Subgroups — #12000

Order 256.

121 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	#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×#334	C₂	1	1	yes	maximal, characteristic
4	#1368	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂	C₄	1	1	yes	characteristic
6	C₂×M₃₂	C₄	1	1	yes	characteristic
7	#431	—	2	1	no	—
8	C₂×C₄⋊C₈	—	2	1	no	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	#346	C₂²	1	2	yes	—
11	C₄×C₁₆	C₄	1	2	yes	—
12	#335	—	2	1	no	—
13	#334	—	2	1	no	—
14	#334	C₂²	1	2	yes	—
15	#334	C₂²	1	2	yes	—
16	C₂²×C₈	—	4	1	no	—
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄²	D₄	1	1	yes	characteristic
20	M₃₂	—	2	2	no	—
21	M₃₂	C₈	1	4	yes	complemented
22	C₂×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₆	C₂×C₄	1	2	yes	—
24	C₄⋊C₈	—	2	2	no	—
25	C₄⋊C₈	—	2	2	no	—
26	C₄⋊C₈	—	4	1	no	—
27	C₂²⋊C₈	—	4	1	no	—
28	C₄×C₈	—	4	1	no	—
29	C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈	D₄	1	1	yes	characteristic
31	C₄×C₈	Q₈	1	1	yes	characteristic
32	C₄×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄	SD₁₆	1	1	yes	characteristic
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	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈	C₂×C₈	1	2	yes	—
43	C₂×C₈	M₁₆	1	2	yes	—
44	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₄²	C₄⋊C₄	1	1	yes	characteristic
47	C₄²	SD₁₆	1	1	yes	characteristic
48	C₄²	SD₁₆	1	1	yes	characteristic
49	C₄²	C₂×D₄	1	1	yes	characteristic
50	C₁₆	—	2	2	no	—
51	C₁₆	C₂×C₈	1	4	yes	complemented
52	C₂³	#106	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	C₄⋊C₈	1	2	yes	—
56	C₂×C₄	#106	1	1	yes	characteristic
57	C₂×C₄	#106	1	1	yes	characteristic
58	C₂×C₄	#106	1	1	yes	characteristic
59	C₂×C₄	#108	1	2	yes	—
60	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₈	—	8	2	no	—
66	C₈	—	8	2	no	—
67	C₈	C₄⋊C₈	1	2	yes	—
68	C₈	C₂²×C₈	1	1	yes	characteristic
69	C₈	C₂×M₁₆	1	1	yes	characteristic
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	#334	1	2	yes	—
73	C₂²	C₂×#106	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	#334	1	2	yes	—
78	C₄	#365	1	2	yes	—
79	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₄	C₂×#108	1	1	yes	characteristic
81	C₂	—	2	2	no	—
82	C₂	C₂×#334	1	1	yes	characteristic
83	C₂	C₂×#365	1	1	yes	characteristic
84	C₂	#2155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#106
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×#106
prime radical	C₂³	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #1368	C₂	semidirect
21. M₃₂	C₈	semidirect
51. C₁₆	C₂×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
#1564	2
C₂×#334	2
#1368	2