Subgroups — #7151

Order 256.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆	—	2	2	no	—
4	C₂×C₄×C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₄⋊C₁₆	—	2	2	no	—
8	C₄⋊C₁₆	C₄	1	4	yes	—
9	C₈²	—	2	2	no	—
10	C₂²×C₈	—	2	1	no	—
11	C₂²×C₈	C₈	1	2	yes	—
12	C₂²×C₈	D₄	1	1	yes	characteristic
13	C₂²×C₈	Q₈	1	1	yes	characteristic
14	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆	—	2	4	no	—
16	C₂×C₁₆	—	4	2	no	—
17	C₂×C₁₆	—	4	2	no	—
18	C₄×C₈	—	2	1	no	—
19	C₄×C₈	—	2	1	no	—
20	C₄×C₈	—	2	1	no	—
21	C₄×C₈	C₈	1	2	yes	—
22	C₄×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₈	D₄	1	2	yes	—
24	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄	M₁₆	1	1	yes	characteristic
27	C₂×C₈	—	2	4	no	—
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	C₁₆	1	4	yes	complemented
34	C₂×C₈	C₂²⋊C₄	1	2	yes	—
35	C₂×C₈	C₄⋊C₄	1	2	yes	—
36	C₂×C₈	D₈	1	1	yes	characteristic
37	C₂×C₈	SD₁₆	1	2	yes	—
38	C₂×C₈	Q₁₆	1	1	yes	characteristic
39	C₄²	C₄²	1	1	yes	characteristic
40	C₄²	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²	C₂×C₈	1	1	yes	characteristic
42	C₄²	M₁₆	1	1	yes	characteristic
43	C₁₆	—	4	4	no	—
44	C₂³	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	#95	1	1	yes	characteristic
47	C₂×C₄	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄	#97	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄	#106	1	1	yes	characteristic
53	C₂×C₄	#107	1	1	yes	characteristic
54	C₂×C₄	#108	1	1	yes	characteristic
55	C₂×C₄	#108	1	1	yes	characteristic
56	C₂×C₄	C₂×C₁₆	1	2	yes	—
57	C₂×C₄	M₃₂	1	2	yes	—
58	C₈	—	2	4	no	—
59	C₈	—	2	2	no	—
60	C₈	—	2	4	no	—
61	C₈	#102	1	2	yes	—
62	C₈	#103	1	2	yes	—
63	C₂²	#334	1	1	yes	characteristic
64	C₂²	#335	1	1	yes	characteristic
65	C₂²	#336	1	1	yes	characteristic
66	C₂²	C₄⋊C₁₆	1	2	yes	—
67	C₂²	#364	1	2	yes	—
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	#334	1	1	yes	characteristic
71	C₄	#335	1	1	yes	characteristic
72	C₄	#340	1	1	yes	characteristic
73	C₄	#341	1	1	yes	characteristic
74	C₄	C₄×C₁₆	1	1	yes	characteristic
75	C₄	#346	1	1	yes	characteristic
76	C₄	C₂²⋊C₁₆	1	2	yes	—
77	C₂	#1277	1	1	yes	characteristic
78	C₂	#1367	1	2	yes	—
79	C₂	#1370	1	2	yes	—
80	C₂	#1374	1	1	yes	characteristic
81	C₂	#1376	1	1	yes	characteristic

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

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₁₆	2
C₂×C₈²	2