Subgroups — C₂²×#121 #3432

Order 128.

1264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄	C₂	1	2	yes	maximal, complemented
2	C₂³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#121	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂⁵	C₂²	1	2	yes	—
8	C₂⁵	C₂²	1	1	yes	characteristic
9	C₂²×D₄	—	2	16	no	—
10	C₂²×D₄	—	2	8	no	—
11	C₂²×D₄	C₂²	1	24	yes	complemented
12	C₂²×D₄	C₂²	1	12	yes	complemented
13	C₂³×C₄	—	2	8	no	—
14	C₂³×C₄	C₂²	1	1	yes	characteristic
15	C₂³×C₄	C₂²	1	2	yes	—
16	C₂³×C₄	C₂²	1	1	yes	characteristic
17	C₂³×C₄	C₂²	1	12	yes	complemented
18	#121	C₂²	1	64	yes	direct product factor
19	C₂×C₄⋊C₄	C₂²	1	12	yes	complemented
20	C₂×C₂²⋊C₄	C₂²	1	24	yes	complemented
21	C₂⁴	—	2	4	no	—
22	C₂⁴	—	2	4	no	—
23	C₂⁴	—	2	8	no	—
24	C₂⁴	—	2	8	no	—
25	C₂⁴	—	2	8	no	—
26	C₂⁴	D₄	1	8	yes	complemented
27	C₂⁴	C₂³	1	12	yes	—
28	C₂⁴	C₂³	1	6	yes	—
29	C₂⁴	C₂³	1	1	yes	characteristic
30	C₂×D₄	—	2	96	no	—
31	C₂×D₄	—	2	48	no	—
32	C₂×D₄	C₂³	1	32	yes	complemented
33	C₂×D₄	C₂³	1	16	yes	complemented
34	C₂²×C₄	—	2	4	no	—
35	C₂²×C₄	—	2	8	no	—
36	C₂²×C₄	—	2	48	no	—
37	C₂²×C₄	D₄	1	8	yes	complemented
38	C₂²×C₄	C₂³	1	6	yes	—
39	C₂²×C₄	C₂³	1	12	yes	—
40	C₂²×C₄	C₂³	1	6	yes	—
41	C₂²×C₄	C₂³	1	16	yes	complemented
42	C₄⋊C₄	C₂³	1	16	yes	complemented
43	C₂²⋊C₄	C₂³	1	32	yes	complemented
44	C₂³	—	2	32	no	—
45	C₂³	—	2	12	no	—
46	C₂³	—	2	12	no	—
47	C₂³	—	2	24	no	—
48	C₂³	—	2	24	no	—
49	C₂³	—	2	24	no	—
50	C₂³	—	4	32	no	—
51	C₂³	C₂×D₄	1	4	yes	—
52	C₂³	C₂×D₄	1	4	yes	—
53	C₂³	C₂×D₄	1	24	yes	complemented
54	C₂³	#41	1	4	yes	—
55	C₂³	C₂⁴	1	8	yes	—
56	C₂³	C₂⁴	1	4	yes	—
57	C₂³	C₂⁴	1	3	yes	—
58	D₄	—	2	64	no	—
59	D₄	—	2	32	no	—
60	C₂×C₄	—	2	12	no	—
61	C₂×C₄	—	2	24	no	—
62	C₂×C₄	—	2	32	no	—
63	C₂×C₄	C₂×D₄	1	24	yes	complemented
64	C₂×C₄	C₂⁴	1	4	yes	—
65	C₂×C₄	C₂⁴	1	8	yes	—
66	C₂×C₄	C₂⁴	1	4	yes	—
67	C₂²	—	2	48	no	—
68	C₂²	—	2	4	no	—
69	C₂²	—	2	4	no	—
70	C₂²	—	2	8	no	—
71	C₂²	—	2	8	no	—
72	C₂²	—	2	8	no	—
73	C₂²	—	4	48	no	—
74	C₂²	#121	1	16	yes	direct product factor
75	C₂²	C₂²×D₄	1	6	yes	—
76	C₂²	C₂²×D₄	1	6	yes	—
77	C₂²	C₂²×D₄	1	8	yes	complemented
78	C₂²	C₂×#41	1	6	yes	—
79	C₂²	C₂⁵	1	1	yes	characteristic
80	C₄	—	2	4	no	—
81	C₄	—	2	8	no	—
82	C₄	C₂²×D₄	1	8	yes	complemented
83	C₂	—	2	8	no	—
84	C₂	—	4	8	no	—
85	C₂	C₂×#121	1	12	yes	direct product factor
86	C₂	C₂³×D₄	1	1	yes	characteristic
87	C₂	C₂³×D₄	1	1	yes	characteristic
88	C₂	C₂²×#41	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵
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×D₄	C₂	semidirect
2. C₂³×D₄	C₂	semidirect
3. C₂⁴×C₄	C₂	semidirect
4. C₂×#121	C₂	direct (N × H)
5. C₂²×C₄⋊C₄	C₂	semidirect
6. C₂²×C₂²⋊C₄	C₂	semidirect
11. C₂²×D₄	C₂²	semidirect
12. C₂²×D₄	C₂²	semidirect
17. C₂³×C₄	C₂²	semidirect
18. #121	C₂²	direct (N × H)
19. C₂×C₄⋊C₄	C₂²	semidirect
20. C₂×C₂²⋊C₄	C₂²	semidirect
26. C₂⁴	D₄	semidirect
32. C₂×D₄	C₂³	semidirect
33. C₂×D₄	C₂³	semidirect
37. C₂²×C₄	D₄	semidirect
41. C₂²×C₄	C₂³	semidirect
42. C₄⋊C₄	C₂³	semidirect
43. C₂²⋊C₄	C₂³	semidirect
53. C₂³	C₂×D₄	semidirect
63. C₂×C₄	C₂×D₄	semidirect
74. C₂²	#121	direct (N × H)
77. C₂²	C₂²×D₄	semidirect
82. C₄	C₂²×D₄	semidirect
85. C₂	C₂×#121	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×D₄	2
C₂⁴×C₄	2
C₂×#121	2
C₂²×C₄⋊C₄	2
C₂²×C₂²⋊C₄	2

Subgroups — C22×#121 #3432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#121 #3432