Subgroups — #7470

Order 256.

89 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic
3	#2108	C₂	1	1	yes	maximal, characteristic
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	1	no	—
8	M₆₄	C₄	1	2	yes	complemented
9	C₂×C₃₂	—	2	1	no	—
10	C₂×C₃₂	—	2	1	no	—
11	C₂×C₃₂	—	2	1	no	—
12	C₂×C₃₂	C₄	1	2	yes	complemented
13	#346	—	2	1	no	—
14	C₄×C₁₆	—	2	1	no	—
15	C₂×M₁₆	C₈	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	#117	C₈	1	1	yes	characteristic
18	M₃₂	—	2	1	no	—
19	M₃₂	C₈	1	2	yes	—
20	C₂×C₁₆	—	2	1	no	—
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆	D₄	1	1	yes	characteristic
24	C₂×C₁₆	D₄	1	1	yes	characteristic
25	C₂×C₁₆	D₄	1	1	yes	characteristic
26	C₂×C₁₆	Q₈	1	1	yes	characteristic
27	#97	—	2	1	no	—
28	C₄×C₈	—	2	1	no	—
29	C₃₂	—	2	2	no	—
30	C₃₂	—	2	2	no	—
31	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
32	M₁₆	—	2	1	no	—
33	M₁₆	C₁₆	1	2	yes	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	C₄²	1	1	yes	characteristic
36	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈	M₁₆	1	1	yes	characteristic
40	C₂×C₈	M₁₆	1	1	yes	characteristic
41	C₄⋊C₄	C₁₆	1	2	yes	—
42	C₂²⋊C₄	—	2	1	no	—
43	C₄²	—	2	1	no	—
44	C₁₆	—	2	2	no	—
45	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₆	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆	C₄⋊C₄	1	1	yes	characteristic
49	C₂³	M₃₂	1	1	yes	characteristic
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄	M₃₂	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₈	#95	1	1	yes	characteristic
59	C₈	#97	1	1	yes	characteristic
60	C₈	C₂²⋊C₈	1	1	yes	characteristic
61	C₈	C₄⋊C₈	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	#346	1	1	yes	characteristic
64	C₂²	C₂²⋊C₁₆	1	1	yes	characteristic
65	C₂²	C₄⋊C₁₆	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	#336	1	1	yes	characteristic
68	C₄	C₄×C₁₆	1	1	yes	characteristic
69	C₄	C₂²⋊C₁₆	1	1	yes	characteristic
70	C₄	C₄⋊C₁₆	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	#1374	1	1	yes	characteristic
74	C₂	#1401	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	D₄
commutator	C₄	C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂²
socle	C₂²	C₂²⋊C₁₆
prime radical	C₂³	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. M₆₄	C₄	semidirect
12. C₂×C₃₂	C₄	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
C₂²×C₃₂	2
#2108	2