Subgroups — #72240504

Order 2048.

At least 6345 conjugacy classes of subgroups; only the 536 normal subgroups are shown.

The subgroup lattice has 240 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	2	yes	maximal, complemented
2	#72240134	C₂	1	4	yes	maximal, complemented
3	#72226038	C₂	1	2	yes	maximal, complemented
4	#72218291	C₂	1	1	yes	maximal, characteristic, complemented
5	#72217875	C₂	1	1	yes	maximal, characteristic
6	#72217789	C₂	1	1	yes	maximal, characteristic, complemented
7	#72213256	C₂	1	4	yes	maximal, complemented
8	C₂²×#3484	C₄	1	1	yes	characteristic, complemented
9	C₂⁴×#120	C₂²	1	1	yes	characteristic
10	C₂×#35346	C₄	1	1	yes	characteristic, complemented
11	C₂×#35346	C₂²	1	1	yes	characteristic
12	C₂×#21746	C₂²	1	2	yes	complemented
13	#616960	C₄	1	2	yes	complemented
14	#616960	C₂²	1	2	yes	—
15	#612424	C₄	1	2	yes	complemented
16	#612424	C₂²	1	2	yes	—
17	#517411	C₄	1	1	yes	characteristic, complemented
18	#515285	C₄	1	1	yes	characteristic, complemented
19	#515285	C₂²	1	1	yes	characteristic
20	#496225	C₂²	1	8	yes	complemented
21	#392113	C₂²	1	2	yes	complemented
22	#392109	C₂²	1	2	yes	complemented
23	#382422	C₂²	1	2	yes	—
24	C₂×#8547	C₂²	1	2	yes	—
25	C₂×#8547	C₂²	1	2	yes	complemented
26	#128073	C₂²	1	8	yes	complemented
27	C₂⁸	D₄	1	1	yes	characteristic
28	C₂³×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂⁵×D₄	C₂×C₄	1	1	yes	characteristic
30	C₂⁵×D₄	C₂×C₄	1	1	yes	characteristic
31	C₂⁵×D₄	C₂³	1	1	yes	characteristic
32	C₂²×#534	C₂×C₄	1	2	yes	—
33	C₂³×#120	C₂×C₄	1	1	yes	characteristic
34	C₂³×#120	C₂×C₄	1	1	yes	characteristic
35	C₂³×#120	C₂×C₄	1	2	yes	complemented
36	C₂³×#120	D₄	1	1	yes	characteristic
37	C₂³×#120	D₄	1	1	yes	characteristic
38	C₂³×#120	D₄	1	1	yes	characteristic
39	C₂³×#120	D₄	1	1	yes	characteristic
40	C₂³×#120	C₂³	1	1	yes	characteristic
41	C₂³×#120	C₂³	1	1	yes	characteristic
42	C₂²×#518	C₂×C₄	1	1	yes	characteristic, complemented
43	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic
44	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic
45	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic
46	C₂×#2403	C₂×C₄	1	2	yes	—
47	C₂×#2403	C₂×C₄	1	2	yes	complemented
48	C₂×#2403	C₂×C₄	1	2	yes	complemented
49	C₂×#2403	C₂³	1	2	yes	—
50	C₂×#2308	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×#2308	C₂×C₄	1	1	yes	characteristic, complemented
52	C₂×#1783	C₂³	1	4	yes	—
53	C₂×#1781	C₂³	1	4	yes	—
54	#15948	D₄	1	8	yes	—
55	#15947	D₄	1	8	yes	—
56	#14679	C₂×C₄	1	2	yes	complemented
57	#14335	C₂×C₄	1	2	yes	—
58	#13867	C₂×C₄	1	2	yes	—
59	#13867	C₂×C₄	1	2	yes	complemented
60	#13867	C₂×C₄	1	2	yes	complemented
61	#13867	C₂³	1	2	yes	—
62	#10965	D₄	1	8	yes	complemented
63	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
64	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
65	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
66	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
67	C₂⁷	C₂²×C₄	1	1	yes	characteristic
68	C₂⁷	C₂×D₄	1	1	yes	characteristic
69	C₂⁷	C₂×D₄	1	1	yes	characteristic
70	C₂⁷	C₂×D₄	1	1	yes	characteristic
71	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
76	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
77	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
78	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
79	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
80	C₂⁴×D₄	C₂²×C₄	1	2	yes	—
81	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
83	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
84	C₂⁵×C₄	C₂²⋊C₄	1	1	yes	characteristic
85	C₂⁵×C₄	C₂²⋊C₄	1	1	yes	characteristic
86	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×#123	C₂²⋊C₄	1	4	yes	complemented
88	C₂²×#120	C₂²⋊C₄	1	4	yes	complemented
89	C₂²×#120	C₂²⋊C₄	1	4	yes	complemented
90	C₂²×#120	C₂×D₄	1	4	yes	—
91	C₂²×#120	C₂×D₄	1	8	yes	—
92	C₂²×#120	C₂×D₄	1	4	yes	—
93	C₂²×#120	#41	1	4	yes	—
94	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₂²⋊C₄	C₂²×C₄	1	2	yes	—
101	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
104	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×#99	C₂×D₄	1	8	yes	complemented
108	C₂×#386	C₂²⋊C₄	1	4	yes	complemented
109	C₂×#386	C₂×D₄	1	4	yes	—
110	C₂×#379	C₂²×C₄	1	2	yes	—
111	C₂×#379	C₂²×C₄	1	2	yes	—
112	#1438	C₂²⋊C₄	1	8	yes	complemented
113	#1438	C₂²⋊C₄	1	8	yes	complemented
114	#1438	C₂×D₄	1	8	yes	—
115	C₂⁶	#99	1	2	yes	—
116	C₂⁶	#99	1	2	yes	—
117	C₂⁶	#99	1	2	yes	—
118	C₂⁶	#99	1	2	yes	—
119	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	complemented
127	C₂⁶	#120	1	4	yes	—
128	C₂⁶	#120	1	2	yes	—
129	C₂⁶	#123	1	2	yes	—
130	C₂⁶	C₂³×C₄	1	1	yes	characteristic
131	C₂⁶	C₂²×D₄	1	1	yes	characteristic
132	C₂³×D₄	#99	1	4	yes	—
133	C₂³×D₄	C₂×C₂²⋊C₄	1	4	yes	—
134	C₂³×D₄	C₂×C₂²⋊C₄	1	4	yes	—
135	C₂³×D₄	C₄×D₄	1	4	yes	—
136	C₂³×D₄	C₄×D₄	1	4	yes	—
137	C₂³×D₄	C₂²×D₄	1	4	yes	—
138	C₂³×D₄	C₂²×D₄	1	2	yes	—
139	C₂³×D₄	C₂×#41	1	2	yes	—
140	C₂³×D₄	H(5,𝔽₂)	1	4	yes	—
141	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	complemented
147	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₂²×C₂²⋊C₄	#99	1	4	yes	—
149	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	complemented
150	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	complemented
151	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	complemented
152	C₂²×C₂²⋊C₄	#120	1	4	yes	—
153	C₂²×C₂²⋊C₄	#120	1	4	yes	—
154	C₂²×C₂²⋊C₄	#120	1	2	yes	—
155	C₂²×C₂²⋊C₄	#120	1	4	yes	—
156	C₂²×C₂²⋊C₄	#121	1	4	yes	—
157	C₂²×C₂²⋊C₄	#121	1	4	yes	—
158	C₂²×C₂²⋊C₄	#123	1	2	yes	—
159	C₂⁵	AΣL(1,𝔽₁₆)	1	8	yes	complemented
160	C₂⁵	#379	1	2	yes	—
161	C₂⁵	#379	1	2	yes	—
162	C₂⁵	#386	1	2	yes	—
163	C₂⁵	#386	1	2	yes	—
164	C₂⁵	C₂×#99	1	2	yes	—
165	C₂⁵	C₂×#99	1	2	yes	—
166	C₂⁵	C₂×#99	1	1	yes	characteristic
167	C₂⁵	C₂×#99	1	1	yes	characteristic
168	C₂⁵	C₂×#99	1	2	yes	—
169	C₂⁵	C₂×#99	1	2	yes	—
170	C₂⁵	C₂×#99	1	1	yes	characteristic
171	C₂⁵	C₂×#99	1	2	yes	—
172	C₂⁵	C₂×#99	1	2	yes	—
173	C₂⁵	C₂×#99	1	2	yes	—
174	C₂⁵	C₂×#99	1	1	yes	characteristic
175	C₂⁵	C₂²×C₂²⋊C₄	1	2	yes	—
176	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
177	C₂⁵	C₂×C₄×D₄	1	2	yes	—
178	C₂⁵	#518	1	2	yes	—
179	C₂⁵	C₂×#120	1	1	yes	characteristic
180	C₂⁵	C₂×#120	1	2	yes	—
181	C₂⁵	C₂×#123	1	1	yes	characteristic
182	C₂⁵	#534	1	2	yes	—
183	C₂²×D₄	D₄²	1	8	yes	—
184	C₂²×D₄	#546	1	8	yes	—
185	C₂³×C₄	#379	1	2	yes	—
186	C₂³×C₄	#379	1	2	yes	—
187	C₂³×C₄	#386	1	2	yes	—
188	C₂³×C₄	#386	1	2	yes	—
189	C₂³×C₄	C₂×#99	1	2	yes	—
190	C₂³×C₄	C₂×#99	1	2	yes	—
191	C₂³×C₄	C₂×#99	1	2	yes	—
192	C₂³×C₄	C₂×#120	1	2	yes	—
193	C₂³×C₄	C₂×#121	1	2	yes	—
194	C₂³×C₄	#534	1	2	yes	—
195	C₂⁴	#1781	1	2	yes	—
196	C₂⁴	#1781	1	2	yes	—
197	C₂⁴	#1781	1	2	yes	—
198	C₂⁴	#1781	1	2	yes	—
199	C₂⁴	#1898	1	4	yes	—
200	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
201	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
202	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
203	C₂⁴	#2121	1	4	yes	—
204	C₂⁴	C₂×#379	1	1	yes	characteristic
205	C₂⁴	C₂×#386	1	1	yes	characteristic
206	C₂⁴	#2308	1	1	yes	characteristic
207	C₂⁴	D₄×C₂²⋊C₄	1	4	yes	—
208	C₂⁴	#2403	1	2	yes	—
209	C₂⁴	#2405	1	2	yes	—
210	C₂⁴	C₂²×#99	1	1	yes	characteristic
211	C₂⁴	C₂²×#99	1	1	yes	characteristic
212	C₂⁴	C₂²×#99	1	1	yes	characteristic
213	C₂⁴	C₂²×#99	1	1	yes	characteristic
214	C₂²×C₄	#1898	1	4	yes	—
215	C₂²×C₄	#2410	1	4	yes	—
216	C₂³	#12687	1	4	yes	—
217	C₂³	#12705	1	4	yes	—
218	C₂³	#13086	1	8	yes	—
219	C₂³	#13867	1	1	yes	characteristic
220	C₂³	C₂×#1781	1	1	yes	characteristic
221	C₂³	C₂×#1781	1	1	yes	characteristic
222	C₂³	C₂×#1898	1	1	yes	characteristic
223	C₂³	C₂×#1898	1	1	yes	characteristic
224	C₂³	#21746	1	1	yes	characteristic
225	C₂³	#21746	1	1	yes	characteristic
226	C₂³	#23895	1	4	yes	—
227	C₂³	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
228	C₂³	C₂×#2121	1	2	yes	—
229	C₂²	#359428	1	1	yes	characteristic
230	C₂²	#359549	1	2	yes	—
231	C₂²	C₂×#12687	1	1	yes	characteristic
232	C₂²	#493357	1	1	yes	characteristic
233	C₂²	C₂×#12705	1	2	yes	—
234	C₂²	C₂×#13086	1	2	yes	—
235	C₂²	#496225	1	2	yes	—
236	C₂	#72218069	1	1	yes	characteristic
237	C₂	#72218180	1	1	yes	characteristic
238	C₂	#72219294	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#359428
commutator	C₂⁶	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72217875	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×#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. #72240134	C₂	semidirect
3. #72226038	C₂	semidirect
4. #72218291	C₂	semidirect
6. #72217789	C₂	semidirect
7. #72213256	C₂	semidirect
8. C₂²×#3484	C₄	semidirect
10. C₂×#35346	C₄	semidirect
12. C₂×#21746	C₂²	semidirect
13. #616960	C₄	semidirect
15. #612424	C₄	semidirect
17. #517411	C₄	semidirect
18. #515285	C₄	semidirect
20. #496225	C₂²	semidirect
21. #392113	C₂²	semidirect
22. #392109	C₂²	semidirect
25. C₂×#8547	C₂²	semidirect
26. #128073	C₂²	semidirect
28. C₂³×H(5,𝔽₂)	C₂×C₄	semidirect
35. C₂³×#120	C₂×C₄	semidirect
42. C₂²×#518	C₂×C₄	semidirect
47. C₂×#2403	C₂×C₄	semidirect
48. C₂×#2403	C₂×C₄	semidirect
50. C₂×#2308	C₂×C₄	semidirect
51. C₂×#2308	C₂×C₄	semidirect
56. #14679	C₂×C₄	semidirect
59. #13867	C₂×C₄	semidirect
60. #13867	C₂×C₄	semidirect
62. #10965	D₄	semidirect
87. C₂²×#123	C₂²⋊C₄	semidirect
88. C₂²×#120	C₂²⋊C₄	semidirect
89. C₂²×#120	C₂²⋊C₄	semidirect
107. C₂²×#99	C₂×D₄	semidirect
108. C₂×#386	C₂²⋊C₄	semidirect
112. #1438	C₂²⋊C₄	semidirect
113. #1438	C₂²⋊C₄	semidirect
126. C₂⁶	C₂×C₂²⋊C₄	semidirect
146. C₂⁴×C₄	C₂×C₂²⋊C₄	semidirect
149. C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	semidirect
150. C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	semidirect
151. C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	semidirect
159. C₂⁵	AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
#72240178	2
#72240134	2
#72226038	2
#72218291	2
#72217875	2
#72217789	2
#72213256	2