Subgroups — #1772

Order 128.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108	C₂	1	1	yes	maximal, characteristic
2	#427	C₂	1	1	yes	maximal, characteristic
3	#405	C₂	1	2	yes	maximal
4	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	#339	C₂	1	2	yes	maximal
6	C₂×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	#117	C₂²	1	2	yes	—
10	C₂×C₄²	C₂²	1	1	yes	characteristic
11	#108	C₄	1	4	yes	complemented
12	#107	C₄	1	2	yes	complemented
13	#106	C₄	1	2	yes	complemented
14	#97	—	2	2	no	—
15	C₄×C₈	—	2	2	no	—
16	C₄×C₈	C₄	1	4	yes	complemented
17	C₂²×C₄	—	2	1	no	—
18	C₂²×C₄	C₂³	1	1	yes	characteristic
19	M₁₆	—	2	2	no	—
20	M₁₆	C₂×C₄	1	4	yes	—
21	C₂×C₈	—	2	2	no	—
22	C₂×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₈	D₄	1	1	yes	characteristic
28	C₂×C₈	D₄	1	1	yes	characteristic
29	C₂×C₈	Q₈	1	2	yes	—
30	C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄	—	2	2	no	—
32	C₄²	—	2	2	no	—
33	C₄²	—	2	1	no	—
34	C₄²	C₂×C₄	1	2	yes	—
35	C₂³	#41	1	1	yes	characteristic
36	C₂×C₄	—	2	4	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄	#41	1	1	yes	characteristic
45	C₈	—	2	4	no	—
46	C₈	C₄²	1	2	yes	—
47	C₈	C₄²	1	2	yes	—
48	C₈	C₄⋊C₄	1	2	yes	—
49	C₈	C₄⋊C₄	1	2	yes	—
50	C₂²	—	2	2	no	—
51	C₂²	#117	1	1	yes	characteristic
52	C₂²	C₄×D₄	1	1	yes	characteristic
53	C₂²	C₄×Q₈	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	—	4	4	no	—
56	C₄	C₂×C₄²	1	1	yes	characteristic
57	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄	C₄×D₄	1	1	yes	characteristic
59	C₄	C₄×Q₈	1	1	yes	characteristic
60	C₂	—	2	1	no	—
61	C₂	—	2	1	no	—
62	C₂	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂	#443	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄
commutator	C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₄×D₄
prime radical	C₂³	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #108	C₄	semidirect
12. #107	C₄	semidirect
13. #106	C₄	semidirect
16. C₄×C₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108	2
#427	2
#405	2
C₂×C₄×C₈	2
#339	2