Subgroups — #7176

Order 256.

123 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	C₄×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	C₂×#108	C₄	1	2	yes	—
6	C₂×C₄⋊C₈	—	2	1	no	—
7	C₄×M₁₆	—	2	1	no	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	#346	C₄	1	2	yes	—
10	C₄×C₁₆	—	2	1	no	—
11	#341	—	2	1	no	—
12	#335	—	2	1	no	—
13	#335	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	—	2	4	no	—
20	C₂×C₁₆	—	2	2	no	—
21	#108	—	2	2	no	—
22	#108	C₈	1	4	yes	complemented
23	C₄⋊C₈	—	2	2	no	—
24	C₄⋊C₈	—	4	1	no	—
25	#97	—	4	1	no	—
26	C₄×C₈	—	4	1	no	—
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	D₄	1	1	yes	characteristic
30	C₄×C₈	Q₈	1	1	yes	characteristic
31	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄	Q₁₆	1	1	yes	characteristic
34	M₁₆	—	2	4	no	—
35	M₁₆	—	4	2	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	—	4	2	no	—
39	C₂×C₈	—	8	2	no	—
40	C₂×C₈	C₄²	1	1	yes	characteristic
41	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈	C₂×C₈	1	2	yes	—
45	C₂×C₈	M₁₆	1	2	yes	—
46	C₄²	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²	C₄⋊C₄	1	1	yes	characteristic
48	C₄²	D₈	1	1	yes	characteristic
49	C₄²	SD₁₆	1	1	yes	characteristic
50	C₁₆	—	2	4	no	—
51	C₂³	#103	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	#95	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄	#102	1	1	yes	characteristic
58	C₂×C₄	#102	1	1	yes	characteristic
59	C₂×C₄	#103	1	1	yes	characteristic
60	C₂×C₄	#104	1	2	yes	—
61	C₂×C₄	#106	1	1	yes	characteristic
62	C₂×C₄	#107	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₈	—	4	4	no	—
66	C₈	—	8	2	no	—
67	C₈	C₄×C₈	1	1	yes	characteristic
68	C₈	#97	1	1	yes	characteristic
69	C₈	C₄⋊C₈	1	2	yes	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	Q₈⋊C₈	1	2	yes	—
73	C₂²	#340	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	D₄⋊C₈	1	2	yes	—
78	C₄	#336	1	1	yes	characteristic
79	C₄	#339	1	1	yes	characteristic
80	C₄	#361	1	1	yes	characteristic
81	C₄	#362	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	#1279	1	1	yes	characteristic
84	C₂	#1387	1	1	yes	characteristic
85	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	#103
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
22. #108	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
C₄×#108	2
C₂×#335	2