Subgroups — #11999

Order 256.

117 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	#1564	C₂	1	2	yes	maximal
3	#1368	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₁₆	C₄	1	1	yes	characteristic
6	#431	—	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	#335	—	2	2	no	—
11	#334	C₂²	1	4	yes	—
12	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈	Q₈	1	1	yes	characteristic
14	C₂×C₄²	D₄	1	1	yes	characteristic
15	C₂×C₁₆	—	2	2	no	—
16	C₂×C₁₆	C₈	1	4	yes	complemented
17	C₂×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₁₆	C₂×C₄	1	2	yes	—
19	C₄⋊C₈	—	2	4	no	—
20	C₂²⋊C₈	—	4	2	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	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈	C₂×C₈	1	2	yes	—
33	C₂×C₈	M₁₆	1	2	yes	—
34	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
36	C₄²	C₄⋊C₄	1	1	yes	characteristic
37	C₄²	SD₁₆	1	2	yes	—
38	C₄²	C₂×D₄	1	1	yes	characteristic
39	C₁₆	—	2	2	no	—
40	C₁₆	C₂×C₈	1	4	yes	—
41	C₂³	#106	1	1	yes	characteristic
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	C₄⋊C₈	1	2	yes	—
44	C₂×C₄	#106	1	2	yes	—
45	C₂×C₄	#106	1	1	yes	characteristic
46	C₂×C₄	#108	1	2	yes	—
47	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄	C₂×SD₁₆	1	2	yes	—
49	C₈	—	2	1	no	—
50	C₈	—	2	1	no	—
51	C₈	—	8	4	no	—
52	C₈	C₄⋊C₈	1	2	yes	—
53	C₈	C₂²×C₈	1	1	yes	characteristic
54	C₈	C₂×M₁₆	1	1	yes	characteristic
55	C₂²	—	2	2	no	—
56	C₂²	#334	1	2	yes	—
57	C₂²	C₂×#106	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	2	no	—
60	C₄	#334	1	2	yes	—
61	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₄	C₂×#108	1	1	yes	characteristic
63	C₂	—	2	2	no	—
64	C₂	C₂×#334	1	1	yes	characteristic
65	C₂	#2155	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#106
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×#106
prime radical	C₂³	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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	semidirect
16. C₂×C₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#346	2
#1564	2
#1368	2