
45	C₄	—	2	1	no	—
46	C₄	#67	1	1	yes	characteristic
47	C₄	C₂²×D₃	1	1	yes	characteristic
48	C₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
49	C₂	—	2	1	no	—
50	C₂	—	4	1	no	—
51	C₂	—	12	2	no	—
52	C₂	C₂×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#67
commutator, cosocle	C₃×C₄	C₂³
Frattini	C₄	C₂²×D₃
Fitting	C₃×#41	C₂
socle	C₂×C₃	C₂×D₄
hypercenter	#41	D₃
hypocenter	C₃	AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₃×#41	C₂	semidirect
2. C₂×D₁₂	C₂	semidirect
3. #215	C₂	semidirect
4. #213	C₂	semidirect
5. #208	C₂	semidirect
8. C₃×Q₈	C₂²	semidirect
10. C₃×D₄	C₂²	semidirect
15. C₃⋊C₈	C₂²	semidirect
16. #41	D₃	semidirect
29. Q₈	C₂×D₃	semidirect
33. D₄	C₂×D₃	semidirect
48. C₃	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
C₃×#41	2
C₂×D₁₂	2
#215	2
#213	2
#208	2
AGL(1,ℤ/8)	3