Subgroups — #13630

Order 256.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	2	yes	maximal
2	#2108	C₂	1	1	yes	maximal, characteristic
3	#1422	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂	C₄	1	1	yes	characteristic
5	C₂²×C₁₆	C₂²	1	1	yes	characteristic
6	#405	C₄	1	1	yes	characteristic
7	M₆₄	—	2	2	no	—
8	M₆₄	C₂²	1	4	yes	—
9	C₂×C₃₂	—	2	2	no	—
10	#346	C₄	1	2	yes	—
11	C₄×C₁₆	C₂²	1	2	yes	—
12	C₂×M₁₆	C₈	1	1	yes	characteristic
13	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
14	#117	C₈	1	1	yes	characteristic
15	M₃₂	—	2	2	no	—
16	C₂×C₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆	D₄	1	2	yes	—
19	C₂×C₁₆	Q₈	1	2	yes	—
20	C₂×C₁₆	C₂³	1	1	yes	characteristic
21	#97	C₈	1	2	yes	—
22	C₄×C₈	C₂×C₄	1	2	yes	—
23	C₃₂	—	2	4	no	—
24	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
25	M₁₆	C₁₆	1	4	yes	—
26	C₂×C₈	C₄⋊C₄	1	2	yes	—
27	C₂×C₈	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈	C₂×C₈	1	2	yes	—
29	C₂×C₈	M₁₆	1	2	yes	—
30	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄	—	2	1	no	—
32	C₂²⋊C₄	—	2	1	no	—
33	C₄²	C₂×C₈	1	2	yes	—
34	C₁₆	—	2	2	no	—
35	C₁₆	C₄⋊C₄	1	2	yes	—
36	C₁₆	C₂×D₄	1	1	yes	characteristic
37	C₁₆	C₂×Q₈	1	1	yes	characteristic
38	C₂³	M₃₂	1	1	yes	characteristic
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	C₄⋊C₈	1	2	yes	—
41	C₂×C₄	C₂×C₁₆	1	2	yes	—
42	C₂×C₄	M₃₂	1	1	yes	characteristic
43	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
44	C₈	C₄⋊C₈	1	2	yes	—
45	C₈	C₂×C₁₆	1	4	yes	—
46	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₈	C₂×M₁₆	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	C₄⋊C₁₆	1	2	yes	—
50	C₂²	C₂×M₃₂	1	1	yes	characteristic
51	C₄	—	4	2	no	—
52	C₄	C₄⋊C₁₆	1	2	yes	—
53	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₄	C₂²×C₁₆	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	—	2	2	no	—
57	C₂	C₂×C₄⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×D₄
commutator	C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂	C₂×C₄⋊C₁₆
prime radical	C₂³	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ 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
3. #1422	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₆₄	2
#2108	2
#1422	2