Subgroups — #3261

Order 128.

161 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#541	C₂	1	2	yes	maximal
2	#479	C₂	1	4	yes	maximal, complemented
3	#470	C₂	1	2	yes	maximal
4	#467	C₂	1	2	yes	maximal
5	#462	C₂	1	4	yes	maximal, complemented
6	#424	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₁₆	C₂²	1	2	yes	—
8	C₂×Q₁₆	C₂²	1	2	yes	—
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈	C₂²	1	2	yes	—
12	#126	—	2	4	no	—
13	#125	—	2	4	no	—
14	#125	C₂²	1	2	yes	—
15	C₂²⋊Q₈	C₂²	1	4	yes	—
16	C₄×Q₈	C₂²	1	4	yes	—
17	#117	C₂²	1	1	yes	characteristic
18	#107	C₂²	1	2	yes	—
19	#106	C₂²	1	2	yes	—
20	C₄⋊C₈	C₂²	1	4	yes	—
21	#103	C₂²	1	4	yes	—
22	#103	C₂²	1	4	yes	—
23	C₂×Q₈	C₂³	1	4	yes	—
24	C₂²×C₄	C₂³	1	1	yes	characteristic
25	Q₁₆	—	4	2	no	—
26	Q₁₆	—	4	2	no	—
27	M₁₆	—	4	1	no	—
28	C₂×C₈	—	4	1	no	—
29	C₂×C₈	C₂³	1	2	yes	—
30	C₂×C₈	C₂³	1	2	yes	—
31	C₄⋊C₄	—	2	4	no	—
32	C₄⋊C₄	—	2	4	no	—
33	C₄⋊C₄	—	2	4	no	—
34	C₄⋊C₄	—	2	4	no	—
35	C₄⋊C₄	D₄	1	2	yes	—
36	C₄⋊C₄	C₂³	1	4	yes	—
37	C₂²⋊C₄	—	2	4	no	—
38	C₂²⋊C₄	D₄	1	2	yes	—
39	C₄²	—	2	4	no	—
40	C₄²	C₂³	1	2	yes	—
41	C₂³	C₂×D₄	1	1	yes	characteristic
42	Q₈	—	2	4	no	—
43	Q₈	—	2	4	no	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	4	no	—
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	C₂×D₄	1	4	yes	—
49	C₂×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄	C₂⁴	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	2	2	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	2	no	—
55	C₂²	C₂²×D₄	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	2	4	no	—
58	C₄	—	4	4	no	—
59	C₄	—	4	4	no	—
60	C₄	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂	—	4	1	no	—
63	C₂	#537	1	1	yes	characteristic
64	C₂	#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #479	C₂	semidirect
5. #462	C₂	semidirect

Maximal subgroups

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

subgroup	index
#541	2
#479	2
#470	2
#467	2
#462	2
#424	2