Subgroups — #7148

Order 256.

151 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2162	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈	—	2	2	no	—
4	C₂×C₄×C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
6	C₄³	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆	C₄	1	4	yes	complemented
8	C₂²⋊C₁₆	—	2	2	no	—
9	C₄×C₁₆	—	2	2	no	—
10	C₂²×C₈	—	2	1	no	—
11	C₂²×C₈	D₄	1	2	yes	—
12	C₂×C₄²	—	2	1	no	—
13	C₂×C₄²	—	2	1	no	—
14	C₂×C₄²	C₈	1	2	yes	—
15	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆	—	2	4	no	—
17	C₄×C₈	—	2	8	no	—
18	C₄×C₈	—	2	1	no	—
19	C₄×C₈	C₈	1	2	yes	—
20	C₄×C₈	C₂×C₄	1	2	yes	—
21	C₄×C₈	D₄	1	1	yes	characteristic
22	C₄×C₈	Q₈	1	1	yes	characteristic
23	C₂²×C₄	—	2	2	no	—
24	C₂²×C₄	C₂²⋊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	4	no	—
30	C₂×C₈	C₂²⋊C₄	1	2	yes	—
31	C₂×C₈	C₄⋊C₄	1	2	yes	—
32	C₄²	—	2	4	no	—
33	C₄²	—	2	4	no	—
34	C₄²	—	2	2	no	—
35	C₄²	C₁₆	1	4	yes	complemented
36	C₄²	C₄²	1	1	yes	characteristic
37	C₄²	C₄⋊C₄	1	1	yes	characteristic
38	C₄²	C₂×C₈	1	1	yes	characteristic
39	C₄²	M₁₆	1	1	yes	characteristic
40	C₁₆	—	4	4	no	—
41	C₂³	C₂²⋊C₈	1	1	yes	characteristic
42	C₂×C₄	—	2	4	no	—
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	#95	1	1	yes	characteristic
48	C₂×C₄	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄	#97	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	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄	C₂×C₁₆	1	2	yes	—
54	C₂×C₄	M₃₂	1	2	yes	—
55	C₈	—	2	4	no	—
56	C₈	—	2	2	no	—
57	C₈	#104	1	4	yes	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	#336	1	1	yes	characteristic
61	C₂²	C₂²⋊C₁₆	1	2	yes	—
62	C₄	—	2	8	no	—
63	C₄	—	2	1	no	—
64	C₄	—	2	1	no	—
65	C₄	#339	1	2	yes	—
66	C₄	C₄×C₁₆	1	1	yes	characteristic
67	C₄	#346	1	1	yes	characteristic
68	C₄	C₄⋊C₁₆	1	2	yes	—
69	C₄	#364	1	2	yes	—
70	C₂	—	2	2	no	—
71	C₂	#1276	1	1	yes	characteristic
72	C₂	#1374	1	1	yes	characteristic
73	C₂	#1376	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	D₄
commutator	C₄	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#336
prime radical	C₂³	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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
7. C₄⋊C₁₆	C₄	semidirect
35. C₄²	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2162	2
C₄²×C₈	2