Subgroups — #72240913

Order 6144.

Only the 176 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 104 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72240896	C₂	1	1	yes	maximal, characteristic, complemented
2	#72240892	C₂	1	1	yes	maximal, characteristic, complemented
3	#72240887	C₂	1	4	yes	maximal, complemented
4	#72240885	C₂	1	1	yes	maximal, characteristic, complemented
5	#72236524	C₂	1	1	yes	maximal, characteristic, complemented
6	#72236409	C₂²	1	2	yes	complemented
7	#72236339	C₂²	1	1	yes	characteristic, complemented
8	C₂²×#87588⋊C₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#72235002	C₂	1	4	yes	maximal, complemented
10	#72234981	C₂²	1	4	yes	complemented
11	#72234980	C₂²	1	4	yes	complemented
12	#72233785	C₂²	1	2	yes	complemented
13	#72233211	C₂	1	1	yes	maximal, characteristic, complemented
14	#72233210	C₂²	1	4	yes	complemented
15	#72233210	C₂²	1	2	yes	complemented
16	#72233208	C₂²	1	1	yes	characteristic, complemented
17	C₂×#72215837	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#11710729	C₂²	1	1	yes	characteristic, complemented
19	C₂²×#85645	C₂²	1	1	yes	characteristic, complemented
20	C₂²×#87588	C₂²	1	2	yes	complemented
21	C₂×#87588⋊C₂	C₂²	1	1	yes	characteristic
22	C₂×#87588⋊C₂	C₂²	1	2	yes	complemented
23	C₂×#87588⋊C₂	C₂²	1	4	yes	complemented
24	C₂×#87588⋊C₂	C₂²	1	1	yes	characteristic, complemented
25	C₂×#87588⋊C₂	C₂²	1	1	yes	characteristic, complemented
26	#72217511	D₃	1	1	yes	characteristic, complemented
27	#72209712	C₂²	1	2	yes	complemented
28	C₂×#87588	C₂³	1	2	yes	—
29	C₂×#87588	C₂³	1	2	yes	complemented
30	C₂×#87588	C₂³	1	2	yes	complemented
31	#11715843	C₂³	1	2	yes	complemented
32	#87588⋊C₂	D₄	1	4	yes	complemented
33	C₂²×C₂²⋊(C₂²⋊A₄)	C₂³	1	1	yes	characteristic, complemented
34	C₂×#85645	C₂³	1	1	yes	characteristic
35	C₂×#85645	C₂³	1	2	yes	complemented
36	C₂×#85645	C₂³	1	1	yes	characteristic, complemented
37	#11709759	C₂³	1	2	yes	complemented
38	C₂×#62817	C₂×D₃	1	1	yes	characteristic, complemented
39	C₂⁴×#120	C₂×D₃	1	1	yes	characteristic, complemented
40	#7641845	C₂×D₃	1	2	yes	complemented
41	C₂²×#2846	C₂×D₃	1	2	yes	complemented
42	#7626202	C₂×D₃	1	1	yes	characteristic, complemented
43	C₂×C₂²⋊(C₂²⋊A₄)	C₂⁴	1	1	yes	characteristic
44	#87588	C₂×D₄	1	4	yes	complemented
45	#85645	C₂×D₄	1	2	yes	complemented
46	C₂⁸	C₂²×D₃	1	1	yes	characteristic, complemented
47	C₂³×#120	C₂²×D₃	1	1	yes	characteristic
48	C₂³×#120	C₂²×D₃	1	1	yes	characteristic, complemented
49	C₂³×#120	C₂²×D₃	1	2	yes	complemented
50	C₂×#2846	S₄	1	1	yes	characteristic, complemented
51	D₄×#120	S₄	1	2	yes	complemented
52	C₂²×#379	C₂²×D₃	1	2	yes	complemented
53	C₂²⋊(C₂²⋊A₄)	C₂²×D₄	1	1	yes	characteristic, complemented
54	C₂⁷	C₂×S₄	1	1	yes	characteristic, complemented
55	C₂⁷	C₂³×D₃	1	1	yes	characteristic
56	C₂⁴×D₄	C₂×S₄	1	2	yes	complemented
57	C₂²×#120	D₃×D₄	1	2	yes	complemented
58	C₂²×#120	C₂×S₄	1	2	yes	complemented
59	C₂²×#120	C₂×S₄	1	1	yes	characteristic, complemented
60	C₂²×#120	C₂×S₄	1	2	yes	complemented
61	#2847	C₂×S₄	1	2	yes	complemented
62	#2846	C₂×S₄	1	2	yes	complemented
63	#2846	C₂×S₄	1	2	yes	complemented
64	#2846	C₂×S₄	1	2	yes	complemented
65	#2846	C₂×S₄	1	2	yes	complemented
66	C₄×#120	C₂×S₄	1	2	yes	complemented
67	C₂×#379	C₂×S₄	1	1	yes	characteristic, complemented
68	C₂⁶	C₂×D₃×D₄	1	1	yes	characteristic, complemented
69	C₂⁶	C₂²×S₄	1	1	yes	characteristic
70	C₂⁶	C₂²×S₄	1	2	yes	—
71	C₂⁶	C₂²×S₄	1	1	yes	characteristic
72	C₂⁶	C₂²×S₄	1	1	yes	characteristic
73	C₂⁶	C₂²×S₄	1	2	yes	complemented
74	C₂³×D₄	#1020	1	1	yes	characteristic, complemented
75	C₂⁴×C₄	C₂²×S₄	1	2	yes	complemented
76	C₂×#120	C₂²×S₄	1	2	yes	—
77	C₂×#120	C₂²×S₄	1	2	yes	—
78	C₂×#120	C₂²×S₄	1	2	yes	—
79	C₂×#120	C₂²×S₄	1	2	yes	complemented
80	#379	C₂²×S₄	1	2	yes	—
81	#379	C₂²×S₄	1	2	yes	complemented
82	C₂⁵	D₄×S₄	1	2	yes	complemented
83	C₂⁵	C₂³×S₄	1	2	yes	—
84	C₂⁵	C₂³×S₄	1	1	yes	characteristic
85	C₂⁵	C₂×#1020	1	1	yes	characteristic
86	C₂⁵	C₂×#1020	1	1	yes	characteristic, complemented
87	C₂²×D₄	C₂×#1020	1	2	yes	—
88	C₂²×D₄	C₂×#1020	1	2	yes	complemented
89	C₂³×C₄	C₂×#1020	1	1	yes	characteristic, complemented
90	#120	D₄×S₄	1	4	yes	complemented
91	C₂⁴	C₂×D₄×S₄	1	2	yes	—
92	C₂⁴	C₂×D₄×S₄	1	1	yes	characteristic
93	C₂⁴	C₂²×#1020	1	1	yes	characteristic
94	C₂⁴	C₂²×#1020	1	1	yes	characteristic
95	C₂⁴	C₂²×#1020	1	2	yes	—
96	C₂⁴	C₂²×#1020	1	1	yes	characteristic
97	C₂²×C₄	C₂²×#1020	1	2	yes	—
98	C₂³	D₄×#1020	1	2	yes	—
99	C₂³	C₂³×#1020	1	1	yes	characteristic
100	C₂²	C₂×D₄×#1020	1	1	yes	characteristic
101	C₂²	#72215837	1	1	yes	characteristic, complemented
102	C₂	C₂×#72215837	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#72215837
commutator, cosocle	C₂×C₂²⋊(C₂²⋊A₄)	C₂⁴
Frattini, socle	C₂³	C₂³×#1020
Fitting	#72217511	D₃
hypercenter	C₂²	#72215837
hypocenter	C₂²⋊(C₂²⋊A₄)	C₂²×D₄
LCS	G ▷ C₂×C₂²⋊(C₂²⋊A₄) ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. #72240896	C₂	semidirect
2. #72240892	C₂	semidirect
3. #72240887	C₂	semidirect
4. #72240885	C₂	semidirect
5. #72236524	C₂	semidirect
6. #72236409	C₂²	semidirect
7. #72236339	C₂²	semidirect
8. C₂²×#87588⋊C₂	C₂	semidirect
9. #72235002	C₂	semidirect
10. #72234981	C₂²	semidirect
11. #72234980	C₂²	semidirect
12. #72233785	C₂²	semidirect
13. #72233211	C₂	semidirect
14. #72233210	C₂²	semidirect
15. #72233210	C₂²	semidirect
16. #72233208	C₂²	semidirect
17. C₂×#72215837	C₂	semidirect
18. C₂×#11710729	C₂²	semidirect
19. C₂²×#85645	C₂²	semidirect
20. C₂²×#87588	C₂²	semidirect
22. C₂×#87588⋊C₂	C₂²	semidirect
23. C₂×#87588⋊C₂	C₂²	semidirect
24. C₂×#87588⋊C₂	C₂²	semidirect
25. C₂×#87588⋊C₂	C₂²	semidirect
26. #72217511	D₃	semidirect
27. #72209712	C₂²	semidirect
29. C₂×#87588	C₂³	semidirect
30. C₂×#87588	C₂³	semidirect
31. #11715843	C₂³	semidirect
32. #87588⋊C₂	D₄	semidirect
33. C₂²×C₂²⋊(C₂²⋊A₄)	C₂³	semidirect
35. C₂×#85645	C₂³	semidirect
36. C₂×#85645	C₂³	semidirect
37. #11709759	C₂³	semidirect
38. C₂×#62817	C₂×D₃	semidirect
39. C₂⁴×#120	C₂×D₃	semidirect
40. #7641845	C₂×D₃	semidirect
41. C₂²×#2846	C₂×D₃	semidirect
42. #7626202	C₂×D₃	semidirect
44. #87588	C₂×D₄	semidirect
45. #85645	C₂×D₄	semidirect
46. C₂⁸	C₂²×D₃	semidirect
48. C₂³×#120	C₂²×D₃	semidirect
49. C₂³×#120	C₂²×D₃	semidirect
50. C₂×#2846	S₄	semidirect
51. D₄×#120	S₄	semidirect
52. C₂²×#379	C₂²×D₃	semidirect
53. C₂²⋊(C₂²⋊A₄)	C₂²×D₄	semidirect
54. C₂⁷	C₂×S₄	semidirect
56. C₂⁴×D₄	C₂×S₄	semidirect
57. C₂²×#120	D₃×D₄	semidirect
58. C₂²×#120	C₂×S₄	semidirect
59. C₂²×#120	C₂×S₄	semidirect
60. C₂²×#120	C₂×S₄	semidirect
61. #2847	C₂×S₄	semidirect
62. #2846	C₂×S₄	semidirect
63. #2846	C₂×S₄	semidirect
64. #2846	C₂×S₄	semidirect
65. #2846	C₂×S₄	semidirect
66. C₄×#120	C₂×S₄	semidirect
67. C₂×#379	C₂×S₄	semidirect
68. C₂⁶	C₂×D₃×D₄	semidirect
73. C₂⁶	C₂²×S₄	semidirect
74. C₂³×D₄	#1020	semidirect
75. C₂⁴×C₄	C₂²×S₄	semidirect
79. C₂×#120	C₂²×S₄	semidirect
81. #379	C₂²×S₄	semidirect
82. C₂⁵	D₄×S₄	semidirect
86. C₂⁵	C₂×#1020	semidirect
88. C₂²×D₄	C₂×#1020	semidirect
89. C₂³×C₄	C₂×#1020	semidirect
90. #120	D₄×S₄	semidirect
101. C₂²	#72215837	semidirect

Maximal subgroups

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

subgroup	index
#72240896	2
#72240892	2
#72240887	2
#72240885	2
#72236524	2
C₂²×#87588⋊C₂	2
#72235002	2
#72233211	2
C₂×#72215837	2
#72225648	3
D₄×#1020⋊C₂	4
C₂×#11713200	4