Subgroups — Q₈×#63216 #14545197

Order 2176.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#63216	C₂	1	3	yes	maximal
2	C₄×#63216	C₂	1	3	yes	maximal
3	Q₈×#3622	C₂	1	1	yes	maximal, characteristic
4	C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
5	C₂×#63216	C₂²	1	3	yes	—
6	C₂×#63216	C₂²	1	1	yes	characteristic
7	C₄⋊#3622	C₄	1	3	yes	—
8	C₄×#3622	C₂²	1	3	yes	—
9	C₂×Q₈×C₁₇	C₈	1	1	yes	characteristic
10	C₄⋊C₄×C₁₇	C₈	1	3	yes	—
11	C₄²×C₁₇	C₂×C₄	1	3	yes	—
12	C₂×#3622	C₂×C₄	1	3	yes	—
13	C₂×#3622	C₂³	1	1	yes	characteristic
14	#63216	—	2	3	no	—
15	#63216	Q₈	1	2	yes	direct product factor
16	Q₈×C₁₇	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₁₇	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₁₇	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
20	#3622	—	2	3	no	—
21	#3622	C₂×Q₈	1	1	yes	characteristic
22	#3622	#41	1	1	yes	characteristic
23	Q₈×C₁₆	—	17	1	no	maximal, Sylow
24	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
25	C₄×C₁₇	—	2	3	no	—
26	C₄×C₁₇	C₂×C₁₆	1	6	yes	—
27	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
28	C₄×C₁₇	#131	1	1	yes	characteristic
29	C₈×Q₈	—	17	1	no	—
30	C₄⋊C₁₆	—	17	3	no	—
31	C₄×C₁₆	—	17	3	no	—
32	C₂×C₁₇	C₈×Q₈	1	1	yes	characteristic
33	C₂×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
34	C₂×C₁₇	#504	1	1	yes	characteristic
35	C₄×Q₈	#595	1	1	yes	characteristic
36	C₂×C₁₆	—	17	1	no	—
37	C₂×C₁₆	—	17	3	no	—
38	C₄⋊C₈	—	17	3	no	—
39	C₄×C₈	—	17	3	no	—
40	C₁₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
41	C₂×Q₈	#3624	1	1	yes	characteristic
42	C₂×C₈	—	17	1	no	—
43	C₂×C₈	—	17	3	no	—
44	C₄⋊C₄	#3624	1	3	yes	—
45	C₄²	C₂×#595	1	3	yes	—
46	C₁₆	—	17	2	no	—
47	C₁₆	—	34	3	no	—
48	Q₈	#63216	1	4	yes	direct product factor
49	C₂×C₄	C₂×#3624	1	3	yes	—
50	C₂×C₄	C₂×#3624	1	3	yes	—
51	C₂×C₄	C₂²×#595	1	1	yes	characteristic
52	C₈	—	17	1	no	—
53	C₈	—	17	1	no	—
54	C₈	—	34	3	no	—
55	C₂²	C₂²×#3624	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×#63216	1	6	yes	—
58	C₄	#10587680	1	1	yes	characteristic
59	C₄	Q₈×#595	1	1	yes	characteristic
60	C₂	Q₈×#3624	1	1	yes	characteristic
61	C₂	C₂²×#63216	1	1	yes	characteristic
62	C₂	#11763867	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#595
commutator	C₂×C₁₇	C₂²×C₁₆
Fitting	C₄×Q₈×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂³
hypercenter	C₄×Q₈	#595
hypocenter	C₁₇	Q₈×C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
15. #63216	Q₈	direct (N × H)
16. Q₈×C₁₇	C₁₆	semidirect
40. C₁₇	Q₈×C₁₆	semidirect
48. Q₈	#63216	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⋊#63216	2
C₄×#63216	2
Q₈×#3622	2
Q₈×C₁₆	17

Subgroups — Q8×#63216 #14545197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#63216 #14545197