Subgroups — #7124

Order 256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1940	C₂	1	1	yes	maximal, characteristic
2	#1377	C₂	1	1	yes	maximal, characteristic
3	#1377	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	C₂×M₃₂	—	2	1	no	—
6	C₂×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₂×#97	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
10	#336	—	2	1	no	—
11	C₂²×C₈	—	2	1	no	—
12	C₂²×C₈	D₄	1	1	yes	characteristic
13	C₂²×C₈	D₄	1	1	yes	characteristic
14	C₂×C₄⋊C₄	—	2	1	no	—
15	C₂×C₄²	—	2	1	no	—
16	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
17	M₃₂	—	2	2	no	—
18	M₃₂	—	2	2	no	—
19	M₃₂	—	4	1	no	—
20	M₃₂	—	4	1	no	—
21	C₂×C₁₆	—	4	1	no	—
22	C₂×C₁₆	—	4	1	no	—
23	C₄⋊C₈	—	4	1	no	—
24	C₄⋊C₈	—	4	1	no	—
25	C₄⋊C₈	—	4	1	no	—
26	#97	—	4	1	no	—
27	#95	C₈	1	1	yes	characteristic
28	#95	C₈	1	1	yes	characteristic
29	C₂²×C₄	—	2	1	no	—
30	C₂²×C₄	—	2	1	no	—
31	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄	M₁₆	1	1	yes	characteristic
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	4	1	no	—
39	C₂×C₈	—	4	1	no	—
40	C₂×C₈	—	4	1	no	—
41	C₂×C₈	D₈	1	1	yes	characteristic
42	C₂×C₈	SD₁₆	1	1	yes	characteristic
43	C₂×C₈	SD₁₆	1	1	yes	characteristic
44	C₂×C₈	Q₁₆	1	1	yes	characteristic
45	C₄⋊C₄	—	4	2	no	—
46	C₄²	—	2	1	no	—
47	C₄²	—	2	1	no	—
48	C₄²	—	4	1	no	—
49	C₄²	—	4	1	no	—
50	C₁₆	—	4	2	no	—
51	C₁₆	—	4	2	no	—
52	C₂³	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	#99	1	1	yes	characteristic
64	C₂×C₄	#101	1	1	yes	characteristic
65	C₂×C₄	#102	1	1	yes	characteristic
66	C₂×C₄	#103	1	1	yes	characteristic
67	C₂×C₄	#104	1	1	yes	characteristic
68	C₂×C₄	#104	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	2	1	no	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	—	8	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	#324	1	1	yes	characteristic
77	C₂²	D₄⋊C₈	1	1	yes	characteristic
78	C₂²	Q₈⋊C₈	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	8	1	no	—
83	C₄	#327	1	1	yes	characteristic
84	C₄	#328	1	1	yes	characteristic
85	C₄	#330	1	1	yes	characteristic
86	C₄	#332	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	2	1	no	—
89	C₂	#1271	1	1	yes	characteristic
90	C₂	#1327	1	1	yes	characteristic
91	C₂	#1328	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#99
commutator	C₂²×C₄	C₂×C₈
Frattini, cosocle	C₂×#97	C₂²
socle	C₂²	#324
prime radical	C₂³	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#1940	2
#1377	2