Subgroups — Q₈×#88034 #15783550

Order 3328.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#88034	C₂	1	3	yes	maximal
2	C₄×#88034	C₂	1	3	yes	maximal
3	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
5	C₁₃×C₄⋊C₁₆	C₄	1	3	yes	—
6	C₄×C₁₃×C₁₆	C₂²	1	3	yes	—
7	C₂×#88034	C₂²	1	3	yes	—
8	C₂×#88034	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
10	C₂×C₁₃×C₁₆	C₂×C₄	1	3	yes	—
11	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
12	C₁₃×C₄⋊C₈	C₈	1	3	yes	—
13	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
14	#88034	—	2	3	no	—
15	#88034	Q₈	1	2	yes	direct product factor
16	Q₈×C₃₂	—	13	1	no	maximal, Sylow
17	C₂×Q₈×C₁₃	C₁₆	1	1	yes	characteristic
18	C₂×C₈×C₁₃	C₂×C₈	1	3	yes	—
19	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
20	C₁₃×C₄⋊C₄	C₁₆	1	3	yes	—
21	C₄²×C₁₃	C₂×C₈	1	3	yes	—
22	C₁₃×C₁₆	—	2	3	no	—
23	C₁₃×C₁₆	C₂×Q₈	1	1	yes	characteristic
24	C₁₃×C₁₆	#41	1	1	yes	characteristic
25	Q₈×C₁₆	D₁₃	1	1	yes	characteristic
26	C₄⋊C₃₂	—	13	3	no	—
27	C₄×C₃₂	—	13	3	no	—
28	Q₈×C₁₃	C₃₂	1	4	yes	complemented
29	C₂×C₄×C₁₃	C₂×C₁₆	1	3	yes	—
30	C₂×C₄×C₁₃	C₂×C₁₆	1	3	yes	—
31	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
32	C₈×C₁₃	—	2	3	no	—
33	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
34	C₈×C₁₃	#131	1	1	yes	characteristic
35	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
36	C₂×C₃₂	—	13	1	no	—
37	C₂×C₃₂	—	13	3	no	—
38	C₄⋊C₁₆	Dic₁₃	1	3	yes	—
39	C₄×C₁₆	C₂×D₁₃	1	3	yes	—
40	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
41	C₄×C₁₃	—	2	3	no	—
42	C₄×C₁₃	C₂×C₃₂	1	6	yes	—
43	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
44	C₄×C₁₃	#504	1	1	yes	characteristic
45	C₄×Q₈	#1055	1	1	yes	characteristic
46	C₂×C₁₆	C₂×Dic₁₃	1	3	yes	—
47	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
48	C₄⋊C₈	#1055	1	3	yes	—
49	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
50	C₃₂	—	13	2	no	—
51	C₃₂	—	26	3	no	—
52	C₂×C₁₃	Q₈×C₁₆	1	1	yes	characteristic
53	C₂×C₁₃	C₂²×C₃₂	1	1	yes	characteristic
54	C₂×C₁₃	#2258	1	1	yes	characteristic
55	C₂×Q₈	#6090	1	1	yes	characteristic
56	C₂×C₈	C₂×#1055	1	3	yes	—
57	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
58	C₄⋊C₄	#6090	1	3	yes	—
59	C₄²	C₂×#1055	1	3	yes	—
60	C₁₆	—	2	3	no	—
61	C₁₆	Q₈×D₁₃	1	1	yes	characteristic
62	C₁₆	#6131	1	1	yes	characteristic
63	C₁₃	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
64	Q₈	#88034	1	4	yes	direct product factor
65	C₂×C₄	C₂×#6090	1	3	yes	—
66	C₂×C₄	C₂×#6090	1	3	yes	—
67	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
68	C₈	—	2	3	no	—
69	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
70	C₈	#88202	1	1	yes	characteristic
71	C₂²	C₂²×#6090	1	1	yes	characteristic
72	C₄	—	2	3	no	—
73	C₄	C₂×#88034	1	6	yes	—
74	C₄	Q₈×#1055	1	1	yes	characteristic
75	C₄	#11718801	1	1	yes	characteristic
76	C₂	Q₈×#6090	1	1	yes	characteristic
77	C₂	C₂²×#88034	1	1	yes	characteristic
78	C₂	#13117130	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₃
commutator	C₂×C₁₃	C₂²×C₃₂
Fitting	Q₈×C₁₃×C₁₆	C₂
socle, prime radical	C₂²×C₁₃	C₂²×C₁₆
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	Q₈×C₁₆	D₁₃
hypocenter	C₁₃	Q₈×C₃₂
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
15. #88034	Q₈	direct (N × H)
28. Q₈×C₁₃	C₃₂	semidirect
63. C₁₃	Q₈×C₃₂	semidirect
64. Q₈	#88034	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄⋊#88034	2
C₄×#88034	2
Q₈×C₁₃×C₁₆	2
Q₈×C₃₂	13

Subgroups — Q8×#88034 #15783550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88034 #15783550