Subgroups — #41×3₋¹⁺² #88764

Order 432.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×3₋¹⁺²	C₂	1	3	yes	maximal, complemented
3	C₂×C₄×3₋¹⁺²	C₂	1	3	yes	maximal, complemented
4	C₃²×#41	C₃	1	1	yes	maximal, characteristic
5	C₉×#41	C₃	1	3	yes	maximal, complemented
6	C₂²×3₋¹⁺²	C₂²	1	3	yes	—
7	C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₄×3₋¹⁺²	C₂²	1	3	yes	—
9	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₃²×D₄	C₂×C₃	1	3	yes	—
11	C₂×C₃²×C₄	C₂×C₃	1	3	yes	—
12	Q₈×C₉	C₂×C₃	1	3	yes	complemented
13	D₄×C₉	C₂×C₃	1	9	yes	complemented
14	C₂×C₄×C₉	C₂×C₃	1	9	yes	complemented
15	C₂×3₋¹⁺²	—	2	3	no	—
16	C₂×3₋¹⁺²	C₂³	1	1	yes	characteristic
17	C₃×#41	—	3	1	no	—
18	C₃×#41	C₃²	1	1	yes	characteristic
19	C₂²×C₃²	C₂²×C₃	1	3	yes	—
20	C₃²×C₄	C₂²×C₃	1	3	yes	—
21	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
22	C₂²×C₉	C₂²×C₃	1	9	yes	—
23	C₄×C₉	C₂²×C₃	1	9	yes	—
24	C₄×C₉	C₂²×C₃	1	3	yes	—
25	3₋¹⁺²	#41	1	1	yes	characteristic, direct product factor, Sylow
26	C₃×Q₈	—	3	1	no	—
27	C₃×Q₈	C₂×C₃²	1	1	yes	characteristic
28	C₃×D₄	—	3	3	no	—
29	C₃×D₄	C₂×C₃²	1	3	yes	—
30	C₂×C₃×C₄	—	3	3	no	—
31	C₂×C₃×C₄	C₂×C₃²	1	3	yes	—
32	C₂×C₃²	—	2	3	no	—
33	C₂×C₃²	C₂³×C₃	1	1	yes	characteristic
34	C₂×C₉	—	2	9	no	—
35	C₂×C₉	C₂³×C₃	1	3	yes	—
36	#41	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
37	C₂²×C₃	—	3	3	no	—
38	C₂²×C₃	C₂²×C₃²	1	3	yes	—
39	C₃×C₄	—	3	1	no	—
40	C₃×C₄	—	3	3	no	—
41	C₃×C₄	C₂²×C₃²	1	3	yes	—
42	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
43	C₃²	C₃×#41	1	1	yes	characteristic
44	C₉	C₃×#41	1	3	yes	complemented
45	Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
46	D₄	C₂×3₋¹⁺²	1	3	yes	complemented
47	C₂×C₄	C₂×3₋¹⁺²	1	3	yes	complemented
48	C₂×C₃	—	2	3	no	—
49	C₂×C₃	—	3	1	no	—
50	C₂×C₃	—	6	3	no	—
51	C₂×C₃	C₂³×C₃²	1	1	yes	characteristic
52	C₂²	C₂²×3₋¹⁺²	1	3	yes	—
53	C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
54	C₄	C₂²×3₋¹⁺²	1	3	yes	—
55	C₃	—	3	1	no	—
56	C₃	C₃²×#41	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂³×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂²×C₃²
commutator, Frattini, socle, cosocle	C₂×C₃	C₂³×C₃²
prime radical	C₃²×#41	C₃
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×3₋¹⁺²	C₂	semidirect
2. D₄×3₋¹⁺²	C₂	semidirect
3. C₂×C₄×3₋¹⁺²	C₂	semidirect
5. C₉×#41	C₃	semidirect
12. Q₈×C₉	C₂×C₃	semidirect
13. D₄×C₉	C₂×C₃	semidirect
14. C₂×C₄×C₉	C₂×C₃	semidirect
25. 3₋¹⁺²	#41	direct (N × H)
36. #41	3₋¹⁺²	direct (N × H)
44. C₉	C₃×#41	semidirect
45. Q₈	C₂×3₋¹⁺²	semidirect
46. D₄	C₂×3₋¹⁺²	semidirect
47. C₂×C₄	C₂×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
Q₈×3₋¹⁺²	2
D₄×3₋¹⁺²	2
C₂×C₄×3₋¹⁺²	2
C₃²×#41	3
C₉×#41	3

Subgroups — #41×3−1+2 #88764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×3₋¹⁺² #88764