Subgroups — Q₈×#4208 #13048935

Order 1296.

371 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#737	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#4208	C₂	1	3	yes	maximal
3	#10619966	C₂	1	3	yes	maximal, complemented
4	Q₈×#272	C₃	1	1	yes	maximal, characteristic
5	D₃×Q₈×C₉	—	3	4	no	maximal
6	C₂×#4208	C₂²	1	1	yes	characteristic
7	C₄×#737	C₂²	1	3	yes	—
8	#66410	C₂²	1	3	yes	—
9	Q₈×3₋¹⁺²	—	2	4	no	—
10	Q₈×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
11	C₃×Q₈×C₉	D₃	1	4	yes	complemented
12	C₄×#272	C₂×C₃	1	3	yes	—
13	#6229	C₂×C₃	1	3	yes	—
14	C₄×D₃×C₉	—	3	12	no	—
15	C₉×Dic₆	—	3	12	no	—
16	C₂×#737	C₂³	1	1	yes	characteristic
17	#4208	Q₈	1	2	yes	direct product factor
18	C₃×D₃×Q₈	—	3	4	no	—
19	C₂×Q₈×C₉	—	9	1	no	—
20	C₂×#272	C₂²×C₃	1	1	yes	characteristic
21	C₂×D₃×C₉	—	3	4	no	—
22	C₄×3₋¹⁺²	—	2	12	no	—
23	C₄×H(3,𝔽₃)	C₂²×C₃	1	3	yes	—
24	C₃×C₄×C₉	C₂×D₃	1	12	yes	—
25	#1084	C₂²×C₃	1	3	yes	—
26	C₉×Dic₃	—	3	12	no	—
27	#737	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
28	C₃²×Q₈	C₃×D₃	1	4	yes	—
29	C₃×C₄×D₃	—	3	12	no	—
30	C₃×Dic₆	—	3	12	no	—
31	Q₈×C₉	—	2	4	no	—
32	Q₈×C₉	—	9	3	no	—
33	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic
34	C₂×C₄×C₉	—	9	3	no	—
35	C₂×3₋¹⁺²	—	2	4	no	—
36	C₂×H(3,𝔽₃)	C₂³×C₃	1	1	yes	characteristic
37	C₂×C₃×C₉	C₂²×D₃	1	4	yes	—
38	#272	C₃×Q₈	1	2	yes	—
39	D₃×C₉	—	3	8	no	—
40	C₂×C₃×Q₈	—	9	1	no	—
41	D₃×Q₈	—	3	4	no	—
42	C₂×C₃×D₃	—	3	4	no	—
43	C₃²×C₄	C₂×C₃×D₃	1	12	yes	—
44	C₃×Dic₃	—	3	12	no	—
45	C₂²×C₉	—	9	1	no	—
46	C₄×C₉	—	2	12	no	—
47	C₄×C₉	—	9	3	no	—
48	C₄×C₉	C₂×C₃⋊D₃	1	3	yes	—
49	3₋¹⁺²	—	2	4	no	—
50	H(3,𝔽₃)	C₂×C₃×Q₈	1	1	yes	characteristic
51	C₃×C₉	D₃×Q₈	1	4	yes	complemented
52	C₃×Q₈	—	3	4	no	—
53	C₃×Q₈	—	9	3	no	—
54	C₃×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
55	C₂×C₃×C₄	—	9	3	no	—
56	C₄×D₃	—	3	12	no	—
57	Dic₆	—	3	12	no	—
58	C₂×C₃²	C₂²×C₃×D₃	1	4	yes	—
59	C₃×D₃	—	3	8	no	—
60	C₂×C₉	—	2	4	no	—
61	C₂×C₉	—	9	2	no	—
62	C₂×C₉	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	C₂×Q₈	—	9	1	no	Sylow
64	C₂²×C₃	—	9	1	no	—
65	C₂×D₃	—	3	4	no	—
66	C₃×C₄	—	3	12	no	—
67	C₃×C₄	—	9	3	no	—
68	C₃×C₄	C₂×C₃×C₃⋊D₃	1	3	yes	—
69	Dic₃	—	3	12	no	—
70	C₃²	C₃×D₃×Q₈	1	4	yes	—
71	C₉	—	2	4	no	—
72	C₉	Q₈×C₃⋊D₃	1	1	yes	characteristic
73	Q₈	—	9	3	no	—
74	Q₈	#4208	1	1	yes	characteristic, direct product factor
75	C₂×C₄	—	9	3	no	—
76	C₂×C₃	—	3	4	no	—
77	C₂×C₃	—	9	2	no	—
78	C₂×C₃	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
79	D₃	—	3	8	no	—
80	C₂²	—	9	1	no	—
81	C₄	—	9	3	no	—
82	C₄	C₂×#4208	1	3	yes	—
83	C₃	—	3	4	no	—
84	C₃	C₃×Q₈×C₃⋊D₃	1	1	yes	characteristic
85	C₂	—	9	2	no	—
86	C₂	C₂²×#4208	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂²×C₃⋊D₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³×C₃
Frattini, socle	C₂×C₃	C₂²×C₃×C₃⋊D₃
Fitting	Q₈×#737	C₂
prime radical	C₂×#272	C₂²×C₃
hypercenter	Q₈×C₉	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×C₃×Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₉ ◁ Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. Q₈×#737	C₂	semidirect
3. #10619966	C₂	semidirect
11. C₃×Q₈×C₉	D₃	semidirect
17. #4208	Q₈	direct (N × H)
27. #737	C₂×Q₈	semidirect
51. C₃×C₉	D₃×Q₈	semidirect
74. Q₈	#4208	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#737	2
C₄×#4208	2
#10619966	2
Q₈×#272	3
D₃×Q₈×C₉	3

Subgroups — Q8×#4208 #13048935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#4208 #13048935