Subgroups — #1374

Order 128.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆	C₂	1	2	yes	maximal
2	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈	C₄	1	1	yes	characteristic
4	C₂²×C₈	C₂²	1	1	yes	characteristic
5	C₂×C₄²	C₄	1	1	yes	characteristic
6	C₂×C₁₆	—	2	2	no	—
7	C₂×C₁₆	—	2	2	no	—
8	C₂×C₁₆	C₄	1	4	yes	complemented
9	C₄×C₈	—	2	2	no	—
10	C₂²×C₄	C₈	1	2	yes	—
11	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈	—	2	1	no	—
13	C₂×C₈	—	2	1	no	—
14	C₂×C₈	C₈	1	2	yes	—
15	C₂×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₈	D₄	1	1	yes	characteristic
17	C₂×C₈	D₄	1	2	yes	—
18	C₂×C₈	Q₈	1	1	yes	characteristic
19	C₄²	—	2	1	no	—
20	C₄²	—	2	1	no	—
21	C₁₆	—	2	4	no	—
22	C₂³	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄	—	2	2	no	—
24	C₂×C₄	—	2	2	no	—
25	C₂×C₄	C₁₆	1	4	yes	complemented
26	C₂×C₄	C₄²	1	1	yes	characteristic
27	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄	M₁₆	1	1	yes	characteristic
31	C₂×C₄	M₁₆	1	1	yes	characteristic
32	C₈	—	2	2	no	—
33	C₈	C₂²⋊C₄	1	2	yes	—
34	C₈	C₄⋊C₄	1	2	yes	—
35	C₂²	C₄×C₈	1	1	yes	characteristic
36	C₂²	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²	C₄⋊C₈	1	1	yes	characteristic
38	C₂²	C₂×C₁₆	1	2	yes	—
39	C₂²	M₃₂	1	2	yes	—
40	C₄	—	2	4	no	—
41	C₄	#95	1	1	yes	characteristic
42	C₄	#97	1	1	yes	characteristic
43	C₄	C₂²⋊C₈	1	1	yes	characteristic
44	C₄	C₄⋊C₈	1	1	yes	characteristic
45	C₂	#336	1	1	yes	characteristic
46	C₂	C₄×C₁₆	1	1	yes	characteristic
47	C₂	#346	1	1	yes	characteristic
48	C₂	C₂²⋊C₁₆	1	2	yes	—
49	C₂	C₄⋊C₁₆	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
8. C₂×C₁₆	C₄	semidirect
25. 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
C₂²×C₁₆	2
C₂×C₄×C₈	2