Subgroups — #72258669

Order 9216.

At least 9121 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

The subgroup lattice has 103 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72228310	C₂	1	1	yes	maximal, characteristic, complemented
2	#72228045	C₂	1	2	yes	maximal, complemented
3	#72226315	C₂	1	2	yes	maximal, complemented
4	#72226303	C₂	1	4	yes	maximal, complemented
5	A₄×#85489	C₂	1	2	yes	maximal, complemented
6	C₂²×#85489	D₃	1	2	yes	complemented
7	#72226218	C₂	1	2	yes	maximal, complemented
8	#72226203	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×A₄⋊S₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#72225974	C₂²	1	1	yes	characteristic, complemented
11	#72225971	C₂²	1	2	yes	complemented
12	A₄×#5912	C₂²	1	2	yes	complemented
13	#72225965	C₂²	1	1	yes	characteristic, complemented
14	#72225963	C₂²	1	4	yes	complemented
15	A₄×#5415	C₂²	1	4	yes	complemented
16	#72225957	C₂²	1	4	yes	complemented
17	A₄×#5414	C₂²	1	4	yes	complemented
18	#72225952	C₂²	1	2	yes	complemented
19	A₄×#5409	C₂²	1	2	yes	complemented
20	#72225947	C₂²	1	2	yes	complemented
21	C₂×C₄⋊(A₄⋊S₄)	C₂²	1	1	yes	characteristic, complemented
22	C₂×D₄×A₄²	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄×A₄⋊S₄	C₂²	1	1	yes	characteristic
24	D₄×A₄⋊S₄	C₂²	1	2	yes	—
25	D₄×A₄⋊S₄	C₂²	1	2	yes	complemented
26	C₂²×A₄⋊S₄	D₄	1	1	yes	characteristic
27	D₄×A₄²	C₂³	1	2	yes	—
28	C₂×C₄×A₄²	C₂³	1	1	yes	characteristic
29	C₂×A₄⋊SL(2,ℤ/4)	D₄	1	1	yes	characteristic
30	C₄⋊(A₄⋊S₄)	C₂³	1	2	yes	—
31	C₄×A₄⋊S₄	D₄	1	1	yes	characteristic
32	C₄×A₄⋊S₄	D₄	1	1	yes	characteristic
33	C₄×A₄⋊S₄	C₂³	1	1	yes	characteristic
34	C₄×A₄⋊S₄	C₂³	1	1	yes	characteristic
35	A₄×C₄⋊S₄	C₂³	1	2	yes	complemented
36	A₄×A₄⋊Q₈	C₂³	1	2	yes	complemented
37	A₄×A₄⋊C₈	C₂³	1	4	yes	complemented
38	C₂²×#5912	C₂×D₃	1	2	yes	complemented
39	C₂²×#5414	C₂×D₃	1	4	yes	complemented
40	C₂²×#5415	C₂×D₃	1	4	yes	complemented
41	C₂²×#5409	C₂×D₃	1	2	yes	complemented
42	C₂³×D₄×A₄	C₂×D₃	1	2	yes	complemented
43	C₂²×A₄²	C₂×D₄	1	1	yes	characteristic
44	C₂×A₄⋊S₄	C₂×D₄	1	1	yes	characteristic
45	C₂×A₄⋊S₄	C₂×D₄	1	1	yes	characteristic
46	C₄×A₄²	C₂×D₄	1	1	yes	characteristic
47	C₄×A₄²	C₂⁴	1	1	yes	characteristic
48	A₄⋊SL(2,ℤ/4)	C₂×D₄	1	1	yes	characteristic
49	A₄⋊SL(2,ℤ/4)	C₂×D₄	1	1	yes	characteristic
50	C₂²×D₄×A₄	C₂²×D₃	1	4	yes	—
51	C₂³×C₄×A₄	C₂²×D₃	1	2	yes	—
52	C₂²×C₄⋊S₄	C₂²×D₃	1	2	yes	complemented
53	C₂²×A₄⋊Q₈	C₂²×D₃	1	2	yes	complemented
54	#85489	S₄	1	2	yes	complemented
55	C₂²×A₄⋊C₈	C₂²×D₃	1	4	yes	complemented
56	C₂×A₄²	C₂²×D₄	1	1	yes	characteristic
57	A₄⋊S₄	AGL(1,ℤ/8)	1	2	yes	complemented
58	C₂⁵×D₄	D₃²	1	1	yes	characteristic, complemented
59	C₂⁴×A₄	D₃×D₄	1	2	yes	—
60	C₂×D₄×A₄	C₂×S₄	1	2	yes	complemented
61	C₂²×C₄×A₄	D₃×D₄	1	2	yes	—
62	C₂²×C₄×A₄	C₂³×D₃	1	2	yes	—
63	#5912	C₂×S₄	1	2	yes	complemented
64	#5415	C₂×S₄	1	4	yes	complemented
65	#5414	C₂×S₄	1	4	yes	complemented
66	#5409	C₂×S₄	1	2	yes	complemented
67	A₄²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented
68	C₂⁴×D₄	C₂×D₃²	1	2	yes	—
69	C₂⁵×C₄	C₂×D₃²	1	1	yes	characteristic
70	C₂³×A₄	C₂×D₃×D₄	1	2	yes	—
71	D₄×A₄	C₂²×S₄	1	4	yes	—
72	C₂×C₄×A₄	C₂²×S₄	1	2	yes	—
73	C₄⋊S₄	C₂²×S₄	1	2	yes	complemented
74	A₄⋊Q₈	C₂²×S₄	1	2	yes	complemented
75	A₄⋊C₈	C₂²×S₄	1	4	yes	complemented
76	C₂⁶	#3811	1	1	yes	characteristic
77	C₂³×D₄	D₃×S₄	1	2	yes	complemented
78	C₂⁴×C₄	#3811	1	1	yes	characteristic
79	C₂⁴×C₄	C₂²×D₃²	1	1	yes	characteristic
80	C₂²×A₄	D₃×AGL(1,ℤ/8)	1	2	yes	complemented
81	C₂²×A₄	D₄×S₄	1	2	yes	—
82	C₄×A₄	D₄×S₄	1	2	yes	—
83	C₄×A₄	C₂³×S₄	1	2	yes	—
84	C₂⁵	C₂×#3811	1	1	yes	characteristic
85	C₂²×D₄	C₂×D₃×S₄	1	4	yes	—
86	C₂³×C₄	C₂×D₃×S₄	1	2	yes	—
87	C₂×A₄	C₂×D₄×S₄	1	2	yes	—
88	C₂⁴	#10595100	1	1	yes	characteristic, complemented
89	C₂⁴	#10596473	1	2	yes	—
90	C₂×D₄	S₄²	1	1	yes	characteristic, complemented
91	C₂²×C₄	#10596473	1	2	yes	—
92	C₂²×C₄	C₂²×D₃×S₄	1	2	yes	—
93	A₄	S₄×AGL(1,ℤ/8)	1	2	yes	complemented
94	C₂³	C₂×#10596473	1	2	yes	—
95	D₄	C₂×S₄²	1	2	yes	—
96	C₂×C₄	C₂×S₄²	1	1	yes	characteristic
97	C₂²	#72215293	1	1	yes	characteristic
98	C₂²	#72258598	1	2	yes	complemented
99	C₄	#72215293	1	1	yes	characteristic
100	C₄	C₂²×S₄²	1	1	yes	characteristic
101	C₂	C₂×#72215293	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#72215293
commutator, cosocle	C₄×A₄²	C₂⁴
Frattini	C₄	C₂²×S₄²
Fitting	C₂⁵×D₄	D₃²
socle	C₂⁵	C₂×#3811
hypercenter	C₂×D₄	S₄²
hypocenter	A₄²	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₄×A₄² ▷ C₂×A₄² ▷ A₄²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #72228310	C₂	semidirect
2. #72228045	C₂	semidirect
3. #72226315	C₂	semidirect
4. #72226303	C₂	semidirect
5. A₄×#85489	C₂	semidirect
6. C₂²×#85489	D₃	semidirect
7. #72226218	C₂	semidirect
8. #72226203	C₂	semidirect
9. C₂×D₄×A₄⋊S₄	C₂	semidirect
10. #72225974	C₂²	semidirect
11. #72225971	C₂²	semidirect
12. A₄×#5912	C₂²	semidirect
13. #72225965	C₂²	semidirect
14. #72225963	C₂²	semidirect
15. A₄×#5415	C₂²	semidirect
16. #72225957	C₂²	semidirect
17. A₄×#5414	C₂²	semidirect
18. #72225952	C₂²	semidirect
19. A₄×#5409	C₂²	semidirect
20. #72225947	C₂²	semidirect
21. C₂×C₄⋊(A₄⋊S₄)	C₂²	semidirect
22. C₂×D₄×A₄²	C₂²	semidirect
25. D₄×A₄⋊S₄	C₂²	semidirect
35. A₄×C₄⋊S₄	C₂³	semidirect
36. A₄×A₄⋊Q₈	C₂³	semidirect
37. A₄×A₄⋊C₈	C₂³	semidirect
38. C₂²×#5912	C₂×D₃	semidirect
39. C₂²×#5414	C₂×D₃	semidirect
40. C₂²×#5415	C₂×D₃	semidirect
41. C₂²×#5409	C₂×D₃	semidirect
42. C₂³×D₄×A₄	C₂×D₃	semidirect
52. C₂²×C₄⋊S₄	C₂²×D₃	semidirect
53. C₂²×A₄⋊Q₈	C₂²×D₃	semidirect
54. #85489	S₄	semidirect
55. C₂²×A₄⋊C₈	C₂²×D₃	semidirect
57. A₄⋊S₄	AGL(1,ℤ/8)	semidirect
58. C₂⁵×D₄	D₃²	semidirect
60. C₂×D₄×A₄	C₂×S₄	semidirect
63. #5912	C₂×S₄	semidirect
64. #5415	C₂×S₄	semidirect
65. #5414	C₂×S₄	semidirect
66. #5409	C₂×S₄	semidirect
67. A₄²	C₂×AGL(1,ℤ/8)	semidirect
73. C₄⋊S₄	C₂²×S₄	semidirect
74. A₄⋊Q₈	C₂²×S₄	semidirect
75. A₄⋊C₈	C₂²×S₄	semidirect
77. C₂³×D₄	D₃×S₄	semidirect
80. C₂²×A₄	D₃×AGL(1,ℤ/8)	semidirect
88. C₂⁴	#10595100	semidirect
90. C₂×D₄	S₄²	semidirect
93. A₄	S₄×AGL(1,ℤ/8)	semidirect
98. C₂²	#72258598	semidirect

Maximal subgroups

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

subgroup	index
#72258631	3
#72258598	4
#72228310	2
#72228045	2
#72226315	2
#72226303	2
A₄×#85489	2
#72226218	2
#72226203	2
C₂×D₄×A₄⋊S₄	2