Subgroups — #1568

Order 128.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#432	C₂	1	2	yes	maximal
2	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	#335	C₂	1	2	yes	maximal, complemented
4	#334	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄²	C₂²	1	1	yes	characteristic
8	C₄⋊C₈	C₂²	1	4	yes	—
9	C₂²⋊C₈	—	2	2	no	—
10	#97	—	2	2	no	—
11	C₄×C₈	C₄	1	1	yes	characteristic
12	C₄×C₈	C₄	1	1	yes	characteristic
13	C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄	D₄	1	1	yes	characteristic
17	C₂²×C₄	Q₈	1	1	yes	characteristic
18	C₂×C₈	—	2	2	no	—
19	C₂×C₈	—	2	2	no	—
20	C₂×C₈	—	2	4	no	—
21	C₂×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄²	C₂×C₄	1	1	yes	characteristic
26	C₄²	D₄	1	1	yes	characteristic
27	C₄²	Q₈	1	1	yes	characteristic
28	C₄²	C₂³	1	1	yes	characteristic
29	C₂³	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄	—	2	2	no	—
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	—	2	2	no	—
34	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
40	C₈	—	2	1	no	—
41	C₈	—	2	1	no	—
42	C₈	—	4	4	no	—
43	C₈	M₁₆	1	2	yes	—
44	C₈	M₁₆	1	2	yes	—
45	C₂²	—	2	1	no	—
46	C₂²	—	2	1	no	—
47	C₂²	#108	1	2	yes	—
48	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	C₂×M₁₆	1	1	yes	characteristic
54	C₄	C₂×M₁₆	1	1	yes	characteristic
55	C₄	#135	1	2	yes	—
56	C₂	—	2	2	no	—
57	C₂	#423	1	1	yes	characteristic
58	C₂	#427	1	1	yes	characteristic
59	C₂	C₂×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₄²	C₂³
socle	C₂²	C₂×C₄⋊C₄
prime radical	C₂³	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #335	C₂	semidirect
4. #334	C₂	semidirect

Maximal subgroups

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

subgroup	index
#432	2
C₂×C₄×C₈	2
#335	2
#334	2