Subgroups — #2860

Order 128.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#126	C₂	1	2	yes	maximal, complemented
2	#390	C₂	1	3	yes	maximal
3	#389	C₂	1	3	yes	maximal, complemented
4	#388	C₂	1	6	yes	maximal, complemented
5	C₄³	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄	C₂²	1	1	yes	characteristic
7	#126	—	2	8	no	—
8	C₂×C₄⋊C₄	C₂²	1	6	yes	—
9	C₂×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
11	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
12	C₂×C₄²	C₂²	1	3	yes	—
13	C₂×C₄²	C₂²	1	3	yes	—
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	#95	C₂²	1	6	yes	—
16	C₂⁴	C₂³	1	2	yes	—
17	C₂×D₄	—	2	6	no	—
18	C₂²×C₄	C₂³	1	6	yes	—
19	C₂²×C₄	C₂³	1	3	yes	—
20	C₂²×C₄	C₂³	1	3	yes	—
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	C₄⋊C₄	—	2	12	no	—
23	C₄⋊C₄	—	2	6	no	—
24	C₂²⋊C₄	—	2	12	no	—
25	C₂²⋊C₄	—	2	12	no	—
26	C₄²	—	2	6	no	—
27	C₄²	—	2	6	no	—
28	C₄²	D₄	1	4	yes	complemented
29	C₂³	—	2	2	no	—
30	C₂³	—	2	6	no	—
31	C₂³	—	2	6	no	—
32	C₂³	C₂⁴	1	1	yes	characteristic
33	D₄	—	4	4	no	—
34	C₂×C₄	—	2	3	no	—
35	C₂×C₄	—	2	3	no	—
36	C₂×C₄	—	2	3	no	—
37	C₂×C₄	—	2	3	no	—
38	C₂×C₄	—	2	6	no	—
39	C₂×C₄	—	2	6	no	—
40	C₂×C₄	—	2	6	no	—
41	C₂×C₄	C₂×D₄	1	6	yes	—
42	C₂×C₄	#41	1	6	yes	—
43	C₂×C₄	#41	1	6	yes	—
44	C₂²	—	4	2	no	—
45	C₂²	—	4	6	no	—
46	C₂²	—	4	6	no	—
47	C₂²	C₂²×D₄	1	1	yes	characteristic
48	C₂²	C₂×#41	1	3	yes	—
49	C₂²	C₂×#41	1	3	yes	—
50	C₄	—	2	6	no	—
51	C₄	—	2	6	no	—
52	C₄	—	4	6	no	—
53	C₄	#126	1	4	yes	complemented
54	C₂	—	8	2	no	—
55	C₂	C₂×#126	1	1	yes	characteristic
56	C₂	#529	1	3	yes	—
57	C₂	#532	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#126	C₂	semidirect
3. #389	C₂	semidirect
4. #388	C₂	semidirect
5. C₄³	C₂	semidirect
28. C₄²	D₄	semidirect
53. C₄	#126	semidirect

Maximal subgroups

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

subgroup	index
C₂×#126	2
#390	2
#389	2
#388	2
C₄³	2