Subgroups — #2736

Order 128.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#397	C₂	1	3	yes	maximal, complemented
2	#393	C₂	1	3	yes	maximal, complemented
3	#388	C₂	1	6	yes	maximal, complemented
4	#379	C₂	1	3	yes	maximal
5	C₂⁵	C₂²	1	1	yes	characteristic
6	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₄⋊C₄	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
9	C₂×C₂²⋊C₄	C₂²	1	3	yes	—
10	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
11	C₂×C₂²⋊C₄	C₂²	1	3	yes	—
12	C₂×C₄²	C₂²	1	2	yes	complemented
13	#95	C₂²	1	3	yes	complemented
14	#95	C₂²	1	6	yes	complemented
15	#95	C₂²	1	1	yes	characteristic, complemented
16	C₂⁴	—	2	6	no	—
17	C₂⁴	—	4	3	no	—
18	C₂⁴	—	4	1	no	—
19	C₂⁴	C₂³	1	3	yes	—
20	C₂×Q₈	—	4	3	no	—
21	C₂²×C₄	C₂³	1	3	yes	—
22	C₂²×C₄	C₂³	1	3	yes	—
23	C₂²×C₄	C₂³	1	6	yes	—
24	C₄⋊C₄	—	4	3	no	—
25	C₂²⋊C₄	—	2	12	no	—
26	C₂²⋊C₄	—	4	3	no	—
27	C₂²⋊C₄	—	4	6	no	—
28	C₂²⋊C₄	—	4	3	no	—
29	C₄²	—	4	2	no	—
30	C₂³	—	2	6	no	—
31	C₂³	—	2	3	no	—
32	C₂³	—	2	3	no	—
33	C₂³	—	2	6	no	—
34	C₂³	—	4	12	no	—
35	C₂³	—	4	12	no	—
36	C₂³	—	4	4	no	—
37	C₂³	#41	1	6	yes	—
38	C₂³	C₂⁴	1	1	yes	characteristic
39	Q₈	—	4	4	no	—
40	C₂×C₄	—	2	6	no	—
41	C₂×C₄	—	2	6	no	—
42	C₂×C₄	—	2	6	no	—
43	C₂×C₄	—	2	6	no	—
44	C₂×C₄	—	2	3	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂²	—	2	6	no	—
48	C₂²	—	2	12	no	—
49	C₂²	—	4	16	no	—
50	C₂²	—	4	6	no	—
51	C₂²	—	4	3	no	—
52	C₂²	—	4	3	no	—
53	C₂²	C₂×#41	1	3	yes	—
54	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂²	H(5,𝔽₂)	1	3	yes	—
56	C₄	—	4	6	no	—
57	C₄	—	4	3	no	—
58	C₄	—	4	3	no	—
59	C₂	—	4	6	no	—
60	C₂	#538	1	3	yes	—
61	C₂	#551	1	3	yes	—
62	C₂	H(3,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁴
prime radical	C₂⁵	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. #397	C₂	semidirect
2. #393	C₂	semidirect
3. #388	C₂	semidirect
6. C₂²×Q₈	C₂²	semidirect
7. C₂×C₄⋊C₄	C₂²	semidirect
12. C₂×C₄²	C₂²	semidirect
13. #95	C₂²	semidirect
14. #95	C₂²	semidirect
15. #95	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#397	2
#393	2
#388	2
#379	2