Subgroups — #1314

Order 128.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂³×C₄	C₄	1	1	yes	characteristic
4	C₂²×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₁₆	—	2	2	no	—
7	C₂²⋊C₈	—	2	2	no	—
8	C₂⁴	C₈	1	1	yes	characteristic
9	C₂²×C₄	—	2	1	no	—
10	C₂²×C₄	—	2	1	no	—
11	C₂²×C₄	—	4	1	no	—
12	C₂²×C₄	—	4	1	no	—
13	C₂²×C₄	C₈	1	1	yes	characteristic
14	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈	—	2	1	no	—
16	C₂×C₈	—	4	1	no	—
17	C₂×C₈	—	4	1	no	—
18	C₂×C₈	D₄	1	2	yes	—
19	C₁₆	—	4	2	no	—
20	C₂³	—	2	1	no	—
21	C₂³	—	2	1	no	—
22	C₂³	—	4	1	no	—
23	C₂³	—	4	1	no	—
24	C₂³	C₁₆	1	2	yes	complemented
25	C₂³	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄	—	2	1	no	—
27	C₂×C₄	—	2	1	no	—
28	C₂×C₄	—	2	1	no	—
29	C₂×C₄	—	4	2	no	—
30	C₂×C₄	—	4	1	no	—
31	C₂×C₄	—	4	1	no	—
32	C₂×C₄	—	4	1	no	—
33	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄	M₁₆	1	1	yes	characteristic
35	C₈	—	2	2	no	—
36	C₈	—	4	1	no	—
37	C₂²	—	2	1	no	—
38	C₂²	—	2	1	no	—
39	C₂²	—	2	2	no	—
40	C₂²	—	4	2	no	—
41	C₂²	—	4	2	no	—
42	C₂²	—	4	1	no	—
43	C₂²	—	4	1	no	—
44	C₂²	C₂²⋊C₈	1	1	yes	characteristic
45	C₂²	C₂×C₁₆	1	1	yes	characteristic
46	C₂²	M₃₂	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	4	1	no	—
49	C₄	#99	1	1	yes	characteristic
50	C₄	#100	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	—	2	1	no	—
53	C₂	—	4	2	no	—
54	C₂	#323	1	1	yes	characteristic
55	C₂	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₂	#349	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄
commutator	C₂²	C₂×C₁₆
Frattini, cosocle	C₂²×C₈	C₂²
socle	C₂²	C₂²⋊C₈
prime radical	C₂⁴	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²⋊C₁₆	C₂	semidirect
24. C₂³	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊C₈	2
C₂²⋊C₁₆	2