Subgroups — #72261728

Order 2048.

At least 6843 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

The subgroup lattice has 203 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72261723	C₂	1	1	yes	maximal, characteristic, complemented
2	#72261653	C₂	1	1	yes	maximal, characteristic, complemented
3	#72261651	C₂	1	1	yes	maximal, characteristic, complemented
4	#72233402	C₂	1	1	yes	maximal, characteristic, complemented
5	#72233391	C₂	1	1	yes	maximal, characteristic, complemented
6	#72233146	C₂	1	1	yes	maximal, characteristic, complemented
7	#72233081	C₂	1	1	yes	maximal, characteristic, complemented
8	#72233076	C₂	1	1	yes	maximal, characteristic, complemented
9	#72233019	C₂	1	1	yes	maximal, characteristic, complemented
10	#72233019	C₂	1	1	yes	maximal, characteristic, complemented
11	#72232982	C₂	1	1	yes	maximal, characteristic, complemented
12	#72232748	C₂	1	1	yes	maximal, characteristic, complemented
13	#72232745	C₂	1	1	yes	maximal, characteristic, complemented
14	#72226063	C₂	1	1	yes	maximal, characteristic, complemented
15	#72226063	C₂	1	1	yes	maximal, characteristic, complemented
16	#6516837	C₂²	1	1	yes	characteristic
17	#4929182	C₂²	1	1	yes	characteristic
18	#3056889	C₂²	1	1	yes	characteristic
19	#2306250	C₂²	1	1	yes	characteristic
20	#2306225	C₂²	1	1	yes	characteristic
21	#1434728	C₂²	1	1	yes	characteristic
22	#1418363	C₂²	1	1	yes	characteristic
23	#1418363	C₂²	1	1	yes	characteristic
24	#347043	C₂²	1	1	yes	characteristic, complemented
25	#347043	C₂²	1	1	yes	characteristic, complemented
26	#346449	C₂²	1	1	yes	characteristic, complemented
27	#346449	C₂²	1	1	yes	characteristic, complemented
28	#186818	C₂²	1	1	yes	characteristic, complemented
29	#185276	C₂²	1	1	yes	characteristic, complemented
30	#139908	C₂²	1	1	yes	characteristic, complemented
31	#139908	C₂²	1	1	yes	characteristic, complemented
32	#139794	C₂²	1	1	yes	characteristic, complemented
33	#139794	C₂²	1	1	yes	characteristic, complemented
34	#139793	C₂²	1	1	yes	characteristic, complemented
35	#139793	C₂²	1	1	yes	characteristic, complemented
36	#139713	C₂²	1	1	yes	characteristic, complemented
37	#139713	C₂²	1	1	yes	characteristic, complemented
38	#139511	C₂²	1	1	yes	characteristic, complemented
39	#139511	C₂²	1	1	yes	characteristic, complemented
40	#139472	C₂²	1	1	yes	characteristic, complemented
41	#139472	C₂²	1	1	yes	characteristic, complemented
42	#135567	C₂²	1	1	yes	characteristic, complemented
43	#135518	C₂²	1	1	yes	characteristic, complemented
44	#135477	C₂²	1	1	yes	characteristic, complemented
45	#135477	C₂²	1	1	yes	characteristic, complemented
46	#134020	C₂²	1	1	yes	characteristic
47	#134020	C₂²	1	1	yes	characteristic
48	#119716	C₂²	1	1	yes	characteristic, complemented
49	#119539	C₂²	1	1	yes	characteristic, complemented
50	#115165	C₂²	1	1	yes	characteristic
51	#62818	C₂³	1	1	yes	characteristic
52	C₂²×D₄²	D₄	1	1	yes	characteristic, complemented
53	C₂²×D₄²	D₄	1	1	yes	characteristic, complemented
54	C₂×#2549	D₄	1	1	yes	characteristic, complemented
55	C₂×#2433	C₂³	1	1	yes	characteristic
56	#21793	C₂³	1	1	yes	characteristic
57	#21752	C₂³	1	1	yes	characteristic
58	#17396	C₂³	1	1	yes	characteristic
59	#17136	D₄	1	1	yes	characteristic, complemented
60	#17082	D₄	1	1	yes	characteristic, complemented
61	#15968	D₄	1	1	yes	characteristic, complemented
62	#15947	D₄	1	1	yes	characteristic, complemented
63	#15947	D₄	1	1	yes	characteristic, complemented
64	#15947	D₄	1	1	yes	characteristic, complemented
65	#15657	D₄	1	1	yes	characteristic, complemented
66	#15636	D₄	1	1	yes	characteristic, complemented
67	#15310	C₂³	1	1	yes	characteristic
68	#15310	C₂³	1	1	yes	characteristic
69	#15267	D₄	1	1	yes	characteristic, complemented
70	#15267	D₄	1	1	yes	characteristic, complemented
71	#15258	D₄	1	1	yes	characteristic, complemented
72	#15258	D₄	1	1	yes	characteristic, complemented
73	#15257	C₂³	1	1	yes	characteristic
74	#15257	C₂³	1	1	yes	characteristic
75	#14571	D₄	1	1	yes	characteristic, complemented
76	#14533	D₄	1	1	yes	characteristic, complemented
77	#14365	D₄	1	1	yes	characteristic, complemented
78	#14358	D₄	1	1	yes	characteristic, complemented
79	#14358	D₄	1	1	yes	characteristic, complemented
80	#14358	D₄	1	1	yes	characteristic, complemented
81	#14271	C₂³	1	1	yes	characteristic
82	#14207	D₄	1	1	yes	characteristic, complemented
83	#14181	D₄	1	1	yes	characteristic, complemented
84	#14152	C₂³	1	1	yes	characteristic
85	#14134	D₄	1	1	yes	characteristic, complemented
86	#13902	D₄	1	1	yes	characteristic, complemented
87	C₂×#1438	D₄	1	1	yes	characteristic, complemented
88	C₂²⋊C₄²	D₄	1	1	yes	characteristic, complemented
89	#13749	D₄	1	1	yes	characteristic, complemented
90	#8549	C₂³	1	1	yes	characteristic
91	#8549	C₂³	1	1	yes	characteristic
92	#8542	C₂³	1	1	yes	characteristic
93	#8542	C₂³	1	1	yes	characteristic
94	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×#127	C₂×D₄	1	1	yes	characteristic
96	C₂²×#127	C₂×D₄	1	1	yes	characteristic
97	C₂²×#121	C₂⁴	1	1	yes	characteristic
98	C₂²×#120	C₂×D₄	1	1	yes	characteristic
99	C₂²×#120	C₂×D₄	1	1	yes	characteristic
100	C₂²×#120	C₂×D₄	1	1	yes	characteristic
101	C₂²×#120	C₂×D₄	1	1	yes	characteristic
102	C₂²×#120	C₂×D₄	1	1	yes	characteristic
103	C₂²×#120	C₂×D₄	1	1	yes	characteristic
104	C₂²×#120	C₂×D₄	1	1	yes	characteristic
105	C₂²×#120	C₂×D₄	1	1	yes	characteristic
106	C₂×#394	C₂×D₄	1	1	yes	characteristic
107	C₂×#394	C₂×D₄	1	1	yes	characteristic
108	C₂×#392	C₂×D₄	1	1	yes	characteristic
109	C₂×#392	C₂×D₄	1	1	yes	characteristic
110	C₂×#380	C₂×D₄	1	1	yes	characteristic
111	C₂×#379	C₂×D₄	1	1	yes	characteristic
112	C₂×#379	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
115	C₂²×#95	C₂×D₄	1	1	yes	characteristic
116	C₂²×#95	C₂×D₄	1	1	yes	characteristic
117	#1445	C₂×D₄	1	1	yes	characteristic
118	#1445	C₂×D₄	1	1	yes	characteristic
119	#1440	C₂×D₄	1	1	yes	characteristic
120	#1440	C₂×D₄	1	1	yes	characteristic
121	#1438	C₂×D₄	1	1	yes	characteristic
122	#1438	C₂×D₄	1	1	yes	characteristic
123	#1438	C₂×D₄	1	1	yes	characteristic
124	#1438	C₂×D₄	1	1	yes	characteristic
125	#1438	C₂×D₄	1	1	yes	characteristic
126	#1438	C₂×D₄	1	1	yes	characteristic
127	#1438	C₂×D₄	1	1	yes	characteristic
128	#1438	C₂×D₄	1	1	yes	characteristic
129	#1437	C₂×D₄	1	1	yes	characteristic
130	#1437	C₂×D₄	1	1	yes	characteristic
131	#1437	C₂×D₄	1	1	yes	characteristic
132	#1437	C₂×D₄	1	1	yes	characteristic
133	#1437	C₂×D₄	1	1	yes	characteristic
134	#1437	C₂×D₄	1	1	yes	characteristic
135	#1436	C₂×D₄	1	1	yes	characteristic
136	#1436	C₂×D₄	1	1	yes	characteristic
137	C₂⁶	#120	1	1	yes	characteristic
138	C₂⁶	#120	1	1	yes	characteristic
139	C₂⁶	#120	1	1	yes	characteristic
140	C₂⁶	#120	1	1	yes	characteristic
141	C₂³×D₄	#120	1	1	yes	characteristic
142	C₂³×D₄	#120	1	1	yes	characteristic
143	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
144	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
145	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
146	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
147	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
148	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
149	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
150	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
151	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
152	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
154	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
155	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
156	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
157	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
158	C₂²×C₄²	#120	1	1	yes	characteristic
159	C₂²×C₄²	#120	1	1	yes	characteristic
160	C₂²×C₄²	#120	1	1	yes	characteristic
161	C₂²×C₄²	#120	1	1	yes	characteristic
162	C₂×#95	#120	1	1	yes	characteristic
163	C₂×#95	#120	1	1	yes	characteristic
164	C₂×#95	#120	1	1	yes	characteristic
165	C₂×#95	#120	1	1	yes	characteristic
166	C₂×#95	#120	1	1	yes	characteristic
167	C₂×#95	#120	1	1	yes	characteristic
168	C₂×#95	#120	1	1	yes	characteristic
169	C₂×#95	#120	1	1	yes	characteristic
170	C₂×#95	#120	1	1	yes	characteristic
171	C₂×#95	#120	1	1	yes	characteristic
172	C₂⁵	C₂×#120	1	1	yes	characteristic
173	C₂⁵	C₂×#120	1	1	yes	characteristic
174	C₂⁵	C₂×#120	1	1	yes	characteristic
175	C₂³×C₄	C₂×#120	1	1	yes	characteristic
176	C₂³×C₄	C₂×#120	1	1	yes	characteristic
177	C₂³×C₄	C₂×#120	1	1	yes	characteristic
178	C₂³×C₄	C₂×#120	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
180	C₂⁴	#2846	1	1	yes	characteristic
181	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
182	C₂⁴	#3026	1	1	yes	characteristic
183	C₂²×C₄	#3025	1	1	yes	characteristic
184	C₂²×C₄	#3025	1	1	yes	characteristic
185	C₂²×C₄	#3027	1	1	yes	characteristic
186	C₂²×C₄	#3027	1	1	yes	characteristic
187	C₂²⋊C₄	#2196	1	4	yes	—
188	C₂³	#32898	1	1	yes	characteristic
189	C₂³	#32899	1	1	yes	characteristic
190	C₂³	#32899	1	1	yes	characteristic
191	C₂³	C₂×#2196	1	2	yes	—
192	C₂²	#347045	1	1	yes	characteristic
193	C₂²	#511771	1	1	yes	characteristic
194	C₂²	#511812	1	1	yes	characteristic
195	C₂²	#511846	1	1	yes	characteristic
196	C₂²	#511849	1	1	yes	characteristic
197	C₂²	#511853	1	1	yes	characteristic
198	C₂²	#511853	1	1	yes	characteristic
199	C₂	#72255837	1	1	yes	characteristic
200	C₂	#72260719	1	1	yes	characteristic
201	C₂	#72261123	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#347045
commutator, Frattini, cosocle	C₂²×#121	C₂⁴
LCS	G ▷ C₂²×#121 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂²×#121 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72261723	C₂	semidirect
2. #72261653	C₂	semidirect
3. #72261651	C₂	semidirect
4. #72233402	C₂	semidirect
5. #72233391	C₂	semidirect
6. #72233146	C₂	semidirect
7. #72233081	C₂	semidirect
8. #72233076	C₂	semidirect
9. #72233019	C₂	semidirect
10. #72233019	C₂	semidirect
11. #72232982	C₂	semidirect
12. #72232748	C₂	semidirect
13. #72232745	C₂	semidirect
14. #72226063	C₂	semidirect
15. #72226063	C₂	semidirect
24. #347043	C₂²	semidirect
25. #347043	C₂²	semidirect
26. #346449	C₂²	semidirect
27. #346449	C₂²	semidirect
28. #186818	C₂²	semidirect
29. #185276	C₂²	semidirect
30. #139908	C₂²	semidirect
31. #139908	C₂²	semidirect
32. #139794	C₂²	semidirect
33. #139794	C₂²	semidirect
34. #139793	C₂²	semidirect
35. #139793	C₂²	semidirect
36. #139713	C₂²	semidirect
37. #139713	C₂²	semidirect
38. #139511	C₂²	semidirect
39. #139511	C₂²	semidirect
40. #139472	C₂²	semidirect
41. #139472	C₂²	semidirect
42. #135567	C₂²	semidirect
43. #135518	C₂²	semidirect
44. #135477	C₂²	semidirect
45. #135477	C₂²	semidirect
48. #119716	C₂²	semidirect
49. #119539	C₂²	semidirect
52. C₂²×D₄²	D₄	semidirect
53. C₂²×D₄²	D₄	semidirect
54. C₂×#2549	D₄	semidirect
59. #17136	D₄	semidirect
60. #17082	D₄	semidirect
61. #15968	D₄	semidirect
62. #15947	D₄	semidirect
63. #15947	D₄	semidirect
64. #15947	D₄	semidirect
65. #15657	D₄	semidirect
66. #15636	D₄	semidirect
69. #15267	D₄	semidirect
70. #15267	D₄	semidirect
71. #15258	D₄	semidirect
72. #15258	D₄	semidirect
75. #14571	D₄	semidirect
76. #14533	D₄	semidirect
77. #14365	D₄	semidirect
78. #14358	D₄	semidirect
79. #14358	D₄	semidirect
80. #14358	D₄	semidirect
82. #14207	D₄	semidirect
83. #14181	D₄	semidirect
85. #14134	D₄	semidirect
86. #13902	D₄	semidirect
87. C₂×#1438	D₄	semidirect
88. C₂²⋊C₄²	D₄	semidirect
89. #13749	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72261723	2
#72261653	2
#72261651	2
#72233402	2
#72233391	2
#72233146	2
#72233081	2
#72233076	2
#72233019	2
#72232982	2
#72232748	2
#72232745	2
#72226063	2