Subgroups — #72339051

Order 4096.

At least 6960 conjugacy classes of subgroups; only the 617 normal subgroups are shown.

The subgroup lattice has 243 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72339049	C₂	1	4	yes	maximal, complemented
2	#72339047	C₂	1	1	yes	maximal, characteristic, complemented
3	#72339046	C₂²	1	2	yes	complemented
4	#72339045	C₂	1	2	yes	maximal, complemented
5	#72339044	C₂	1	1	yes	maximal, characteristic, complemented
6	#72339042	C₂²	1	2	yes	complemented
7	#72339025	C₂	1	4	yes	maximal, complemented
8	#72339024	C₂²	1	2	yes	complemented
9	#72339023	C₂	1	4	yes	maximal, complemented
10	#72339022	C₂²	1	8	yes	complemented
11	#72339021	C₂²	1	4	yes	complemented
12	#72339020	C₂²	1	2	yes	complemented
13	#72339018	C₂	1	4	yes	maximal, complemented
14	#72339017	C₂²	1	4	yes	complemented
15	#72339016	C₂	1	4	yes	maximal, complemented
16	#72339015	C₂²	1	4	yes	complemented
17	#72339014	C₂²	1	4	yes	complemented
18	#72339014	C₂²	1	4	yes	complemented
19	#72339013	C₂²	1	4	yes	complemented
20	#72339012	C₂²	1	2	yes	complemented
21	#72338562	C₂²	1	2	yes	complemented
22	#72338537	C₂	1	2	yes	maximal, complemented
23	#72338536	C₂²	1	2	yes	—
24	#72338535	C₂²	1	4	yes	complemented
25	#72338534	C₂²	1	4	yes	complemented
26	#72318844	C₂²	1	2	yes	complemented
27	#72285817	C₂²	1	4	yes	complemented
28	#72285797	C₂²	1	4	yes	complemented
29	#72285651	C₂	1	2	yes	maximal, complemented
30	#72285635	C₂²	1	1	yes	characteristic, complemented
31	#72285611	C₂²	1	4	yes	complemented
32	#72285220	C₂²	1	8	yes	complemented
33	#72284925	C₂²	1	8	yes	complemented
34	#72284884	C₂²	1	8	yes	complemented
35	#72284342	C₂²	1	8	yes	complemented
36	#72283656	C₂²	1	4	yes	complemented
37	#72283476	C₂²	1	4	yes	complemented
38	#72278795	C₂²	1	4	yes	complemented
39	#72236986	C₂	1	1	yes	maximal, characteristic, complemented
40	#72236979	C₂²	1	1	yes	characteristic, complemented
41	#72236975	C₂²	1	2	yes	complemented
42	#72236974	C₂²	1	4	yes	—
43	#72236974	C₂²	1	4	yes	complemented
44	#72236189	C₂²	1	1	yes	characteristic, complemented
45	#72213834	C₂	1	2	yes	maximal, complemented
46	#72213804	C₂²	1	4	yes	complemented
47	#72213786	C₂²	1	4	yes	complemented
48	#72213776	C₂²	1	8	yes	complemented
49	#72213758	C₂²	1	2	yes	complemented
50	#72213734	C₂²	1	4	yes	complemented
51	#72213423	C₂²	1	1	yes	characteristic, complemented
52	#72213224	C₂²	1	4	yes	—
53	#72210524	C₂²	1	1	yes	characteristic
54	#72210439	C₂²	1	2	yes	—
55	#10575560	D₄	1	2	yes	complemented
56	C₂×#60388	D₄	1	2	yes	—
57	#8934366	C₂³	1	2	yes	—
58	#8195808	C₂³	1	4	yes	—
59	#7672751	C₂³	1	1	yes	characteristic
60	#6631462	C₂³	1	1	yes	characteristic, complemented
61	#6628901	C₂³	1	1	yes	characteristic
62	#6437630	C₂³	1	1	yes	characteristic, complemented
63	#6435070	C₂³	1	2	yes	—
64	#6432077	D₄	1	2	yes	complemented
65	#485688	C₂³	1	8	yes	—
66	#485608	C₂³	1	8	yes	—
67	#480719	C₂³	1	4	yes	—
68	#477175	C₂³	1	2	yes	complemented
69	#453898	C₂³	1	2	yes	complemented
70	#449492	D₄	1	1	yes	characteristic
71	#449492	D₄	1	1	yes	characteristic
72	#441113	D₄	1	2	yes	complemented
73	#441113	C₂³	1	2	yes	—
74	#432844	C₂³	1	2	yes	complemented
75	#422772	D₄	1	2	yes	complemented
76	#422772	C₂³	1	2	yes	—
77	#418884	C₂³	1	2	yes	complemented
78	#417934	D₄	1	2	yes	complemented
79	#417934	C₂³	1	2	yes	—
80	#417807	D₄	1	1	yes	characteristic
81	#417807	D₄	1	1	yes	characteristic
82	#417798	D₄	1	2	yes	complemented
83	#417798	C₂³	1	2	yes	—
84	#417566	C₂³	1	2	yes	—
85	#417566	C₂³	1	2	yes	—
86	#391096	C₂³	1	2	yes	complemented
87	#388991	C₂³	1	2	yes	complemented
88	#374260	C₂³	1	2	yes	—
89	#374009	C₂³	1	1	yes	characteristic
90	#361634	D₄	1	2	yes	complemented
91	#360027	D₄	1	2	yes	complemented
92	#359605	D₄	1	2	yes	—
93	#357286	C₂³	1	1	yes	characteristic, complemented
94	#356056	C₂³	1	1	yes	characteristic, complemented
95	#251828	C₂³	1	4	yes	—
96	#251828	C₂³	1	4	yes	—
97	#213860	C₂³	1	2	yes	—
98	#150616	C₂³	1	8	yes	complemented
99	#150561	C₂³	1	8	yes	complemented
100	#149742	C₂³	1	4	yes	complemented
101	#149742	C₂³	1	4	yes	complemented
102	#134094	C₂³	1	4	yes	—
103	#134056	C₂³	1	4	yes	—
104	#133905	C₂³	1	4	yes	—
105	#133429	C₂³	1	8	yes	—
106	#130068	C₂³	1	8	yes	—
107	#130068	C₂³	1	8	yes	—
108	#94484	C₂³	1	8	yes	complemented
109	#94173	C₂³	1	8	yes	complemented
110	#94169	C₂³	1	8	yes	complemented
111	C₂×H(7,𝔽₂)	C₂×D₄	1	1	yes	characteristic
112	C₂×#3498	C₂×D₄	1	2	yes	—
113	C₂×#3498	C₂×D₄	1	2	yes	—
114	#62686	C₂×D₄	1	2	yes	complemented
115	#60644	C₂×D₄	1	2	yes	—
116	#60644	C₂×D₄	1	2	yes	complemented
117	#60619	C₂×D₄	1	2	yes	—
118	#60619	C₂×D₄	1	2	yes	complemented
119	C₂×#3019	C₂×D₄	1	2	yes	—
120	C₂×#3019	C₂×D₄	1	2	yes	—
121	C₂²×#453	C₂×D₄	1	2	yes	—
122	C₂²×#453	C₂×D₄	1	2	yes	—
123	#60125	C₂×D₄	1	2	yes	complemented
124	C₂×#2888	C₂×D₄	1	2	yes	—
125	#46384	C₂⁴	1	4	yes	—
126	C₂×#2867	C₂×D₄	1	1	yes	characteristic
127	#35527	C₂×D₄	1	1	yes	characteristic
128	#35527	C₂×D₄	1	2	yes	—
129	#35527	C₂×D₄	1	1	yes	characteristic
130	#35527	C₂×D₄	1	1	yes	characteristic
131	#35527	C₂⁴	1	1	yes	characteristic
132	#35272	C₂×D₄	1	2	yes	—
133	#35272	C₂⁴	1	2	yes	—
134	C₄×#532	C₂×D₄	1	2	yes	—
135	#23560	C₂×D₄	1	2	yes	complemented
136	#23498	C₂×D₄	1	2	yes	complemented
137	#23469	C₂×D₄	1	2	yes	complemented
138	#23469	C₂⁴	1	2	yes	—
139	#23461	C₂×D₄	1	2	yes	—
140	#23445	C₂×D₄	1	2	yes	complemented
141	#23445	C₂⁴	1	2	yes	—
142	#23406	C₂×D₄	1	2	yes	—
143	#20632	C₂×D₄	1	2	yes	complemented
144	#20586	C₂×D₄	1	1	yes	characteristic
145	#20586	C₂⁴	1	1	yes	characteristic
146	#20586	C₂⁴	1	1	yes	characteristic
147	C₄×#413	C₂×D₄	1	2	yes	complemented
148	#11503	C₂⁴	1	8	yes	—
149	#11167	C₂⁴	1	2	yes	—
150	#11006	C₂×D₄	1	2	yes	—
151	#11006	C₂×D₄	1	2	yes	—
152	#11006	C₂×D₄	1	2	yes	complemented
153	#11006	C₂×D₄	1	2	yes	complemented
154	#10989	C₂×D₄	1	2	yes	—
155	#10989	C₂×D₄	1	2	yes	—
156	#10981	C₂×D₄	1	1	yes	characteristic
157	#10981	C₂×D₄	1	1	yes	characteristic
158	#10975	C₂×D₄	1	2	yes	—
159	#10975	C₂×D₄	1	2	yes	—
160	#10959	C₂×D₄	1	2	yes	—
161	#10959	C₂×D₄	1	2	yes	—
162	#10939	C₂×D₄	1	2	yes	complemented
163	#10939	C₂×D₄	1	2	yes	complemented
164	#10939	C₂⁴	1	2	yes	—
165	#10939	C₂⁴	1	2	yes	—
166	#10893	C₂×D₄	1	2	yes	complemented
167	#10893	C₂×D₄	1	2	yes	complemented
168	#8748	C₂⁴	1	4	yes	—
169	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
170	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
171	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
172	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
173	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
174	#3458	C₂²×D₄	1	2	yes	—
175	#3449	C₂²×D₄	1	2	yes	—
176	C₂×#535	#120	1	2	yes	—
177	C₂×#535	C₂²×D₄	1	2	yes	—
178	C₂×#535	C₂²×D₄	1	2	yes	—
179	C₂×#535	C₂²×D₄	1	2	yes	—
180	C₂×#532	#120	1	2	yes	—
181	C₂²×#127	#120	1	2	yes	—
182	C₂²×#127	#120	1	1	yes	characteristic
183	C₂²×#127	C₂²×D₄	1	2	yes	—
184	C₂²×#127	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×#41	#120	1	2	yes	—
186	C₂×C₄×#41	C₂²×D₄	1	2	yes	—
187	C₂²×#100	C₂²×D₄	1	2	yes	—
188	C₂²×#100	C₂²×D₄	1	2	yes	—
189	#2873	C₂²×D₄	1	2	yes	—
190	#2873	C₂²×D₄	1	2	yes	—
191	#2842	H(5,𝔽₂)	1	4	yes	—
192	#2403	H(5,𝔽₂)	1	4	yes	—
193	#2309	#120	1	1	yes	characteristic
194	#2309	C₂²×D₄	1	1	yes	characteristic
195	#2309	C₂²×D₄	1	1	yes	characteristic
196	#2309	C₂²×D₄	1	2	yes	—
197	#2309	C₂⁵	1	1	yes	characteristic
198	#2297	#120	1	2	yes	—
199	#2297	C₂²×D₄	1	2	yes	—
200	#2285	C₂²×D₄	1	2	yes	—
201	C₂×C₄³	#120	1	1	yes	characteristic
202	#1742	C₂²×D₄	1	2	yes	—
203	#1742	C₂²×D₄	1	2	yes	—
204	#1740	C₂²×D₄	1	2	yes	—
205	#1740	C₂²×D₄	1	2	yes	—
206	C₂²×#41	C₂×#120	1	2	yes	—
207	C₂²×#41	C₂×#120	1	2	yes	—
208	C₂³×D₄	C₂×#120	1	2	yes	—
209	C₂³×D₄	C₂×#120	1	1	yes	characteristic
210	C₂³×D₄	C₂³×D₄	1	1	yes	characteristic
211	C₂×C₄⋊Q₈	B(4,𝔽₂)	1	2	yes	—
212	C₂×#127	B(4,𝔽₂)	1	2	yes	—
213	C₂×#124	B(4,𝔽₂)	1	2	yes	—
214	C₂×#124	B(4,𝔽₂)	1	2	yes	—
215	C₂×#117	C₂×#120	1	2	yes	—
216	C₂×#117	C₂³×D₄	1	2	yes	—
217	C₂²×C₄²	C₂×#120	1	2	yes	—
218	C₂²×C₄²	C₂×#120	1	1	yes	characteristic
219	#379	C₂×H(5,𝔽₂)	1	4	yes	—
220	C₂⁵	#3484	1	2	yes	—
221	C₂²×Q₈	C₂×B(4,𝔽₂)	1	2	yes	—
222	C₂²×Q₈	C₂×B(4,𝔽₂)	1	2	yes	—
223	C₂²×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
224	C₂²×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
225	C₂³×C₄	C₂²×#120	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	#3484	1	4	yes	—
227	C₂×C₄²	C₂×B(4,𝔽₂)	1	2	yes	—
228	C₂×C₄²	C₂×B(4,𝔽₂)	1	2	yes	—
229	C₂⁴	#35346	1	1	yes	characteristic
230	C₂×D₄	#33551	1	4	yes	—
231	C₂²×C₄	C₂²×B(4,𝔽₂)	1	2	yes	—
232	C₄²	#33551	1	2	yes	—
233	C₄²	#33551	1	2	yes	—
234	C₂³	#6499855	1	1	yes	characteristic
235	C₂³	#6499856	1	1	yes	characteristic
236	C₂³	#6499867	1	1	yes	characteristic
237	C₂×C₄	C₂×#33551	1	2	yes	—
238	C₂×C₄	C₂×#33551	1	2	yes	—
239	C₂²	#72233499	1	2	yes	—
240	C₂²	#72256319	1	1	yes	characteristic
241	C₂	#72262797	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72262797
commutator, Frattini, cosocle	#2309	C₂⁵
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #7672751 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72339049	C₂	semidirect
2. #72339047	C₂	semidirect
3. #72339046	C₂²	semidirect
4. #72339045	C₂	semidirect
5. #72339044	C₂	semidirect
6. #72339042	C₂²	semidirect
7. #72339025	C₂	semidirect
8. #72339024	C₂²	semidirect
9. #72339023	C₂	semidirect
10. #72339022	C₂²	semidirect
11. #72339021	C₂²	semidirect
12. #72339020	C₂²	semidirect
13. #72339018	C₂	semidirect
14. #72339017	C₂²	semidirect
15. #72339016	C₂	semidirect
16. #72339015	C₂²	semidirect
17. #72339014	C₂²	semidirect
18. #72339014	C₂²	semidirect
19. #72339013	C₂²	semidirect
20. #72339012	C₂²	semidirect
21. #72338562	C₂²	semidirect
22. #72338537	C₂	semidirect
24. #72338535	C₂²	semidirect
25. #72338534	C₂²	semidirect
26. #72318844	C₂²	semidirect
27. #72285817	C₂²	semidirect
28. #72285797	C₂²	semidirect
29. #72285651	C₂	semidirect
30. #72285635	C₂²	semidirect
31. #72285611	C₂²	semidirect
32. #72285220	C₂²	semidirect
33. #72284925	C₂²	semidirect
34. #72284884	C₂²	semidirect
35. #72284342	C₂²	semidirect
36. #72283656	C₂²	semidirect
37. #72283476	C₂²	semidirect
38. #72278795	C₂²	semidirect
39. #72236986	C₂	semidirect
40. #72236979	C₂²	semidirect
41. #72236975	C₂²	semidirect
43. #72236974	C₂²	semidirect
44. #72236189	C₂²	semidirect
45. #72213834	C₂	semidirect
46. #72213804	C₂²	semidirect
47. #72213786	C₂²	semidirect
48. #72213776	C₂²	semidirect
49. #72213758	C₂²	semidirect
50. #72213734	C₂²	semidirect
51. #72213423	C₂²	semidirect
55. #10575560	D₄	semidirect
60. #6631462	C₂³	semidirect
62. #6437630	C₂³	semidirect
64. #6432077	D₄	semidirect
68. #477175	C₂³	semidirect
69. #453898	C₂³	semidirect
72. #441113	D₄	semidirect
74. #432844	C₂³	semidirect
75. #422772	D₄	semidirect
77. #418884	C₂³	semidirect
78. #417934	D₄	semidirect
82. #417798	D₄	semidirect
86. #391096	C₂³	semidirect
87. #388991	C₂³	semidirect
90. #361634	D₄	semidirect
91. #360027	D₄	semidirect
93. #357286	C₂³	semidirect
94. #356056	C₂³	semidirect
98. #150616	C₂³	semidirect
99. #150561	C₂³	semidirect
100. #149742	C₂³	semidirect
101. #149742	C₂³	semidirect
108. #94484	C₂³	semidirect
109. #94173	C₂³	semidirect
110. #94169	C₂³	semidirect
114. #62686	C₂×D₄	semidirect
116. #60644	C₂×D₄	semidirect
118. #60619	C₂×D₄	semidirect
123. #60125	C₂×D₄	semidirect
135. #23560	C₂×D₄	semidirect
136. #23498	C₂×D₄	semidirect
137. #23469	C₂×D₄	semidirect
140. #23445	C₂×D₄	semidirect
143. #20632	C₂×D₄	semidirect
147. C₄×#413	C₂×D₄	semidirect
152. #11006	C₂×D₄	semidirect
153. #11006	C₂×D₄	semidirect
162. #10939	C₂×D₄	semidirect
163. #10939	C₂×D₄	semidirect
166. #10893	C₂×D₄	semidirect
167. #10893	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72339049	2
#72339047	2
#72339045	2
#72339044	2
#72339025	2
#72339023	2
#72339018	2
#72339016	2
#72338537	2
#72285651	2
#72236986	2
#72213834	2