Subgroups — C₂×3₋¹⁺²⋊Q₈ #88705

Order 432.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×3₋¹⁺²⋊C₄	C₂	1	2	yes	maximal
3	3₋¹⁺²⋊Q₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×Dic₆	—	3	1	no	maximal
5	C₂×Dic₁₈	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₄×3₋¹⁺²	C₂²	1	2	yes	—
8	3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
9	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
10	C₂×C₃×Dic₃	—	3	2	no	—
11	C₃×Dic₆	—	3	4	no	—
12	C₂×C₄×C₉	—	2	1	no	—
13	C₂×C₄×C₉	C₂×C₃	1	1	yes	characteristic
14	C₂×Dic₉	C₂×C₃	1	2	yes	—
15	Dic₁₈	C₂×C₃	1	4	yes	complemented
16	C₂×3₋¹⁺²	Q₈	1	2	yes	complemented
17	C₂×3₋¹⁺²	C₂³	1	1	yes	characteristic
18	C₂×C₃×Q₈	—	9	1	no	—
19	C₂×Dic₆	—	3	1	no	—
20	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
21	C₃²×C₄	C₂×D₃	1	2	yes	—
22	C₃×Dic₃	—	3	4	no	—
23	C₂²×C₉	—	2	1	no	—
24	C₂²×C₉	C₂²×C₃	1	1	yes	characteristic
25	C₄×C₉	—	2	2	no	—
26	C₄×C₉	C₂²×C₃	1	2	yes	—
27	Dic₉	C₂²×C₃	1	4	yes	—
28	3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
29	C₃×Q₈	—	9	4	no	—
30	C₂×C₃×C₄	—	3	1	no	—
31	C₂×C₃×C₄	—	9	2	no	—
32	C₂×C₃×C₄	C₃×D₃	1	1	yes	characteristic
33	C₂×Dic₃	—	3	2	no	—
34	Dic₆	—	3	4	no	—
35	C₂×C₃²	Dic₆	1	2	yes	—
36	C₂×C₃²	C₂²×D₃	1	1	yes	characteristic
37	C₂×C₉	—	2	2	no	—
38	C₂×C₉	—	2	1	no	—
39	C₂×C₉	C₃×Q₈	1	2	yes	complemented
40	C₂×C₉	C₂³×C₃	1	1	yes	characteristic
41	C₂×Q₈	—	9	1	no	Sylow
42	C₂²×C₃	—	3	1	no	—
43	C₂²×C₃	C₂×C₃×D₃	1	1	yes	characteristic
44	C₃×C₄	—	3	2	no	—
45	C₃×C₄	—	9	4	no	—
46	C₃×C₄	C₂×C₃×D₃	1	2	yes	—
47	Dic₃	—	3	4	no	—
48	C₃²	C₂×Dic₆	1	1	yes	characteristic
49	C₉	—	2	1	no	—
50	C₉	C₂×C₃×Q₈	1	1	yes	characteristic, complemented
51	Q₈	—	9	4	no	—
52	C₂×C₄	—	9	2	no	—
53	C₂×C₄	AGL(1,ℤ/9)	1	1	yes	characteristic
54	C₂×C₃	—	3	2	no	—
55	C₂×C₃	—	3	1	no	—
56	C₂×C₃	C₃×Dic₆	1	2	yes	—
57	C₂×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
58	C₂²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
59	C₄	—	9	4	no	—
60	C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
61	C₃	—	3	1	no	—
62	C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
63	C₂	3₋¹⁺²⋊Q₈	1	2	yes	direct product factor
64	C₂	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₉	C₂³×C₃
Frattini	C₂×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×D₃
prime radical	C₂²×C₃²	C₂×D₃
hypercenter	C₂×C₄	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×Q₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²⋊Q₈	C₂	direct (N × H)
5. C₂×Dic₁₈	C₃	semidirect
15. Dic₁₈	C₂×C₃	semidirect
16. C₂×3₋¹⁺²	Q₈	semidirect
28. 3₋¹⁺²	C₂×Q₈	semidirect
39. C₂×C₉	C₃×Q₈	semidirect
50. C₉	C₂×C₃×Q₈	semidirect
63. C₂	3₋¹⁺²⋊Q₈	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×C₄×3₋¹⁺²	2
C₂×3₋¹⁺²⋊C₄	2
3₋¹⁺²⋊Q₈	2
C₂×C₃×Dic₆	3
C₂×Dic₁₈	3

Subgroups — C2×3−1+2⋊Q8 #88705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×3₋¹⁺²⋊Q₈ #88705