Subgroups — #7162

Order 256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	1	yes	maximal, characteristic
2	C₂×#364	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	C₂×M₃₂	—	2	1	no	—
6	C₂×M₃₂	C₄	1	1	yes	characteristic
7	C₂×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	#364	—	2	1	no	—
10	#364	—	2	1	no	—
11	#364	C₄	1	2	yes	—
12	#364	C₄	1	2	yes	—
13	#346	—	2	1	no	—
14	C₄×C₁₆	—	2	1	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂×C₄²	Q₈	1	1	yes	characteristic
18	M₃₂	—	2	1	no	—
19	M₃₂	—	2	1	no	—
20	M₃₂	—	2	2	no	—
21	M₃₂	—	2	2	no	—
22	M₃₂	—	4	1	no	—
23	M₃₂	—	4	1	no	—
24	M₃₂	C₈	1	2	yes	—
25	M₃₂	C₈	1	2	yes	—
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	—	2	1	no	—
28	C₂×C₁₆	—	4	1	no	—
29	C₂×C₁₆	—	4	1	no	—
30	C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₄×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₄	—	2	1	no	—
35	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈	C₄²	1	1	yes	characteristic
37	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈	M₁₆	1	1	yes	characteristic
43	C₂×C₈	M₁₆	1	1	yes	characteristic
44	C₂×C₈	D₈	1	1	yes	characteristic
45	C₂×C₈	SD₁₆	1	1	yes	characteristic
46	C₂×C₈	SD₁₆	1	1	yes	characteristic
47	C₂×C₈	Q₁₆	1	1	yes	characteristic
48	C₄²	—	2	1	no	—
49	C₄²	C₄⋊C₄	1	1	yes	characteristic
50	C₄²	C₄⋊C₄	1	1	yes	characteristic
51	C₁₆	—	2	2	no	—
52	C₁₆	—	2	2	no	—
53	C₁₆	—	4	2	no	—
54	C₁₆	—	4	2	no	—
55	C₂³	#108	1	1	yes	characteristic
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	#95	1	1	yes	characteristic
60	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄	#106	1	1	yes	characteristic
63	C₂×C₄	#107	1	1	yes	characteristic
64	C₂×C₄	#108	1	1	yes	characteristic
65	C₈	C₄×C₈	1	1	yes	characteristic
66	C₈	#97	1	1	yes	characteristic
67	C₈	C₂²⋊C₈	1	1	yes	characteristic
68	C₈	C₂²⋊C₈	1	1	yes	characteristic
69	C₈	#102	1	1	yes	characteristic
70	C₈	#102	1	1	yes	characteristic
71	C₈	#103	1	1	yes	characteristic
72	C₈	#103	1	1	yes	characteristic
73	C₂²	—	4	1	no	—
74	C₂²	#334	1	1	yes	characteristic
75	C₂²	#335	1	1	yes	characteristic
76	C₂²	#341	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	4	1	no	—
79	C₄	#334	1	1	yes	characteristic
80	C₄	#335	1	1	yes	characteristic
81	C₄	#336	1	1	yes	characteristic
82	C₄	#340	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	—	2	1	no	—
85	C₂	—	2	1	no	—
86	C₂	#1277	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#102
commutator	C₈	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂	#1277
prime radical	C₂³	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₂×#364	2
C₄×M₃₂	2