Subgroups — #914

Order 96.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#239	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#215	C₂	1	1	yes	maximal, characteristic, complemented
5	#213	C₂	1	1	yes	maximal, characteristic, complemented
6	D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
7	ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
8	AGL(1,ℤ/8)	—	3	1	no	maximal, Sylow
9	C₂²×D₃	—	2	1	no	—
10	C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₃×D₄	C₂²	1	1	yes	characteristic
12	#67	—	2	1	no	—
13	D₁₂	—	2	1	no	—
14	D₁₂	C₂²	1	1	yes	characteristic
15	D₁₂	C₂²	1	1	yes	characteristic, complemented
16	C₄×D₃	—	2	1	no	—
17	C₄×D₃	C₂²	1	1	yes	characteristic
18	C₃×C₈	C₂²	1	1	yes	characteristic, complemented
19	C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
20	#41	—	3	1	no	—
21	C₂×D₄	—	3	1	no	—
22	SD₁₆	—	3	1	no	—
23	SD₁₆	D₃	1	1	yes	characteristic, complemented
24	D₈	—	3	1	no	—
25	D₈	—	3	1	no	—
26	M₁₆	—	3	1	no	—
27	C₂²×C₃	—	2	1	no	—
28	C₂×D₃	—	2	1	no	—
29	C₂×D₃	—	2	1	no	—
30	C₂×D₃	—	4	2	no	—
31	C₂×D₃	D₄	1	1	yes	characteristic
32	C₃×C₄	—	2	1	no	—
33	C₃×C₄	C₂³	1	1	yes	characteristic
34	Dic₃	D₄	1	1	yes	characteristic
35	C₂³	—	6	1	no	—
36	Q₈	C₂×D₃	1	1	yes	characteristic, complemented
37	D₄	—	3	1	no	—
38	D₄	—	3	1	no	—
39	D₄	—	6	1	no	—
40	D₄	—	6	1	no	—
41	D₄	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄	—	3	1	no	—
43	C₂×C₄	—	6	1	no	—
44	C₈	—	3	1	no	—
45	C₈	C₂×D₃	1	1	yes	characteristic, complemented
46	C₂×C₃	—	4	1	no	—
47	C₂×C₃	C₂×D₄	1	1	yes	characteristic
48	D₃	—	2	1	no	—
49	D₃	—	4	1	no	—
50	D₃	—	4	1	no	—
51	C₂²	—	2	1	no	—
52	C₂²	—	3	1	no	—
53	C₂²	—	6	1	no	—
54	C₂²	—	6	1	no	—
55	C₂²	—	12	2	no	—
56	C₄	—	2	1	no	—
57	C₄	—	3	1	no	—
58	C₄	C₂²×D₃	1	1	yes	characteristic
59	C₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
60	C₂	—	4	1	no	—
61	C₂	—	6	1	no	—
62	C₂	—	12	1	no	—
63	C₂	—	12	1	no	—
64	C₂	D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×D₄
commutator, cosocle	C₃×C₄	C₂³
Frattini	C₄	C₂²×D₃
Fitting	C₃×SD₁₆	C₂
socle	C₂×C₃	C₂×D₄
hypercenter	SD₁₆	D₃
hypocenter	C₃	AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. #239	C₂	semidirect
2. D₃×D₄	C₂	semidirect
3. C₃×SD₁₆	C₂	semidirect
4. #215	C₂	semidirect
5. #213	C₂	semidirect
6. D₂₄	C₂	semidirect
7. ΓL(1,𝔽₂₅)	C₂	semidirect
10. C₃×Q₈	C₂²	semidirect
15. D₁₂	C₂²	semidirect
18. C₃×C₈	C₂²	semidirect
19. C₃⋊C₈	C₂²	semidirect
23. SD₁₆	D₃	semidirect
36. Q₈	C₂×D₃	semidirect
45. C₈	C₂×D₃	semidirect
59. 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
#239	2
D₃×D₄	2
C₃×SD₁₆	2
#215	2
#213	2
D₂₄	2
ΓL(1,𝔽₂₅)	2
AGL(1,ℤ/8)	3