Subgroups — #7158

Order 256.

111 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	#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	C₄	1	2	yes	—
5	#431	—	2	1	no	—
6	C₂×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	#346	—	2	1	no	—
9	C₄×C₁₆	—	2	1	no	—
10	#335	C₄	1	2	yes	—
11	#334	—	2	1	no	—
12	#334	—	2	1	no	—
13	#334	C₄	1	2	yes	—
14	C₂²×C₈	—	4	1	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	Q₈	1	1	yes	characteristic
17	C₂×C₄²	D₄	1	1	yes	characteristic
18	M₃₂	—	2	2	no	—
19	M₃₂	C₈	1	4	yes	complemented
20	C₂×C₁₆	—	2	2	no	—
21	C₄⋊C₈	—	2	2	no	—
22	C₄⋊C₈	—	2	2	no	—
23	C₄⋊C₈	—	4	1	no	—
24	C₂²⋊C₈	—	4	1	no	—
25	C₄×C₈	—	4	1	no	—
26	C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄	D₈	1	1	yes	characteristic
32	C₂²×C₄	Q₁₆	1	1	yes	characteristic
33	C₂×C₈	—	2	2	no	—
34	C₂×C₈	—	4	2	no	—
35	C₂×C₈	—	4	2	no	—
36	C₂×C₈	—	8	2	no	—
37	C₂×C₈	C₄²	1	1	yes	characteristic
38	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈	C₂×C₈	1	2	yes	—
42	C₂×C₈	M₁₆	1	2	yes	—
43	C₄²	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²	SD₁₆	1	1	yes	characteristic
46	C₄²	SD₁₆	1	1	yes	characteristic
47	C₁₆	—	2	4	no	—
48	C₂³	#107	1	1	yes	characteristic
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	#95	1	1	yes	characteristic
52	C₂×C₄	#102	1	1	yes	characteristic
53	C₂×C₄	#102	1	1	yes	characteristic
54	C₂×C₄	#103	1	1	yes	characteristic
55	C₂×C₄	#103	1	1	yes	characteristic
56	C₂×C₄	C₄⋊C₈	1	2	yes	—
57	C₂×C₄	#106	1	1	yes	characteristic
58	C₂×C₄	#108	1	2	yes	—
59	C₈	—	2	1	no	—
60	C₈	—	2	1	no	—
61	C₈	—	8	2	no	—
62	C₈	—	8	2	no	—
63	C₈	C₄×C₈	1	1	yes	characteristic
64	C₈	#97	1	1	yes	characteristic
65	C₈	C₂²⋊C₈	1	2	yes	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	#335	1	2	yes	—
69	C₂²	#340	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	2	1	no	—
73	C₄	#334	1	2	yes	—
74	C₄	#336	1	1	yes	characteristic
75	C₄	#341	1	1	yes	characteristic
76	C₄	#360	1	2	yes	—
77	C₂	—	2	2	no	—
78	C₂	#1277	1	1	yes	characteristic
79	C₂	#1386	1	1	yes	characteristic
80	C₂	#1388	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#102
commutator	C₈	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#340
prime radical	C₂³	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
19. M₃₂	C₈	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
#1564	2
C₂×#334	2