Subgroups — #72262778

Order 4096.

At least 10475 conjugacy classes of subgroups; only the 813 normal subgroups are shown.

The subgroup lattice has 322 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262772	C₂	1	1	yes	maximal, characteristic, complemented
2	#72262730	C₂	1	2	yes	maximal, complemented
3	#72262704	C₂	1	1	yes	maximal, characteristic, complemented
4	#72262693	C₂	1	2	yes	maximal, complemented
5	#72262683	C₂	1	2	yes	maximal, complemented
6	#72262671	C₂²	1	1	yes	characteristic, complemented
7	#72262671	C₂²	1	1	yes	characteristic, complemented
8	#72256300	C₂	1	4	yes	maximal, complemented
9	#72256250	C₂	1	4	yes	maximal, complemented
10	#72256064	C₂	1	4	yes	maximal, complemented
11	#72256050	C₂²	1	4	yes	complemented
12	#72256041	C₂²	1	4	yes	complemented
13	#72256033	C₂²	1	4	yes	complemented
14	#72255894	C₂	1	4	yes	maximal, complemented
15	#72255893	C₂²	1	4	yes	complemented
16	#72255891	C₂²	1	4	yes	complemented
17	#72255871	C₂²	1	4	yes	complemented
18	#72255838	C₂²	1	4	yes	complemented
19	#72255826	C₂²	1	4	yes	complemented
20	#72255777	C₂²	1	4	yes	complemented
21	#72255774	C₂²	1	4	yes	complemented
22	C₂×#144380	C₂²	1	2	yes	complemented
23	#72251552	C₂²	1	4	yes	complemented
24	#72249600	C₂	1	2	yes	maximal, complemented
25	#72249592	C₂²	1	2	yes	complemented
26	#72249592	C₂²	1	2	yes	complemented
27	#72249587	C₂²	1	4	yes	complemented
28	#72249584	C₂²	1	2	yes	complemented
29	#72240083	C₂²	1	2	yes	complemented
30	#72238974	C₂²	1	2	yes	complemented
31	#72238598	C₂	1	1	yes	maximal, characteristic, complemented
32	#72237900	C₂²	1	4	yes	complemented
33	#72237480	C₂	1	2	yes	maximal, complemented
34	#72237476	C₂²	1	4	yes	complemented
35	#72237473	C₂²	1	4	yes	complemented
36	#72237472	C₂²	1	4	yes	complemented
37	#72237467	C₂²	1	4	yes	complemented
38	#72237039	C₂	1	2	yes	maximal, complemented
39	#72237037	C₂²	1	4	yes	complemented
40	#72237034	C₂²	1	2	yes	complemented
41	#72237033	C₂²	1	2	yes	complemented
42	#72236636	C₂²	1	4	yes	complemented
43	#72236237	C₂²	1	1	yes	characteristic, complemented
44	#72235238	C₂²	1	4	yes	—
45	#72235234	C₂²	1	2	yes	complemented
46	#72234953	C₂²	1	4	yes	complemented
47	#72234668	C₂²	1	1	yes	characteristic
48	#72234668	C₂²	1	1	yes	characteristic
49	#72234596	C₂²	1	4	yes	complemented
50	C₂×#511771	C₂²	1	4	yes	complemented
51	C₂×#188035	C₂²	1	1	yes	characteristic, complemented
52	#72225744	C₂²	1	2	yes	—
53	#72225712	C₂²	1	2	yes	complemented
54	#72225665	C₂²	1	1	yes	characteristic
55	#72225635	C₂²	1	2	yes	—
56	#72225635	C₂²	1	2	yes	complemented
57	#72224136	C₂²	1	2	yes	—
58	#72219646	C₂²	1	2	yes	complemented
59	#72219424	C₂²	1	2	yes	complemented
60	#72219314	C₂²	1	2	yes	—
61	#72219158	C₂²	1	4	yes	complemented
62	#72218838	C₂²	1	4	yes	—
63	#72218104	C₂²	1	4	yes	complemented
64	#72218104	C₂²	1	4	yes	complemented
65	C₂×#144378	C₂²	1	2	yes	complemented
66	C₂×#139731	C₂²	1	4	yes	complemented
67	#10586404	C₂³	1	2	yes	—
68	C₂³×D₄²	C₂³	1	1	yes	characteristic
69	C₂×#34980	C₂³	1	1	yes	characteristic
70	#7626230	C₂³	1	2	yes	—
71	#7626202	C₂³	1	1	yes	characteristic
72	#7626202	C₂³	1	2	yes	—
73	#7626202	C₂³	1	1	yes	characteristic
74	#6442728	D₄	1	4	yes	complemented
75	C₂×#23895	C₂³	1	2	yes	—
76	#6437621	C₂³	1	4	yes	—
77	C₂×#23900	C₂³	1	4	yes	—
78	C₂×#23900	C₂³	1	4	yes	—
79	C₂×#23906	C₂³	1	4	yes	—
80	#6431795	C₂³	1	2	yes	—
81	#6374128	C₂³	1	2	yes	—
82	C₂×#21746	C₂³	1	1	yes	characteristic
83	C₂×#21746	C₂³	1	1	yes	characteristic
84	C₂²×#1781	D₄	1	2	yes	complemented
85	#4420586	D₄	1	4	yes	complemented
86	#663621	D₄	1	2	yes	complemented
87	#647656	D₄	1	2	yes	complemented
88	#617710	C₂³	1	2	yes	—
89	#616960	C₂³	1	2	yes	—
90	#556752	C₂³	1	1	yes	characteristic
91	C₂×#12707	C₂³	1	4	yes	—
92	C₂×#12705	C₂³	1	4	yes	—
93	C₂×#12693	C₂³	1	4	yes	—
94	#493357	C₂³	1	4	yes	—
95	C₂×#12687	C₂³	1	4	yes	—
96	C₂×#12687	C₂³	1	4	yes	—
97	#392030	C₂³	1	2	yes	—
98	#382546	C₂³	1	2	yes	—
99	#382544	C₂³	1	2	yes	—
100	#382544	C₂³	1	2	yes	—
101	#382434	C₂³	1	4	yes	—
102	#364532	D₄	1	8	yes	complemented
103	#364173	D₄	1	4	yes	complemented
104	#359705	D₄	1	4	yes	complemented
105	#359558	C₂³	1	4	yes	—
106	#359549	C₂³	1	4	yes	—
107	#359549	C₂³	1	4	yes	—
108	#359442	C₂³	1	4	yes	—
109	#359442	C₂³	1	4	yes	—
110	#355913	D₄	1	2	yes	complemented
111	#188109	C₂³	1	2	yes	—
112	#188049	C₂³	1	2	yes	—
113	C₂×#8546	C₂³	1	2	yes	—
114	C₂×#8541	C₂³	1	2	yes	—
115	#187944	C₂³	1	4	yes	—
116	#186835	C₂³	1	4	yes	—
117	#146479	C₂³	1	4	yes	—
118	#146475	C₂³	1	4	yes	—
119	#146475	C₂³	1	4	yes	—
120	#128033	C₂³	1	4	yes	—
121	C₂×#7316	C₂³	1	4	yes	—
122	C₂×#7314	C₂³	1	4	yes	—
123	C₂×#7311	C₂³	1	2	yes	—
124	C₂×#7310	C₂³	1	4	yes	—
125	C₂×#7310	C₂³	1	2	yes	—
126	#128009	C₂³	1	4	yes	—
127	#107114	C₂³	1	4	yes	—
128	#107114	C₂³	1	4	yes	—
129	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
130	#62842	C₂×D₄	1	2	yes	complemented
131	C₂×#3484	C₂⁴	1	4	yes	—
132	C₂²×D₄²	C₂×D₄	1	2	yes	—
133	C₂²×D₄²	C₂×D₄	1	2	yes	—
134	C₂²×#534	C₂⁴	1	2	yes	—
135	C₂³×#127	C₂⁴	1	1	yes	characteristic
136	C₂³×#120	C₂⁴	1	1	yes	characteristic
137	C₂³×#120	C₂⁴	1	1	yes	characteristic
138	C₂²×B(4,𝔽₂)	C₂×D₄	1	8	yes	—
139	C₂×#2883	C₂×D₄	1	4	yes	complemented
140	C₂×#2849	C₂×D₄	1	1	yes	characteristic
141	C₂×#2849	#41	1	1	yes	characteristic
142	C₂×#2846	C₂×D₄	1	1	yes	characteristic
143	C₂×#2846	#41	1	1	yes	characteristic
144	C₂×#2613	C₂×D₄	1	2	yes	—
145	C₂×#2613	#41	1	2	yes	—
146	#34198	C₂×D₄	1	4	yes	—
147	#34198	C₂×D₄	1	4	yes	—
148	#34176	C₂×D₄	1	2	yes	—
149	#34171	C₂⁴	1	2	yes	—
150	#34171	C₂⁴	1	2	yes	—
151	C₂×#2308	C₂×D₄	1	2	yes	—
152	C₂²×#379	C₂×D₄	1	2	yes	—
153	C₂²×#379	C₂×D₄	1	2	yes	—
154	C₂²×#379	C₂×D₄	1	1	yes	characteristic
155	C₂×#1889	C₂×D₄	1	8	yes	—
156	C₂×#1819	C₂×D₄	1	2	yes	—
157	#20633	C₂×D₄	1	4	yes	complemented
158	#20609	C₂×D₄	1	8	yes	complemented
159	C₂×#1783	C₂⁴	1	4	yes	—
160	C₂×#1781	C₂×D₄	1	8	yes	—
161	C₂×#1781	C₂⁴	1	4	yes	—
162	C₂×#1781	C₂⁴	1	2	yes	—
163	C₂×#1781	C₂⁴	1	2	yes	—
164	C₂×#1781	C₂⁴	1	4	yes	—
165	#20354	C₂×D₄	1	2	yes	complemented
166	#15948	C₂×D₄	1	4	yes	—
167	#15947	C₂×D₄	1	4	yes	—
168	#15261	C₂×D₄	1	4	yes	—
169	#15258	C₂×D₄	1	4	yes	—
170	#14657	C₂×D₄	1	2	yes	—
171	#14335	C₂⁴	1	2	yes	—
172	#14134	C₂×D₄	1	2	yes	—
173	#10768	C₂×D₄	1	4	yes	complemented
174	#10754	C₂×D₄	1	8	yes	complemented
175	#10735	C₂×D₄	1	8	yes	—
176	C₂⁷	#121	1	1	yes	characteristic, complemented
177	C₂⁷	C₂²×D₄	1	1	yes	characteristic
178	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂²×H(5,𝔽₂)	C₂²×D₄	1	4	yes	—
180	C₂⁴×D₄	#121	1	2	yes	complemented
181	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
182	C₂⁴×D₄	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂⁴×D₄	C₂⁵	1	1	yes	characteristic
184	C₂⁵×C₄	#121	1	1	yes	characteristic, complemented
185	C₂²×#127	C₂²×D₄	1	2	yes	—
186	C₂²×#127	H(5,𝔽₂)	1	2	yes	—
187	C₂²×#123	C₂²×D₄	1	2	yes	—
188	C₂²×#123	C₂×#41	1	2	yes	—
189	C₂²×#120	C₂²×D₄	1	4	yes	—
190	C₂²×#120	C₂²×D₄	1	2	yes	—
191	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
192	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
193	C₂²×#120	C₂²×D₄	1	2	yes	—
194	C₂²×#120	C₂×#41	1	2	yes	—
195	C₂²×#120	C₂×#41	1	1	yes	characteristic
196	C₂²×#120	C₂×#41	1	1	yes	characteristic
197	C₂²×#120	H(5,𝔽₂)	1	2	yes	—
198	C₂²×#120	H(5,𝔽₂)	1	4	yes	—
199	C₂×#518	C₂²×D₄	1	4	yes	—
200	C₂×#518	C₂²×D₄	1	2	yes	—
201	C₂×#518	C₂²×D₄	1	2	yes	—
202	C₂×#518	H(5,𝔽₂)	1	2	yes	—
203	C₂×#518	H(5,𝔽₂)	1	2	yes	—
204	C₂³×C₂²⋊C₄	#120	1	4	yes	—
205	C₂³×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
206	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
207	C₂³×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂³×C₄²	H(5,𝔽₂)	1	1	yes	characteristic
209	C₂²×#99	C₂²×D₄	1	8	yes	—
210	#2315	#120	1	4	yes	—
211	#2308	C₂²×D₄	1	4	yes	—
212	C₂×#381	#121	1	2	yes	complemented
213	C₂×#381	#121	1	2	yes	complemented
214	C₂×#379	#120	1	2	yes	—
215	C₂×#379	#120	1	2	yes	—
216	C₂×#379	#121	1	1	yes	characteristic, complemented
217	C₂×#379	#121	1	2	yes	complemented
218	C₂×#379	#121	1	1	yes	characteristic, complemented
219	C₂×#379	#121	1	2	yes	complemented
220	C₂×#379	#121	1	2	yes	complemented
221	C₂×#379	C₂²×D₄	1	4	yes	—
222	C₂×#379	C₂²×D₄	1	4	yes	—
223	#2273	#120	1	2	yes	—
224	#2273	#120	1	2	yes	—
225	#1438	C₂²×D₄	1	4	yes	—
226	C₂⁶	C₂×#120	1	2	yes	—
227	C₂⁶	C₂×#121	1	1	yes	characteristic
228	C₂⁶	C₂×#121	1	1	yes	characteristic
229	C₂⁶	C₂×#121	1	2	yes	—
230	C₂⁶	#534	1	2	yes	—
231	C₂⁶	#535	1	2	yes	—
232	C₂⁶	C₂³×D₄	1	1	yes	characteristic
233	C₂⁶	C₂×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂³×D₄	#534	1	2	yes	—
235	C₂³×D₄	C₂³×D₄	1	1	yes	characteristic
236	C₂³×D₄	C₂³×D₄	1	2	yes	—
237	C₂³×D₄	C₂²×#41	1	1	yes	characteristic
238	C₂³×D₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
239	C₂³×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
240	C₂³×D₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂³×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
242	C₂³×D₄	#585	1	2	yes	—
243	C₂⁴×C₄	C₂×#121	1	2	yes	—
244	C₂⁴×C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
245	C₂×#120	D₄²	1	4	yes	—
246	C₂×#120	D₄²	1	4	yes	—
247	C₂×#120	#546	1	4	yes	—
248	C₂×#120	#546	1	4	yes	—
249	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
250	C₂²×C₂²⋊C₄	C₂×#120	1	4	yes	—
251	C₂²×C₂²⋊C₄	C₂×#120	1	4	yes	—
252	C₂²×C₂²⋊C₄	C₂×#121	1	2	yes	—
253	C₂²×C₂²⋊C₄	C₂×#121	1	2	yes	—
254	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
255	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
256	C₂²×C₂²⋊C₄	#534	1	2	yes	—
257	C₂²×C₂²⋊C₄	#535	1	2	yes	—
258	C₂²×C₄²	#534	1	2	yes	—
259	#379	B(4,𝔽₂)	1	4	yes	complemented
260	#379	B(4,𝔽₂)	1	4	yes	complemented
261	C₂⁵	#2408	1	2	yes	—
262	C₂⁵	#2408	1	2	yes	—
263	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
264	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
265	C₂⁵	#3025	1	2	yes	—
266	C₂⁵	#3025	1	2	yes	—
267	C₂⁵	C₂²×#120	1	1	yes	characteristic
268	C₂⁵	C₂²×#121	1	1	yes	characteristic
269	C₂⁵	C₂×#534	1	1	yes	characteristic
270	C₂⁵	C₂×#535	1	1	yes	characteristic
271	C₂⁵	#3484	1	1	yes	characteristic
272	C₂⁵	#3484	1	2	yes	—
273	C₂⁵	#3484	1	1	yes	characteristic
274	C₂⁵	#3484	1	1	yes	characteristic
275	C₂⁵	#3485	1	1	yes	characteristic
276	C₂⁵	#3490	1	1	yes	characteristic
277	C₂⁵	#3534	1	2	yes	—
278	C₂²×D₄	#3459	1	2	yes	—
279	C₂²×D₄	C₂×D₄²	1	2	yes	—
280	C₂²×D₄	C₂×#546	1	2	yes	—
281	C₂²×D₄	#3488	1	2	yes	—
282	C₂²×D₄	#3488	1	2	yes	—
283	C₂²×D₄	#3496	1	2	yes	—
284	C₂²×D₄	#3537	1	2	yes	—
285	C₂²×D₄	#3537	1	2	yes	—
286	C₂³×C₄	C₂×#534	1	1	yes	characteristic
287	C₂³×C₄	#3484	1	2	yes	—
288	C₂³×C₄	#3490	1	1	yes	characteristic
289	C₂³×C₄	#3534	1	2	yes	—
290	C₂×C₂²⋊C₄	#2408	1	2	yes	—
291	C₂×C₂²⋊C₄	#2408	1	2	yes	—
292	C₂×C₂²⋊C₄	C₂×B(4,𝔽₂)	1	8	yes	—
293	C₂⁴	#24260	1	2	yes	—
294	C₂⁴	#24260	1	2	yes	—
295	C₂⁴	#24260	1	2	yes	—
296	C₂⁴	#24260	1	2	yes	—
297	C₂⁴	C₂×#2408	1	1	yes	characteristic
298	C₂⁴	#34981	1	1	yes	characteristic
299	C₂⁴	#34984	1	1	yes	characteristic
300	C₂⁴	#35346	1	1	yes	characteristic
301	C₂⁴	#35512	1	1	yes	characteristic
302	C₂⁴	#36717	1	1	yes	characteristic
303	C₂⁴	#36721	1	1	yes	characteristic
304	C₂⁴	#37672	1	2	yes	—
305	C₂⁴	#37744	1	2	yes	—
306	C₂⁴	C₂²×B(4,𝔽₂)	1	2	yes	—
307	C₂⁴	C₂×#3025	1	2	yes	—
308	C₂²×C₄	#37672	1	2	yes	—
309	C₂²×C₄	#37744	1	2	yes	—
310	C₂³	#625760	1	1	yes	characteristic
311	C₂³	C₂×#24260	1	2	yes	—
312	C₂³	#6499867	1	1	yes	characteristic
313	C₂³	#6499869	1	1	yes	characteristic
314	C₂³	#6499874	1	4	yes	—
315	C₂³	#6512843	1	2	yes	—
316	C₂²	#72210439	1	2	yes	—
317	C₂²	#72225647	1	2	yes	—
318	C₂²	#72240633	1	2	yes	—
319	C₂²	#72251064	1	1	yes	characteristic
320	C₂	#72251598	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72251598
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁵
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ 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. #72262772	C₂	semidirect
2. #72262730	C₂	semidirect
3. #72262704	C₂	semidirect
4. #72262693	C₂	semidirect
5. #72262683	C₂	semidirect
6. #72262671	C₂²	semidirect
7. #72262671	C₂²	semidirect
8. #72256300	C₂	semidirect
9. #72256250	C₂	semidirect
10. #72256064	C₂	semidirect
11. #72256050	C₂²	semidirect
12. #72256041	C₂²	semidirect
13. #72256033	C₂²	semidirect
14. #72255894	C₂	semidirect
15. #72255893	C₂²	semidirect
16. #72255891	C₂²	semidirect
17. #72255871	C₂²	semidirect
18. #72255838	C₂²	semidirect
19. #72255826	C₂²	semidirect
20. #72255777	C₂²	semidirect
21. #72255774	C₂²	semidirect
22. C₂×#144380	C₂²	semidirect
23. #72251552	C₂²	semidirect
24. #72249600	C₂	semidirect
25. #72249592	C₂²	semidirect
26. #72249592	C₂²	semidirect
27. #72249587	C₂²	semidirect
28. #72249584	C₂²	semidirect
29. #72240083	C₂²	semidirect
30. #72238974	C₂²	semidirect
31. #72238598	C₂	semidirect
32. #72237900	C₂²	semidirect
33. #72237480	C₂	semidirect
34. #72237476	C₂²	semidirect
35. #72237473	C₂²	semidirect
36. #72237472	C₂²	semidirect
37. #72237467	C₂²	semidirect
38. #72237039	C₂	semidirect
39. #72237037	C₂²	semidirect
40. #72237034	C₂²	semidirect
41. #72237033	C₂²	semidirect
42. #72236636	C₂²	semidirect
43. #72236237	C₂²	semidirect
45. #72235234	C₂²	semidirect
46. #72234953	C₂²	semidirect
49. #72234596	C₂²	semidirect
50. C₂×#511771	C₂²	semidirect
51. C₂×#188035	C₂²	semidirect
53. #72225712	C₂²	semidirect
56. #72225635	C₂²	semidirect
58. #72219646	C₂²	semidirect
59. #72219424	C₂²	semidirect
61. #72219158	C₂²	semidirect
63. #72218104	C₂²	semidirect
64. #72218104	C₂²	semidirect
65. C₂×#144378	C₂²	semidirect
66. C₂×#139731	C₂²	semidirect
74. #6442728	D₄	semidirect
84. C₂²×#1781	D₄	semidirect
85. #4420586	D₄	semidirect
86. #663621	D₄	semidirect
87. #647656	D₄	semidirect
102. #364532	D₄	semidirect
103. #364173	D₄	semidirect
104. #359705	D₄	semidirect
110. #355913	D₄	semidirect
130. #62842	C₂×D₄	semidirect
139. C₂×#2883	C₂×D₄	semidirect
157. #20633	C₂×D₄	semidirect
158. #20609	C₂×D₄	semidirect
165. #20354	C₂×D₄	semidirect
173. #10768	C₂×D₄	semidirect
174. #10754	C₂×D₄	semidirect
176. C₂⁷	#121	semidirect
180. C₂⁴×D₄	#121	semidirect
184. C₂⁵×C₄	#121	semidirect
212. C₂×#381	#121	semidirect
213. C₂×#381	#121	semidirect
216. C₂×#379	#121	semidirect
217. C₂×#379	#121	semidirect
218. C₂×#379	#121	semidirect
219. C₂×#379	#121	semidirect
220. C₂×#379	#121	semidirect
259. #379	B(4,𝔽₂)	semidirect
260. #379	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72262772	2
#72262730	2
#72262704	2
#72262693	2
#72262683	2
#72256300	2
#72256250	2
#72256064	2
#72255894	2
#72249600	2
#72238598	2
#72237480	2
#72237039	2