Subgroups — D₈×#731 #13046891

Order 1296.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#731	C₂	1	2	yes	maximal, complemented
2	C₈×#731	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
4	D₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×C₉×D₈	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×#731	—	2	2	no	—
7	C₄×#731	C₂²	1	1	yes	characteristic
8	D₄×3₋¹⁺²	C₂×C₃	1	4	yes	complemented
9	D₄×H(3,𝔽₃)	C₂×C₃	1	2	yes	—
10	C₃×D₄×C₉	C₂×C₃	1	2	yes	complemented
11	C₈×3₋¹⁺²	C₂×C₃	1	2	yes	complemented
12	C₈×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
13	C₃×C₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂×#731	—	4	2	no	—
15	C₂×#731	D₄	1	1	yes	characteristic
16	C₃²×D₈	—	3	1	no	—
17	C₃²×D₈	C₃²	1	1	yes	characteristic
18	C₉×D₈	—	3	1	no	—
19	C₉×D₈	—	3	2	no	—
20	C₂²×3₋¹⁺²	—	2	4	no	—
21	C₂²×H(3,𝔽₃)	—	2	2	no	—
22	C₂²×C₃×C₉	—	2	2	no	—
23	C₄×3₋¹⁺²	C₂²×C₃	1	2	yes	—
24	C₄×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
25	C₃×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
26	#731	D₈	1	1	yes	characteristic, direct product factor, Sylow
27	C₃²×D₄	—	3	2	no	—
28	C₃²×D₄	C₂×C₃²	1	2	yes	—
29	C₃²×C₈	—	3	1	no	—
30	C₃²×C₈	C₂×C₃²	1	1	yes	characteristic
31	D₄×C₉	—	3	2	no	—
32	D₄×C₉	—	3	4	no	—
33	C₈×C₉	—	3	1	no	—
34	C₈×C₉	—	3	2	no	—
35	C₂×3₋¹⁺²	—	4	4	no	—
36	C₂×3₋¹⁺²	C₃×D₄	1	2	yes	—
37	C₂×H(3,𝔽₃)	—	4	2	no	—
38	C₂×H(3,𝔽₃)	C₃×D₄	1	1	yes	characteristic
39	C₂×C₃×C₉	—	4	2	no	—
40	C₂×C₃×C₉	C₃×D₄	1	1	yes	characteristic
41	C₃×D₈	—	3	1	no	—
42	C₃×D₈	—	9	1	no	—
43	C₃×D₈	H(3,𝔽₃)	1	1	yes	characteristic
44	C₂²×C₃²	—	2	2	no	—
45	C₂²×C₃²	—	6	2	no	—
46	C₃²×C₄	—	3	1	no	—
47	C₃²×C₄	C₂²×C₃²	1	1	yes	characteristic
48	C₂²×C₉	—	6	2	no	—
49	C₂²×C₉	—	6	4	no	—
50	C₄×C₉	—	3	1	no	—
51	C₄×C₉	—	3	2	no	—
52	3₋¹⁺²	C₃×D₈	1	2	yes	complemented
53	H(3,𝔽₃)	C₃×D₈	1	1	yes	characteristic
54	C₃×C₉	C₃×D₈	1	1	yes	characteristic, complemented
55	C₃×D₄	—	3	2	no	—
56	C₃×D₄	—	9	2	no	—
57	C₃×D₄	C₂×H(3,𝔽₃)	1	2	yes	—
58	C₃×C₈	—	3	1	no	—
59	C₃×C₈	—	9	1	no	—
60	C₃×C₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic
61	C₂×C₃²	—	3	1	no	—
62	C₂×C₃²	—	4	2	no	—
63	C₂×C₃²	—	12	2	no	—
64	C₂×C₃²	C₃²×D₄	1	1	yes	characteristic
65	C₂×C₉	—	3	1	no	—
66	C₂×C₉	—	3	2	no	—
67	C₂×C₉	—	12	2	no	—
68	C₂×C₉	—	12	4	no	—
69	D₈	#731	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²×C₃	—	2	2	no	—
71	C₂²×C₃	—	6	2	no	—
72	C₂²×C₃	—	18	2	no	—
73	C₃×C₄	—	3	1	no	—
74	C₃×C₄	—	9	1	no	—
75	C₃×C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
76	C₃²	—	3	1	no	—
77	C₃²	C₃²×D₈	1	1	yes	characteristic
78	C₉	—	3	1	no	—
79	C₉	—	3	2	no	—
80	D₄	C₂×#731	1	2	yes	complemented
81	C₈	C₂×#731	1	1	yes	characteristic, complemented
82	C₂×C₃	—	3	1	no	—
83	C₂×C₃	—	4	2	no	—
84	C₂×C₃	—	9	1	no	—
85	C₂×C₃	—	12	2	no	—
86	C₂×C₃	—	36	2	no	—
87	C₂×C₃	D₄×H(3,𝔽₃)	1	1	yes	characteristic
88	C₂²	—	2	2	no	—
89	C₄	C₂²×#731	1	1	yes	characteristic
90	C₃	—	3	1	no	—
91	C₃	—	9	1	no	—
92	C₃	D₈×H(3,𝔽₃)	1	1	yes	characteristic
93	C₂	—	4	2	no	—
94	C₂	D₄×#731	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	D₄×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²×C₄	C₂²×C₃²
prime radical	D₈×H(3,𝔽₃)	C₃
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. D₄×#731	C₂	semidirect
2. C₈×#731	C₂	semidirect
3. D₈×3₋¹⁺²	C₃	semidirect
5. C₃×C₉×D₈	C₃	semidirect
8. D₄×3₋¹⁺²	C₂×C₃	semidirect
10. C₃×D₄×C₉	C₂×C₃	semidirect
11. C₈×3₋¹⁺²	C₂×C₃	semidirect
13. C₃×C₈×C₉	C₂×C₃	semidirect
26. #731	D₈	direct (N × H)
52. 3₋¹⁺²	C₃×D₈	semidirect
54. C₃×C₉	C₃×D₈	semidirect
69. D₈	#731	direct (N × H)
80. D₄	C₂×#731	semidirect
81. C₈	C₂×#731	semidirect

Maximal subgroups

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

subgroup	index
D₄×#731	2
C₈×#731	2
D₈×3₋¹⁺²	3
D₈×H(3,𝔽₃)	3
C₃×C₉×D₈	3

Subgroups — D8×#731 #13046891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×#731 #13046891