Subgroups — #13647

Order 256.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2258	C₂	1	2	yes	maximal, complemented
2	#2170	C₂	1	1	yes	maximal, characteristic
3	#1422	C₂	1	1	yes	maximal, characteristic, complemented
4	#1401	C₂	1	2	yes	maximal, complemented
5	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#504	C₂²	1	2	yes	—
7	#443	C₄	1	1	yes	characteristic
8	M₆₄	—	2	2	no	—
9	M₆₄	C₂²	1	2	yes	—
10	C₂×C₃₂	—	2	2	no	—
11	C₂×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₃₂	C₂²	1	1	yes	characteristic
13	#364	C₄	1	1	yes	characteristic
14	#350	C₄	1	2	yes	—
15	C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#135	C₈	1	1	yes	characteristic
17	#131	C₂×C₄	1	2	yes	—
18	M₃₂	—	2	2	no	—
19	M₃₂	C₂×C₄	1	2	yes	—
20	C₂×C₁₆	—	2	2	no	—
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂³	1	1	yes	characteristic
23	#108	C₈	1	1	yes	characteristic
24	#104	C₈	1	2	yes	—
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃₂	—	2	2	no	—
27	C₃₂	—	2	1	no	—
28	C₃₂	D₄	1	1	yes	characteristic
29	C₃₂	D₄	1	1	yes	characteristic
30	#41	C₂×C₈	1	2	yes	—
31	Q₁₆	C₁₆	1	1	yes	characteristic
32	SD₁₆	C₁₆	1	2	yes	—
33	D₈	C₁₆	1	1	yes	characteristic
34	M₁₆	—	2	2	no	—
35	M₁₆	C₂×C₈	1	2	yes	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₄²	C₂×C₈	1	1	yes	characteristic
40	C₁₆	—	2	2	no	—
41	C₁₆	—	2	1	no	—
42	C₁₆	C₂×D₄	1	1	yes	characteristic
43	C₁₆	#41	1	1	yes	characteristic
44	Q₈	C₂×C₁₆	1	2	yes	—
45	D₄	—	2	2	no	—
46	D₄	C₂×C₁₆	1	2	yes	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	C₂×C₁₆	1	1	yes	characteristic
52	C₈	C₂×C₁₆	1	1	yes	characteristic
53	C₈	C₄×D₄	1	1	yes	characteristic
54	C₈	#131	1	1	yes	characteristic
55	C₂²	—	2	2	no	—
56	C₂²	#504	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	2	2	no	—
59	C₄	C₈×D₄	1	1	yes	characteristic
60	C₄	C₂²×C₁₆	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	—	4	2	no	—
63	C₂	D₄×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	D₄
commutator	C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂	D₄×C₁₆
prime radical	#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ 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
1. #2258	C₂	semidirect
3. #1422	C₂	semidirect
4. #1401	C₂	semidirect
5. C₄×C₃₂	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2258	2
#2170	2
#1422	2
#1401	2
C₄×C₃₂	2