Subgroups — #2972

Order 128.

250 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₈×D₄	C₂	1	8	yes	maximal, complemented
3	#431	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂²×D₄	C₄	1	1	yes	characteristic
6	C₂³×C₄	C₂²	1	2	yes	—
7	C₂²×C₈	C₂²	1	8	yes	complemented
8	C₄×D₄	C₄	1	4	yes	—
9	C₄×D₄	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊C₄	C₄	1	2	yes	—
12	C₂×C₄²	C₂²	1	1	yes	characteristic
13	C₄⋊C₈	C₂²	1	4	yes	complemented
14	C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂²⋊C₈	C₂²	1	8	yes	complemented
16	C₄×C₈	C₂²	1	4	yes	complemented
17	C₂⁴	C₂×C₄	1	2	yes	—
18	C₂×D₄	C₈	1	8	yes	complemented
19	C₂×D₄	C₂×C₄	1	4	yes	—
20	C₂²×C₄	—	4	4	no	—
21	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₄	C₂×C₄	1	4	yes	—
24	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄	C₂³	1	1	yes	characteristic
26	C₂²×C₄	C₂³	1	4	yes	—
27	C₂×C₈	—	4	8	no	—
28	C₂×C₈	C₂³	1	8	yes	—
29	C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂²⋊C₄	C₂×C₄	1	8	yes	—
31	C₄²	C₂×C₄	1	2	yes	—
32	C₄²	C₂³	1	2	yes	—
33	C₂³	—	4	2	no	—
34	C₂³	—	4	2	no	—
35	C₂³	C₂×C₈	1	8	yes	complemented
36	C₂³	C₂²×C₄	1	4	yes	—
37	C₂³	C₂²×C₄	1	1	yes	characteristic
38	D₄	C₂×C₈	1	16	yes	complemented
39	C₂×C₄	—	2	8	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	4	4	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	C₂×C₈	1	4	yes	complemented
45	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄	C₂²×C₄	1	2	yes	—
47	C₂×C₄	C₂²×C₄	1	4	yes	—
48	C₂×C₄	C₂²×C₄	1	2	yes	—
49	C₂×C₄	C₂⁴	1	1	yes	characteristic
50	C₈	—	2	8	no	—
51	C₂²	—	2	1	no	—
52	C₂²	—	2	4	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	4	no	—
55	C₂²	—	4	8	no	—
56	C₂²	C₂²×C₈	1	2	yes	—
57	C₂²	C₂²×C₈	1	8	yes	—
58	C₂²	C₂³×C₄	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	4	no	—
61	C₄	—	2	2	no	—
62	C₄	C₂²×C₈	1	4	yes	—
63	C₄	H(5,𝔽₂)	1	2	yes	—
64	C₂	—	2	2	no	—
65	C₂	—	2	8	no	—
66	C₂	#518	1	1	yes	characteristic
67	C₂	C₂³×C₈	1	1	yes	characteristic
68	C₂	#568	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator	C₂	C₂³×C₈
socle	C₂²	C₂³×C₄
prime radical	C₂²×D₄	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₈×D₄	C₂	semidirect
3. #431	C₂	semidirect
4. C₂×C₂²⋊C₈	C₂	semidirect
7. C₂²×C₈	C₂²	semidirect
13. C₄⋊C₈	C₂²	semidirect
15. C₂²⋊C₈	C₂²	semidirect
16. C₄×C₈	C₂²	semidirect
18. C₂×D₄	C₈	semidirect
35. C₂³	C₂×C₈	semidirect
38. D₄	C₂×C₈	semidirect
44. C₂×C₄	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄	2
C₈×D₄	2
#431	2
C₂×C₂²⋊C₈	2