Subgroups — D₁₂×#3861 #72245884

Order 3528.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×#3861	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×#3861	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇²×D₁₂	C₃	1	1	yes	maximal, characteristic, complemented
4	D₄×#3861	—	3	1	no	maximal
5	C₂×C₃×#3861	C₂²	1	1	yes	characteristic
6	D₃×#3861	—	2	2	no	—
7	C₂×D₃×C₇²	C₂×C₃	1	2	yes	complemented
8	C₂²×#3861	—	3	2	no	—
9	C₃×C₄×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
10	C₄×#3861	—	2	1	no	—
11	C₄×#3861	D₃	1	1	yes	characteristic, complemented
12	ΓL(1,𝔽₈)×D₁₂	—	7	8	no	maximal
13	C₃×#3861	D₄	1	1	yes	characteristic, complemented, Hall
14	C₇²×D₄	—	3	1	no	Hall
15	C₂×C₃×C₇²	C₂²×C₃	1	1	yes	characteristic
16	D₃×C₇²	—	2	2	no	—
17	C₂×#3861	—	2	1	no	—
18	C₂×#3861	—	6	2	no	—
19	C₂×#3861	C₂×D₃	1	1	yes	characteristic
20	C₂×D₃×ΓL(1,𝔽₈)	—	7	16	no	—
21	C₃×C₄×ΓL(1,𝔽₈)	—	7	8	no	—
22	C₂²×C₇²	—	3	2	no	—
23	C₄×C₇²	C₃×D₃	1	1	yes	characteristic, complemented
24	C₇×D₁₂	ΓL(1,𝔽₈)	1	8	yes	complemented
25	D₄×ΓL(1,𝔽₈)	—	21	8	no	—
26	C₃×C₇²	C₃×D₄	1	1	yes	characteristic, complemented
27	#3861	—	2	1	no	—
28	#3861	D₁₂	1	1	yes	characteristic, direct product factor
29	C₂×C₃×ΓL(1,𝔽₈)	—	7	8	no	—
30	D₃×ΓL(1,𝔽₈)	—	14	16	no	—
31	C₂×C₇²	—	6	2	no	—
32	C₂×C₇²	C₂×C₃×D₃	1	1	yes	characteristic
33	C₂×D₃×C₇	C₂×ΓL(1,𝔽₈)	1	16	yes	complemented
34	C₂²×ΓL(1,𝔽₈)	—	21	16	no	—
35	C₃×C₄×C₇	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
36	C₄×ΓL(1,𝔽₈)	—	7	8	no	—
37	C₄×ΓL(1,𝔽₈)	—	14	8	no	—
38	C₃×D₁₂	—	49	1	no	Hall
39	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
40	C₇×D₄	—	3	8	no	—
41	C₇²	C₃×D₁₂	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₃×C₇	C₂²×ΓL(1,𝔽₈)	1	8	yes	—
43	D₃×C₇	—	2	16	no	—
44	C₂×ΓL(1,𝔽₈)	—	7	8	no	—
45	C₂×ΓL(1,𝔽₈)	—	14	8	no	—
46	C₂×ΓL(1,𝔽₈)	—	42	16	no	—
47	C₂×C₃×D₃	—	49	2	no	—
48	C₃²×C₄	—	49	1	no	—
49	C₂²×C₇	—	3	16	no	—
50	C₄×C₇	D₃×ΓL(1,𝔽₈)	1	8	yes	complemented
51	C₃×D₄	—	147	1	no	—
52	D₁₂	#3861	1	1	yes	characteristic, direct product factor
53	C₃×C₇	D₄×ΓL(1,𝔽₈)	1	8	yes	complemented
54	ΓL(1,𝔽₈)	—	7	8	no	—
55	ΓL(1,𝔽₈)	—	14	8	no	—
56	C₂×C₃²	—	49	1	no	—
57	C₃×D₃	—	98	2	no	—
58	C₂×C₇	—	6	16	no	—
59	C₂×C₇	C₂×D₃×ΓL(1,𝔽₈)	1	8	yes	—
60	C₂²×C₃	—	147	2	no	—
61	C₂×D₃	C₂×#3861	1	2	yes	complemented
62	C₃×C₄	—	49	1	no	—
63	C₃×C₄	—	98	1	no	—
64	C₃×C₄	C₂×#3861	1	1	yes	characteristic, complemented
65	C₃²	—	49	1	no	Sylow
66	D₄	—	3	1	no	Sylow
67	C₇	ΓL(1,𝔽₈)×D₁₂	1	8	yes	complemented
68	C₂×C₃	—	294	2	no	—
69	C₂×C₃	—	49	1	no	—
70	C₂×C₃	—	98	1	no	—
71	C₂×C₃	C₂²×#3861	1	1	yes	characteristic
72	D₃	—	2	2	no	—
73	C₂²	—	3	2	no	—
74	C₄	D₃×#3861	1	1	yes	characteristic, complemented
75	C₃	—	49	1	no	—
76	C₃	—	98	1	no	—
77	C₃	D₄×#3861	1	1	yes	characteristic, complemented
78	C₂	—	6	2	no	—
79	C₂	C₂×D₃×#3861	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×#3861
commutator, socle, cosocle	C₂×C₃×C₇²	C₂²×C₃
Fitting	C₃×C₄×C₇²	C₂×C₃
hypercenter	C₄	D₃×#3861
hypocenter	C₃×C₇²	C₃×D₄
LCS	G ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×#3861	C₂	semidirect
2. C₃×C₄×#3861	C₂	semidirect
3. C₇²×D₁₂	C₃	semidirect
7. C₂×D₃×C₇²	C₂×C₃	semidirect
9. C₃×C₄×C₇²	C₂×C₃	semidirect
11. C₄×#3861	D₃	semidirect
13. C₃×#3861	D₄	semidirect
23. C₄×C₇²	C₃×D₃	semidirect
24. C₇×D₁₂	ΓL(1,𝔽₈)	semidirect
26. C₃×C₇²	C₃×D₄	semidirect
28. #3861	D₁₂	direct (N × H)
33. C₂×D₃×C₇	C₂×ΓL(1,𝔽₈)	semidirect
35. C₃×C₄×C₇	C₂×ΓL(1,𝔽₈)	semidirect
41. C₇²	C₃×D₁₂	semidirect
50. C₄×C₇	D₃×ΓL(1,𝔽₈)	semidirect
52. D₁₂	#3861	direct (N × H)
53. C₃×C₇	D₄×ΓL(1,𝔽₈)	semidirect
61. C₂×D₃	C₂×#3861	semidirect
64. C₃×C₄	C₂×#3861	semidirect
67. C₇	ΓL(1,𝔽₈)×D₁₂	semidirect
74. C₄	D₃×#3861	semidirect
77. C₃	D₄×#3861	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×#3861	2
C₃×C₄×#3861	2
C₇²×D₁₂	3
D₄×#3861	3
ΓL(1,𝔽₈)×D₁₂	7

Subgroups — D12×#3861 #72245884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₂×#3861 #72245884