Subgroups — #72314629

Order 8192.

Only the 336 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 287 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314621	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314603	C₂	1	1	yes	maximal, characteristic, complemented
3	#72314595	C₂	1	1	yes	maximal, characteristic, complemented
4	#72314568	C₂	1	1	yes	maximal, characteristic
5	#72314516	C₄	1	2	yes	complemented
6	#72314498	C₄	1	2	yes	complemented
7	#72312950	C₂	1	1	yes	maximal, characteristic
8	#72310419	C₄	1	1	yes	characteristic, complemented
9	#72302118	C₂	1	1	yes	maximal, characteristic
10	#72302078	C₄	1	1	yes	characteristic, complemented
11	#72302062	C₄	1	1	yes	characteristic, complemented
12	#72302047	C₄	1	1	yes	characteristic, complemented
13	#72302019	C₄	1	1	yes	characteristic, complemented
14	#72265785	C₂²	1	1	yes	characteristic
15	#72265785	C₂²	1	1	yes	characteristic
16	#72265779	C₂×C₄	1	2	yes	—
17	#72237032	C₄	1	1	yes	characteristic, complemented
18	#72237027	C₄	1	1	yes	characteristic, complemented
19	#72234688	C₂²	1	1	yes	characteristic
20	#72234665	C₂²	1	1	yes	characteristic
21	#72234663	D₄	1	1	yes	characteristic
22	#72234619	C₂×C₄	1	1	yes	characteristic
23	#72234619	C₂×C₄	1	1	yes	characteristic
24	#72234591	Q₈	1	1	yes	characteristic
25	#72234591	Q₈	1	1	yes	characteristic
26	#72232884	D₄	1	1	yes	characteristic
27	#72226365	C₂	1	1	yes	maximal, characteristic, complemented
28	#72225781	C₂²	1	1	yes	characteristic
29	#72225778	D₄	1	2	yes	—
30	#72225777	C₂×C₄	1	1	yes	characteristic, complemented
31	#72225776	C₂×C₄	1	1	yes	characteristic, complemented
32	#72225776	C₂×C₄	1	1	yes	characteristic, complemented
33	#72225776	C₂×C₄	1	1	yes	characteristic, complemented
34	#72225655	C₂×C₄	1	1	yes	characteristic
35	#72225655	C₂×C₄	1	1	yes	characteristic
36	#72225629	C₂²	1	1	yes	characteristic
37	#72224136	D₄	1	1	yes	characteristic
38	#72224136	D₄	1	1	yes	characteristic
39	#72219397	C₂²	1	1	yes	characteristic
40	#72219340	D₄	1	2	yes	—
41	#72219325	C₂×C₄	1	2	yes	—
42	#72219311	D₄	1	1	yes	characteristic
43	#72219310	D₄	1	1	yes	characteristic
44	#72219247	C₄	1	1	yes	characteristic, complemented
45	#72219227	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×#7670203	C₂³	1	1	yes	characteristic
47	#72218518	C₂×C₄	1	1	yes	characteristic, complemented
48	#72218518	D₄	1	1	yes	characteristic
49	#72218375	C₂×C₄	1	1	yes	characteristic, complemented
50	#72218291	C₂×C₄	1	1	yes	characteristic, complemented
51	#72218291	D₄	1	1	yes	characteristic
52	#72217511	C₂×C₄	1	1	yes	characteristic
53	#72217511	C₂×C₄	1	1	yes	characteristic
54	#72216064	D₄	1	1	yes	characteristic
55	#72216064	D₄	1	1	yes	characteristic
56	#10586404	C₂²⋊C₄	1	1	yes	characteristic
57	#10586404	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×#62817	C₄⋊C₄	1	1	yes	characteristic
59	C₂×#62817	C₄⋊C₄	1	1	yes	characteristic
60	#10586066	C₂²⋊C₄	1	1	yes	characteristic
61	#10586066	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
63	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
64	C₂⁴×#120	C₂²×C₄	1	1	yes	characteristic
65	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
66	#7642244	C₄²	1	1	yes	characteristic
67	#7642244	C₄²	1	1	yes	characteristic
68	#7642244	C₄²	1	1	yes	characteristic
69	#7642244	C₄²	1	1	yes	characteristic
70	C₂²×#2846	#41	1	1	yes	characteristic
71	#7630939	C₂²⋊C₄	1	1	yes	characteristic
72	#7630939	C₂²⋊C₄	1	1	yes	characteristic
73	#7630939	C₂²⋊C₄	1	1	yes	characteristic
74	#7630939	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×#35390	C₂×D₄	1	1	yes	characteristic
76	C₂×#35348	C₂²×C₄	1	1	yes	characteristic
77	C₂×#35348	#41	1	1	yes	characteristic
78	C₂×#35346	C₂²×C₄	1	1	yes	characteristic
79	C₂×#34981	C₂×D₄	1	1	yes	characteristic
80	C₂×#34981	#41	1	1	yes	characteristic
81	C₂×#34980	C₂×D₄	1	1	yes	characteristic
82	C₂×#34980	C₂×D₄	1	1	yes	characteristic
83	#7626202	C₄⋊C₄	1	1	yes	characteristic
84	#7626202	C₄⋊C₄	1	1	yes	characteristic
85	C₂×#34198	C₂×Q₈	1	1	yes	characteristic
86	C₂×#34198	#41	1	1	yes	characteristic
87	#6374210	C₄⋊C₄	1	1	yes	characteristic
88	#6374210	C₄⋊C₄	1	1	yes	characteristic
89	#6374210	C₄⋊C₄	1	1	yes	characteristic
90	#6374210	C₄⋊C₄	1	1	yes	characteristic
91	#6374128	C₂²⋊C₄	1	1	yes	characteristic
92	#6374128	C₂²⋊C₄	1	1	yes	characteristic
93	#6371504	C₄⋊C₄	1	1	yes	characteristic
94	#6371504	C₄⋊C₄	1	1	yes	characteristic
95	#6371504	C₄⋊C₄	1	1	yes	characteristic
96	#6371504	C₄⋊C₄	1	1	yes	characteristic
97	#6370693	C₂²⋊C₄	1	1	yes	characteristic
98	#6370693	C₂²⋊C₄	1	1	yes	characteristic
99	#190147	C₂²⋊C₄	1	2	yes	complemented
100	#190116	C₂²⋊C₄	1	2	yes	complemented
101	C₂×#8547	C₂²⋊C₄	1	2	yes	complemented
102	C₂×#8541	C₂²⋊C₄	1	2	yes	complemented
103	C₂⁸	C₄×D₄	1	1	yes	characteristic
104	C₂×H(7,𝔽₂)	#99	1	1	yes	characteristic
105	C₂×H(7,𝔽₂)	#99	1	1	yes	characteristic
106	C₂³×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂⁵×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂⁵×D₄	#120	1	1	yes	characteristic
109	#62862	#99	1	4	yes	—
110	#62842	#99	1	1	yes	characteristic
111	#62842	#99	1	1	yes	characteristic
112	#62842	#99	1	1	yes	characteristic
113	#62842	#99	1	1	yes	characteristic
114	C₂×#3484	#95	1	1	yes	characteristic
115	C₂×#3484	#95	1	1	yes	characteristic
116	C₂×#3484	#95	1	1	yes	characteristic
117	C₂×#3484	#95	1	1	yes	characteristic
118	C₂²×#534	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×#534	#120	1	1	yes	characteristic
120	C₂²×#534	#120	1	1	yes	characteristic
121	C₂²×#534	#121	1	1	yes	characteristic
122	C₂²×#534	#121	1	1	yes	characteristic
123	C₂³×#120	C₂×C₄²	1	1	yes	characteristic
124	C₂³×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂³×#120	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×#120	C₄×D₄	1	1	yes	characteristic
127	C₂³×#120	C₄×D₄	1	1	yes	characteristic
128	C₂³×#120	C₄×D₄	1	1	yes	characteristic
129	C₂³×#120	C₄×D₄	1	1	yes	characteristic
130	C₂³×#120	#120	1	1	yes	characteristic
131	C₂³×#120	#121	1	1	yes	characteristic
132	C₂³×#120	#121	1	1	yes	characteristic
133	C₂³×#120	#123	1	1	yes	characteristic
134	#62652	#99	1	1	yes	characteristic
135	#62652	#99	1	1	yes	characteristic
136	C₂²×#518	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×#518	#123	1	1	yes	characteristic
138	C₂⁴×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
139	C₂⁴×C₂²⋊C₄	C₂²⋊Q₈	1	1	yes	characteristic
140	#36614	#99	1	2	yes	—
141	C₂×#2846	#99	1	2	yes	—
142	#35222	#99	1	2	yes	—
143	#34980	#99	1	2	yes	—
144	#34171	#95	1	1	yes	characteristic
145	#34171	#95	1	1	yes	characteristic
146	#34171	#95	1	1	yes	characteristic
147	#34171	#95	1	1	yes	characteristic
148	C₂×#2315	#117	1	1	yes	characteristic
149	C₂×#2315	C₂²⋊Q₈	1	1	yes	characteristic
150	C₂×#2308	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×#2308	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×#2308	C₄×D₄	1	1	yes	characteristic
153	C₂×#2308	C₄×D₄	1	1	yes	characteristic
154	C₂×#2308	#123	1	1	yes	characteristic
155	C₂²×#379	#117	1	1	yes	characteristic
156	C₂²×#379	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×#379	#123	1	1	yes	characteristic
158	C₂×#2273	C₂²⋊Q₈	1	1	yes	characteristic
159	#21749	#99	1	4	yes	—
160	C₂⁷	C₄×C₂²⋊C₄	1	1	yes	characteristic
161	C₂⁷	#385	1	1	yes	characteristic
162	C₂⁷	#386	1	1	yes	characteristic
163	C₂⁷	C₂×#99	1	1	yes	characteristic
164	C₂⁷	B(4,𝔽₂)	1	1	yes	characteristic
165	C₂²×H(5,𝔽₂)	#342	1	1	yes	characteristic
166	C₂²×H(5,𝔽₂)	#342	1	1	yes	characteristic
167	C₂²×H(5,𝔽₂)	#342	1	1	yes	characteristic
168	C₂²×H(5,𝔽₂)	#342	1	1	yes	characteristic
169	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
170	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
171	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
172	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
173	C₂⁴×D₄	C₂×#95	1	1	yes	characteristic
174	C₂⁴×D₄	#379	1	1	yes	characteristic
175	C₂⁴×D₄	#386	1	1	yes	characteristic
176	C₂⁴×D₄	#386	1	1	yes	characteristic
177	C₂⁴×D₄	#386	1	1	yes	characteristic
178	C₂⁴×D₄	C₂×#99	1	1	yes	characteristic
179	C₂⁴×D₄	C₂×#99	1	1	yes	characteristic
180	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
181	C₂⁴×D₄	#458	1	1	yes	characteristic
182	C₂⁵×C₄	C₂×#99	1	1	yes	characteristic
183	C₂⁵×C₄	#458	1	1	yes	characteristic
184	C₂×#535	#342	1	2	yes	—
185	C₂×#534	#342	1	2	yes	—
186	C₂×#534	C₂×#99	1	2	yes	—
187	C₂×#534	C₂×#99	1	2	yes	—
188	C₂×#534	B(4,𝔽₂)	1	2	yes	—
189	C₂×#534	#458	1	2	yes	—
190	C₂×#525	#342	1	2	yes	—
191	C₂²×#120	#342	1	2	yes	—
192	C₂×#518	#342	1	1	yes	characteristic
193	C₂×#518	#342	1	1	yes	characteristic
194	C₂×#518	#342	1	1	yes	characteristic
195	C₂×#518	#342	1	1	yes	characteristic
196	C₂³×C₂²⋊C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
197	C₂³×C₂²⋊C₄	#380	1	1	yes	characteristic
198	C₂³×C₂²⋊C₄	#380	1	1	yes	characteristic
199	C₂³×C₂²⋊C₄	#381	1	1	yes	characteristic
200	C₂³×C₂²⋊C₄	#385	1	1	yes	characteristic
201	C₂³×C₂²⋊C₄	#385	1	1	yes	characteristic
202	C₂³×C₂²⋊C₄	#385	1	1	yes	characteristic
203	#2309	#342	1	2	yes	—
204	#2308	#342	1	2	yes	—
205	#2280	#342	1	2	yes	—
206	C₂×#379	#342	1	2	yes	—
207	C₂⁶	C₂²⋊#95	1	1	yes	characteristic
208	C₂⁶	C₂×#342	1	1	yes	characteristic
209	C₂⁶	#1790	1	1	yes	characteristic
210	C₂⁶	#1867	1	1	yes	characteristic
211	C₂⁶	#1867	1	1	yes	characteristic
212	C₂⁶	#1889	1	1	yes	characteristic
213	C₂⁶	#1898	1	1	yes	characteristic
214	C₂⁶	#1898	1	1	yes	characteristic
215	C₂⁶	#1898	1	1	yes	characteristic
216	C₂⁶	#1913	1	1	yes	characteristic
217	C₂²×#41	#1304	1	2	yes	—
218	C₂³×D₄	#1304	1	2	yes	—
219	C₂³×D₄	C₂×#342	1	1	yes	characteristic
220	C₂³×D₄	C₂×#342	1	1	yes	characteristic
221	C₂³×D₄	C₂×#342	1	1	yes	characteristic
222	C₂³×D₄	C₂×#342	1	1	yes	characteristic
223	C₂³×D₄	#1781	1	2	yes	—
224	C₂³×D₄	#1790	1	2	yes	—
225	C₂³×D₄	#1889	1	1	yes	characteristic
226	C₂³×D₄	#1890	1	1	yes	characteristic
227	C₂³×D₄	#1898	1	1	yes	characteristic
228	C₂³×D₄	#1898	1	1	yes	characteristic
229	C₂³×D₄	#1898	1	1	yes	characteristic
230	C₂³×D₄	#1898	1	1	yes	characteristic
231	C₂³×D₄	#1913	1	1	yes	characteristic
232	C₂³×D₄	#1913	1	1	yes	characteristic
233	C₂⁴×C₄	C₂×#342	1	1	yes	characteristic
234	C₂⁴×C₄	#1790	1	1	yes	characteristic
235	C₂⁴×C₄	#1890	1	1	yes	characteristic
236	C₂⁴×C₄	#1898	1	1	yes	characteristic
237	C₂⁴×C₄	#1913	1	1	yes	characteristic
238	C₂×#117	#1304	1	2	yes	—
239	C₂²×C₂²⋊C₄	#1304	1	2	yes	—
240	C₂²×C₂²⋊C₄	#1819	1	2	yes	—
241	C₂²×C₂²⋊C₄	#1867	1	2	yes	—
242	C₂⁵	#7310	1	2	yes	complemented
243	C₂⁵	C₂×#1304	1	1	yes	characteristic
244	C₂⁵	#8101	1	1	yes	characteristic
245	C₂⁵	#8534	1	2	yes	—
246	C₂⁵	#9495	1	1	yes	characteristic
247	C₂⁵	#9495	1	1	yes	characteristic
248	C₂⁵	#9495	1	1	yes	characteristic
249	C₂⁵	#9499	1	1	yes	characteristic
250	C₂⁵	#10725	1	1	yes	characteristic
251	C₂⁵	#10735	1	1	yes	characteristic
252	C₂⁵	#10735	1	1	yes	characteristic
253	C₂⁵	#10745	1	1	yes	characteristic
254	C₂⁵	#10745	1	1	yes	characteristic
255	C₂⁵	#10754	1	1	yes	characteristic
256	C₂⁵	#10754	1	1	yes	characteristic
257	C₂²×Q₈	#7310	1	2	yes	complemented
258	C₂²×D₄	#7310	1	2	yes	complemented
259	C₂³×C₄	#7310	1	2	yes	complemented
260	C₂³×C₄	C₂×#1304	1	1	yes	characteristic
261	C₂³×C₄	#8101	1	1	yes	characteristic
262	C₂³×C₄	#8534	1	2	yes	—
263	C₂³×C₄	#9495	1	1	yes	characteristic
264	C₂³×C₄	#9499	1	1	yes	characteristic
265	C₂³×C₄	#9499	1	1	yes	characteristic
266	C₂³×C₄	#9499	1	1	yes	characteristic
267	C₂⁴	C₂²⋊#1304	1	1	yes	characteristic
268	C₂⁴	#106542	1	1	yes	characteristic
269	C₂⁴	#106542	1	1	yes	characteristic
270	C₂⁴	#106766	1	1	yes	characteristic
271	C₂⁴	#106766	1	1	yes	characteristic
272	C₂⁴	#107114	1	1	yes	characteristic
273	C₂⁴	#108222	1	1	yes	characteristic
274	C₂⁴	C₂×#7310	1	2	yes	—
275	C₂⁴	#139743	1	2	yes	—
276	C₂²×C₄	C₂×#7310	1	2	yes	—
277	C₂³	#72218381	1	1	yes	characteristic
278	C₂³	#72218381	1	1	yes	characteristic
279	C₂³	#72218854	1	1	yes	characteristic
280	C₂³	#72246287	1	1	yes	characteristic
281	C₂³	#72255793	1	1	yes	characteristic
282	C₂²	#72258227	1	1	yes	characteristic
283	C₂²	#72258384	1	1	yes	characteristic
284	C₂²	#72265757	1	1	yes	characteristic
285	C₂	#72266203	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266203
commutator	C₂³×#120	C₂×C₄²
Frattini, cosocle	C₂×#7670203	C₂³
prime radical	#72234665	C₂²
LCS	G ▷ C₂³×#120 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#7670203 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72314621	C₂	semidirect
2. #72314603	C₂	semidirect
3. #72314595	C₂	semidirect
5. #72314516	C₄	semidirect
6. #72314498	C₄	semidirect
8. #72310419	C₄	semidirect
10. #72302078	C₄	semidirect
11. #72302062	C₄	semidirect
12. #72302047	C₄	semidirect
13. #72302019	C₄	semidirect
17. #72237032	C₄	semidirect
18. #72237027	C₄	semidirect
27. #72226365	C₂	semidirect
30. #72225777	C₂×C₄	semidirect
31. #72225776	C₂×C₄	semidirect
32. #72225776	C₂×C₄	semidirect
33. #72225776	C₂×C₄	semidirect
44. #72219247	C₄	semidirect
45. #72219227	C₂×C₄	semidirect
47. #72218518	C₂×C₄	semidirect
49. #72218375	C₂×C₄	semidirect
50. #72218291	C₂×C₄	semidirect
99. #190147	C₂²⋊C₄	semidirect
100. #190116	C₂²⋊C₄	semidirect
101. C₂×#8547	C₂²⋊C₄	semidirect
102. C₂×#8541	C₂²⋊C₄	semidirect
242. C₂⁵	#7310	semidirect
257. C₂²×Q₈	#7310	semidirect
258. C₂²×D₄	#7310	semidirect
259. C₂³×C₄	#7310	semidirect

Maximal subgroups

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

subgroup	index
#72314621	2
#72314603	2
#72314595	2
#72314568	2
#72312950	2
#72302118	2
#72226365	2