Subgroups — C₂×#1368 #11998

Order 256.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×#334	C₂	1	2	yes	maximal
3	#1368	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈	—	2	2	no	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	#346	C₄	1	2	yes	—
9	#346	C₂²	1	2	yes	—
10	#334	—	2	2	no	—
11	#334	C₂²	1	4	yes	—
12	C₂²×C₈	—	4	2	no	—
13	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈	Q₈	1	1	yes	characteristic
15	C₂×C₄²	D₄	1	1	yes	characteristic
16	C₂×C₁₆	—	2	2	no	—
17	C₂×C₁₆	C₈	1	4	yes	complemented
18	C₂×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₆	C₂×C₄	1	2	yes	—
20	C₄⋊C₈	—	2	4	no	—
21	C₄⋊C₈	—	4	2	no	—
22	C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈	D₄	1	1	yes	characteristic
24	C₄×C₈	Q₈	1	1	yes	characteristic
25	C₄×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄	SD₁₆	1	2	yes	—
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	—	4	4	no	—
30	C₂×C₈	—	8	4	no	—
31	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈	C₂×C₈	1	2	yes	—
34	C₂×C₈	M₁₆	1	2	yes	—
35	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
37	C₄²	C₄⋊C₄	1	1	yes	characteristic
38	C₄²	SD₁₆	1	2	yes	—
39	C₄²	C₂×D₄	1	1	yes	characteristic
40	C₁₆	—	2	2	no	—
41	C₁₆	C₂×C₈	1	4	yes	complemented
42	C₂³	#106	1	1	yes	characteristic
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	C₄⋊C₈	1	2	yes	—
45	C₂×C₄	#106	1	2	yes	—
46	C₂×C₄	#106	1	1	yes	characteristic
47	C₂×C₄	#108	1	2	yes	—
48	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄	C₂×SD₁₆	1	2	yes	—
50	C₈	—	2	1	no	—
51	C₈	—	2	1	no	—
52	C₈	—	8	4	no	—
53	C₈	C₄⋊C₈	1	2	yes	—
54	C₈	C₂²×C₈	1	1	yes	characteristic
55	C₈	C₂×M₁₆	1	1	yes	characteristic
56	C₂²	#334	1	2	yes	—
57	C₂²	#365	1	4	yes	—
58	C₂²	C₂×#106	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	—	2	2	no	—
61	C₄	#334	1	2	yes	—
62	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₄	C₂×#108	1	1	yes	characteristic
64	C₂	#1368	1	4	yes	direct product factor
65	C₂	C₂×#334	1	1	yes	characteristic
66	C₂	C₂×#365	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	#106
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1368	C₂	direct (N × H)
17. C₂×C₁₆	C₈	semidirect
41. C₁₆	C₂×C₈	semidirect
64. C₂	#1368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#346	2
C₂×#334	2
#1368	2

Subgroups — C2×#1368 #11998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1368 #11998