Subgroups — C₂²×#493 #60629

Order 256.

2319 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈	C₂	1	4	yes	maximal, complemented
2	C₂²×#127	C₂	1	2	yes	maximal, complemented
3	C₂×#493	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄	—	2	4	no	—
6	C₂³×D₄	C₂²	1	4	yes	—
7	C₂²×D₈	—	2	32	no	—
8	C₂²×D₈	C₂²	1	48	yes	complemented
9	C₂³×C₈	C₂²	1	2	yes	—
10	C₂×#127	—	2	12	no	—
11	C₂×#127	C₂²	1	24	yes	complemented
12	C₂²×C₄²	C₂²	1	1	yes	characteristic
13	#493	C₂²	1	64	yes	direct product factor
14	C₂×C₄×C₈	C₂²	1	12	yes	complemented
15	C₂⁵	—	2	4	no	—
16	C₂²×D₄	—	2	32	no	—
17	C₂²×D₄	—	2	12	no	—
18	C₂²×D₄	—	2	48	no	—
19	C₂²×D₄	—	4	32	no	—
20	C₂²×D₄	C₂³	1	24	yes	—
21	C₂³×C₄	D₄	1	2	yes	—
22	C₂³×C₄	C₂³	1	1	yes	characteristic
23	C₂×D₈	—	2	192	no	—
24	C₂×D₈	C₂³	1	64	yes	complemented
25	C₂²×C₈	D₄	1	16	yes	complemented
26	C₂²×C₈	C₂³	1	12	yes	—
27	#127	—	2	48	no	—
28	#127	C₂³	1	32	yes	complemented
29	C₂×C₄²	D₄	1	6	yes	—
30	C₂×C₄²	C₂³	1	6	yes	—
31	C₄×C₈	C₂³	1	16	yes	complemented
32	C₂⁴	—	2	24	no	—
33	C₂⁴	—	4	16	no	—
34	C₂⁴	—	4	32	no	—
35	C₂⁴	C₂×D₄	1	1	yes	characteristic
36	C₂×D₄	—	2	96	no	—
37	C₂×D₄	—	2	24	no	—
38	C₂×D₄	—	2	64	no	—
39	C₂×D₄	—	4	48	no	—
40	C₂×D₄	—	4	192	no	—
41	C₂×D₄	C₂⁴	1	16	yes	—
42	C₂²×C₄	D₈	1	16	yes	complemented
43	C₂²×C₄	C₂×D₄	1	3	yes	—
44	C₂²×C₄	C₂×D₄	1	8	yes	—
45	C₂²×C₄	C₂×D₄	1	12	yes	—
46	C₂²×C₄	C₂⁴	1	3	yes	—
47	D₈	—	2	128	no	—
48	C₂×C₈	C₂×D₄	1	48	yes	complemented
49	C₂×C₈	C₂⁴	1	8	yes	—
50	C₄²	C₂×D₄	1	12	yes	—
51	C₄²	C₂⁴	1	4	yes	—
52	C₂³	—	2	16	no	—
53	C₂³	—	4	48	no	—
54	C₂³	—	4	96	no	—
55	C₂³	—	8	64	no	—
56	C₂³	#127	1	4	yes	—
57	C₂³	C₂×D₈	1	8	yes	—
58	C₂³	C₂²×D₄	1	3	yes	—
59	D₄	—	2	32	no	—
60	D₄	—	4	48	no	—
61	D₄	—	4	128	no	—
62	C₂×C₄	#127	1	12	yes	—
63	C₂×C₄	C₂×D₈	1	48	yes	complemented
64	C₂×C₄	C₂²×D₄	1	3	yes	—
65	C₂×C₄	C₂²×D₄	1	12	yes	—
66	C₂×C₄	C₂²×D₄	1	8	yes	—
67	C₂×C₄	C₂⁵	1	1	yes	characteristic
68	C₈	C₂²×D₄	1	16	yes	complemented
69	C₂²	—	4	16	no	—
70	C₂²	—	4	32	no	—
71	C₂²	—	8	96	no	—
72	C₂²	#493	1	16	yes	direct product factor
73	C₂²	C₂×#127	1	6	yes	—
74	C₂²	C₂²×D₈	1	12	yes	—
75	C₂²	C₂³×D₄	1	1	yes	characteristic
76	C₄	C₂×#127	1	6	yes	—
77	C₄	C₂²×D₈	1	16	yes	complemented
78	C₄	C₂³×D₄	1	2	yes	—
79	C₂	—	8	16	no	—
80	C₂	C₂×#493	1	12	yes	direct product factor
81	C₂	C₂²×#127	1	1	yes	characteristic
82	C₂	C₂³×D₈	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×D₈	C₂	semidirect
2. C₂²×#127	C₂	semidirect
3. C₂×#493	C₂	direct (N × H)
4. C₂²×C₄×C₈	C₂	semidirect
8. C₂²×D₈	C₂²	semidirect
11. C₂×#127	C₂²	semidirect
13. #493	C₂²	direct (N × H)
14. C₂×C₄×C₈	C₂²	semidirect
24. C₂×D₈	C₂³	semidirect
25. C₂²×C₈	D₄	semidirect
28. #127	C₂³	semidirect
31. C₄×C₈	C₂³	semidirect
42. C₂²×C₄	D₈	semidirect
48. C₂×C₈	C₂×D₄	semidirect
63. C₂×C₄	C₂×D₈	semidirect
68. C₈	C₂²×D₄	semidirect
72. C₂²	#493	direct (N × H)
77. C₄	C₂²×D₈	semidirect
80. C₂	C₂×#493	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₈	2
C₂²×#127	2
C₂×#493	2
C₂²×C₄×C₈	2

Subgroups — C22×#493 #60629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#493 #60629