Subgroups — C₂²×#124 #3436

Order 128.

940 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#124	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁵	C₂²	1	2	yes	—
7	C₂²×Q₈	—	2	8	no	—
8	C₂²×Q₈	C₂²	1	12	yes	complemented
9	C₂²×D₄	—	2	8	no	—
10	C₂²×D₄	C₂²	1	12	yes	—
11	C₂³×C₄	C₂²	1	2	yes	—
12	C₂³×C₄	C₂²	1	2	yes	—
13	C₂³×C₄	C₂²	1	1	yes	characteristic
14	#124	C₂²	1	64	yes	direct product factor
15	C₂×C₂²⋊C₄	C₂²	1	48	yes	complemented
16	C₂×C₄²	C₂²	1	12	yes	complemented
17	C₂⁴	—	2	8	no	—
18	C₂⁴	—	2	16	no	—
19	C₂⁴	C₂³	1	12	yes	—
20	C₂⁴	C₂³	1	1	yes	characteristic
21	C₂×Q₈	—	2	48	no	—
22	C₂×Q₈	C₂³	1	16	yes	complemented
23	C₂×D₄	—	2	48	no	—
24	C₂×D₄	C₂³	1	16	yes	—
25	C₂²×C₄	—	2	8	no	—
26	C₂²×C₄	—	2	8	no	—
27	C₂²×C₄	D₄	1	8	yes	—
28	C₂²×C₄	C₂³	1	12	yes	—
29	C₂²×C₄	C₂³	1	12	yes	—
30	C₂²×C₄	C₂³	1	6	yes	—
31	C₂²⋊C₄	C₂³	1	64	yes	complemented
32	C₄²	C₂³	1	16	yes	complemented
33	C₂³	—	2	24	no	—
34	C₂³	—	2	48	no	—
35	C₂³	—	4	32	no	—
36	C₂³	C₂×D₄	1	4	yes	—
37	C₂³	#41	1	8	yes	—
38	C₂³	C₂⁴	1	8	yes	—
39	C₂³	C₂⁴	1	3	yes	—
40	Q₈	—	2	32	no	—
41	D₄	—	2	32	no	—
42	C₂×C₄	—	2	24	no	—
43	C₂×C₄	—	2	24	no	—
44	C₂×C₄	C₂×D₄	1	24	yes	—
45	C₂×C₄	C₂⁴	1	8	yes	—
46	C₂×C₄	C₂⁴	1	8	yes	—
47	C₂×C₄	C₂⁴	1	4	yes	—
48	C₂²	—	2	8	no	—
49	C₂²	—	2	16	no	—
50	C₂²	—	4	48	no	—
51	C₂²	#124	1	16	yes	direct product factor
52	C₂²	C₂²×D₄	1	6	yes	—
53	C₂²	C₂×#41	1	12	yes	—
54	C₂²	C₂⁵	1	1	yes	characteristic
55	C₄	—	2	8	no	—
56	C₄	—	2	8	no	—
57	C₄	C₂²×D₄	1	8	yes	—
58	C₂	—	4	8	no	—
59	C₂	C₂×#124	1	12	yes	direct product factor
60	C₂	C₂³×D₄	1	1	yes	characteristic
61	C₂	C₂²×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂³×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈	C₂	semidirect
3. C₂×#124	C₂	direct (N × H)
4. C₂²×C₂²⋊C₄	C₂	semidirect
5. C₂²×C₄²	C₂	semidirect
8. C₂²×Q₈	C₂²	semidirect
14. #124	C₂²	direct (N × H)
15. C₂×C₂²⋊C₄	C₂²	semidirect
16. C₂×C₄²	C₂²	semidirect
22. C₂×Q₈	C₂³	semidirect
31. C₂²⋊C₄	C₂³	semidirect
32. C₄²	C₂³	semidirect
51. C₂²	#124	direct (N × H)
59. C₂	C₂×#124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈	2
C₂³×D₄	2
C₂×#124	2
C₂²×C₂²⋊C₄	2
C₂²×C₄²	2

Subgroups — C22×#124 #3436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#124 #3436