Subgroups — C₈×2₋¹⁺⁴ #60329

Order 256.

1095 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	#2988	C₂	1	10	yes	maximal, complemented
3	#2982	C₂	1	5	yes	maximal, complemented
4	C₈×#41	C₂	1	10	yes	maximal, complemented
5	C₂×C₈×Q₈	C₂	1	5	yes	maximal, complemented
6	C₂×2₋¹⁺⁴	C₄	1	1	yes	characteristic
7	#520	C₄	1	10	yes	—
8	#519	C₄	1	5	yes	—
9	C₄×#41	C₂²	1	10	yes	—
10	C₂×C₄×Q₈	C₂²	1	5	yes	—
11	C₈×Q₈	C₂²	1	10	yes	—
12	C₈×Q₈	C₂²	1	20	yes	—
13	C₈×Q₈	C₂²	1	10	yes	—
14	C₈×D₄	C₂²	1	10	yes	—
15	C₈×D₄	C₂²	1	30	yes	—
16	#431	C₂²	1	30	yes	—
17	C₂×C₄⋊C₈	C₂²	1	15	yes	—
18	C₂×C₄×C₈	C₂²	1	15	yes	—
19	2₋¹⁺⁴	C₈	1	16	yes	direct product factor
20	C₂×#41	C₂×C₄	1	10	yes	—
21	C₂²×Q₈	C₂×C₄	1	5	yes	—
22	C₂²×C₈	C₂³	1	5	yes	—
23	C₂²×C₈	C₂³	1	15	yes	—
24	C₄×Q₈	C₂×C₄	1	20	yes	—
25	C₄×Q₈	C₂×C₄	1	10	yes	—
26	C₄×Q₈	C₂³	1	10	yes	—
27	C₄×D₄	C₂×C₄	1	30	yes	—
28	C₄×D₄	C₂³	1	10	yes	—
29	#117	C₂×C₄	1	30	yes	—
30	C₂×C₄⋊C₄	C₂×C₄	1	15	yes	—
31	C₂×C₄²	C₂³	1	15	yes	—
32	C₄⋊C₈	C₂³	1	30	yes	—
33	C₄⋊C₈	C₂³	1	10	yes	—
34	C₂²⋊C₈	C₂³	1	20	yes	—
35	C₄×C₈	C₂³	1	30	yes	—
36	C₄×C₈	C₂³	1	10	yes	—
37	#41	C₂×C₈	1	80	yes	complemented
38	C₂×Q₈	C₂×C₈	1	40	yes	complemented
39	C₂×Q₈	C₂²×C₄	1	10	yes	—
40	C₂×D₄	C₂²×C₄	1	10	yes	—
41	C₂²×C₄	C₂²×C₄	1	15	yes	—
42	C₂²×C₄	C₂²×C₄	1	15	yes	—
43	C₂²×C₄	C₂⁴	1	5	yes	—
44	C₂×C₈	—	2	10	no	—
45	C₂×C₈	—	4	15	no	—
46	C₂×C₈	C₂⁴	1	5	yes	—
47	C₂×C₈	C₂⁴	1	10	yes	—
48	C₂×C₈	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄	C₂²×C₄	1	30	yes	—
50	C₄⋊C₄	C₂²×C₄	1	10	yes	—
51	C₂²⋊C₄	C₂²×C₄	1	20	yes	—
52	C₄²	C₂²×C₄	1	30	yes	—
53	C₄²	C₂⁴	1	10	yes	—
54	C₂³	C₂³×C₄	1	5	yes	—
55	Q₈	C₂²×C₈	1	40	yes	—
56	D₄	C₂²×C₈	1	40	yes	—
57	C₂×C₄	—	2	5	no	—
58	C₂×C₄	—	2	5	no	—
59	C₂×C₄	—	4	15	no	—
60	C₂×C₄	C₂²×C₈	1	60	yes	—
61	C₂×C₄	C₂³×C₄	1	5	yes	—
62	C₂×C₄	C₂³×C₄	1	10	yes	—
63	C₂×C₄	C₂³×C₄	1	10	yes	—
64	C₂×C₄	C₂⁵	1	1	yes	characteristic
65	C₈	—	2	5	no	—
66	C₈	—	2	10	no	—
67	C₈	2₋¹⁺⁴	1	2	yes	direct product factor
68	C₂²	—	2	5	no	—
69	C₂²	—	2	5	no	—
70	C₂²	C₂³×C₈	1	10	yes	—
71	C₂²	C₂⁴×C₄	1	1	yes	characteristic
72	C₄	—	2	5	no	—
73	C₄	—	2	10	no	—
74	C₄	C₂³×C₈	1	20	yes	—
75	C₄	C₂×2₋¹⁺⁴	1	1	yes	characteristic
76	C₄	#585	1	1	yes	characteristic
77	C₂	—	2	10	no	—
78	C₂	C₄×2₋¹⁺⁴	1	1	yes	characteristic
79	C₂	C₂⁴×C₈	1	1	yes	characteristic
80	C₂	#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂⁴
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂²	C₂⁴×C₄
prime radical	C₂×2₋¹⁺⁴	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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. #2988	C₂	semidirect
3. #2982	C₂	semidirect
4. C₈×#41	C₂	semidirect
5. C₂×C₈×Q₈	C₂	semidirect
19. 2₋¹⁺⁴	C₈	direct (N × H)
37. #41	C₂×C₈	semidirect
38. C₂×Q₈	C₂×C₈	semidirect
67. C₈	2₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×2₋¹⁺⁴	2
#2988	2
#2982	2
C₈×#41	2
C₂×C₈×Q₈	2

Subgroups — C8×2−1+4 #60329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×2₋¹⁺⁴ #60329