Subgroups — C₂²×C₈×Q₈ #60185

Order 256.

1239 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₈	C₂	1	3	yes	maximal
4	C₂²×C₄×C₈	C₂	1	3	yes	maximal
5	C₂³×Q₈	C₄	1	1	yes	characteristic
6	C₂³×C₈	C₂²	1	3	yes	—
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈	C₄	1	12	yes	—
9	C₂×C₄×Q₈	C₂²	1	12	yes	—
10	C₂²×C₄⋊C₄	C₄	1	3	yes	—
11	C₂²×C₄²	C₂²	1	3	yes	—
12	C₈×Q₈	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊C₈	C₂²	1	36	yes	—
14	C₂×C₄×C₈	C₂²	1	36	yes	—
15	C₂²×Q₈	C₈	1	16	yes	direct product factor
16	C₂²×Q₈	C₂×C₄	1	12	yes	—
17	C₂³×C₄	C₂×C₄	1	3	yes	—
18	C₂³×C₄	C₂×C₄	1	3	yes	—
19	C₂³×C₄	C₂³	1	1	yes	characteristic
20	C₂²×C₈	—	2	12	no	—
21	C₂²×C₈	Q₈	1	8	yes	direct product factor
22	C₂²×C₈	C₂³	1	18	yes	—
23	C₂²×C₈	C₂³	1	6	yes	—
24	C₄×Q₈	C₂×C₄	1	48	yes	—
25	C₄×Q₈	C₂³	1	16	yes	—
26	C₂×C₄⋊C₄	C₂×C₄	1	36	yes	—
27	C₂×C₄²	C₂×C₄	1	18	yes	—
28	C₂×C₄²	C₂³	1	18	yes	—
29	C₄⋊C₈	C₂³	1	48	yes	—
30	C₄×C₈	C₂³	1	48	yes	—
31	C₂⁴	C₂²×C₄	1	1	yes	characteristic
32	C₂×Q₈	C₂×C₈	1	96	yes	direct product factor
33	C₂×Q₈	C₂²×C₄	1	16	yes	—
34	C₂²×C₄	—	2	12	no	—
35	C₂²×C₄	C₂×C₈	1	24	yes	—
36	C₂²×C₄	C₂²×C₄	1	18	yes	—
37	C₂²×C₄	C₂²×C₄	1	3	yes	—
38	C₂²×C₄	C₂²×C₄	1	18	yes	—
39	C₂²×C₄	C₂×Q₈	1	4	yes	—
40	C₂²×C₄	#41	1	4	yes	—
41	C₂²×C₄	C₂⁴	1	3	yes	—
42	C₂×C₈	—	2	36	no	—
43	C₂×C₈	C₂×Q₈	1	24	yes	direct product factor
44	C₂×C₈	C₂⁴	1	12	yes	—
45	C₂×C₈	C₂⁴	1	4	yes	—
46	C₄⋊C₄	C₂²×C₄	1	48	yes	—
47	C₄²	C₂²×C₄	1	36	yes	—
48	C₄²	C₂⁴	1	12	yes	—
49	C₂³	C₄×Q₈	1	4	yes	—
50	C₂³	C₂²×C₈	1	4	yes	—
51	C₂³	#131	1	4	yes	—
52	C₂³	C₂³×C₄	1	3	yes	—
53	Q₈	C₂²×C₈	1	64	yes	direct product factor
54	C₂×C₄	—	2	36	no	—
55	C₂×C₄	C₄×Q₈	1	12	yes	—
56	C₂×C₄	C₂²×C₈	1	72	yes	—
57	C₂×C₄	C₂³×C₄	1	12	yes	—
58	C₂×C₄	C₂³×C₄	1	3	yes	—
59	C₂×C₄	C₂³×C₄	1	12	yes	—
60	C₂×C₄	C₂²×Q₈	1	6	yes	—
61	C₂×C₄	C₂×#41	1	6	yes	—
62	C₂×C₄	C₂⁵	1	1	yes	characteristic
63	C₈	—	2	12	no	—
64	C₈	C₂²×Q₈	1	8	yes	direct product factor
65	C₂²	C₈×Q₈	1	16	yes	direct product factor
66	C₂²	C₂×C₄×Q₈	1	6	yes	—
67	C₂²	C₂³×C₈	1	6	yes	—
68	C₂²	C₂×#131	1	6	yes	—
69	C₂²	C₂⁴×C₄	1	1	yes	characteristic
70	C₄	—	2	12	no	—
71	C₄	C₂×C₄×Q₈	1	6	yes	—
72	C₄	C₂³×C₈	1	24	yes	—
73	C₄	C₂³×Q₈	1	1	yes	characteristic
74	C₄	C₂²×#41	1	1	yes	characteristic
75	C₂	C₂×C₈×Q₈	1	12	yes	direct product factor
76	C₂	C₂²×C₄×Q₈	1	1	yes	characteristic
77	C₂	C₂⁴×C₈	1	1	yes	characteristic
78	C₂	C₂²×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂²
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₂×C₄	C₂⁵
socle, prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈	C₂	direct (N × H)
12. C₈×Q₈	C₂²	direct (N × H)
15. C₂²×Q₈	C₈	direct (N × H)
21. C₂²×C₈	Q₈	direct (N × H)
32. C₂×Q₈	C₂×C₈	direct (N × H)
43. C₂×C₈	C₂×Q₈	direct (N × H)
53. Q₈	C₂²×C₈	direct (N × H)
64. C₈	C₂²×Q₈	direct (N × H)
65. C₂²	C₈×Q₈	direct (N × H)
75. C₂	C₂×C₈×Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈	2
C₂×C₈×Q₈	2
C₂²×C₄⋊C₈	2
C₂²×C₄×C₈	2

Subgroups — C22×C8×Q8 #60185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×Q₈ #60185