Subgroups — C₂²×C₃×#126 #87322

Order 384.

1320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#126	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₃×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₂²×C₃×C₂²⋊C₄	C₂	1	3	yes	maximal
4	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#126	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂⁵×C₃	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄	C₂²	1	3	yes	—
8	C₂³×C₃×C₄	C₂²	1	3	yes	—
9	C₃×#126	C₂²	1	64	yes	direct product factor
10	C₂×C₃×C₄⋊C₄	C₂²	1	36	yes	complemented
11	C₂×C₃×C₂²⋊C₄	C₂²	1	36	yes	—
12	C₂×C₃×C₄²	C₂²	1	12	yes	complemented
13	C₂×#126	C₂×C₃	1	24	yes	direct product factor
14	C₂²×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
15	C₂²×C₂²⋊C₄	C₂×C₃	1	3	yes	—
16	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂⁴×C₃	—	2	12	no	—
18	C₂⁴×C₃	C₂³	1	6	yes	—
19	C₂⁴×C₃	C₂³	1	1	yes	characteristic
20	C₂²×C₃×C₄	—	2	12	no	—
21	C₂²×C₃×C₄	—	2	12	no	—
22	C₂²×C₃×C₄	C₂³	1	18	yes	—
23	C₂²×C₃×C₄	C₂³	1	18	yes	—
24	C₃×C₄⋊C₄	C₂³	1	48	yes	complemented
25	C₃×C₂²⋊C₄	C₂³	1	48	yes	—
26	C₃×C₄²	C₂³	1	16	yes	complemented
27	C₂⁵	C₂²×C₃	1	1	yes	characteristic
28	C₂³×C₄	C₂²×C₃	1	3	yes	—
29	C₂³×C₄	C₂²×C₃	1	3	yes	—
30	#126	C₂²×C₃	1	64	yes	direct product factor
31	C₂×C₄⋊C₄	C₂²×C₃	1	36	yes	complemented
32	C₂×C₂²⋊C₄	C₂²×C₃	1	36	yes	—
33	C₂×C₄²	C₂²×C₃	1	12	yes	complemented
34	C₂³×C₃	—	2	36	no	—
35	C₂³×C₃	—	4	16	no	—
36	C₂³×C₃	#41	1	12	yes	—
37	C₂³×C₃	C₂⁴	1	4	yes	—
38	C₂³×C₃	C₂⁴	1	3	yes	—
39	C₂×C₃×C₄	—	2	36	no	—
40	C₂×C₃×C₄	—	2	36	no	—
41	C₂×C₃×C₄	C₂⁴	1	12	yes	—
42	C₂×C₃×C₄	C₂⁴	1	12	yes	—
43	C₂⁴	—	2	12	no	—
44	C₂⁴	C₂³×C₃	1	6	yes	—
45	C₂⁴	C₂³×C₃	1	1	yes	characteristic
46	C₂²×C₄	—	2	12	no	—
47	C₂²×C₄	—	2	12	no	—
48	C₂²×C₄	C₂³×C₃	1	18	yes	—
49	C₂²×C₄	C₂³×C₃	1	18	yes	—
50	C₄⋊C₄	C₂³×C₃	1	48	yes	complemented
51	C₂²⋊C₄	C₂³×C₃	1	48	yes	—
52	C₄²	C₂³×C₃	1	16	yes	complemented
53	C₂²×C₃	—	2	12	no	—
54	C₂²×C₃	—	4	24	no	—
55	C₂²×C₃	#126	1	16	yes	direct product factor
56	C₂²×C₃	C₂×#41	1	18	yes	—
57	C₂²×C₃	C₂⁵	1	1	yes	characteristic
58	C₃×C₄	—	2	12	no	—
59	C₃×C₄	—	2	12	no	—
60	C₂³	—	2	36	no	—
61	C₂³	—	4	16	no	—
62	C₂³	C₃×#41	1	12	yes	—
63	C₂³	C₂⁴×C₃	1	4	yes	—
64	C₂³	C₂⁴×C₃	1	3	yes	—
65	C₂×C₄	—	2	36	no	—
66	C₂×C₄	—	2	36	no	—
67	C₂×C₄	C₂⁴×C₃	1	12	yes	—
68	C₂×C₄	C₂⁴×C₃	1	12	yes	—
69	C₂×C₃	—	4	4	no	—
70	C₂×C₃	C₂×#126	1	12	yes	direct product factor
71	C₂×C₃	C₂²×#41	1	3	yes	—
72	C₂²	—	2	12	no	—
73	C₂²	—	4	24	no	—
74	C₂²	C₃×#126	1	16	yes	direct product factor
75	C₂²	C₂×C₃×#41	1	18	yes	—
76	C₂²	C₂⁵×C₃	1	1	yes	characteristic
77	C₄	—	2	12	no	—
78	C₄	—	2	12	no	—
79	C₃	C₂²×#126	1	1	yes	characteristic, direct product factor, Sylow
80	C₂	—	4	4	no	—
81	C₂	C₂×C₃×#126	1	12	yes	direct product factor
82	C₂	C₂²×C₃×#41	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#126	C₂	direct (N × H)
2. C₂²×C₃×C₄⋊C₄	C₂	semidirect
4. C₂²×C₃×C₄²	C₂	semidirect
5. C₂²×#126	C₃	direct (N × H)
9. C₃×#126	C₂²	direct (N × H)
10. C₂×C₃×C₄⋊C₄	C₂²	semidirect
12. C₂×C₃×C₄²	C₂²	semidirect
13. C₂×#126	C₂×C₃	direct (N × H)
14. C₂²×C₄⋊C₄	C₂×C₃	semidirect
16. C₂²×C₄²	C₂×C₃	semidirect
24. C₃×C₄⋊C₄	C₂³	semidirect
26. C₃×C₄²	C₂³	semidirect
30. #126	C₂²×C₃	direct (N × H)
31. C₂×C₄⋊C₄	C₂²×C₃	semidirect
33. C₂×C₄²	C₂²×C₃	semidirect
50. C₄⋊C₄	C₂³×C₃	semidirect
52. C₄²	C₂³×C₃	semidirect
55. C₂²×C₃	#126	direct (N × H)
70. C₂×C₃	C₂×#126	direct (N × H)
74. C₂²	C₃×#126	direct (N × H)
79. C₃	C₂²×#126	direct (N × H)
81. C₂	C₂×C₃×#126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#126	2
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₂²⋊C₄	2
C₂²×C₃×C₄²	2
C₂²×#126	3

Subgroups — C22×C3×#126 #87322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#126 #87322