Subgroups — #7477

Order 256.

89 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic
2	#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×#367	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆	C₂²	1	1	yes	characteristic
5	#427	—	2	1	no	—
6	C₂×#107	—	2	1	no	—
7	M₆₄	—	2	1	no	—
8	M₆₄	C₄	1	2	yes	complemented
9	C₂×C₃₂	—	2	1	no	—
10	#367	—	2	1	no	—
11	#367	—	2	1	no	—
12	#367	C₄	1	2	yes	complemented
13	#366	C₄	1	2	yes	complemented
14	C₂²×C₈	D₄	1	1	yes	characteristic
15	#117	—	4	1	no	—
16	C₂×C₄⋊C₄	—	4	1	no	—
17	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₆	Q₈	1	1	yes	characteristic
23	#107	—	2	2	no	—
24	#107	—	2	2	no	—
25	#107	—	4	1	no	—
26	#106	—	4	1	no	—
27	C₃₂	—	2	2	no	—
28	C₂²×C₄	—	8	1	no	—
29	C₂²×C₄	D₈	1	1	yes	characteristic
30	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈	SD₁₆	1	1	yes	characteristic
34	C₂×C₈	SD₁₆	1	1	yes	characteristic
35	C₂×C₈	Q₁₆	1	1	yes	characteristic
36	C₄⋊C₄	—	4	2	no	—
37	C₄⋊C₄	—	4	2	no	—
38	C₄⋊C₄	—	8	1	no	—
39	C₂²⋊C₄	—	8	1	no	—
40	C₄²	—	8	1	no	—
41	C₁₆	C₄²	1	1	yes	characteristic
42	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
43	C₁₆	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆	C₄⋊C₄	1	1	yes	characteristic
45	C₂³	D₁₆	1	1	yes	characteristic
46	C₂×C₄	—	8	2	no	—
47	C₂×C₄	—	8	2	no	—
48	C₂×C₄	—	16	2	no	—
49	C₂×C₄	#102	1	1	yes	characteristic
50	C₂×C₄	#102	1	1	yes	characteristic
51	C₂×C₄	#107	1	1	yes	characteristic
52	C₂×C₄	SD₃₂	1	1	yes	characteristic
53	C₂×C₄	SD₃₂	1	1	yes	characteristic
54	C₂×C₄	Q₃₂	1	1	yes	characteristic
55	C₈	#95	1	1	yes	characteristic
56	C₈	#103	1	1	yes	characteristic
57	C₈	#103	1	1	yes	characteristic
58	C₈	#106	1	1	yes	characteristic
59	C₂²	—	2	2	no	—
60	C₂²	#357	1	1	yes	characteristic
61	C₂²	#357	1	1	yes	characteristic
62	C₂²	#366	1	1	yes	characteristic
63	C₄	—	16	2	no	—
64	C₄	—	16	2	no	—
65	C₄	#340	1	1	yes	characteristic
66	C₄	#358	1	1	yes	characteristic
67	C₄	#358	1	1	yes	characteristic
68	C₄	#367	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	#1380	1	1	yes	characteristic
72	C₂	#1418	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#357
commutator	C₁₆	C₄²
Frattini, cosocle	C₂²×C₁₆	C₂²
prime radical	C₂³	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. M₆₄	C₄	semidirect
12. #367	C₄	semidirect
13. #366	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₆₄	2
#2158	2
C₂×#367	2