Subgroups — Q₈×#1088 #11722452

Order 864.

181 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#272	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#1088	C₂	1	3	yes	maximal
3	C₄×#1088	C₂	1	3	yes	maximal
4	C₂×#1088	C₂²	1	3	yes	—
5	C₂×#1088	C₂²	1	1	yes	characteristic
6	Q₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic, complemented
7	C₄×#272	C₂²	1	3	yes	—
8	#6229	C₄	1	3	yes	complemented
9	C₃×D₃×Q₈	—	6	2	no	—
10	C₂×#272	C₂³	1	1	yes	characteristic
11	#1088	—	2	3	no	—
12	#1088	Q₈	1	2	yes	direct product factor
13	C₄×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
14	#1084	C₂×C₄	1	3	yes	—
15	C₃×C₄×Q₈	—	9	1	no	maximal
16	C₃²×Q₈	—	2	2	no	—
17	C₃×C₄×D₃	—	6	6	no	—
18	C₃×Dic₆	—	6	6	no	—
19	C₂×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
20	#272	C₂×Q₈	1	1	yes	characteristic
21	#272	#41	1	1	yes	characteristic
22	C₂×C₃×Q₈	—	9	1	no	—
23	D₃×Q₈	—	6	2	no	—
24	C₃×C₄⋊C₄	—	9	3	no	—
25	C₃×C₄²	—	9	3	no	—
26	C₂×C₃×D₃	—	6	2	no	—
27	C₃²×C₄	—	2	6	no	—
28	C₃×Dic₃	—	6	6	no	—
29	C₄×Q₈	—	9	1	no	Sylow
30	H(3,𝔽₃)	C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
31	C₃×Q₈	—	6	2	no	—
32	C₃×Q₈	—	9	3	no	—
33	C₃×Q₈	#157	1	1	yes	characteristic
34	C₂×C₃×C₄	—	9	1	no	—
35	C₂×C₃×C₄	—	9	3	no	—
36	C₂×C₃×C₄	—	9	3	no	—
37	C₄×D₃	—	6	6	no	—
38	Dic₆	—	6	6	no	—
39	C₂×C₃²	—	2	2	no	—
40	C₃×D₃	—	6	2	no	—
41	C₃×D₃	—	6	2	no	—
42	C₂×Q₈	—	9	1	no	—
43	C₄⋊C₄	—	9	3	no	—
44	C₄²	—	9	3	no	—
45	C₂²×C₃	—	9	1	no	—
46	C₂×D₃	—	6	2	no	—
47	C₃×C₄	—	6	6	no	—
48	C₃×C₄	—	9	2	no	—
49	C₃×C₄	—	9	3	no	—
50	C₃×C₄	—	18	3	no	—
51	C₃×C₄	C₂×#157	1	3	yes	—
52	Dic₃	—	6	6	no	—
53	C₃²	—	2	2	no	—
54	Q₈	—	9	3	no	—
55	Q₈	#1088	1	1	yes	characteristic, direct product factor
56	C₂×C₄	—	9	1	no	—
57	C₂×C₄	—	9	3	no	—
58	C₂×C₄	—	9	3	no	—
59	C₂×C₃	—	6	2	no	—
60	C₂×C₃	—	9	1	no	—
61	C₂×C₃	—	9	1	no	—
62	C₂×C₃	C₂²×#157	1	1	yes	characteristic
63	D₃	—	6	2	no	—
64	D₃	—	6	2	no	—
65	C₂²	—	9	1	no	—
66	C₄	—	9	2	no	—
67	C₄	—	9	3	no	—
68	C₄	—	18	3	no	—
69	C₄	C₂×#1088	1	3	yes	—
70	C₃	—	6	2	no	—
71	C₃	Q₈×#157	1	1	yes	characteristic
72	C₂	—	9	1	no	—
73	C₂	—	9	1	no	—
74	C₂	C₂²×#1088	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂²×#157
commutator	C₂×H(3,𝔽₃)	C₂²×C₄
Fitting	Q₈×H(3,𝔽₃)	C₄
prime radical, cosocle	C₂×#272	C₂³
hypercenter	C₃×Q₈	#157
hypocenter	H(3,𝔽₃)	C₄×Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×H(3,𝔽₃)	C₄	semidirect
8. #6229	C₄	semidirect
12. #1088	Q₈	direct (N × H)
30. H(3,𝔽₃)	C₄×Q₈	semidirect
55. Q₈	#1088	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×#272	2
C₄⋊#1088	2
C₄×#1088	2
C₃×C₄×Q₈	9

Subgroups — Q8×#1088 #11722452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#1088 #11722452