Subgroups — C₄×#11715828 #72341272

Order 3072.

At least 6703 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

The subgroup lattice has 127 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#87522	C₂	1	1	yes	maximal, characteristic, complemented
2	#72341268	C₂	1	2	yes	maximal, complemented
3	#72341264	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#87516	C₂	1	2	yes	maximal, complemented
5	#72227857	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#87547	C₂	1	1	yes	maximal, characteristic, complemented
7	#72227324	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#87520	C₂	1	1	yes	maximal, characteristic, complemented
9	#72226121	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#5935	C₂	1	2	yes	maximal, complemented
11	C₂×#11715828	C₂	1	1	yes	maximal, characteristic
12	#11715828	C₄	1	8	yes	direct product factor
13	C₂×#87522	C₂²	1	1	yes	characteristic
14	C₂×#87516	C₂²	1	2	yes	—
15	C₂×#87520	C₂²	1	1	yes	characteristic
16	C₂²×#5935	C₂²	1	2	yes	—
17	#11715680	C₂²	1	2	yes	complemented
18	#11715679	C₂²	1	2	yes	complemented
19	#11715679	C₂²	1	2	yes	complemented
20	C₄×#5933	C₂²	1	1	yes	characteristic, complemented
21	C₄×#5933	C₂²	1	1	yes	characteristic, complemented
22	C₄×#5935	C₂²	1	2	yes	complemented
23	C₄×#5935	C₂²	1	2	yes	complemented
24	C₄×#5935	C₂²	1	1	yes	characteristic, complemented
25	C₄×#5935	C₂²	1	2	yes	complemented
26	C₄×#5935	C₂²	1	1	yes	characteristic, complemented
27	#11715665	C₂²	1	1	yes	characteristic, complemented
28	#11715664	C₂²	1	2	yes	complemented
29	#11715663	C₂²	1	1	yes	characteristic, complemented
30	C₂×#85565	C₂²	1	2	yes	complemented
31	C₂×#87547	C₂²	1	1	yes	characteristic
32	#11710604	C₂²	1	1	yes	characteristic, complemented
33	#11710602	C₂²	1	2	yes	complemented
34	C₄×#5950	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₄×#997	C₂²	1	2	yes	complemented
36	C₄×H(7,𝔽₂)	D₃	1	1	yes	characteristic, complemented
37	#87547	C₂×C₄	1	4	yes	complemented
38	C₂×#5950	C₂³	1	1	yes	characteristic
39	C₂²×#997	C₂³	1	2	yes	—
40	#87522	C₂×C₄	1	4	yes	complemented
41	#87520	C₂×C₄	1	4	yes	complemented
42	#87516	C₂×C₄	1	8	yes	complemented
43	C₂×#5935	C₂×C₄	1	8	yes	complemented
44	C₂×#5935	C₂³	1	1	yes	characteristic
45	C₂×#5935	C₂³	1	2	yes	—
46	C₂×#5933	C₂³	1	1	yes	characteristic
47	C₄×#997	C₂³	1	1	yes	characteristic
48	C₄×#997	C₂³	1	2	yes	—
49	C₄×#997	C₂³	1	1	yes	characteristic
50	#85565	C₂³	1	1	yes	characteristic
51	#85565	C₂³	1	2	yes	—
52	#85565	C₂³	1	1	yes	characteristic
53	C₂×H(7,𝔽₂)	C₂×D₃	1	1	yes	characteristic
54	#62654	C₂×D₃	1	1	yes	characteristic, complemented
55	#62652	C₂×D₃	1	2	yes	complemented
56	C₄×#585	C₂×D₃	1	1	yes	characteristic, complemented
57	C₂×C₄×H(5,𝔽₂)	C₂×D₃	1	2	yes	complemented
58	#5950	C₂²×C₄	1	2	yes	complemented
59	C₂×#997	C₂²×C₄	1	4	yes	complemented
60	C₂×#997	C₂⁴	1	1	yes	characteristic
61	#5935	C₂²×C₄	1	2	yes	complemented
62	#5935	C₂²×C₄	1	4	yes	complemented
63	#5933	C₂²×C₄	1	2	yes	complemented
64	H(7,𝔽₂)	C₄×D₃	1	4	yes	complemented
65	C₂×#585	C₂²×D₃	1	1	yes	characteristic
66	C₂²×H(5,𝔽₂)	C₂²×D₃	1	2	yes	—
67	C₄×2₋¹⁺⁴	S₄	1	2	yes	complemented
68	C₄×H(5,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
69	C₄×H(5,𝔽₂)	C₂²×D₃	1	2	yes	—
70	C₄×H(5,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
71	C₂²×C₄×D₄	S₄	1	1	yes	characteristic
72	#997	C₂³×C₄	1	1	yes	characteristic
73	#585	C₂×C₄×D₃	1	2	yes	complemented
74	C₂×2₋¹⁺⁴	C₂×S₄	1	2	yes	—
75	C₂×H(5,𝔽₂)	C₂×C₄×D₃	1	4	yes	complemented
76	C₂×H(5,𝔽₂)	C₂³×D₃	1	1	yes	characteristic
77	C₂³×D₄	C₂×S₄	1	1	yes	characteristic
78	C₂⁴×C₄	C₂×S₄	1	2	yes	—
79	#520	C₂×S₄	1	2	yes	complemented
80	#519	C₂×S₄	1	4	yes	complemented
81	C₄×#41	C₂×S₄	1	2	yes	complemented
82	C₂×C₄×Q₈	C₂×S₄	1	4	yes	complemented
83	C₂²×C₄⋊C₄	C₂×S₄	1	1	yes	characteristic
84	C₂²×C₂²⋊C₄	C₂×S₄	1	2	yes	—
85	C₂²×C₄²	C₂×S₄	1	1	yes	characteristic
86	C₂⁵	C₂²×S₄	1	2	yes	—
87	2₋¹⁺⁴	C₄×S₄	1	8	yes	complemented
88	H(5,𝔽₂)	C₂²×C₄×D₃	1	1	yes	characteristic
89	C₂×#41	C₂²×S₄	1	2	yes	—
90	C₂²×Q₈	C₂²×S₄	1	4	yes	—
91	C₂²×D₄	C₄×S₄	1	4	yes	—
92	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
93	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
94	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
95	C₂³×C₄	C₂²×S₄	1	2	yes	—
96	C₄×Q₈	C₂²×S₄	1	2	yes	—
97	C₄×Q₈	C₂²×S₄	1	2	yes	—
98	C₄×Q₈	C₂²×S₄	1	4	yes	—
99	C₄×D₄	#1020	1	1	yes	characteristic
100	C₂⁴	C₂×C₄×S₄	1	4	yes	—
101	C₂⁴	C₂³×S₄	1	1	yes	characteristic
102	#41	C₂×C₄×S₄	1	4	yes	complemented
103	C₂×Q₈	C₂×C₄×S₄	1	8	yes	complemented
104	C₂×Q₈	C₂³×S₄	1	2	yes	—
105	C₂×D₄	C₂×#1020	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₄×S₄	1	2	yes	—
107	C₂²×C₄	C₂×#1020	1	2	yes	—
108	C₄⋊C₄	C₂×#1020	1	1	yes	characteristic
109	C₂²⋊C₄	C₂×#1020	1	2	yes	—
110	C₄²	C₂×#1020	1	1	yes	characteristic
111	C₂³	C₂²×C₄×S₄	1	1	yes	characteristic
112	C₂³	C₂²×#1020	1	2	yes	—
113	Q₈	C₂²×C₄×S₄	1	2	yes	—
114	D₄	C₄×#1020	1	4	yes	—
115	C₂×C₄	C₂²×#1020	1	2	yes	—
116	C₂×C₄	C₂²×#1020	1	1	yes	characteristic
117	C₂×C₄	C₂²×#1020	1	1	yes	characteristic
118	C₂×C₄	C₂²×#1020	1	1	yes	characteristic
119	C₂²	C₂×C₄×#1020	1	4	yes	—
120	C₂²	C₂³×#1020	1	1	yes	characteristic
121	C₄	C₂×C₄×#1020	1	2	yes	—
122	C₄	#11715828	1	2	yes	direct product factor
123	C₂	C₂×#11715828	1	1	yes	characteristic
124	C₂	#72215767	1	1	yes	characteristic
125	C₂	C₂²×C₄×#1020	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#1020
commutator, hypocenter	#997	C₂³×C₄
Frattini, socle	C₂²	C₂³×#1020
Fitting	C₄×H(7,𝔽₂)	D₃
prime radical	C₂×#11715828	C₂
cosocle	C₂×#997	C₂⁴
hypercenter	C₄×D₄	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₄×#87522	C₂	semidirect
2. #72341268	C₂	semidirect
3. #72341264	C₂	semidirect
4. C₄×#87516	C₂	semidirect
5. #72227857	C₂	semidirect
6. C₄×#87547	C₂	semidirect
7. #72227324	C₂	semidirect
8. C₄×#87520	C₂	semidirect
9. #72226121	C₂	semidirect
10. C₂×C₄×#5935	C₂	semidirect
12. #11715828	C₄	direct (N × H)
17. #11715680	C₂²	semidirect
18. #11715679	C₂²	semidirect
19. #11715679	C₂²	semidirect
20. C₄×#5933	C₂²	semidirect
21. C₄×#5933	C₂²	semidirect
22. C₄×#5935	C₂²	semidirect
23. C₄×#5935	C₂²	semidirect
24. C₄×#5935	C₂²	semidirect
25. C₄×#5935	C₂²	semidirect
26. C₄×#5935	C₂²	semidirect
27. #11715665	C₂²	semidirect
28. #11715664	C₂²	semidirect
29. #11715663	C₂²	semidirect
30. C₂×#85565	C₂²	semidirect
32. #11710604	C₂²	semidirect
33. #11710602	C₂²	semidirect
34. C₄×#5950	C₂²	semidirect
35. C₂×C₄×#997	C₂²	semidirect
36. C₄×H(7,𝔽₂)	D₃	semidirect
37. #87547	C₂×C₄	semidirect
40. #87522	C₂×C₄	semidirect
41. #87520	C₂×C₄	semidirect
42. #87516	C₂×C₄	semidirect
43. C₂×#5935	C₂×C₄	semidirect
54. #62654	C₂×D₃	semidirect
55. #62652	C₂×D₃	semidirect
56. C₄×#585	C₂×D₃	semidirect
57. C₂×C₄×H(5,𝔽₂)	C₂×D₃	semidirect
58. #5950	C₂²×C₄	semidirect
59. C₂×#997	C₂²×C₄	semidirect
61. #5935	C₂²×C₄	semidirect
62. #5935	C₂²×C₄	semidirect
63. #5933	C₂²×C₄	semidirect
64. H(7,𝔽₂)	C₄×D₃	semidirect
67. C₄×2₋¹⁺⁴	S₄	semidirect
73. #585	C₂×C₄×D₃	semidirect
75. C₂×H(5,𝔽₂)	C₂×C₄×D₃	semidirect
79. #520	C₂×S₄	semidirect
80. #519	C₂×S₄	semidirect
81. C₄×#41	C₂×S₄	semidirect
82. C₂×C₄×Q₈	C₂×S₄	semidirect
87. 2₋¹⁺⁴	C₄×S₄	semidirect
102. #41	C₂×C₄×S₄	semidirect
103. C₂×Q₈	C₂×C₄×S₄	semidirect
122. C₄	#11715828	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#87522	2
#72341268	2
#72341264	2
C₄×#60388	3
C₄×#87516	2
#72227857	2
C₄×#87547	2
#72227324	2
C₄×#87520	2
#72226121	2
C₂×C₄×#5935	2
C₂×#11715828	2
C₄×#5926	4
C₄×D₄×S₄	4

Subgroups — C4×#11715828 #72341272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11715828 #72341272