Subgroups — #13642

Order 256.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#1422	C₂	1	2	yes	maximal, complemented
3	#1400	C₂	1	2	yes	maximal
4	#1398	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂	C₄	1	1	yes	characteristic
6	C₂×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈	C₄	1	1	yes	characteristic
8	M₆₄	C₂²	1	4	yes	—
9	#346	C₄	1	1	yes	characteristic
10	#346	C₂²	1	1	yes	characteristic
11	C₄×C₁₆	C₄	1	1	yes	characteristic
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈	C₈	1	1	yes	characteristic
14	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²	C₈	1	1	yes	characteristic
16	M₃₂	—	2	2	no	—
17	M₃₂	—	2	2	no	—
18	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂³	1	1	yes	characteristic
22	C₄×C₈	C₈	1	1	yes	characteristic
23	C₄×C₈	C₈	1	1	yes	characteristic
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃₂	—	2	4	no	—
27	C₂²×C₄	—	2	1	no	—
28	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₄²	—	2	1	no	—
37	C₄²	C₂×C₈	1	1	yes	characteristic
38	C₄²	C₂×C₈	1	1	yes	characteristic
39	C₁₆	—	2	1	no	—
40	C₁₆	—	2	1	no	—
41	C₁₆	M₁₆	1	2	yes	—
42	C₁₆	#41	1	2	yes	—
43	C₂³	M₃₂	1	1	yes	characteristic
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	4	2	no	—
49	C₂×C₄	M₃₂	1	1	yes	characteristic
50	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	—	2	1	no	—
53	C₈	M₃₂	1	2	yes	—
54	C₈	#117	1	1	yes	characteristic
55	C₈	C₂×M₁₆	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂×M₃₂	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	#431	1	1	yes	characteristic
63	C₄	C₂×M₃₂	1	1	yes	characteristic
64	C₂	—	2	1	no	—
65	C₂	—	4	1	no	—
66	C₂	#2163	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#117
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂	#2163
prime radical	C₂³	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #346 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1422	C₂	semidirect
4. #1398	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂	2
#1422	2
#1400	2
#1398	2