Subgroups — #7280

Order 256.

113 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2150	C₂	1	1	yes	maximal, characteristic
2	#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	C₂×#108	—	2	1	no	—
6	C₂×#108	C₄	1	2	yes	—
7	#423	—	2	1	no	—
8	C₂×C₄⋊C₈	—	2	1	no	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊C₁₆	—	2	1	no	—
11	C₄⋊C₁₆	C₄	1	2	yes	—
12	#334	—	2	1	no	—
13	#334	C₄	1	2	yes	—
14	C₂×M₁₆	—	2	2	no	—
15	C₂²×C₈	—	4	1	no	—
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂×C₄²	D₄	1	1	yes	characteristic
19	M₃₂	—	4	1	no	—
20	M₃₂	—	4	1	no	—
21	C₂×C₁₆	—	2	2	no	—
22	#108	—	4	2	no	—
23	#108	—	4	1	no	—
24	#108	—	4	1	no	—
25	C₄⋊C₈	—	2	2	no	—
26	C₄⋊C₈	—	4	1	no	—
27	C₄⋊C₈	—	4	1	no	—
28	C₄⋊C₈	—	4	1	no	—
29	C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈	D₄	1	1	yes	characteristic
32	C₄×C₈	Q₈	1	1	yes	characteristic
33	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄	Q₁₆	1	1	yes	characteristic
36	M₁₆	—	4	2	no	—
37	M₁₆	—	4	2	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	4	2	no	—
41	C₂×C₈	—	4	2	no	—
42	C₂×C₈	—	8	2	no	—
43	C₂×C₈	C₄²	1	1	yes	characteristic
44	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈	D₈	1	1	yes	characteristic
48	C₂×C₈	SD₁₆	1	1	yes	characteristic
49	C₂×C₈	SD₁₆	1	1	yes	characteristic
50	C₂×C₈	Q₁₆	1	1	yes	characteristic
51	C₄²	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²	C₄⋊C₄	1	1	yes	characteristic
53	C₄²	D₈	1	1	yes	characteristic
54	C₄²	SD₁₆	1	1	yes	characteristic
55	C₁₆	—	4	2	no	—
56	C₂³	#103	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	#95	1	1	yes	characteristic
60	C₂×C₄	#100	1	1	yes	characteristic
61	C₂×C₄	#101	1	1	yes	characteristic
62	C₂×C₄	#102	1	1	yes	characteristic
63	C₂×C₄	#102	1	1	yes	characteristic
64	C₂×C₄	#102	1	1	yes	characteristic
65	C₂×C₄	#103	1	1	yes	characteristic
66	C₂×C₄	#103	1	1	yes	characteristic
67	C₂×C₄	#106	1	1	yes	characteristic
68	C₂×C₄	#107	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	2	1	no	—
71	C₈	—	8	2	no	—
72	C₈	—	8	2	no	—
73	C₈	#102	1	1	yes	characteristic
74	C₈	#103	1	1	yes	characteristic
75	C₈	#106	1	1	yes	characteristic
76	C₈	#107	1	1	yes	characteristic
77	C₂²	—	2	1	no	—
78	C₂²	—	2	1	no	—
79	C₂²	#332	1	1	yes	characteristic
80	C₂²	#333	1	1	yes	characteristic
81	C₂²	#340	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	#331	1	1	yes	characteristic
86	C₄	#332	1	1	yes	characteristic
87	C₄	#340	1	1	yes	characteristic
88	C₄	#343	1	1	yes	characteristic
89	C₄	#365	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	2	1	no	—
92	C₂	#1296	1	1	yes	characteristic
93	C₂	#1383	1	1	yes	characteristic
94	C₂	#1388	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#340
commutator	C₂×C₈	C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂²
prime radical	C₂³	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
#2150	2
#1848	2
C₂×#334	2