Subgroups — #2162

Order 128.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₁₆	C₂	1	2	yes	maximal
4	C₄×C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄²	C₄	1	1	yes	characteristic
8	C₂×C₁₆	C₂²	1	4	yes	—
9	C₄×C₈	C₄	1	1	yes	characteristic
10	C₄×C₈	C₄	1	1	yes	characteristic
11	C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄	C₈	1	1	yes	characteristic
14	C₂²×C₄	C₈	1	1	yes	characteristic
15	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈	—	2	2	no	—
17	C₂×C₈	—	2	2	no	—
18	C₂×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈	C₂³	1	1	yes	characteristic
22	C₄²	C₈	1	1	yes	characteristic
23	C₄²	C₈	1	1	yes	characteristic
24	C₄²	C₂×C₄	1	1	yes	characteristic
25	C₄²	C₂×C₄	1	1	yes	characteristic
26	C₁₆	—	2	4	no	—
27	C₂³	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄	—	2	2	no	—
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	—	2	1	no	—
31	C₂×C₄	C₁₆	1	4	yes	complemented
32	C₂×C₄	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	—	2	1	no	—
40	C₈	M₁₆	1	2	yes	—
41	C₈	#41	1	2	yes	—
42	C₂²	—	2	1	no	—
43	C₂²	—	2	1	no	—
44	C₂²	C₂×C₁₆	1	2	yes	—
45	C₂²	C₂²×C₈	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	—	2	1	no	—
48	C₄	C₂×C₁₆	1	2	yes	—
49	C₄	C₂×C₁₆	1	2	yes	—
50	C₄	M₃₂	1	2	yes	—
51	C₄	#117	1	1	yes	characteristic
52	C₄	C₂×M₁₆	1	1	yes	characteristic
53	C₂	—	2	2	no	—
54	C₂	#431	1	1	yes	characteristic
55	C₂	C₂²×C₁₆	1	1	yes	characteristic
56	C₂	C₂×M₃₂	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₁₆	C₂	semidirect
4. C₄×C₁₆	C₂	semidirect
31. C₂×C₄	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈	2
C₄⋊C₁₆	2
C₂²⋊C₁₆	2
C₄×C₁₆	2