Subgroups — C₂×#2246 #33709

Order 256.

511 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆	C₂	1	4	yes	maximal, complemented
2	C₂×#493	C₂	1	2	yes	maximal, complemented
3	#2246	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈	—	2	2	no	—
6	C₂²×D₈	C₂²	1	4	yes	—
7	C₂×#127	—	2	2	no	—
8	C₂×D₁₆	—	2	16	no	—
9	C₂×D₁₆	C₂²	1	16	yes	complemented
10	C₂²×C₁₆	C₂²	1	2	yes	—
11	#493	—	2	4	no	—
12	#493	C₂²	1	8	yes	complemented
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₁₆	C₂²	1	4	yes	complemented
15	C₂²×D₄	—	2	4	no	—
16	C₂²×D₄	—	4	4	no	—
17	C₂×D₈	—	2	16	no	—
18	C₂×D₈	—	2	8	no	—
19	C₂×D₈	—	2	4	no	—
20	C₂×D₈	—	4	8	no	—
21	C₂×D₈	C₂³	1	8	yes	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	#127	—	2	8	no	—
25	#127	—	4	4	no	—
26	C₂×C₄²	D₄	1	1	yes	characteristic
27	D₁₆	—	2	32	no	—
28	C₂×C₁₆	D₄	1	8	yes	complemented
29	C₂×C₁₆	C₂³	1	4	yes	—
30	C₄×C₈	D₄	1	2	yes	—
31	C₄×C₈	C₂³	1	2	yes	—
32	C₂⁴	—	4	4	no	—
33	C₂×D₄	—	2	8	no	—
34	C₂×D₄	—	4	8	no	—
35	C₂×D₄	—	4	8	no	—
36	C₂×D₄	—	4	16	no	—
37	C₂×D₄	—	4	4	no	—
38	C₂×D₄	—	8	16	no	—
39	C₂²×C₄	D₈	1	2	yes	—
40	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
41	D₈	—	2	16	no	—
42	D₈	—	4	8	no	—
43	D₈	—	4	16	no	—
44	C₂×C₈	D₈	1	4	yes	—
45	C₂×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈	C₂×D₄	1	4	yes	—
47	C₂×C₈	C₂×D₄	1	2	yes	—
48	C₂×C₈	C₂⁴	1	1	yes	characteristic
49	C₄²	D₈	1	2	yes	—
50	C₄²	C₂×D₄	1	2	yes	—
51	C₁₆	C₂×D₄	1	8	yes	complemented
52	C₂³	—	4	8	no	—
53	C₂³	—	8	8	no	—
54	C₂³	—	8	16	no	—
55	C₂³	C₂×D₈	1	1	yes	characteristic
56	D₄	—	4	8	no	—
57	D₄	—	4	8	no	—
58	D₄	—	8	8	no	—
59	D₄	—	8	32	no	—
60	C₂×C₄	D₁₆	1	8	yes	complemented
61	C₂×C₄	#127	1	2	yes	—
62	C₂×C₄	C₂×D₈	1	2	yes	—
63	C₂×C₄	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄	C₂×D₈	1	4	yes	—
65	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
66	C₈	#127	1	2	yes	—
67	C₈	C₂×D₈	1	4	yes	—
68	C₈	C₂²×D₄	1	2	yes	—
69	C₂²	—	8	8	no	—
70	C₂²	—	8	16	no	—
71	C₂²	—	16	16	no	—
72	C₂²	#493	1	2	yes	—
73	C₂²	C₂×D₁₆	1	4	yes	—
74	C₂²	C₂²×D₈	1	1	yes	characteristic
75	C₄	#493	1	2	yes	—
76	C₄	C₂×D₁₆	1	8	yes	complemented
77	C₄	C₂×#127	1	1	yes	characteristic
78	C₄	C₂²×D₈	1	1	yes	characteristic
79	C₂	—	16	8	no	—
80	C₂	#2246	1	4	yes	direct product factor
81	C₂	C₂×#493	1	1	yes	characteristic
82	C₂	C₂²×D₁₆	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₆	C₂	semidirect
2. C₂×#493	C₂	semidirect
3. #2246	C₂	direct (N × H)
4. C₂×C₄×C₁₆	C₂	semidirect
9. C₂×D₁₆	C₂²	semidirect
12. #493	C₂²	semidirect
14. C₄×C₁₆	C₂²	semidirect
28. C₂×C₁₆	D₄	semidirect
51. C₁₆	C₂×D₄	semidirect
60. C₂×C₄	D₁₆	semidirect
76. C₄	C₂×D₁₆	semidirect
80. C₂	#2246	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×D₁₆	2
C₂×#493	2
#2246	2
C₂×C₄×C₁₆	2

Subgroups — C2×#2246 #33709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2246 #33709