Subgroups — #72262263

Order 4096.

At least 11712 conjugacy classes of subgroups; only the 501 normal subgroups are shown.

The subgroup lattice has 268 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72237908	C₂	1	2	yes	maximal, complemented
2	#72237813	C₂	1	4	yes	maximal, complemented
3	#72237770	C₂	1	4	yes	maximal, complemented
4	#72237747	C₂²	1	8	yes	complemented
5	#72237630	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237622	C₂²	1	2	yes	complemented
7	#72237596	C₂	1	2	yes	maximal, complemented
8	#72237594	C₂²	1	8	yes	complemented
9	#72235718	C₂²	1	2	yes	—
10	#72234688	C₂	1	1	yes	maximal, characteristic, complemented
11	#72234592	C₂²	1	2	yes	complemented
12	#72225873	C₂²	1	2	yes	complemented
13	#72225776	C₂²	1	2	yes	complemented
14	#72225738	C₂	1	1	yes	maximal, characteristic
15	#72225736	C₄	1	1	yes	characteristic, complemented
16	#72225735	C₄	1	1	yes	characteristic, complemented
17	#72225733	C₂²	1	2	yes	—
18	#72225732	C₄	1	2	yes	complemented
19	#72225732	C₂²	1	2	yes	—
20	#72225713	C₄	1	1	yes	characteristic, complemented
21	#72225713	C₂²	1	1	yes	characteristic
22	#72224136	C₄	1	1	yes	characteristic, complemented
23	#72224136	C₂²	1	1	yes	characteristic
24	#72219311	C₂²	1	2	yes	—
25	C₂×#7670203	C₂²	1	1	yes	characteristic
26	#72217643	C₄	1	2	yes	complemented
27	#10586404	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂²×#3484	D₄	1	1	yes	characteristic
29	C₂²×#3484	D₄	1	1	yes	characteristic
30	C₂⁴×#120	D₄	1	1	yes	characteristic
31	C₂⁴×#120	D₄	1	1	yes	characteristic
32	#7642273	C₂×C₄	1	2	yes	—
33	C₂²×#2846	C₂×C₄	1	1	yes	characteristic
34	C₂×#35390	C₂×C₄	1	1	yes	characteristic
35	C₂×#35348	D₄	1	1	yes	characteristic
36	C₂×#35348	D₄	1	1	yes	characteristic
37	C₂×#35346	C₂³	1	1	yes	characteristic
38	C₂×#34981	D₄	1	1	yes	characteristic
39	C₂×#34981	D₄	1	1	yes	characteristic
40	C₂×#34980	C₂×C₄	1	1	yes	characteristic
41	C₂×#34980	C₂³	1	1	yes	characteristic
42	C₂×#34198	C₂×C₄	1	1	yes	characteristic
43	C₂×#34198	C₂³	1	1	yes	characteristic
44	#6451874	C₂×C₄	1	2	yes	—
45	#6451874	C₂³	1	2	yes	—
46	C₂×#23984	C₂×C₄	1	2	yes	complemented
47	C₂×#23984	C₂×C₄	1	2	yes	complemented
48	#6374251	C₂×C₄	1	1	yes	characteristic, complemented
49	#6374128	C₂×C₄	1	1	yes	characteristic, complemented
50	#6374128	C₂×C₄	1	1	yes	characteristic, complemented
51	#6372488	C₂×C₄	1	2	yes	complemented
52	#480250	C₂³	1	4	yes	—
53	#417896	D₄	1	4	yes	—
54	#417896	D₄	1	4	yes	—
55	#417824	D₄	1	4	yes	—
56	#417824	D₄	1	4	yes	—
57	#392368	C₂×C₄	1	2	yes	complemented
58	#382422	C₂×C₄	1	2	yes	—
59	#359756	C₂×C₄	1	2	yes	complemented
60	#359756	C₂×C₄	1	2	yes	complemented
61	#359705	C₂×C₄	1	2	yes	—
62	#359705	C₂³	1	2	yes	—
63	#190116	C₂³	1	4	yes	—
64	#115690	D₄	1	4	yes	complemented
65	#115631	D₄	1	4	yes	complemented
66	C₂⁸	C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
68	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
70	C₂×#3490	C₂²×C₄	1	2	yes	—
71	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×#534	C₂²×C₄	1	1	yes	characteristic
75	C₂²×#534	C₂×D₄	1	1	yes	characteristic
76	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
77	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
79	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
80	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
81	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
82	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
83	C₂³×#120	C₂×D₄	1	1	yes	characteristic
84	C₂³×#120	C₂×D₄	1	1	yes	characteristic
85	C₂³×#120	C₂×D₄	1	1	yes	characteristic
86	C₂³×#120	C₂×D₄	1	1	yes	characteristic
87	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×#518	C₂×D₄	1	1	yes	characteristic
89	C₂⁴×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
90	C₂⁴×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×#2849	#41	1	2	yes	—
92	C₂×#2846	C₂×D₄	1	2	yes	—
93	#35390	C₂²⋊C₄	1	4	yes	complemented
94	#35367	C₂²⋊C₄	1	4	yes	complemented
95	#35346	C₂×D₄	1	4	yes	—
96	#35346	C₂×D₄	1	4	yes	—
97	C₂×#2613	C₂×D₄	1	2	yes	—
98	C₂×#2613	#41	1	2	yes	—
99	#34176	C₂²×C₄	1	2	yes	—
100	C₂×#2315	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×#2315	C₂×D₄	1	1	yes	characteristic
102	C₂×#2308	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#2308	C₂²×C₄	1	1	yes	characteristic
104	C₂×#2308	C₂×D₄	1	1	yes	characteristic
105	C₂×#2308	C₂×D₄	1	1	yes	characteristic
106	C₂×#2308	C₂⁴	1	1	yes	characteristic
107	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×#379	C₂²×C₄	1	1	yes	characteristic
109	C₂²×#379	C₂²×C₄	1	1	yes	characteristic
110	C₂×#2273	C₂²×C₄	1	1	yes	characteristic
111	#23468	C₂²⋊C₄	1	2	yes	complemented
112	#23468	C₂²⋊C₄	1	2	yes	complemented
113	#23468	C₂×D₄	1	4	yes	—
114	#23444	C₂²⋊C₄	1	2	yes	complemented
115	#23444	C₂²⋊C₄	1	2	yes	complemented
116	#23444	C₂×D₄	1	4	yes	—
117	C₂×#1819	C₂²×C₄	1	2	yes	—
118	C₂×#1819	C₂²×C₄	1	2	yes	—
119	#10965	C₂²⋊C₄	1	4	yes	complemented
120	#10965	C₂²⋊C₄	1	4	yes	complemented
121	#10938	C₂²⋊C₄	1	4	yes	complemented
122	#10938	C₂²⋊C₄	1	4	yes	complemented
123	#10938	C₂×D₄	1	4	yes	—
124	#10938	C₂×D₄	1	4	yes	—
125	#8101	C₂×D₄	1	8	yes	complemented
126	C₂⁷	#99	1	1	yes	characteristic
127	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂⁷	#120	1	1	yes	characteristic
131	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
132	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
133	C₂⁴×D₄	#99	1	1	yes	characteristic
134	C₂⁴×D₄	#99	1	1	yes	characteristic
135	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂⁴×D₄	#120	1	1	yes	characteristic
140	C₂⁴×D₄	#123	1	1	yes	characteristic
141	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
142	C₂⁵×C₄	#99	1	1	yes	characteristic
143	C₂⁵×C₄	#123	1	1	yes	characteristic
144	C₂×#543	C₂×C₂²⋊C₄	1	2	yes	—
145	C₂×#534	#99	1	2	yes	—
146	C₂×#534	#99	1	2	yes	—
147	C₂×#534	#99	1	4	yes	—
148	C₂×#534	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂²×#123	#99	1	2	yes	—
150	C₂²×#123	C₄×D₄	1	2	yes	—
151	C₂²×#123	#123	1	2	yes	—
152	C₂²×#120	#99	1	2	yes	—
153	C₂²×#120	C₂×C₂²⋊C₄	1	4	yes	—
154	C₂²×#120	C₄×D₄	1	2	yes	—
155	C₂²×#120	#120	1	2	yes	—
156	C₂²×#120	#120	1	2	yes	—
157	C₂²×#120	#120	1	4	yes	—
158	C₂²×#120	#120	1	2	yes	—
159	C₂²×#120	#121	1	2	yes	—
160	C₂²×#120	#121	1	2	yes	—
161	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂³×C₂²⋊C₄	C₂³×C₄	1	1	yes	characteristic
167	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
168	#2308	C₂×C₂²⋊C₄	1	2	yes	—
169	#2308	C₂×C₂²⋊C₄	1	2	yes	—
170	#2308	#120	1	4	yes	—
171	#2308	C₂²×D₄	1	4	yes	—
172	#2308	H(5,𝔽₂)	1	4	yes	—
173	C₂×#381	#121	1	2	yes	—
174	C₂×#381	C₂×#41	1	2	yes	—
175	C₂×#379	C₄×D₄	1	2	yes	—
176	C₂×#379	C₄×D₄	1	2	yes	—
177	C₂×#379	#121	1	2	yes	—
178	C₂×#379	C₂²×D₄	1	2	yes	—
179	#2273	C₂×C₂²⋊C₄	1	4	yes	—
180	C₂×#342	C₂×C₂²⋊C₄	1	4	yes	—
181	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂⁶	#379	1	2	yes	—
184	C₂⁶	#386	1	1	yes	characteristic
185	C₂⁶	#386	1	1	yes	characteristic
186	C₂⁶	#386	1	1	yes	characteristic
187	C₂⁶	C₂×#99	1	1	yes	characteristic
188	C₂⁶	C₂×#99	1	1	yes	characteristic
189	C₂⁶	C₂×#99	1	1	yes	characteristic
190	C₂⁶	C₂×#99	1	1	yes	characteristic
191	C₂⁶	C₂×#99	1	1	yes	characteristic
192	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
193	C₂⁶	C₂×#120	1	1	yes	characteristic
194	C₂³×D₄	#379	1	1	yes	characteristic
195	C₂³×D₄	#379	1	1	yes	characteristic
196	C₂³×D₄	#379	1	2	yes	—
197	C₂³×D₄	#386	1	2	yes	—
198	C₂³×D₄	C₂×#99	1	2	yes	—
199	C₂³×D₄	C₂×#99	1	2	yes	—
200	C₂³×D₄	C₂×#99	1	2	yes	—
201	C₂³×D₄	C₂×#99	1	1	yes	characteristic
202	C₂³×D₄	C₂×#99	1	2	yes	—
203	C₂³×D₄	C₂×#99	1	1	yes	characteristic
204	C₂³×D₄	C₂×#120	1	2	yes	—
205	C₂³×D₄	#534	1	2	yes	—
206	C₂⁴×C₄	#386	1	1	yes	characteristic
207	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
208	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
209	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
210	C₂⁴×C₄	C₂×#123	1	1	yes	characteristic
211	C₂²×C₂²⋊C₄	#379	1	2	yes	—
212	C₂²×C₂²⋊C₄	#386	1	2	yes	—
213	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
214	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
215	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
216	C₂²×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	2	yes	—
217	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	2	yes	—
218	C₂²×C₂²⋊C₄	#518	1	2	yes	—
219	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
220	C₂²×C₂²⋊C₄	C₂×#121	1	2	yes	—
221	C₂²×C₂²⋊C₄	#534	1	2	yes	—
222	#381	#546	1	8	yes	—
223	#379	D₄²	1	8	yes	—
224	C₂⁵	#1781	1	1	yes	characteristic
225	C₂⁵	#1781	1	2	yes	—
226	C₂⁵	#1781	1	1	yes	characteristic
227	C₂⁵	#1898	1	2	yes	—
228	C₂⁵	#1898	1	2	yes	—
229	C₂⁵	C₂×#379	1	1	yes	characteristic
230	C₂⁵	C₂×#386	1	1	yes	characteristic
231	C₂⁵	#2308	1	1	yes	characteristic
232	C₂⁵	#2403	1	2	yes	—
233	C₂⁵	C₂²×#99	1	1	yes	characteristic
234	C₂⁵	C₂²×#99	1	1	yes	characteristic
235	C₂⁵	C₂²×#99	1	1	yes	characteristic
236	C₂⁵	C₂²×#99	1	1	yes	characteristic
237	C₂³×C₄	#1781	1	2	yes	—
238	C₂³×C₄	#1781	1	1	yes	characteristic
239	C₂³×C₄	#1781	1	1	yes	characteristic
240	C₂³×C₄	#1898	1	2	yes	—
241	C₂³×C₄	#1898	1	2	yes	—
242	C₂³×C₄	#2405	1	2	yes	—
243	C₂×C₂²⋊C₄	D₄×C₂²⋊C₄	1	4	yes	—
244	C₂×C₂²⋊C₄	#2410	1	4	yes	—
245	C₂⁴	#12707	1	2	yes	—
246	C₂⁴	#13867	1	1	yes	characteristic
247	C₂⁴	C₂×#1781	1	1	yes	characteristic
248	C₂⁴	C₂×#1781	1	1	yes	characteristic
249	C₂⁴	C₂×#1898	1	1	yes	characteristic
250	C₂⁴	C₂×#1898	1	1	yes	characteristic
251	C₂⁴	#21746	1	1	yes	characteristic
252	C₂⁴	#21746	1	1	yes	characteristic
253	C₂⁴	#23897	1	4	yes	—
254	C₂⁴	#33348	1	4	yes	—
255	C₂²×C₄	#12707	1	2	yes	—
256	C₂³	#359428	1	1	yes	characteristic
257	C₂³	#359558	1	2	yes	—
258	C₂³	C₂×#12707	1	1	yes	characteristic
259	C₂³	C₂×#12707	1	1	yes	characteristic
260	C₂³	#493424	1	2	yes	—
261	C₂³	#494700	1	2	yes	—
262	C₂³	#496240	1	2	yes	—
263	C₂²	#72218057	1	1	yes	characteristic
264	C₂²	#72218455	1	1	yes	characteristic
265	C₂²	#72235479	1	1	yes	characteristic
266	C₂	#72237718	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72237718
commutator	C₂³×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂×#2308	C₂⁴
prime radical	#72225738	C₂
LCS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72237908	C₂	semidirect
2. #72237813	C₂	semidirect
3. #72237770	C₂	semidirect
4. #72237747	C₂²	semidirect
5. #72237630	C₂	semidirect
6. #72237622	C₂²	semidirect
7. #72237596	C₂	semidirect
8. #72237594	C₂²	semidirect
10. #72234688	C₂	semidirect
11. #72234592	C₂²	semidirect
12. #72225873	C₂²	semidirect
13. #72225776	C₂²	semidirect
15. #72225736	C₄	semidirect
16. #72225735	C₄	semidirect
18. #72225732	C₄	semidirect
20. #72225713	C₄	semidirect
22. #72224136	C₄	semidirect
26. #72217643	C₄	semidirect
27. #10586404	C₂×C₄	semidirect
46. C₂×#23984	C₂×C₄	semidirect
47. C₂×#23984	C₂×C₄	semidirect
48. #6374251	C₂×C₄	semidirect
49. #6374128	C₂×C₄	semidirect
50. #6374128	C₂×C₄	semidirect
51. #6372488	C₂×C₄	semidirect
57. #392368	C₂×C₄	semidirect
59. #359756	C₂×C₄	semidirect
60. #359756	C₂×C₄	semidirect
64. #115690	D₄	semidirect
65. #115631	D₄	semidirect
93. #35390	C₂²⋊C₄	semidirect
94. #35367	C₂²⋊C₄	semidirect
111. #23468	C₂²⋊C₄	semidirect
112. #23468	C₂²⋊C₄	semidirect
114. #23444	C₂²⋊C₄	semidirect
115. #23444	C₂²⋊C₄	semidirect
119. #10965	C₂²⋊C₄	semidirect
120. #10965	C₂²⋊C₄	semidirect
121. #10938	C₂²⋊C₄	semidirect
122. #10938	C₂²⋊C₄	semidirect
125. #8101	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72237908	2
#72237813	2
#72237770	2
#72237630	2
#72237596	2
#72234688	2
#72225738	2