Subgroups — C₂²×#519 #62646

Order 256.

5615 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#519	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×Q₈	C₂	1	8	yes	maximal, complemented
4	C₂²×#117	C₂	1	6	yes	maximal
5	C₂³×Q₈	C₄	1	32	yes	complemented
6	C₂³×Q₈	C₂²	1	4	yes	—
7	C₂³×Q₈	C₂²	1	24	yes	—
8	C₂⁴×C₄	C₂²	1	3	yes	—
9	#519	C₂²	1	256	yes	direct product factor
10	C₂×C₄×Q₈	C₂²	1	192	yes	complemented
11	C₂×#117	C₂²	1	144	yes	—
12	C₂²×C₄⋊C₄	C₂²	1	12	yes	—
13	C₂²×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂²×C₄²	C₂²	1	12	yes	—
15	C₂⁵	C₂³	1	1	yes	characteristic
16	C₂²×Q₈	C₂×C₄	1	64	yes	—
17	C₂²×Q₈	C₂×C₄	1	384	yes	complemented
18	C₂²×Q₈	C₂³	1	48	yes	—
19	C₂²×Q₈	C₂³	1	64	yes	—
20	C₂³×C₄	C₂×C₄	1	48	yes	—
21	C₂³×C₄	C₂³	1	8	yes	—
22	C₂³×C₄	C₂³	1	36	yes	—
23	C₂³×C₄	C₂³	1	6	yes	—
24	C₄×Q₈	C₂³	1	512	yes	complemented
25	#117	C₂³	1	384	yes	—
26	C₂×C₄⋊C₄	C₂³	1	144	yes	—
27	C₂×C₂²⋊C₄	C₂³	1	48	yes	—
28	C₂×C₄²	C₂³	1	144	yes	—
29	C₂⁴	—	2	8	no	—
30	C₂⁴	C₂²×C₄	1	8	yes	—
31	C₂⁴	C₂⁴	1	6	yes	—
32	C₂⁴	C₂⁴	1	1	yes	characteristic
33	C₂×Q₈	C₂²×C₄	1	384	yes	—
34	C₂×Q₈	C₂²×C₄	1	512	yes	complemented
35	C₂×Q₈	C₂⁴	1	64	yes	—
36	C₂²×C₄	—	2	32	no	—
37	C₂²×C₄	C₂²×C₄	1	48	yes	—
38	C₂²×C₄	C₂²×C₄	1	288	yes	—
39	C₂²×C₄	C₂⁴	1	48	yes	—
40	C₂²×C₄	C₂⁴	1	36	yes	—
41	C₂²×C₄	C₂⁴	1	48	yes	—
42	C₄⋊C₄	C₂⁴	1	192	yes	—
43	C₂²⋊C₄	C₂⁴	1	64	yes	—
44	C₄²	C₂⁴	1	192	yes	—
45	C₂³	—	2	32	no	—
46	C₂³	—	2	24	no	—
47	C₂³	C₂³×C₄	1	24	yes	—
48	C₂³	C₂³×C₄	1	4	yes	—
49	C₂³	2₋¹⁺⁴	1	8	yes	—
50	C₂³	C₂⁵	1	4	yes	—
51	C₂³	C₂⁵	1	3	yes	—
52	Q₈	C₂³×C₄	1	256	yes	—
53	C₂×C₄	—	2	96	no	—
54	C₂×C₄	C₂³×C₄	1	144	yes	—
55	C₂×C₄	C₂³×C₄	1	192	yes	—
56	C₂×C₄	C₂⁵	1	32	yes	—
57	C₂×C₄	C₂⁵	1	24	yes	—
58	C₂²	—	2	8	no	—
59	C₂²	—	2	48	no	—
60	C₂²	#519	1	16	yes	direct product factor
61	C₂²	C₂⁴×C₄	1	8	yes	—
62	C₂²	C₂⁴×C₄	1	6	yes	—
63	C₂²	C₂×2₋¹⁺⁴	1	12	yes	—
64	C₂²	C₂⁶	1	1	yes	characteristic
65	C₄	—	2	32	no	—
66	C₄	C₂⁴×C₄	1	48	yes	—
67	C₂	—	2	8	no	—
68	C₂	C₂×#519	1	12	yes	direct product factor
69	C₂	C₂⁵×C₄	1	1	yes	characteristic
70	C₂	C₂²×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂⁵	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#519	C₂	direct (N × H)
3. C₂²×C₄×Q₈	C₂	semidirect
5. C₂³×Q₈	C₄	semidirect
9. #519	C₂²	direct (N × H)
10. C₂×C₄×Q₈	C₂²	semidirect
17. C₂²×Q₈	C₂×C₄	semidirect
24. C₄×Q₈	C₂³	semidirect
34. C₂×Q₈	C₂²×C₄	semidirect
60. C₂²	#519	direct (N × H)
68. C₂	C₂×#519	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈	2
C₂×#519	2
C₂²×C₄×Q₈	2
C₂²×#117	2

Subgroups — C22×#519 #62646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#519 #62646