Subgroups — Q₈×#226 #85427

Order 384.

195 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#5391	C₂	1	3	yes	maximal
3	C₄×#226	C₂	1	3	yes	maximal
4	Q₈×Q₁₆	—	3	1	no	maximal, Sylow
5	C₂×#226	C₂²	1	1	yes	characteristic
6	Q₈×Dic₃	—	4	1	no	maximal
7	C₄×SL(2,𝔽₃)	C₂²	1	3	yes	—
8	#859	C₂²	1	3	yes	—
9	Q₈²	—	3	1	no	—
10	Q₈²	D₃	1	1	yes	characteristic
11	#500	—	3	3	no	—
12	#477	—	3	3	no	—
13	#477	—	3	3	no	—
14	C₈×Q₈	—	3	1	no	—
15	C₄×Q₁₆	—	3	3	no	—
16	C₂×C₃×Q₈	—	4	1	no	—
17	C₂×SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
18	#226	Q₈	1	2	yes	direct product factor
19	C₄⋊Dic₃	—	4	3	no	—
20	C₄×Dic₃	—	4	3	no	—
21	C₂×Q₁₆	—	3	1	no	—
22	C₄⋊Q₈	—	3	3	no	—
23	C₄⋊Q₈	—	3	3	no	—
24	C₄⋊Q₈	—	6	3	no	—
25	C₄×Q₈	—	3	1	no	—
26	C₄×Q₈	—	3	3	no	—
27	C₄×Q₈	—	6	1	no	—
28	C₄×Q₈	C₂×D₃	1	3	yes	—
29	#107	—	3	3	no	—
30	#107	—	3	6	no	—
31	C₄⋊C₈	—	3	3	no	—
32	#103	—	3	3	no	—
33	#103	—	3	3	no	—
34	C₄×C₈	—	3	3	no	—
35	C₃×Q₈	—	4	4	no	—
36	C₂×C₃×C₄	—	4	3	no	—
37	C₂×Dic₃	—	4	1	no	—
38	C₂×Dic₃	—	4	3	no	—
39	SL(2,𝔽₃)	C₂×Q₈	1	1	yes	characteristic
40	C₂×Q₈	—	3	1	no	—
41	C₂×Q₈	—	6	1	no	—
42	C₂×Q₈	—	6	3	no	—
43	C₂×Q₈	S₄	1	1	yes	characteristic
44	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
45	Q₁₆	—	3	2	no	—
46	Q₁₆	—	3	2	no	—
47	C₂×C₈	—	3	1	no	—
48	C₂×C₈	—	3	3	no	—
49	C₄⋊C₄	—	3	3	no	—
50	C₄⋊C₄	—	3	3	no	—
51	C₄⋊C₄	—	3	3	no	—
52	C₄⋊C₄	—	6	3	no	—
53	C₄⋊C₄	—	6	3	no	—
54	C₄²	—	3	3	no	—
55	C₄²	—	6	3	no	—
56	C₂²×C₃	—	4	1	no	—
57	C₃×C₄	—	4	6	no	—
58	Dic₃	—	4	2	no	—
59	Dic₃	—	8	3	no	—
60	Q₈	—	3	1	no	—
61	Q₈	—	3	2	no	—
62	Q₈	—	6	1	no	—
63	Q₈	—	24	1	no	—
64	Q₈	—	24	3	no	—
65	Q₈	#226	1	4	yes	direct product factor
66	Q₈	D₃×Q₈	1	1	yes	characteristic
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	3	3	no	—
69	C₂×C₄	—	6	1	no	—
70	C₂×C₄	—	6	3	no	—
71	C₂×C₄	C₂×S₄	1	3	yes	—
72	C₈	—	3	2	no	—
73	C₈	—	6	3	no	—
74	C₂×C₃	—	4	1	no	—
75	C₂×C₃	—	4	1	no	—
76	C₂×C₃	—	4	1	no	—
77	C₂²	C₂²×S₄	1	1	yes	characteristic
78	C₄	—	3	1	no	—
79	C₄	—	3	1	no	—
80	C₄	—	6	2	no	—
81	C₄	—	6	3	no	—
82	C₄	—	12	3	no	—
83	C₄	C₂×#226	1	6	yes	—
84	C₃	—	4	1	no	Sylow
85	C₂	C₂²×#226	1	1	yes	characteristic
86	C₂	Q₈×S₄	1	1	yes	characteristic
87	C₂	#5925	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂²	C₂²×S₄
commutator, prime radical, cosocle	C₂×SL(2,𝔽₃)	C₂³
Fitting	Q₈²	D₃
hypercenter	C₂×Q₈	S₄
hypocenter	SL(2,𝔽₃)	C₂×Q₈
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
18. #226	Q₈	direct (N × H)
65. Q₈	#226	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×SL(2,𝔽₃)	2
#5391	2
C₄×#226	2
Q₈×Q₁₆	3
Q₈×Dic₃	4

Subgroups — Q8×#226 #85427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#226 #85427