Subgroups — C₂×#500 #3159

Order 128.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈	C₂	1	2	yes	maximal
2	#500	C₂	1	8	yes	maximal, direct product factor
3	C₂×#107	C₂	1	4	yes	maximal
4	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈	—	2	2	no	—
6	C₂²×C₈	C₂²	1	2	yes	—
7	C₄⋊Q₈	—	2	4	no	—
8	C₄⋊Q₈	C₂²	1	8	yes	—
9	C₂×C₄⋊C₄	—	2	2	no	—
10	C₂×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄²	C₂²	1	1	yes	characteristic
12	#107	C₂²	1	16	yes	—
13	C₄×C₈	C₂²	1	4	yes	—
14	C₂×Q₈	—	2	8	no	—
15	C₂×Q₈	—	4	8	no	—
16	C₂²×C₄	—	2	4	no	—
17	C₂²×C₄	D₄	1	1	yes	characteristic
18	C₂²×C₄	D₄	1	1	yes	characteristic
19	C₂²×C₄	C₂³	1	1	yes	characteristic
20	C₂×C₈	Q₈	1	8	yes	complemented
21	C₂×C₈	C₂³	1	4	yes	—
22	C₄⋊C₄	—	2	8	no	—
23	C₄⋊C₄	—	2	4	no	—
24	C₄⋊C₄	C₂³	1	8	yes	—
25	C₄²	D₄	1	2	yes	—
26	C₄²	C₂³	1	2	yes	—
27	C₂³	C₂×D₄	1	1	yes	characteristic
28	Q₈	—	4	16	no	—
29	C₂×C₄	—	2	8	no	—
30	C₂×C₄	—	4	8	no	—
31	C₂×C₄	D₈	1	4	yes	—
32	C₂×C₄	Q₁₆	1	4	yes	—
33	C₂×C₄	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄	C₂×D₄	1	2	yes	—
35	C₂×C₄	C₂×D₄	1	2	yes	—
36	C₂×C₄	C₂×Q₈	1	4	yes	—
37	C₂×C₄	C₂⁴	1	1	yes	characteristic
38	C₈	C₂×Q₈	1	8	yes	complemented
39	C₂²	C₄⋊Q₈	1	2	yes	—
40	C₂²	C₂×D₈	1	2	yes	—
41	C₂²	C₂×Q₁₆	1	2	yes	—
42	C₂²	C₂²×D₄	1	1	yes	characteristic
43	C₄	—	4	8	no	—
44	C₄	C₄⋊Q₈	1	2	yes	—
45	C₄	C₂×D₈	1	4	yes	—
46	C₄	C₂×Q₁₆	1	4	yes	—
47	C₄	C₂²×Q₈	1	2	yes	—
48	C₂	#500	1	4	yes	direct product factor
49	C₂	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂	C₂²×D₈	1	1	yes	characteristic
51	C₂	C₂²×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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	direct (N × H)
20. C₂×C₈	Q₈	semidirect
38. C₈	C₂×Q₈	semidirect
48. C₂	#500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈	2
#500	2
C₂×#107	2
C₂×C₄×C₈	2

Subgroups — C2×#500 #3159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#500 #3159