Subgroups — Q₈×#97 #20522

Order 256.

477 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈	C₂	1	2	yes	maximal
2	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
3	#1770	C₂	1	6	yes	maximal
4	#1767	C₂	1	3	yes	maximal
5	C₄×#97	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈	C₄	1	2	yes	—
7	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
8	C₈×Q₈	C₄	1	16	yes	complemented
9	C₂×C₄⋊C₈	C₂²	1	6	yes	—
10	C₂×#97	C₂²	1	3	yes	—
11	C₂×#97	C₂²	1	6	yes	—
12	C₂×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	6	yes	—
14	C₄×C₄⋊C₄	C₄	1	6	yes	—
15	C₄×C₄⋊C₄	C₂²	1	3	yes	—
16	C₄³	C₂²	1	3	yes	—
17	#336	C₂²	1	6	yes	—
18	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	6	yes	—
20	C₂²×C₈	C₂³	1	2	yes	—
21	C₄×Q₈	—	2	8	no	—
22	C₄×Q₈	C₂×C₄	1	8	yes	—
23	C₂×C₄⋊C₄	C₂×C₄	1	6	yes	—
24	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
25	C₂×C₄²	C₂×C₄	1	6	yes	—
26	C₂×C₄²	C₂×C₄	1	6	yes	—
27	C₂×C₄²	C₂³	1	3	yes	—
28	C₂×C₄²	C₂³	1	3	yes	—
29	C₂×C₄²	C₂³	1	1	yes	characteristic
30	C₄⋊C₈	C₂×C₄	1	24	yes	—
31	#97	—	2	12	no	—
32	#97	—	2	6	no	—
33	#97	Q₈	1	4	yes	direct product factor
34	C₄×C₈	C₂×C₄	1	24	yes	—
35	#95	C₂×C₄	1	6	yes	—
36	C₂×Q₈	C₄²	1	4	yes	—
37	C₂×Q₈	M₁₆	1	8	yes	—
38	C₂²×C₄	C₂²×C₄	1	6	yes	—
39	C₂²×C₄	C₂²×C₄	1	3	yes	—
40	C₂²×C₄	C₂²×C₄	1	2	yes	—
41	C₂²×C₄	C₂²×C₄	1	3	yes	—
42	C₂²×C₄	C₂⁴	1	1	yes	characteristic
43	C₂×C₈	—	2	6	no	—
44	C₂×C₈	—	2	6	no	—
45	C₂×C₈	C₂²×C₄	1	12	yes	—
46	C₂×C₈	C₂²×C₄	1	4	yes	—
47	C₂×C₈	C₂×Q₈	1	4	yes	—
48	C₂×C₈	#41	1	4	yes	—
49	C₄⋊C₄	—	2	12	no	—
50	C₄⋊C₄	C₄²	1	12	yes	—
51	C₄²	—	2	12	no	—
52	C₄²	—	2	6	no	—
53	C₄²	—	4	6	no	—
54	C₄²	C₂²×C₄	1	12	yes	—
55	C₄²	C₂×Q₈	1	2	yes	—
56	C₄²	#41	1	2	yes	—
57	C₂³	C₂³×C₄	1	1	yes	characteristic
58	Q₈	#97	1	16	yes	direct product factor
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	6	no	—
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	2	6	no	—
65	C₂×C₄	C₂×C₄²	1	6	yes	—
66	C₂×C₄	C₂×C₄²	1	6	yes	—
67	C₂×C₄	C₄×Q₈	1	4	yes	—
68	C₂×C₄	C₂×M₁₆	1	12	yes	—
69	C₂×C₄	C₂³×C₄	1	2	yes	—
70	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄	C₂×#41	1	2	yes	—
72	C₂×C₄	C₂×#41	1	1	yes	characteristic
73	C₈	—	2	12	no	—
74	C₈	C₄×Q₈	1	8	yes	complemented
75	C₂²	C₂²×C₄²	1	1	yes	characteristic
76	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
77	C₂²	C₄×#41	1	1	yes	characteristic
78	C₂²	C₂²×M₁₆	1	2	yes	—
79	C₂²	#568	1	2	yes	—
80	C₄	—	2	6	no	—
81	C₄	—	2	4	no	—
82	C₄	—	4	6	no	—
83	C₄	C₂×#97	1	12	yes	—
84	C₄	C₂×C₄×Q₈	1	2	yes	—
85	C₄	C₄×#41	1	2	yes	—
86	C₂	C₄²×Q₈	1	1	yes	characteristic
87	C₂	C₂²×#97	1	1	yes	characteristic
88	C₂	#2875	1	1	yes	characteristic
89	C₂	Q₈×M₁₆	1	2	yes	—
90	C₂	#2971	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator	C₂²	C₂²×C₄²
socle, prime radical	C₂³	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈	C₄	semidirect
33. #97	Q₈	direct (N × H)
58. Q₈	#97	direct (N × H)
74. C₈	C₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈	2
C₄²×Q₈	2
#1770	2
#1767	2
C₄×#97	2

Subgroups — Q8×#97 #20522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#97 #20522