Subgroups — #72240179

Order 2048.

At least 7842 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

The subgroup lattice has 260 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72240178	C₂	1	1	yes	maximal, characteristic, complemented
2	#72240177	C₂	1	1	yes	maximal, characteristic, complemented
3	#72237045	C₂	1	1	yes	maximal, characteristic, complemented
4	#72233904	C₂	1	1	yes	maximal, characteristic, complemented
5	#72233895	C₂	1	1	yes	maximal, characteristic, complemented
6	#72233800	C₂	1	1	yes	maximal, characteristic, complemented
7	#72233098	C₂	1	1	yes	maximal, characteristic, complemented
8	#72233093	C₂	1	1	yes	maximal, characteristic, complemented
9	#72233007	C₂	1	1	yes	maximal, characteristic, complemented
10	#72233000	C₂	1	1	yes	maximal, characteristic, complemented
11	#72219038	C₂	1	1	yes	maximal, characteristic, complemented
12	#72213293	C₂	1	1	yes	maximal, characteristic, complemented
13	#72213274	C₂	1	1	yes	maximal, characteristic, complemented
14	#72210479	C₂	1	1	yes	maximal, characteristic, complemented
15	#72210462	C₂	1	1	yes	maximal, characteristic, complemented
16	#7642244	C₂²	1	1	yes	characteristic
17	#6470225	C₂²	1	1	yes	characteristic
18	#6437445	C₂²	1	1	yes	characteristic
19	#4894596	C₂²	1	1	yes	characteristic
20	#4312024	C₂²	1	1	yes	characteristic
21	#635984	C₂²	1	1	yes	characteristic
22	#616961	C₂²	1	1	yes	characteristic
23	#604823	C₂²	1	1	yes	characteristic
24	#581603	C₂²	1	1	yes	characteristic
25	#515285	C₂²	1	1	yes	characteristic
26	#496225	C₂²	1	1	yes	characteristic
27	#496225	C₂²	1	1	yes	characteristic
28	#493370	C₂²	1	1	yes	characteristic
29	#493370	C₂²	1	1	yes	characteristic
30	#382422	C₂²	1	1	yes	characteristic
31	#359509	C₂²	1	1	yes	characteristic
32	#201664	C₂²	1	1	yes	characteristic
33	C₂×#8547	C₂²	1	1	yes	characteristic
34	C₂×#8541	C₂²	1	1	yes	characteristic
35	#187928	C₂²	1	1	yes	characteristic
36	#187928	C₂²	1	1	yes	characteristic
37	#187928	C₂²	1	1	yes	characteristic
38	#186835	C₂²	1	1	yes	characteristic
39	#128073	C₂²	1	1	yes	characteristic
40	#128073	C₂²	1	1	yes	characteristic
41	#128073	C₂²	1	1	yes	characteristic
42	#128073	C₂²	1	1	yes	characteristic
43	#128073	C₂²	1	1	yes	characteristic
44	#128073	C₂²	1	1	yes	characteristic
45	#128009	C₂²	1	1	yes	characteristic
46	#128009	C₂²	1	1	yes	characteristic
47	#108238	C₂²	1	1	yes	characteristic
48	#106766	C₂²	1	1	yes	characteristic
49	#106766	C₂²	1	1	yes	characteristic
50	#106766	C₂²	1	1	yes	characteristic
51	C₂×#3484	C₂³	1	1	yes	characteristic
52	C₂²×#535	C₂³	1	1	yes	characteristic
53	C₂³×#120	D₄	1	1	yes	characteristic
54	C₂³×#120	C₂³	1	1	yes	characteristic
55	C₂²×#518	D₄	1	1	yes	characteristic, complemented
56	C₂×#2404	D₄	1	1	yes	characteristic, complemented
57	C₂×#2404	D₄	1	1	yes	characteristic, complemented
58	C₂×#2405	D₄	1	1	yes	characteristic
59	C₂×#2403	C₂³	1	1	yes	characteristic
60	#34171	C₂³	1	1	yes	characteristic
61	C₂×#2308	D₄	1	1	yes	characteristic, complemented
62	C₂×#1783	C₂³	1	1	yes	characteristic
63	C₂×#1783	C₂³	1	1	yes	characteristic
64	C₂×#1781	C₂³	1	1	yes	characteristic
65	C₂×#1781	C₂³	1	1	yes	characteristic
66	C₂×#1781	C₂³	1	1	yes	characteristic
67	C₂×#1781	C₂³	1	1	yes	characteristic
68	#17273	D₄	1	1	yes	characteristic, complemented
69	#17272	D₄	1	1	yes	characteristic, complemented
70	#17137	D₄	1	1	yes	characteristic, complemented
71	#17136	D₄	1	1	yes	characteristic, complemented
72	#15784	D₄	1	1	yes	characteristic, complemented
73	#15313	D₄	1	1	yes	characteristic, complemented
74	#15259	D₄	1	1	yes	characteristic, complemented
75	#15258	D₄	1	1	yes	characteristic, complemented
76	#14679	D₄	1	1	yes	characteristic
77	#14666	D₄	1	1	yes	characteristic, complemented
78	#14662	D₄	1	1	yes	characteristic, complemented
79	#14516	D₄	1	1	yes	characteristic, complemented
80	#14419	D₄	1	1	yes	characteristic, complemented
81	#14419	D₄	1	1	yes	characteristic, complemented
82	#14358	D₄	1	1	yes	characteristic, complemented
83	#14335	C₂³	1	1	yes	characteristic
84	#14169	D₄	1	1	yes	characteristic, complemented
85	#14134	D₄	1	1	yes	characteristic, complemented
86	#13869	D₄	1	1	yes	characteristic
87	#13868	D₄	1	1	yes	characteristic, complemented
88	#13868	D₄	1	1	yes	characteristic, complemented
89	#13867	D₄	1	1	yes	characteristic, complemented
90	#13867	D₄	1	1	yes	characteristic, complemented
91	#13867	C₂³	1	1	yes	characteristic
92	#13867	C₂³	1	1	yes	characteristic
93	#13867	C₂³	1	1	yes	characteristic
94	C₂⁷	C₂×D₄	1	1	yes	characteristic
95	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic, complemented
96	C₂³×#41	C₂×D₄	1	1	yes	characteristic
97	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
98	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
99	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
100	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
101	C₂²×#127	C₂×D₄	1	1	yes	characteristic
102	C₂²×#127	C₂×D₄	1	1	yes	characteristic
103	C₂²×#124	C₂×D₄	1	1	yes	characteristic
104	C₂²×#124	C₂×D₄	1	1	yes	characteristic
105	C₂²×#120	C₂×D₄	1	1	yes	characteristic
106	C₂²×#120	C₂×D₄	1	1	yes	characteristic
107	C₂×#518	#41	1	1	yes	characteristic, complemented
108	C₂²×#117	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
111	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
115	C₂²×#99	#41	1	2	yes	complemented
116	#2309	#41	1	2	yes	complemented
117	C₂×#386	C₂×D₄	1	1	yes	characteristic
118	C₂×#386	C₂×D₄	1	1	yes	characteristic
119	C₂×#381	C₂×D₄	1	1	yes	characteristic
120	C₂×#381	C₂×D₄	1	1	yes	characteristic
121	C₂×#379	C₂×D₄	1	1	yes	characteristic
122	C₂×#379	C₂×D₄	1	1	yes	characteristic
123	C₂×#379	C₂×D₄	1	1	yes	characteristic
124	C₂×#379	C₂×D₄	1	1	yes	characteristic
125	#1783	#41	1	2	yes	complemented
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	#1438	C₂×D₄	1	1	yes	characteristic
130	#1438	C₂×D₄	1	1	yes	characteristic
131	#1438	C₂×D₄	1	1	yes	characteristic
132	#1438	C₂×D₄	1	1	yes	characteristic
133	#1438	C₂×D₄	1	1	yes	characteristic
134	#1438	C₂×D₄	1	1	yes	characteristic
135	#1438	C₂×D₄	1	1	yes	characteristic
136	#1438	C₂×D₄	1	1	yes	characteristic
137	#1438	C₂×D₄	1	1	yes	characteristic
138	#1438	C₂×D₄	1	1	yes	characteristic
139	#1438	C₂×D₄	1	1	yes	characteristic
140	#1438	C₂×D₄	1	1	yes	characteristic
141	#1438	C₂×D₄	1	1	yes	characteristic
142	C₂⁶	#120	1	1	yes	characteristic
143	C₂⁶	#120	1	1	yes	characteristic
144	C₂⁶	#120	1	1	yes	characteristic
145	C₂⁶	C₂²×D₄	1	1	yes	characteristic
146	C₂⁶	C₂²×D₄	1	1	yes	characteristic
147	C₂³×Q₈	#120	1	1	yes	characteristic
148	C₂³×D₄	#120	1	1	yes	characteristic
149	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
150	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
151	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
152	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
153	C₂³×D₄	C₂×#41	1	1	yes	characteristic
154	C₂³×D₄	C₂×#41	1	1	yes	characteristic
155	C₂³×D₄	C₂×#41	1	1	yes	characteristic
156	C₂³×D₄	C₂×#41	1	1	yes	characteristic
157	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂³×D₄	2₋¹⁺⁴	1	1	yes	characteristic
161	C₂⁴×C₄	#120	1	1	yes	characteristic
162	C₂⁴×C₄	#120	1	1	yes	characteristic
163	C₂⁴×C₄	#120	1	1	yes	characteristic
164	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
165	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
166	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
167	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
168	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
169	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
170	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
171	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
172	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
173	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
174	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
175	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
176	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
177	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
179	C₂²×C₄²	#120	1	1	yes	characteristic
180	C₂²×C₄²	#120	1	1	yes	characteristic
181	C₂×#95	#120	1	1	yes	characteristic
182	C₂×#95	#120	1	1	yes	characteristic
183	C₂×#95	#120	1	1	yes	characteristic
184	C₂×#95	#120	1	1	yes	characteristic
185	C₂⁵	B(4,𝔽₂)	1	2	yes	—
186	C₂⁵	C₂×#120	1	1	yes	characteristic
187	C₂⁵	C₂×#120	1	1	yes	characteristic
188	C₂⁵	C₂×#120	1	1	yes	characteristic
189	C₂⁵	C₂×#120	1	1	yes	characteristic
190	C₂⁵	C₂×#120	1	1	yes	characteristic
191	C₂⁵	#525	1	1	yes	characteristic
192	C₂⁵	#532	1	1	yes	characteristic
193	C₂⁵	#535	1	1	yes	characteristic
194	C₂⁵	#537	1	1	yes	characteristic
195	C₂⁵	#546	1	1	yes	characteristic
196	C₂⁵	#546	1	1	yes	characteristic
197	C₂⁵	#546	1	1	yes	characteristic
198	C₂⁵	#547	1	1	yes	characteristic
199	C₂²×D₄	#525	1	1	yes	characteristic
200	C₂²×D₄	#525	1	1	yes	characteristic
201	C₂²×D₄	#532	1	1	yes	characteristic
202	C₂²×D₄	#532	1	1	yes	characteristic
203	C₂²×D₄	#535	1	1	yes	characteristic
204	C₂²×D₄	#535	1	1	yes	characteristic
205	C₂²×D₄	#537	1	1	yes	characteristic
206	C₂²×D₄	#537	1	1	yes	characteristic
207	C₂²×D₄	#546	1	1	yes	characteristic
208	C₂²×D₄	#546	1	1	yes	characteristic
209	C₂²×D₄	#546	1	1	yes	characteristic
210	C₂²×D₄	#547	1	1	yes	characteristic
211	C₂²×D₄	#548	1	1	yes	characteristic
212	C₂²×D₄	#550	1	1	yes	characteristic
213	C₂²×D₄	#550	1	1	yes	characteristic
214	C₂²×D₄	#550	1	1	yes	characteristic
215	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
216	C₂³×C₄	C₂×#120	1	1	yes	characteristic
217	C₂³×C₄	C₂×#120	1	1	yes	characteristic
218	C₂³×C₄	#546	1	1	yes	characteristic
219	C₂³×C₄	#546	1	1	yes	characteristic
220	C₂³×C₄	#546	1	1	yes	characteristic
221	C₂³×C₄	#547	1	1	yes	characteristic
222	C₂⁴	#2388	1	1	yes	characteristic
223	C₂⁴	#2404	1	1	yes	characteristic
224	C₂⁴	#2404	1	1	yes	characteristic
225	C₂⁴	#2407	1	1	yes	characteristic
226	C₂⁴	#2407	1	1	yes	characteristic
227	C₂⁴	#2430	1	1	yes	characteristic
228	C₂⁴	#2430	1	1	yes	characteristic
229	C₂⁴	#2431	1	1	yes	characteristic
230	C₂⁴	#2431	1	1	yes	characteristic
231	C₂⁴	#2615	1	1	yes	characteristic
232	C₂⁴	#2846	1	1	yes	characteristic
233	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
234	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
235	C₂⁴	#3026	1	1	yes	characteristic
236	C₂⁴	#3026	1	1	yes	characteristic
237	C₂²×C₄	#2404	1	1	yes	characteristic
238	C₂²×C₄	#2407	1	1	yes	characteristic
239	C₂²×C₄	#2430	1	1	yes	characteristic
240	C₂²×C₄	#2431	1	1	yes	characteristic
241	C₂³	#14885	1	1	yes	characteristic
242	C₂³	#23984	1	1	yes	characteristic
243	C₂³	#23987	1	1	yes	characteristic
244	C₂³	#23996	1	2	yes	—
245	C₂³	#25563	1	1	yes	characteristic
246	C₂³	#25563	1	1	yes	characteristic
247	C₂³	#32898	1	1	yes	characteristic
248	C₂³	#32898	1	1	yes	characteristic
249	C₂²	#403943	1	1	yes	characteristic
250	C₂²	#403952	1	1	yes	characteristic
251	C₂²	#404015	1	1	yes	characteristic
252	C₂²	#511790	1	1	yes	characteristic
253	C₂²	#511790	1	1	yes	characteristic
254	C₂²	#511791	1	1	yes	characteristic
255	C₂²	#511793	1	1	yes	characteristic
256	C₂	#72234958	1	1	yes	characteristic
257	C₂	#72240105	1	1	yes	characteristic
258	C₂	#72240122	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#403952
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72240178	C₂	semidirect
2. #72240177	C₂	semidirect
3. #72237045	C₂	semidirect
4. #72233904	C₂	semidirect
5. #72233895	C₂	semidirect
6. #72233800	C₂	semidirect
7. #72233098	C₂	semidirect
8. #72233093	C₂	semidirect
9. #72233007	C₂	semidirect
10. #72233000	C₂	semidirect
11. #72219038	C₂	semidirect
12. #72213293	C₂	semidirect
13. #72213274	C₂	semidirect
14. #72210479	C₂	semidirect
15. #72210462	C₂	semidirect
55. C₂²×#518	D₄	semidirect
56. C₂×#2404	D₄	semidirect
57. C₂×#2404	D₄	semidirect
61. C₂×#2308	D₄	semidirect
68. #17273	D₄	semidirect
69. #17272	D₄	semidirect
70. #17137	D₄	semidirect
71. #17136	D₄	semidirect
72. #15784	D₄	semidirect
73. #15313	D₄	semidirect
74. #15259	D₄	semidirect
75. #15258	D₄	semidirect
77. #14666	D₄	semidirect
78. #14662	D₄	semidirect
79. #14516	D₄	semidirect
80. #14419	D₄	semidirect
81. #14419	D₄	semidirect
82. #14358	D₄	semidirect
84. #14169	D₄	semidirect
85. #14134	D₄	semidirect
87. #13868	D₄	semidirect
88. #13868	D₄	semidirect
89. #13867	D₄	semidirect
90. #13867	D₄	semidirect
95. C₂²×H(5,𝔽₂)	#41	semidirect
107. C₂×#518	#41	semidirect
115. C₂²×#99	#41	semidirect
116. #2309	#41	semidirect
125. #1783	#41	semidirect

Maximal subgroups

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

subgroup	index
#72240178	2
#72240177	2
#72237045	2
#72233904	2
#72233895	2
#72233800	2
#72233098	2
#72233093	2
#72233007	2
#72233000	2
#72219038	2
#72213293	2
#72213274	2
#72210479	2
#72210462	2