Subgroups — #3160

Order 128.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	2	yes	maximal
2	#500	C₂	1	2	yes	maximal, complemented
3	#499	C₂	1	4	yes	maximal, complemented
4	#498	C₂	1	2	yes	maximal, complemented
5	#427	C₂	1	4	yes	maximal
6	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₈	C₂²	1	2	yes	—
8	C₄⋊Q₈	C₂²	1	4	yes	—
9	#125	C₂²	1	4	yes	—
10	C₂²⋊Q₈	—	2	4	no	—
11	C₄×Q₈	—	2	4	no	—
12	#117	C₂²	1	4	yes	—
13	C₂×C₄²	C₂²	1	1	yes	characteristic
14	#107	C₂²	1	8	yes	—
15	#106	C₂²	1	8	yes	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	C₄×C₈	C₂²	1	2	yes	—
18	C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×Q₈	—	2	4	no	—
20	C₂²×C₄	D₄	1	2	yes	—
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₈	Q₈	1	8	yes	complemented
23	C₂×C₈	C₂³	1	2	yes	—
24	C₂×C₈	C₂³	1	2	yes	—
25	C₄⋊C₄	—	2	8	no	—
26	C₄⋊C₄	—	2	4	no	—
27	C₄⋊C₄	C₂³	1	8	yes	—
28	C₂²⋊C₄	—	2	4	no	—
29	C₄²	—	2	4	no	—
30	C₄²	D₄	1	2	yes	—
31	C₄²	C₂³	1	1	yes	characteristic
32	C₄²	C₂³	1	1	yes	characteristic
33	C₂³	C₂×D₄	1	1	yes	characteristic
34	Q₈	—	4	8	no	—
35	C₂×C₄	—	2	4	no	—
36	C₂×C₄	—	2	8	no	—
37	C₂×C₄	C₂×D₄	1	2	yes	—
38	C₂×C₄	C₂×D₄	1	2	yes	—
39	C₂×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄	C₂×Q₈	1	4	yes	—
41	C₂×C₄	C₂⁴	1	1	yes	characteristic
42	C₈	C₂×Q₈	1	4	yes	—
43	C₈	C₂×Q₈	1	4	yes	—
44	C₂²	—	2	2	no	—
45	C₂²	C₄⋊Q₈	1	2	yes	—
46	C₂²	C₂²×D₄	1	1	yes	characteristic
47	C₄	—	2	2	no	—
48	C₄	—	4	8	no	—
49	C₄	C₄⋊Q₈	1	2	yes	—
50	C₄	#135	1	4	yes	—
51	C₄	C₂²×Q₈	1	2	yes	—
52	C₂	—	2	2	no	—
53	C₂	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂	C₂×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂²	C₂²×D₄
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #500	C₂	semidirect
3. #499	C₂	semidirect
4. #498	C₂	semidirect
22. C₂×C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#500	2
#499	2
#498	2
#427	2
C₂×C₄×C₈	2