Subgroups — C₂×#3556 #62760

Order 256.

3169 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3556	C₂	1	32	yes	maximal, direct product factor
2	C₂×#564	C₂	1	4	yes	maximal, complemented
3	C₂×#557	C₂	1	12	yes	maximal, complemented
4	C₂×#541	C₂	1	12	yes	maximal
5	C₂×#533	C₂	1	3	yes	maximal
6	#564	C₂²	1	64	yes	complemented
7	#557	C₂²	1	192	yes	complemented
8	#541	C₂²	1	192	yes	—
9	#533	C₂²	1	48	yes	—
10	C₂×C₄⋊Q₈	C₂²	1	6	yes	—
11	C₂×C₄⋊Q₈	C₂²	1	12	yes	—
12	C₂×#126	C₂²	1	16	yes	—
13	C₂×#125	C₂²	1	6	yes	—
14	C₂×#125	C₂²	1	12	yes	—
15	C₂×#125	C₂²	1	48	yes	—
16	C₂×C₂²⋊Q₈	C₂²	1	12	yes	—
17	C₂×C₄×Q₈	C₂²	1	12	yes	—
18	C₂×C₄×Q₈	C₂²	1	24	yes	—
19	C₂×#117	C₂²	1	6	yes	—
20	C₂²×C₄²	C₂²	1	1	yes	characteristic
21	C₂²×Q₈	C₂³	1	12	yes	—
22	C₂³×C₄	C₂³	1	3	yes	—
23	C₄⋊Q₈	C₂³	1	48	yes	—
24	C₄⋊Q₈	C₂³	1	96	yes	—
25	#126	C₂³	1	128	yes	—
26	#125	C₂³	1	48	yes	—
27	#125	C₂³	1	96	yes	—
28	#125	C₂³	1	384	yes	—
29	C₂²⋊Q₈	C₂³	1	96	yes	—
30	C₄×Q₈	C₂³	1	96	yes	—
31	C₄×Q₈	C₂³	1	192	yes	—
32	#117	C₂³	1	48	yes	—
33	C₂×C₄⋊C₄	C₂³	1	12	yes	—
34	C₂×C₄⋊C₄	C₂³	1	12	yes	—
35	C₂×C₄⋊C₄	C₂³	1	24	yes	—
36	C₂×C₄⋊C₄	C₂³	1	48	yes	—
37	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
38	C₂×C₄²	C₂³	1	12	yes	—
39	C₂×C₄²	C₂³	1	3	yes	—
40	C₂×C₄²	C₂³	1	16	yes	—
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₂×C₄²	C₂³	1	8	yes	—
43	C₂⁴	C₂⁴	1	1	yes	characteristic
44	C₂×Q₈	—	2	48	no	—
45	C₂×Q₈	C₂⁴	1	48	yes	—
46	C₂²×C₄	—	2	12	no	—
47	C₂²×C₄	C₂⁴	1	12	yes	—
48	C₂²×C₄	C₂⁴	1	24	yes	—
49	C₂²×C₄	C₂⁴	1	3	yes	—
50	C₂²×C₄	C₂⁴	1	3	yes	—
51	C₄⋊C₄	C₂⁴	1	48	yes	—
52	C₄⋊C₄	C₂⁴	1	48	yes	—
53	C₄⋊C₄	C₂⁴	1	96	yes	—
54	C₄⋊C₄	C₂⁴	1	192	yes	—
55	C₂²⋊C₄	C₂⁴	1	48	yes	—
56	C₄²	C₂⁴	1	12	yes	—
57	C₄²	C₂⁴	1	4	yes	—
58	C₄²	C₂⁴	1	48	yes	—
59	C₄²	C₂⁴	1	64	yes	—
60	C₂³	—	2	6	no	—
61	C₂³	C₂⁵	1	2	yes	—
62	C₂³	C₂⁵	1	1	yes	characteristic
63	Q₈	—	2	96	no	—
64	C₂×C₄	—	2	6	no	—
65	C₂×C₄	—	2	48	no	—
66	C₂×C₄	—	2	24	no	—
67	C₂×C₄	2₋¹⁺⁴	1	12	yes	—
68	C₂×C₄	C₂⁵	1	48	yes	—
69	C₂×C₄	C₂⁵	1	6	yes	—
70	C₂×C₄	C₂⁵	1	6	yes	—
71	C₂²	—	2	6	no	—
72	C₂²	—	4	4	no	—
73	C₂²	C₂×2₋¹⁺⁴	1	6	yes	—
74	C₂²	C₂⁶	1	1	yes	characteristic
75	C₄	—	2	6	no	—
76	C₄	—	2	48	no	—
77	C₄	C₂×2₋¹⁺⁴	1	12	yes	—
78	C₂	—	4	2	no	—
79	C₂	#3556	1	4	yes	direct product factor
80	C₂	C₂²×2₋¹⁺⁴	1	3	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂⁴	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3556	C₂	direct (N × H)
2. C₂×#564	C₂	semidirect
3. C₂×#557	C₂	semidirect
6. #564	C₂²	semidirect
7. #557	C₂²	semidirect
79. C₂	#3556	direct (N × H)

Maximal subgroups

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

subgroup	index
#3556	2
C₂×#564	2
C₂×#557	2
C₂×#541	2
C₂×#533	2

Subgroups — C2×#3556 #62760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3556 #62760