Subgroups — #72252031

Order 2048.

At least 9027 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

The subgroup lattice has 288 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251773	C₂	1	1	yes	maximal, characteristic, complemented
2	#72251725	C₂	1	1	yes	maximal, characteristic, complemented
3	#72237534	C₂	1	1	yes	maximal, characteristic, complemented
4	#72235082	C₂	1	1	yes	maximal, characteristic, complemented
5	#72232354	C₂	1	1	yes	maximal, characteristic, complemented
6	#72231752	C₂	1	1	yes	maximal, characteristic, complemented
7	#72231750	C₂	1	1	yes	maximal, characteristic, complemented
8	#72231748	C₂	1	1	yes	maximal, characteristic, complemented
9	#72231745	C₂	1	1	yes	maximal, characteristic, complemented
10	#72219501	C₂	1	1	yes	maximal, characteristic, complemented
11	#72219497	C₂	1	1	yes	maximal, characteristic, complemented
12	#72215545	C₂	1	1	yes	maximal, characteristic, complemented
13	#72215527	C₂	1	1	yes	maximal, characteristic, complemented
14	#72213659	C₂	1	1	yes	maximal, characteristic, complemented
15	#72210564	C₂	1	1	yes	maximal, characteristic, complemented
16	#10575442	C₂²	1	1	yes	characteristic
17	C₂×#46355	C₂²	1	1	yes	characteristic
18	C₂×#35222	C₂²	1	1	yes	characteristic
19	#6470228	C₂²	1	1	yes	characteristic, complemented
20	#6470025	C₂²	1	1	yes	characteristic, complemented
21	#6437616	C₂²	1	1	yes	characteristic, complemented
22	#6437449	C₂²	1	1	yes	characteristic, complemented
23	#4312017	C₂²	1	1	yes	characteristic
24	#4311734	C₂²	1	1	yes	characteristic
25	#850465	C₂²	1	1	yes	characteristic
26	#850278	C₂²	1	1	yes	characteristic
27	#576038	C₂²	1	1	yes	characteristic
28	#496228	C₂²	1	1	yes	characteristic, complemented
29	#496228	C₂²	1	1	yes	characteristic, complemented
30	#496208	C₂²	1	1	yes	characteristic, complemented
31	#496190	C₂²	1	1	yes	characteristic, complemented
32	#493385	C₂²	1	1	yes	characteristic, complemented
33	#493372	C₂²	1	1	yes	characteristic, complemented
34	#493352	C₂²	1	1	yes	characteristic, complemented
35	#493352	C₂²	1	1	yes	characteristic, complemented
36	#382419	C₂²	1	1	yes	characteristic
37	#359513	C₂²	1	1	yes	characteristic, complemented
38	#359507	C₂²	1	1	yes	characteristic, complemented
39	#201819	C₂²	1	1	yes	characteristic, complemented
40	#201091	C₂²	1	1	yes	characteristic, complemented
41	#187931	C₂²	1	1	yes	characteristic, complemented
42	#187756	C₂²	1	1	yes	characteristic
43	#187756	C₂²	1	1	yes	characteristic
44	#186814	C₂²	1	1	yes	characteristic, complemented
45	#128074	C₂²	1	1	yes	characteristic, complemented
46	#128074	C₂²	1	1	yes	characteristic, complemented
47	#128059	C₂²	1	1	yes	characteristic, complemented
48	#128057	C₂²	1	1	yes	characteristic, complemented
49	#106776	C₂²	1	1	yes	characteristic, complemented
50	#106740	C₂²	1	1	yes	characteristic, complemented
51	C₂⁵×D₄	D₄	1	1	yes	characteristic, complemented
52	#62811	C₂³	1	1	yes	characteristic
53	C₂²×#535	C₂³	1	1	yes	characteristic
54	C₂²×#535	C₂³	1	1	yes	characteristic
55	C₂²×#534	D₄	1	1	yes	characteristic, complemented
56	C₂²×#525	C₂³	1	1	yes	characteristic
57	#60518	D₄	1	2	yes	complemented
58	#60369	D₄	1	2	yes	complemented
59	#60064	C₂³	1	1	yes	characteristic
60	C₂×#2815	D₄	1	1	yes	characteristic, complemented
61	C₂×#2539	D₄	1	1	yes	characteristic, complemented
62	C₂×#2404	C₂³	1	1	yes	characteristic
63	C₂×#2431	C₂³	1	1	yes	characteristic
64	C₂×#2377	D₄	1	1	yes	characteristic, complemented
65	C₂×#2364	D₄	1	1	yes	characteristic, complemented
66	C₂×#2333	D₄	1	1	yes	characteristic, complemented
67	C₂×#2309	D₄	1	1	yes	characteristic, complemented
68	C₂×#2309	D₄	1	1	yes	characteristic, complemented
69	C₂×#2280	D₄	1	1	yes	characteristic, complemented
70	#20591	C₂³	1	1	yes	characteristic
71	#20591	C₂³	1	1	yes	characteristic
72	#20591	C₂³	1	1	yes	characteristic
73	#20584	C₂³	1	1	yes	characteristic
74	#20584	C₂³	1	1	yes	characteristic
75	#20584	C₂³	1	1	yes	characteristic
76	#16094	D₄	1	1	yes	characteristic, complemented
77	#16094	D₄	1	1	yes	characteristic, complemented
78	#15983	D₄	1	1	yes	characteristic, complemented
79	#15948	D₄	1	1	yes	characteristic
80	#15947	D₄	1	1	yes	characteristic
81	#15763	D₄	1	1	yes	characteristic, complemented
82	#15763	D₄	1	1	yes	characteristic, complemented
83	#15463	D₄	1	1	yes	characteristic
84	#15313	D₄	1	1	yes	characteristic, complemented
85	#15313	D₄	1	1	yes	characteristic, complemented
86	#15281	D₄	1	1	yes	characteristic, complemented
87	#15266	D₄	1	1	yes	characteristic
88	#15261	D₄	1	1	yes	characteristic, complemented
89	#15258	D₄	1	1	yes	characteristic, complemented
90	#15258	D₄	1	1	yes	characteristic, complemented
91	#15258	D₄	1	1	yes	characteristic, complemented
92	#14290	D₄	1	1	yes	characteristic, complemented
93	#14271	D₄	1	1	yes	characteristic, complemented
94	#13956	C₂³	1	1	yes	characteristic
95	#13937	C₂³	1	1	yes	characteristic
96	#10984	D₄	1	4	yes	complemented
97	#8516	D₄	1	4	yes	complemented
98	C₂×#585	C₂×D₄	1	1	yes	characteristic, complemented
99	C₂³×#41	C₂⁴	1	1	yes	characteristic
100	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
101	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
102	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×#535	C₂×D₄	1	2	yes	—
104	C₂×#535	C₂×D₄	1	2	yes	—
105	C₂×#535	C₂×D₄	1	2	yes	—
106	C₂×#535	#41	1	2	yes	—
107	C₂²×#127	C₂×D₄	1	1	yes	characteristic
108	C₂²×#127	C₂×D₄	1	1	yes	characteristic
109	C₂²×#127	C₂×D₄	1	1	yes	characteristic
110	C₂²×#124	C₂×D₄	1	1	yes	characteristic
111	C₂²×#124	C₂×D₄	1	1	yes	characteristic
112	C₂²×#124	C₂×D₄	1	1	yes	characteristic
113	C₂²×#123	C₂×D₄	1	1	yes	characteristic
114	C₂²×#123	C₂×D₄	1	1	yes	characteristic
115	C₂²×#121	C₂×D₄	1	1	yes	characteristic
116	C₂²×#121	C₂×D₄	1	1	yes	characteristic
117	C₂²×#121	C₂×D₄	1	1	yes	characteristic
118	C₂²×#121	C₂×D₄	1	1	yes	characteristic
119	C₂²×#121	C₂×D₄	1	1	yes	characteristic
120	C₂²×#120	C₂×D₄	1	1	yes	characteristic
121	C₂²×#120	C₂×D₄	1	1	yes	characteristic
122	C₂²×#120	C₂×D₄	1	1	yes	characteristic
123	C₂²×#120	C₂×D₄	1	1	yes	characteristic
124	#3428	C₂×D₄	1	1	yes	characteristic, complemented
125	C₂²×#117	C₂×D₄	1	1	yes	characteristic
126	C₂²×#117	C₂×D₄	1	1	yes	characteristic
127	C₂²×#117	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
129	C₂×#413	C₂×D₄	1	2	yes	complemented
130	C₂×#410	C₂×D₄	1	2	yes	complemented
131	#2879	C₂×D₄	1	2	yes	complemented
132	C₂×#391	C₂×D₄	1	1	yes	characteristic
133	C₂×#390	C₂×D₄	1	1	yes	characteristic
134	C₂×#386	C₂×D₄	1	1	yes	characteristic
135	C₂×#386	C₂×D₄	1	1	yes	characteristic
136	C₂×#381	C₂×D₄	1	1	yes	characteristic
137	C₂×#381	C₂×D₄	1	1	yes	characteristic
138	C₂×#380	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
141	C₂²×#95	C₂×D₄	1	1	yes	characteristic
142	#1785	C₂×D₄	1	2	yes	complemented
143	#1782	C₂×D₄	1	2	yes	complemented
144	#1446	C₂×D₄	1	1	yes	characteristic
145	#1446	C₂×D₄	1	1	yes	characteristic
146	#1438	C₂×D₄	1	1	yes	characteristic
147	#1438	C₂×D₄	1	1	yes	characteristic
148	#1438	C₂×D₄	1	1	yes	characteristic
149	#1438	C₂×D₄	1	1	yes	characteristic
150	#1438	C₂×D₄	1	1	yes	characteristic
151	#1438	C₂×D₄	1	1	yes	characteristic
152	C₂⁶	#120	1	1	yes	characteristic
153	C₂⁶	#120	1	1	yes	characteristic
154	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
155	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
156	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
157	C₂²×#41	C₂×#41	1	1	yes	characteristic
158	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic
163	C₂³×D₄	#120	1	2	yes	—
164	C₂³×D₄	#120	1	1	yes	characteristic
165	C₂³×D₄	#120	1	1	yes	characteristic
166	C₂³×D₄	#120	1	1	yes	characteristic
167	C₂³×D₄	#120	1	1	yes	characteristic
168	C₂³×D₄	#120	1	1	yes	characteristic
169	C₂³×D₄	#121	1	2	yes	—
170	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
171	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
172	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
173	C₂⁴×C₄	#120	1	1	yes	characteristic
174	C₂⁴×C₄	#120	1	1	yes	characteristic
175	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
176	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
177	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
178	C₂×#121	#120	1	2	yes	—
179	C₂×#121	#120	1	2	yes	—
180	C₂×#121	#121	1	2	yes	—
181	C₂×#121	#121	1	2	yes	—
182	C₂×#121	#121	1	2	yes	—
183	C₂×#121	#121	1	2	yes	—
184	C₂×#121	#127	1	2	yes	—
185	C₂×#121	#127	1	2	yes	—
186	C₂×#117	#120	1	2	yes	—
187	C₂×#117	#121	1	2	yes	—
188	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
189	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
190	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
191	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
192	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
193	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
194	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
195	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
196	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
197	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
198	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
199	C₂²×C₄²	#120	1	1	yes	characteristic
200	C₂²×C₄²	#120	1	1	yes	characteristic
201	C₂²×C₄²	#120	1	1	yes	characteristic
202	C₂²×C₄²	#120	1	1	yes	characteristic
203	C₂×#95	#120	1	1	yes	characteristic
204	C₂×#95	#120	1	1	yes	characteristic
205	C₂×#95	#120	1	1	yes	characteristic
206	C₂×#95	#120	1	1	yes	characteristic
207	C₂⁵	C₂×#120	1	1	yes	characteristic
208	C₂⁵	C₂×#120	1	1	yes	characteristic
209	C₂⁵	C₂×#120	1	1	yes	characteristic
210	C₂²×Q₈	D₄²	1	1	yes	characteristic
211	C₂²×Q₈	D₄²	1	1	yes	characteristic
212	C₂²×Q₈	D₄²	1	1	yes	characteristic
213	C₂²×Q₈	#546	1	1	yes	characteristic
214	C₂²×D₄	#392	1	2	yes	—
215	C₂²×D₄	#392	1	2	yes	—
216	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
217	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
218	C₂²×D₄	D₄²	1	1	yes	characteristic
219	C₂²×D₄	D₄²	1	1	yes	characteristic
220	C₂²×D₄	D₄²	1	1	yes	characteristic
221	C₂²×D₄	D₄²	1	1	yes	characteristic
222	C₂²×D₄	D₄²	1	1	yes	characteristic
223	C₂²×D₄	D₄²	1	1	yes	characteristic
224	C₂²×D₄	D₄²	1	1	yes	characteristic
225	C₂²×D₄	D₄²	1	1	yes	characteristic
226	C₂²×D₄	D₄²	1	1	yes	characteristic
227	C₂²×D₄	#546	1	1	yes	characteristic
228	C₂²×D₄	#546	1	1	yes	characteristic
229	C₂²×D₄	#546	1	1	yes	characteristic
230	C₂³×C₄	C₂×#120	1	1	yes	characteristic
231	C₂³×C₄	C₂×#120	1	1	yes	characteristic
232	C₂³×C₄	C₂×#120	1	1	yes	characteristic
233	C₂³×C₄	C₂×#120	1	1	yes	characteristic
234	C₂³×C₄	C₂×#120	1	1	yes	characteristic
235	C₂³×C₄	C₂×#120	1	1	yes	characteristic
236	C₂³×C₄	C₂×#121	1	1	yes	characteristic
237	C₂³×C₄	C₂×#121	1	1	yes	characteristic
238	C₂³×C₄	C₂×#121	1	1	yes	characteristic
239	C₂³×C₄	C₂×#127	1	1	yes	characteristic
240	C₂³×C₄	#534	1	1	yes	characteristic
241	C₂³×C₄	#534	1	1	yes	characteristic
242	C₂³×C₄	#534	1	1	yes	characteristic
243	C₂³×C₄	#534	1	1	yes	characteristic
244	C₂³×C₄	#535	1	1	yes	characteristic
245	C₂³×C₄	#535	1	1	yes	characteristic
246	C₂×C₄⋊C₄	#392	1	2	yes	—
247	C₂×C₄⋊C₄	#392	1	2	yes	—
248	C₂⁴	#2403	1	1	yes	characteristic
249	C₂⁴	#2403	1	1	yes	characteristic
250	C₂⁴	#2408	1	1	yes	characteristic
251	C₂⁴	#2410	1	1	yes	characteristic
252	C₂⁴	#2410	1	1	yes	characteristic
253	C₂⁴	#2433	1	1	yes	characteristic
254	C₂⁴	#2846	1	1	yes	characteristic
255	C₂×D₄	#2021	1	4	yes	—
256	C₂²×C₄	C₂×#392	1	1	yes	characteristic
257	C₂²×C₄	#2403	1	1	yes	characteristic
258	C₂²×C₄	#2403	1	1	yes	characteristic
259	C₂²×C₄	#2408	1	1	yes	characteristic
260	C₂²×C₄	#2410	1	1	yes	characteristic
261	C₂²×C₄	#2410	1	1	yes	characteristic
262	C₂²×C₄	#2433	1	1	yes	characteristic
263	C₂²×C₄	#2613	1	1	yes	characteristic
264	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
265	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
266	C₂²×C₄	#3026	1	1	yes	characteristic
267	C₂²×C₄	#3026	1	1	yes	characteristic
268	C₂³	#14715	1	1	yes	characteristic
269	C₂³	#23895	1	2	yes	—
270	C₂³	#32898	1	1	yes	characteristic
271	C₂³	#32898	1	1	yes	characteristic
272	C₂×C₄	C₂×#2021	1	1	yes	characteristic
273	C₂×C₄	#23444	1	1	yes	characteristic
274	C₂×C₄	#23895	1	2	yes	—
275	C₂×C₄	#29634	1	1	yes	characteristic
276	C₂×C₄	#29634	1	1	yes	characteristic
277	C₂²	#393929	1	1	yes	characteristic
278	C₂²	#394064	1	1	yes	characteristic
279	C₂²	#394064	1	1	yes	characteristic
280	C₂²	#511819	1	1	yes	characteristic
281	C₂²	#511819	1	1	yes	characteristic
282	C₂²	#511825	1	1	yes	characteristic
283	C₂²	#511827	1	1	yes	characteristic
284	C₂	#72214565	1	1	yes	characteristic
285	C₂	#72225610	1	1	yes	characteristic
286	C₂	#72238397	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#393929
commutator, Frattini, cosocle	C₂³×#41	C₂⁴
LCS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#535 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72251773	C₂	semidirect
2. #72251725	C₂	semidirect
3. #72237534	C₂	semidirect
4. #72235082	C₂	semidirect
5. #72232354	C₂	semidirect
6. #72231752	C₂	semidirect
7. #72231750	C₂	semidirect
8. #72231748	C₂	semidirect
9. #72231745	C₂	semidirect
10. #72219501	C₂	semidirect
11. #72219497	C₂	semidirect
12. #72215545	C₂	semidirect
13. #72215527	C₂	semidirect
14. #72213659	C₂	semidirect
15. #72210564	C₂	semidirect
19. #6470228	C₂²	semidirect
20. #6470025	C₂²	semidirect
21. #6437616	C₂²	semidirect
22. #6437449	C₂²	semidirect
28. #496228	C₂²	semidirect
29. #496228	C₂²	semidirect
30. #496208	C₂²	semidirect
31. #496190	C₂²	semidirect
32. #493385	C₂²	semidirect
33. #493372	C₂²	semidirect
34. #493352	C₂²	semidirect
35. #493352	C₂²	semidirect
37. #359513	C₂²	semidirect
38. #359507	C₂²	semidirect
39. #201819	C₂²	semidirect
40. #201091	C₂²	semidirect
41. #187931	C₂²	semidirect
44. #186814	C₂²	semidirect
45. #128074	C₂²	semidirect
46. #128074	C₂²	semidirect
47. #128059	C₂²	semidirect
48. #128057	C₂²	semidirect
49. #106776	C₂²	semidirect
50. #106740	C₂²	semidirect
51. C₂⁵×D₄	D₄	semidirect
55. C₂²×#534	D₄	semidirect
57. #60518	D₄	semidirect
58. #60369	D₄	semidirect
60. C₂×#2815	D₄	semidirect
61. C₂×#2539	D₄	semidirect
64. C₂×#2377	D₄	semidirect
65. C₂×#2364	D₄	semidirect
66. C₂×#2333	D₄	semidirect
67. C₂×#2309	D₄	semidirect
68. C₂×#2309	D₄	semidirect
69. C₂×#2280	D₄	semidirect
76. #16094	D₄	semidirect
77. #16094	D₄	semidirect
78. #15983	D₄	semidirect
81. #15763	D₄	semidirect
82. #15763	D₄	semidirect
84. #15313	D₄	semidirect
85. #15313	D₄	semidirect
86. #15281	D₄	semidirect
88. #15261	D₄	semidirect
89. #15258	D₄	semidirect
90. #15258	D₄	semidirect
91. #15258	D₄	semidirect
92. #14290	D₄	semidirect
93. #14271	D₄	semidirect
96. #10984	D₄	semidirect
97. #8516	D₄	semidirect
98. C₂×#585	C₂×D₄	semidirect
124. #3428	C₂×D₄	semidirect
129. C₂×#413	C₂×D₄	semidirect
130. C₂×#410	C₂×D₄	semidirect
131. #2879	C₂×D₄	semidirect
142. #1785	C₂×D₄	semidirect
143. #1782	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72251773	2
#72251725	2
#72237534	2
#72235082	2
#72232354	2
#72231752	2
#72231750	2
#72231748	2
#72231745	2
#72219501	2
#72219497	2
#72215545	2
#72215527	2
#72213659	2
#72210564	2