Subgroups — #72310499

Order 8192.

Only the 416 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 313 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310498	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310485	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310462	C₂	1	1	yes	maximal, characteristic, complemented
4	#72310091	C₂	1	1	yes	maximal, characteristic
5	#72309929	C₄	1	1	yes	characteristic, complemented
6	#72309776	C₄	1	1	yes	characteristic, complemented
7	#72309645	C₄	1	1	yes	characteristic, complemented
8	#72309421	C₄	1	1	yes	characteristic, complemented
9	#72309383	C₂	1	1	yes	maximal, characteristic
10	#72309378	C₄	1	2	yes	complemented
11	#72309376	C₄	1	2	yes	complemented
12	#72302118	C₂	1	1	yes	maximal, characteristic
13	#72302078	C₄	1	1	yes	characteristic, complemented
14	#72302062	C₄	1	1	yes	characteristic, complemented
15	#72302047	C₄	1	1	yes	characteristic, complemented
16	#72302019	C₄	1	1	yes	characteristic, complemented
17	#72259029	C₂²	1	1	yes	characteristic
18	#72259029	C₂²	1	1	yes	characteristic
19	#72259021	C₂×C₄	1	2	yes	—
20	#72259008	C₂×C₄	1	1	yes	characteristic, complemented
21	#72259008	D₄	1	1	yes	characteristic
22	#72235847	C₂×C₄	1	1	yes	characteristic, complemented
23	#72235847	D₄	1	1	yes	characteristic
24	#72234688	C₂²	1	1	yes	characteristic
25	#72234665	C₂²	1	1	yes	characteristic
26	#72234663	D₄	1	1	yes	characteristic
27	#72234619	Q₈	1	1	yes	characteristic
28	#72234619	Q₈	1	1	yes	characteristic
29	#72234591	C₂×C₄	1	1	yes	characteristic
30	#72234591	C₂×C₄	1	1	yes	characteristic
31	#72232884	D₄	1	1	yes	characteristic
32	#72226365	C₂	1	1	yes	maximal, characteristic, complemented
33	#72225781	C₂²	1	1	yes	characteristic
34	#72225777	C₂×C₄	1	1	yes	characteristic, complemented
35	#72225776	C₂×C₄	1	1	yes	characteristic, complemented
36	#72225776	C₂×C₄	1	1	yes	characteristic, complemented
37	#72225776	C₂×C₄	1	1	yes	characteristic, complemented
38	#72225655	C₂×C₄	1	1	yes	characteristic
39	#72225655	C₂×C₄	1	1	yes	characteristic
40	#72225629	C₂²	1	1	yes	characteristic
41	C₂×#111462	D₄	1	2	yes	—
42	#72224136	C₂×C₄	1	1	yes	characteristic
43	#72224136	C₂×C₄	1	1	yes	characteristic
44	#72219397	C₂²	1	1	yes	characteristic
45	#72219311	C₂×C₄	1	1	yes	characteristic, complemented
46	#72219310	C₂×C₄	1	1	yes	characteristic, complemented
47	#72219227	D₄	1	1	yes	characteristic
48	C₂×#106076	C₂×C₄	1	2	yes	—
49	C₂×#7670203	C₂³	1	1	yes	characteristic
50	#72218375	D₄	1	1	yes	characteristic
51	#72217511	D₄	1	1	yes	characteristic
52	#72217511	D₄	1	1	yes	characteristic
53	C₂×#115690	D₄	1	2	yes	—
54	#72216064	D₄	1	1	yes	characteristic
55	#72216064	D₄	1	1	yes	characteristic
56	#10586404	C₄⋊C₄	1	1	yes	characteristic
57	#10586404	C₄⋊C₄	1	1	yes	characteristic
58	C₂×#62817	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×#62817	C₂²⋊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	C₂×D₄	1	1	yes	characteristic
65	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
66	#7642244	C₄⋊C₄	1	1	yes	characteristic
67	#7642244	C₄⋊C₄	1	1	yes	characteristic
68	#7642244	C₄⋊C₄	1	1	yes	characteristic
69	#7642244	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×#2846	C₂×D₄	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	#41	1	1	yes	characteristic
76	C₂×#35348	C₂×Q₈	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	C₂²×C₄	1	1	yes	characteristic
82	C₂×#34980	C₂×D₄	1	1	yes	characteristic
83	#7626202	C₂²⋊C₄	1	1	yes	characteristic
84	#7626202	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×#34198	C₂²×C₄	1	1	yes	characteristic
86	C₂×#34198	#41	1	1	yes	characteristic
87	#6374210	C₄²	1	1	yes	characteristic
88	#6374210	C₄²	1	1	yes	characteristic
89	#6374210	C₄²	1	1	yes	characteristic
90	#6374210	C₄²	1	1	yes	characteristic
91	#6374128	C₄⋊C₄	1	1	yes	characteristic
92	#6374128	C₄⋊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	C₂×#10725	C₂²⋊C₄	1	2	yes	complemented
100	C₂×#8101	C₂²⋊C₄	1	2	yes	complemented
101	C₂×#8101	C₂²⋊C₄	1	2	yes	complemented
102	C₂×#7555	C₂²⋊C₄	1	2	yes	complemented
103	C₂⁸	#120	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	#62842	#99	1	1	yes	characteristic
110	#62842	#99	1	1	yes	characteristic
111	#62842	#99	1	1	yes	characteristic
112	#62842	#99	1	1	yes	characteristic
113	C₂×#3484	#95	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×#3532	#99	1	4	yes	—
118	C₂²×#534	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×#534	C₄×D₄	1	1	yes	characteristic
120	C₂²×#534	#121	1	1	yes	characteristic
121	C₂²×#534	#121	1	1	yes	characteristic
122	C₂²×#534	#123	1	1	yes	characteristic
123	C₂³×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂³×#120	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂³×#120	C₄×D₄	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	#120	1	1	yes	characteristic
129	C₂³×#120	#120	1	1	yes	characteristic
130	C₂³×#120	#121	1	1	yes	characteristic
131	C₂³×#120	#121	1	1	yes	characteristic
132	C₂³×#120	C₂²⋊Q₈	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂⁴×C₂²⋊C₄	#123	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	#34356	#99	1	4	yes	—
145	#34171	#95	1	1	yes	characteristic
146	#34171	#95	1	1	yes	characteristic
147	#34171	#95	1	1	yes	characteristic
148	#34171	#95	1	1	yes	characteristic
149	C₂×#2315	#117	1	1	yes	characteristic
150	C₂×#2315	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×#2308	C₂×C₄²	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	C₄×D₄	1	1	yes	characteristic
155	C₂×#2308	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×#379	C₄×D₄	1	1	yes	characteristic
158	C₂²×#379	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×#2273	#117	1	1	yes	characteristic
160	C₂⁷	#379	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	#385	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×H(3,𝔽₄)	#354	1	4	yes	—
185	C₂×#560	#354	1	4	yes	—
186	C₂×#551	#353	1	4	yes	—
187	C₂×D₄²	#353	1	4	yes	—
188	C₂×#535	#342	1	2	yes	—
189	C₂×#534	#342	1	2	yes	—
190	C₂×#525	#342	1	2	yes	—
191	C₂²×#120	#342	1	2	yes	—
192	C₂²×#120	C₂×#99	1	2	yes	—
193	C₂²×#120	C₂×#99	1	2	yes	—
194	C₂²×#120	B(4,𝔽₂)	1	2	yes	—
195	C₂²×#120	#458	1	2	yes	—
196	C₂×#518	#342	1	1	yes	characteristic
197	C₂×#518	#342	1	1	yes	characteristic
198	C₂×#518	#342	1	1	yes	characteristic
199	C₂×#518	#342	1	1	yes	characteristic
200	C₂³×C₂²⋊C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
201	C₂³×C₂²⋊C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
202	C₂³×C₂²⋊C₄	#380	1	1	yes	characteristic
203	C₂³×C₂²⋊C₄	#380	1	1	yes	characteristic
204	C₂³×C₂²⋊C₄	#381	1	1	yes	characteristic
205	C₂³×C₂²⋊C₄	#385	1	1	yes	characteristic
206	C₂³×C₂²⋊C₄	#385	1	1	yes	characteristic
207	#2309	#342	1	2	yes	—
208	#2308	#342	1	2	yes	—
209	#2280	#342	1	2	yes	—
210	C₂×#379	#342	1	2	yes	—
211	C₂⁶	C₂²⋊#95	1	1	yes	characteristic
212	C₂⁶	C₂×#342	1	1	yes	characteristic
213	C₂⁶	#1781	1	2	yes	—
214	C₂⁶	#1790	1	1	yes	characteristic
215	C₂⁶	#1790	1	1	yes	characteristic
216	C₂⁶	#1790	1	2	yes	—
217	C₂⁶	#1867	1	1	yes	characteristic
218	C₂⁶	#1889	1	1	yes	characteristic
219	C₂⁶	#1898	1	1	yes	characteristic
220	C₂⁶	#1898	1	1	yes	characteristic
221	C₂⁶	#1898	1	1	yes	characteristic
222	C₂⁶	#1913	1	1	yes	characteristic
223	C₂²×#41	#1304	1	2	yes	—
224	C₂³×D₄	#1304	1	2	yes	—
225	C₂³×D₄	C₂×#342	1	1	yes	characteristic
226	C₂³×D₄	C₂×#342	1	1	yes	characteristic
227	C₂³×D₄	C₂×#342	1	1	yes	characteristic
228	C₂³×D₄	C₂×#342	1	1	yes	characteristic
229	C₂³×D₄	#1889	1	1	yes	characteristic
230	C₂³×D₄	#1890	1	1	yes	characteristic
231	C₂³×D₄	#1898	1	1	yes	characteristic
232	C₂³×D₄	#1898	1	1	yes	characteristic
233	C₂³×D₄	#1898	1	1	yes	characteristic
234	C₂³×D₄	#1898	1	1	yes	characteristic
235	C₂³×D₄	#1913	1	1	yes	characteristic
236	C₂³×D₄	#1913	1	1	yes	characteristic
237	C₂⁴×C₄	C₂×#342	1	1	yes	characteristic
238	C₂⁴×C₄	#1867	1	1	yes	characteristic
239	C₂⁴×C₄	#1890	1	1	yes	characteristic
240	C₂⁴×C₄	#1898	1	1	yes	characteristic
241	C₂⁴×C₄	#1913	1	1	yes	characteristic
242	C₂×#127	#1393	1	4	yes	—
243	C₂×#124	#1393	1	4	yes	—
244	C₂×#123	#1393	1	4	yes	—
245	C₂×#120	#1393	1	4	yes	—
246	C₂×#120	C₂×#353	1	4	yes	—
247	C₂×#120	C₂×#354	1	4	yes	—
248	C₂×#117	#1304	1	2	yes	—
249	C₂²×C₂²⋊C₄	#1304	1	2	yes	—
250	C₂²×C₂²⋊C₄	#1819	1	2	yes	—
251	C₂²×C₂²⋊C₄	#1867	1	2	yes	—
252	C₂⁵	#7307	1	2	yes	complemented
253	C₂⁵	C₂×#1304	1	1	yes	characteristic
254	C₂⁵	#8101	1	1	yes	characteristic
255	C₂⁵	#8534	1	2	yes	—
256	C₂⁵	#8534	1	2	yes	—
257	C₂⁵	#9495	1	1	yes	characteristic
258	C₂⁵	#9495	1	1	yes	characteristic
259	C₂⁵	#9495	1	1	yes	characteristic
260	C₂⁵	#9499	1	1	yes	characteristic
261	C₂⁵	#10725	1	1	yes	characteristic
262	C₂⁵	#10735	1	1	yes	characteristic
263	C₂⁵	#10735	1	1	yes	characteristic
264	C₂⁵	#10745	1	1	yes	characteristic
265	C₂⁵	#10745	1	1	yes	characteristic
266	C₂⁵	#10754	1	1	yes	characteristic
267	C₂⁵	#10754	1	1	yes	characteristic
268	C₂⁵	#12693	1	2	yes	—
269	C₂⁵	#12694	1	2	yes	—
270	C₂⁵	#12707	1	2	yes	—
271	C₂⁵	#12707	1	2	yes	—
272	C₂²×Q₈	#7307	1	2	yes	complemented
273	C₂²×D₄	#7307	1	2	yes	complemented
274	C₂²×D₄	C₂×#1393	1	4	yes	—
275	C₂³×C₄	#7307	1	2	yes	complemented
276	C₂³×C₄	C₂×#1304	1	1	yes	characteristic
277	C₂³×C₄	#8101	1	1	yes	characteristic
278	C₂³×C₄	#9495	1	1	yes	characteristic
279	C₂³×C₄	#9499	1	1	yes	characteristic
280	C₂³×C₄	#9499	1	1	yes	characteristic
281	C₂³×C₄	#9499	1	1	yes	characteristic
282	C₂×C₂²⋊C₄	#7307	1	4	yes	complemented
283	C₂×C₂²⋊C₄	#12856	1	4	yes	—
284	C₂×C₄²	#7307	1	4	yes	complemented
285	C₂⁴	C₂²⋊#1304	1	1	yes	characteristic
286	C₂⁴	#106542	1	1	yes	characteristic
287	C₂⁴	#106542	1	1	yes	characteristic
288	C₂⁴	#106766	1	1	yes	characteristic
289	C₂⁴	#106766	1	1	yes	characteristic
290	C₂⁴	#107114	1	1	yes	characteristic
291	C₂⁴	#108222	1	1	yes	characteristic
292	C₂⁴	C₂×#7307	1	2	yes	—
293	C₂⁴	#133290	1	2	yes	—
294	C₂⁴	#133290	1	2	yes	—
295	C₂⁴	#133291	1	2	yes	—
296	C₂⁴	#133291	1	2	yes	—
297	C₂⁴	#139743	1	2	yes	—
298	C₂⁴	#146465	1	2	yes	—
299	C₂⁴	#146479	1	2	yes	—
300	C₂²×C₄	C₂×#7307	1	2	yes	—
301	C₂³	#72218498	1	1	yes	characteristic
302	C₂³	#72218854	1	1	yes	characteristic
303	C₂³	#72219248	1	1	yes	characteristic
304	C₂³	#72236978	1	2	yes	—
305	C₂³	#72246287	1	1	yes	characteristic
306	C₂³	#72255793	1	1	yes	characteristic
307	C₂³	#72258001	1	2	yes	—
308	C₂²	#72258075	1	1	yes	characteristic
309	C₂²	#72258227	1	1	yes	characteristic
310	C₂²	#72258384	1	1	yes	characteristic
311	C₂	#72266157	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266157
commutator	C₂×#2308	C₂×C₄²
Frattini, cosocle	C₂×#7670203	C₂³
prime radical	#72234665	C₂²
LCS	G ▷ C₂×#2308 ▷ 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. #72310498	C₂	semidirect
2. #72310485	C₂	semidirect
3. #72310462	C₂	semidirect
5. #72309929	C₄	semidirect
6. #72309776	C₄	semidirect
7. #72309645	C₄	semidirect
8. #72309421	C₄	semidirect
10. #72309378	C₄	semidirect
11. #72309376	C₄	semidirect
13. #72302078	C₄	semidirect
14. #72302062	C₄	semidirect
15. #72302047	C₄	semidirect
16. #72302019	C₄	semidirect
20. #72259008	C₂×C₄	semidirect
22. #72235847	C₂×C₄	semidirect
32. #72226365	C₂	semidirect
34. #72225777	C₂×C₄	semidirect
35. #72225776	C₂×C₄	semidirect
36. #72225776	C₂×C₄	semidirect
37. #72225776	C₂×C₄	semidirect
45. #72219311	C₂×C₄	semidirect
46. #72219310	C₂×C₄	semidirect
99. C₂×#10725	C₂²⋊C₄	semidirect
100. C₂×#8101	C₂²⋊C₄	semidirect
101. C₂×#8101	C₂²⋊C₄	semidirect
102. C₂×#7555	C₂²⋊C₄	semidirect
252. C₂⁵	#7307	semidirect
272. C₂²×Q₈	#7307	semidirect
273. C₂²×D₄	#7307	semidirect
275. C₂³×C₄	#7307	semidirect
282. C₂×C₂²⋊C₄	#7307	semidirect
284. C₂×C₄²	#7307	semidirect

Maximal subgroups

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

subgroup	index
#72310498	2
#72310485	2
#72310462	2
#72310091	2
#72309383	2
#72302118	2
#72226365	2