Subgroups — #7269

Order 256.

119 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	2	yes	maximal
2	#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂	—	2	2	no	—
4	C₂×#107	—	2	2	no	—
5	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
6	#384	C₄	1	2	yes	—
7	#364	—	2	2	no	—
8	#364	C₄	1	4	yes	complemented
9	C₂²×C₈	D₄	1	2	yes	—
10	C₂×C₄⋊C₄	—	2	2	no	—
11	C₂×C₄⋊C₄	—	4	2	no	—
12	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
13	M₃₂	—	2	4	no	—
14	M₃₂	—	4	2	no	—
15	C₂×C₁₆	—	4	2	no	—
16	#107	—	4	2	no	—
17	#107	—	4	2	no	—
18	C₄×C₈	C₂×C₄	1	2	yes	—
19	C₄×C₈	D₄	1	1	yes	characteristic
20	C₄×C₈	Q₈	1	1	yes	characteristic
21	C₂²×C₄	—	2	1	no	—
22	C₂²×C₄	—	4	2	no	—
23	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈	C₄⋊C₄	1	2	yes	—
26	C₂×C₈	D₈	1	2	yes	—
27	C₂×C₈	SD₁₆	1	2	yes	—
28	C₂×C₈	SD₁₆	1	2	yes	—
29	C₂×C₈	Q₁₆	1	2	yes	—
30	C₄⋊C₄	—	4	4	no	—
31	C₄⋊C₄	—	8	4	no	—
32	C₄²	—	2	1	no	—
33	C₄²	C₄²	1	1	yes	characteristic
34	C₄²	C₄⋊C₄	1	1	yes	characteristic
35	C₁₆	—	4	4	no	—
36	C₂³	#100	1	1	yes	characteristic
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	8	4	no	—
42	C₂×C₄	—	8	2	no	—
43	C₂×C₄	#95	1	1	yes	characteristic
44	C₂×C₄	#101	1	1	yes	characteristic
45	C₂×C₄	#102	1	2	yes	—
46	C₂×C₄	#103	1	2	yes	—
47	C₈	#102	1	2	yes	—
48	C₈	#103	1	2	yes	—
49	C₈	#106	1	2	yes	—
50	C₈	#107	1	2	yes	—
51	C₂²	—	2	1	no	—
52	C₂²	—	2	1	no	—
53	C₂²	#331	1	1	yes	characteristic
54	C₂²	#333	1	1	yes	characteristic
55	C₂²	#343	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	2	2	no	—
58	C₄	—	16	2	no	—
59	C₄	#332	1	2	yes	—
60	C₄	#340	1	2	yes	—
61	C₂	—	2	1	no	—
62	C₂	—	2	1	no	—
63	C₂	#1296	1	1	yes	characteristic
64	C₂	#1357	1	1	yes	characteristic
65	C₂	#1358	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#331
commutator	C₄²	C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂²
prime radical	C₂³	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ 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
8. #364	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#364	2
#1845	2