Subgroups — #72258939

Order 2048.

At least 6155 conjugacy classes of subgroups; only the 480 normal subgroups are shown.

The subgroup lattice has 251 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72258744	C₂	1	4	yes	maximal, complemented
2	#72255630	C₂	1	2	yes	maximal, complemented
3	#72225665	C₂	1	1	yes	maximal, characteristic
4	#72225621	C₂	1	1	yes	maximal, characteristic, complemented
5	#72219457	C₂	1	2	yes	maximal, complemented
6	#72219377	C₂	1	1	yes	maximal, characteristic, complemented
7	#72218387	C₂	1	4	yes	maximal, complemented
8	C₂³×D₄²	C₂²	1	1	yes	characteristic
9	C₂×#34980	C₄	1	1	yes	characteristic, complemented
10	C₂²×#2403	C₄	1	1	yes	characteristic, complemented
11	C₂²×#2403	C₂²	1	1	yes	characteristic
12	C₂²×#1781	C₂²	1	2	yes	—
13	#636791	C₄	1	2	yes	complemented
14	#635984	C₂²	1	2	yes	—
15	#613191	C₄	1	2	yes	complemented
16	#612424	C₂²	1	2	yes	—
17	#556752	C₄	1	1	yes	characteristic, complemented
18	#517411	C₄	1	1	yes	characteristic, complemented
19	#517411	C₂²	1	1	yes	characteristic
20	#392109	C₂²	1	2	yes	—
21	#392109	C₂²	1	2	yes	complemented
22	#382422	C₂²	1	2	yes	complemented
23	#359428	C₂²	1	8	yes	complemented
24	C₂×#10965	C₂²	1	2	yes	complemented
25	#154094	C₂²	1	2	yes	complemented
26	C₂²⋊#1304	C₂²	1	8	yes	complemented
27	C₂⁵×D₄	C₂×C₄	1	1	yes	characteristic
28	C₂⁵×D₄	C₂×C₄	1	1	yes	characteristic
29	C₂⁵×D₄	D₄	1	1	yes	characteristic
30	C₂⁵×D₄	C₂³	1	1	yes	characteristic
31	C₂⁵×D₄	C₂³	1	1	yes	characteristic
32	C₂³×#127	D₄	1	1	yes	characteristic
33	C₂³×#121	C₂×C₄	1	1	yes	characteristic
34	C₂³×#121	D₄	1	1	yes	characteristic
35	C₂³×#121	D₄	1	1	yes	characteristic
36	C₂³×#121	D₄	1	1	yes	characteristic
37	C₂³×#120	C₂×C₄	1	1	yes	characteristic
38	C₂³×#120	D₄	1	1	yes	characteristic
39	C₂³×#120	D₄	1	1	yes	characteristic
40	C₂³×#120	D₄	1	2	yes	—
41	C₂³×#120	C₂³	1	1	yes	characteristic
42	C₂³×#120	C₂³	1	2	yes	—
43	C₂³×C₄×D₄	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×D₄	D₄	1	1	yes	characteristic
45	C₂×#2405	C₂×C₄	1	2	yes	—
46	C₂×#2403	C₂×C₄	1	2	yes	—
47	C₂×#2403	C₂×C₄	1	2	yes	—
48	C₂×#2403	C₂×C₄	1	2	yes	—
49	C₂²×#392	D₄	1	4	yes	—
50	C₂×#2308	D₄	1	2	yes	—
51	C₂×#2308	D₄	1	2	yes	—
52	C₂²×#386	C₂×C₄	1	2	yes	—
53	C₂²×#386	D₄	1	2	yes	—
54	C₂²×#379	C₂×C₄	1	1	yes	characteristic
55	C₂²×#379	C₂×C₄	1	1	yes	characteristic
56	C₂²×#379	C₂×C₄	1	1	yes	characteristic
57	C₂²×#379	C₂³	1	4	yes	—
58	#14681	C₂×C₄	1	2	yes	—
59	#14679	C₂³	1	4	yes	—
60	#14679	C₂³	1	2	yes	—
61	#14335	D₄	1	4	yes	—
62	#13869	C₂×C₄	1	2	yes	—
63	#13867	C₂×C₄	1	2	yes	—
64	#13867	C₂×C₄	1	2	yes	—
65	#13867	C₂×C₄	1	2	yes	—
66	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
67	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic, complemented
68	C₂⁷	C₂²×C₄	1	1	yes	characteristic
69	C₂⁷	C₂×D₄	1	1	yes	characteristic
70	C₂⁷	C₂⁴	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	2	yes	complemented
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	C₂×D₄	1	1	yes	characteristic
84	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
85	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
86	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
87	C₂⁴×D₄	C₂×D₄	1	2	yes	—
88	C₂⁴×D₄	C₂×D₄	1	2	yes	—
89	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
90	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂×D₄	1	4	yes	—
92	C₂⁴×D₄	C₂×D₄	1	2	yes	—
93	C₂⁴×D₄	C₂×D₄	1	2	yes	—
94	C₂⁵×C₄	C₂²⋊C₄	1	1	yes	characteristic
95	C₂⁵×C₄	C₂²⋊C₄	1	1	yes	characteristic
96	C₂⁵×C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
97	C₂⁵×C₄	C₂²×C₄	1	1	yes	characteristic
98	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
107	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	4	yes	complemented
108	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
109	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
110	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
111	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
112	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
113	C₂³×C₂²⋊C₄	C₂²×C₄	1	2	yes	—
114	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	C₂×D₄	1	4	yes	—
117	C₂³×C₄²	C₂²⋊C₄	1	1	yes	characteristic
118	C₂×#381	C₂²⋊C₄	1	4	yes	complemented
119	C₂×#381	C₂²⋊C₄	1	4	yes	complemented
120	C₂×#379	C₂²⋊C₄	1	2	yes	complemented
121	C₂×#379	C₂²⋊C₄	1	2	yes	complemented
122	C₂×#379	C₂²×C₄	1	2	yes	—
123	C₂×#379	C₂²×C₄	1	2	yes	—
124	C₂×#379	C₂×D₄	1	4	yes	—
125	C₂×#379	C₂×D₄	1	4	yes	—
126	C₂²×#95	C₂²⋊C₄	1	4	yes	complemented
127	C₂⁶	#99	1	4	yes	complemented
128	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂⁶	C₂³×C₄	1	1	yes	characteristic
139	C₂⁶	C₂²×D₄	1	1	yes	characteristic
140	C₂⁶	C₂²×D₄	1	2	yes	—
141	C₂⁶	C₂²×D₄	1	1	yes	characteristic
142	C₂⁶	C₂²×D₄	1	2	yes	—
143	C₂⁶	H(5,𝔽₂)	1	4	yes	—
144	C₂⁶	H(5,𝔽₂)	1	4	yes	—
145	C₂³×D₄	#99	1	4	yes	complemented
146	C₂³×D₄	#120	1	2	yes	—
147	C₂³×D₄	#120	1	2	yes	—
148	C₂³×D₄	#120	1	2	yes	—
149	C₂³×D₄	#120	1	2	yes	—
150	C₂³×D₄	#120	1	2	yes	—
151	C₂³×D₄	#120	1	2	yes	—
152	C₂³×D₄	#120	1	2	yes	—
153	C₂³×D₄	#120	1	2	yes	—
154	C₂⁴×C₄	#99	1	4	yes	complemented
155	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂⁴×C₄	C₂×C₂²⋊C₄	1	4	yes	—
166	C₂²×C₂²⋊C₄	#99	1	4	yes	complemented
167	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
168	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
170	C₂⁵	#379	1	1	yes	characteristic
171	C₂⁵	#379	1	1	yes	characteristic
172	C₂⁵	#379	1	1	yes	characteristic
173	C₂⁵	#379	1	1	yes	characteristic
174	C₂⁵	C₂×#99	1	2	yes	—
175	C₂⁵	C₂×#99	1	2	yes	—
176	C₂⁵	C₂×#99	1	2	yes	—
177	C₂⁵	C₂×#99	1	2	yes	—
178	C₂⁵	C₂×#99	1	2	yes	—
179	C₂⁵	C₂×#99	1	2	yes	—
180	C₂⁵	C₂×#99	1	2	yes	—
181	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
182	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
183	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
184	C₂⁵	#518	1	2	yes	—
185	C₂⁵	#518	1	2	yes	—
186	C₂⁵	C₂×#120	1	2	yes	—
187	C₂⁵	C₂×#120	1	2	yes	—
188	C₂⁵	#534	1	2	yes	—
189	C₂⁵	#534	1	2	yes	—
190	C₂⁵	#534	1	2	yes	—
191	C₂⁵	#534	1	8	yes	—
192	C₂⁵	#534	1	2	yes	—
193	C₂⁵	#535	1	8	yes	—
194	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
195	C₂²×D₄	#458	1	4	yes	—
196	C₂³×C₄	#379	1	2	yes	—
197	C₂³×C₄	#379	1	2	yes	—
198	C₂³×C₄	C₂×#99	1	2	yes	—
199	C₂³×C₄	C₂×#99	1	2	yes	—
200	C₂³×C₄	C₂×#99	1	2	yes	—
201	C₂³×C₄	C₂×#99	1	2	yes	—
202	C₂³×C₄	C₂×#99	1	2	yes	—
203	C₂⁴	#1781	1	2	yes	—
204	C₂⁴	#1781	1	2	yes	—
205	C₂⁴	#1781	1	2	yes	—
206	C₂⁴	#1781	1	2	yes	—
207	C₂⁴	#1790	1	4	yes	—
208	C₂⁴	C₂×#379	1	1	yes	characteristic
209	C₂⁴	#2308	1	2	yes	—
210	C₂⁴	#2308	1	1	yes	characteristic
211	C₂⁴	#2308	1	1	yes	characteristic
212	C₂⁴	#2309	1	2	yes	—
213	C₂⁴	#2613	1	4	yes	—
214	C₂⁴	#2613	1	4	yes	—
215	C₂⁴	C₂²×#99	1	1	yes	characteristic
216	C₂⁴	C₂²×#99	1	1	yes	characteristic
217	C₂⁴	C₂²×#99	1	1	yes	characteristic
218	C₂⁴	C₂²×#99	1	1	yes	characteristic
219	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
220	C₂⁴	C₂×#458	1	2	yes	—
221	C₂⁴	#3026	1	4	yes	—
222	C₂²×C₄	#1790	1	4	yes	—
223	C₂³	#8534	1	4	yes	—
224	C₂³	#14335	1	1	yes	characteristic
225	C₂³	C₂×#1781	1	1	yes	characteristic
226	C₂³	C₂×#1781	1	1	yes	characteristic
227	C₂³	C₂×#1790	1	2	yes	—
228	C₂³	#20609	1	2	yes	—
229	C₂³	#21746	1	1	yes	characteristic
230	C₂³	#21746	1	1	yes	characteristic
231	C₂³	#21746	1	1	yes	characteristic
232	C₂³	#21746	1	1	yes	characteristic
233	C₂³	#21776	1	4	yes	—
234	C₂³	#21776	1	4	yes	—
235	C₂³	#29634	1	4	yes	—
236	C₂³	#29635	1	4	yes	—
237	C₂²	C₂×#8534	1	1	yes	characteristic
238	C₂²	#187928	1	2	yes	—
239	C₂²	#187944	1	1	yes	characteristic
240	C₂²	#213828	1	2	yes	—
241	C₂²	#213829	1	2	yes	—
242	C₂²	#382387	1	2	yes	—
243	C₂²	#382422	1	1	yes	characteristic
244	C₂²	#382434	1	2	yes	—
245	C₂²	#382434	1	2	yes	—
246	C₂	#72219311	1	1	yes	characteristic
247	C₂	#72233032	1	4	yes	—
248	C₂	#72236929	1	1	yes	characteristic
249	C₂	#72251607	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#14335
commutator	C₂⁶	C₂³×C₄
Frattini, cosocle	C₂⁷	C₂⁴
prime radical	#72225665	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁷ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72258744	C₂	semidirect
2. #72255630	C₂	semidirect
4. #72225621	C₂	semidirect
5. #72219457	C₂	semidirect
6. #72219377	C₂	semidirect
7. #72218387	C₂	semidirect
9. C₂×#34980	C₄	semidirect
10. C₂²×#2403	C₄	semidirect
13. #636791	C₄	semidirect
15. #613191	C₄	semidirect
17. #556752	C₄	semidirect
18. #517411	C₄	semidirect
21. #392109	C₂²	semidirect
22. #382422	C₂²	semidirect
23. #359428	C₂²	semidirect
24. C₂×#10965	C₂²	semidirect
25. #154094	C₂²	semidirect
26. C₂²⋊#1304	C₂²	semidirect
67. C₂⁷	C₂²⋊C₄	semidirect
76. C₂⁴×D₄	C₂²⋊C₄	semidirect
96. C₂⁵×C₄	C₂²⋊C₄	semidirect
106. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
107. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
108. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
109. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
110. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
111. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
118. C₂×#381	C₂²⋊C₄	semidirect
119. C₂×#381	C₂²⋊C₄	semidirect
120. C₂×#379	C₂²⋊C₄	semidirect
121. C₂×#379	C₂²⋊C₄	semidirect
126. C₂²×#95	C₂²⋊C₄	semidirect
127. C₂⁶	#99	semidirect
145. C₂³×D₄	#99	semidirect
154. C₂⁴×C₄	#99	semidirect
166. C₂²×C₂²⋊C₄	#99	semidirect

Maximal subgroups

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

subgroup	index
#72258744	2
#72255630	2
#72225665	2
#72225621	2
#72219457	2
#72219377	2
#72218387	2