Subgroups — #908

Order 96.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#235	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	D₂₄	C₂	1	2	yes	maximal, complemented
5	#204	C₂	1	2	yes	maximal, complemented
6	AGL(1,ℤ/8)	—	3	1	no	maximal, Sylow
7	C₂²×D₃	—	2	1	no	—
8	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
9	#67	—	2	1	no	—
10	D₁₂	—	2	1	no	—
11	D₁₂	C₂²	1	2	yes	—
12	D₁₂	C₂²	1	1	yes	characteristic, complemented
13	C₄×D₃	—	2	1	no	—
14	Dic₆	C₂²	1	1	yes	characteristic, complemented
15	C₃×C₈	C₂²	1	2	yes	complemented
16	#41	—	3	1	no	—
17	C₂×D₄	—	3	1	no	—
18	SD₁₆	—	3	2	no	—
19	D₈	—	3	2	no	—
20	M₁₆	D₃	1	1	yes	characteristic, complemented
21	C₂²×C₃	D₄	1	1	yes	characteristic
22	C₂×D₃	—	2	1	no	—
23	C₂×D₃	—	2	2	no	—
24	C₂×D₃	—	4	2	no	—
25	C₃×C₄	D₄	1	1	yes	characteristic
26	C₃×C₄	C₂³	1	1	yes	characteristic
27	Dic₃	—	2	1	no	—
28	C₂³	—	6	1	no	—
29	Q₈	—	3	1	no	—
30	D₄	—	3	1	no	—
31	D₄	—	3	2	no	—
32	D₄	—	6	1	no	—
33	D₄	—	6	1	no	—
34	C₂×C₄	—	6	1	no	—
35	C₂×C₄	C₂×D₃	1	1	yes	characteristic
36	C₈	C₂×D₃	1	2	yes	complemented
37	C₂×C₃	—	2	1	no	—
38	C₂×C₃	C₂×D₄	1	1	yes	characteristic
39	D₃	—	4	1	no	—
40	D₃	—	4	2	no	—
41	C₂²	—	6	1	no	—
42	C₂²	—	6	2	no	—
43	C₂²	—	12	2	no	—
44	C₂²	D₁₂	1	1	yes	characteristic
45	C₄	—	6	1	no	—
46	C₄	D₁₂	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	—	12	2	no	—
51	C₂	—	12	1	no	—
52	C₂	C₂×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₂
commutator, cosocle	C₃×C₄	C₂³
Frattini	C₄	C₂²×D₃
Fitting	C₃×M₁₆	C₂
socle	C₂×C₃	C₂×D₄
hypercenter	M₁₆	D₃
hypocenter	C₃	AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #235	C₂	semidirect
2. C₂×D₁₂	C₂	semidirect
3. C₃×M₁₆	C₂	semidirect
4. D₂₄	C₂	semidirect
5. #204	C₂	semidirect
12. D₁₂	C₂²	semidirect
14. Dic₆	C₂²	semidirect
15. C₃×C₈	C₂²	semidirect
20. M₁₆	D₃	semidirect
36. C₈	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
#235	2
C₂×D₁₂	2
C₃×M₁₆	2
D₂₄	2
#204	2
AGL(1,ℤ/8)	3