Subgroups — #72317138

Order 4096.

At least 13202 conjugacy classes of subgroups; only the 461 normal subgroups are shown.

The subgroup lattice has 318 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72317133	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314482	C₂	1	1	yes	maximal, characteristic, complemented
3	#72312430	C₂	1	1	yes	maximal, characteristic, complemented
4	#72310388	C₂	1	2	yes	maximal, complemented
5	#72310380	C₂	1	1	yes	maximal, characteristic, complemented
6	#72309591	C₂	1	1	yes	maximal, characteristic, complemented
7	#72309574	C₂	1	2	yes	maximal, complemented
8	#72309379	C₂	1	1	yes	maximal, characteristic, complemented
9	#72266812	C₂	1	1	yes	maximal, characteristic
10	#72266811	C₄	1	1	yes	characteristic, complemented
11	#72266808	C₄	1	1	yes	characteristic, complemented
12	#72266808	C₄	1	1	yes	characteristic, complemented
13	#72266808	C₄	1	2	yes	complemented
14	#72266073	C₂²	1	1	yes	characteristic, complemented
15	#72266028	C₂²	1	1	yes	characteristic, complemented
16	#72264815	C₂	1	2	yes	maximal, complemented
17	#72251565	C₂	1	2	yes	maximal, complemented
18	#72251103	C₄	1	2	yes	complemented
19	#72240522	C₄	1	1	yes	characteristic, complemented
20	#72237398	C₂²	1	1	yes	characteristic
21	#72234663	C₂²	1	1	yes	characteristic
22	#72225619	C₂²	1	2	yes	—
23	#72225619	C₂²	1	1	yes	characteristic
24	#72225619	C₂²	1	1	yes	characteristic
25	#72219418	C₂²	1	1	yes	characteristic, complemented
26	#72219415	C₂²	1	1	yes	characteristic, complemented
27	#72219414	C₂²	1	2	yes	complemented
28	#72219243	C₂²	1	2	yes	complemented
29	#72219237	C₂²	1	2	yes	complemented
30	#72219229	C₂²	1	1	yes	characteristic, complemented
31	#72219228	C₂²	1	1	yes	characteristic, complemented
32	C₂²×#8534	C₂²	1	1	yes	characteristic
33	#72218455	C₂²	1	2	yes	complemented
34	#72218306	C₂²	1	2	yes	—
35	#72218180	C₂²	1	2	yes	complemented
36	#72218069	C₂²	1	2	yes	complemented
37	#72218069	C₂²	1	2	yes	complemented
38	#72218057	C₂²	1	2	yes	complemented
39	#72218057	C₂²	1	2	yes	complemented
40	#72216064	C₂²	1	2	yes	—
41	#10586066	C₂×C₄	1	2	yes	—
42	C₂²×#3484	C₂³	1	1	yes	characteristic
43	C₂×#60399	D₄	1	2	yes	—
44	C₂×#60399	D₄	1	2	yes	—
45	#7630939	C₂³	1	2	yes	—
46	#7630939	C₂³	1	2	yes	—
47	#7626202	D₄	1	1	yes	characteristic
48	#7626202	D₄	1	1	yes	characteristic
49	C₂×#23895	C₂×C₄	1	2	yes	—
50	C₂×#23895	C₂×C₄	1	1	yes	characteristic
51	C₂×#23895	C₂×C₄	1	1	yes	characteristic
52	#6431795	D₄	1	2	yes	—
53	#6431795	D₄	1	2	yes	—
54	#6431795	D₄	1	2	yes	—
55	#6431795	D₄	1	2	yes	—
56	#6374251	D₄	1	1	yes	characteristic
57	#6374251	D₄	1	1	yes	characteristic
58	#6374128	D₄	1	1	yes	characteristic
59	#6374128	D₄	1	1	yes	characteristic
60	C₂×#21746	D₄	1	1	yes	characteristic
61	C₂×#21746	D₄	1	1	yes	characteristic
62	#6371558	D₄	1	2	yes	—
63	#6371558	D₄	1	2	yes	—
64	#6371504	C₂×C₄	1	2	yes	—
65	#6371504	C₂×C₄	1	2	yes	—
66	#6371007	C₂×C₄	1	1	yes	characteristic
67	#6371007	C₂×C₄	1	1	yes	characteristic
68	#6370693	C₂×C₄	1	2	yes	—
69	#6370693	C₂×C₄	1	1	yes	characteristic
70	#6370693	C₂×C₄	1	1	yes	characteristic
71	C₂²×#1781	C₂³	1	1	yes	characteristic
72	C₂²×#1781	C₂³	1	1	yes	characteristic
73	C₂²×#1781	C₂³	1	1	yes	characteristic
74	C₂²×#1781	C₂³	1	1	yes	characteristic
75	#517411	C₂³	1	1	yes	characteristic
76	C₂²×#1438	C₂³	1	1	yes	characteristic
77	#392113	C₂³	1	2	yes	—
78	#392109	C₂³	1	2	yes	—
79	#373637	C₂×C₄	1	2	yes	—
80	#373637	C₂×C₄	1	1	yes	characteristic
81	#373637	C₂×C₄	1	1	yes	characteristic
82	#359549	C₂×C₄	1	2	yes	—
83	#359549	C₂×C₄	1	2	yes	—
84	#359549	C₂×C₄	1	2	yes	—
85	#359549	C₂×C₄	1	2	yes	—
86	C₂×H(7,𝔽₂)	C₂²⋊C₄	1	2	yes	—
87	C₂³×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
88	C₂⁵×D₄	C₂⁴	1	1	yes	characteristic
89	#62842	C₂²⋊C₄	1	2	yes	—
90	#62842	C₂²⋊C₄	1	2	yes	—
91	C₂×#3484	C₂²×C₄	1	2	yes	—
92	C₂×#3484	C₂²×C₄	1	2	yes	—
93	C₂×#3484	C₂×D₄	1	2	yes	—
94	C₂×#3484	C₂×D₄	1	2	yes	—
95	C₂×#3484	C₂×D₄	1	2	yes	—
96	C₂×#3484	C₂×D₄	1	2	yes	—
97	C₂×#3484	C₂×D₄	1	2	yes	—
98	C₂×#3484	C₂×D₄	1	2	yes	—
99	C₂×#3484	C₂×D₄	1	2	yes	—
100	C₂×#3484	C₂×D₄	1	2	yes	—
101	C₂²×#534	C₂×D₄	1	1	yes	characteristic
102	C₂²×#534	C₂×D₄	1	1	yes	characteristic
103	C₂³×#120	C₂×D₄	1	1	yes	characteristic
104	C₂³×#120	C₂×D₄	1	1	yes	characteristic
105	#62652	C₂²⋊C₄	1	2	yes	—
106	C₂²×#518	C₂²×C₄	1	1	yes	characteristic
107	C₂×#2883	C₂²⋊C₄	1	2	yes	—
108	C₂×#2883	C₂²⋊C₄	1	2	yes	—
109	C₂×#2883	C₂²⋊C₄	1	2	yes	—
110	C₂×#2883	C₂²⋊C₄	1	2	yes	—
111	C₂×#2883	C₂²⋊C₄	1	2	yes	—
112	C₂×#2883	C₂²⋊C₄	1	2	yes	—
113	C₂×#2883	C₂²⋊C₄	1	2	yes	—
114	C₂×#2883	C₂²⋊C₄	1	2	yes	—
115	#34176	C₂²⋊C₄	1	1	yes	characteristic
116	#34176	C₂²⋊C₄	1	1	yes	characteristic
117	#34176	C₂²⋊C₄	1	1	yes	characteristic
118	#34176	C₂²⋊C₄	1	1	yes	characteristic
119	#34171	C₂²⋊C₄	1	1	yes	characteristic
120	#34171	C₂²⋊C₄	1	1	yes	characteristic
121	#34171	C₂²⋊C₄	1	1	yes	characteristic
122	#34171	C₂²⋊C₄	1	1	yes	characteristic
123	#34171	C₂²×C₄	1	2	yes	—
124	#34171	C₂²×C₄	1	2	yes	—
125	#34171	C₂×D₄	1	2	yes	—
126	#34171	C₂×D₄	1	2	yes	—
127	#34171	C₂×D₄	1	2	yes	—
128	#34171	C₂×D₄	1	2	yes	—
129	C₂²×#379	C₂×D₄	1	1	yes	characteristic
130	C₂²×#379	C₂×D₄	1	1	yes	characteristic
131	C₂²×#379	C₂×D₄	1	1	yes	characteristic
132	C₂²×#379	C₂×D₄	1	1	yes	characteristic
133	C₂×#1819	C₂²⋊C₄	1	1	yes	characteristic
134	C₂×#1819	C₂²⋊C₄	1	1	yes	characteristic
135	C₂×#1819	C₂²⋊C₄	1	1	yes	characteristic
136	C₂×#1819	C₂²⋊C₄	1	1	yes	characteristic
137	C₂×#1781	C₂²⋊C₄	1	1	yes	characteristic
138	C₂×#1781	C₂²⋊C₄	1	1	yes	characteristic
139	C₂×#1781	C₂²⋊C₄	1	1	yes	characteristic
140	C₂×#1781	C₂²⋊C₄	1	1	yes	characteristic
141	C₂×#1781	C₂²×C₄	1	1	yes	characteristic
142	C₂×#1781	C₂²×C₄	1	1	yes	characteristic
143	C₂×#1781	C₂²×C₄	1	1	yes	characteristic
144	C₂×#1781	C₂²×C₄	1	1	yes	characteristic
145	C₂×#1781	C₂×D₄	1	1	yes	characteristic
146	C₂×#1781	C₂×D₄	1	1	yes	characteristic
147	C₂×#1781	C₂×D₄	1	1	yes	characteristic
148	C₂×#1781	C₂×D₄	1	1	yes	characteristic
149	#20354	C₂²⋊C₄	1	2	yes	—
150	#20354	C₂²⋊C₄	1	2	yes	—
151	#20354	C₂²⋊C₄	1	2	yes	—
152	#20354	C₂²⋊C₄	1	2	yes	—
153	C₂⁷	C₂²×D₄	1	1	yes	characteristic
154	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
156	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
158	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
159	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂⁴×D₄	C₂³×C₄	1	1	yes	characteristic
162	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
163	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
164	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
165	C₂⁴×D₄	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂⁴×D₄	H(5,𝔽₂)	1	2	yes	—
167	C₂⁴×D₄	H(5,𝔽₂)	1	2	yes	—
168	C₂⁴×D₄	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂⁵×C₄	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×#534	#120	1	2	yes	—
171	C₂×#534	#120	1	2	yes	—
172	C₂×#534	#120	1	2	yes	—
173	C₂×#534	#120	1	2	yes	—
174	C₂²×#120	#120	1	1	yes	characteristic
175	C₂²×#120	#120	1	1	yes	characteristic
176	C₂²×#120	#120	1	2	yes	—
177	C₂²×#120	#120	1	2	yes	—
178	C₂²×#120	#120	1	1	yes	characteristic
179	C₂²×#120	#120	1	1	yes	characteristic
180	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
185	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂×#518	C₂×C₂²⋊C₄	1	2	yes	—
188	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	#2309	#99	1	2	yes	complemented
191	#2308	#99	1	2	yes	complemented
192	#2280	#99	1	2	yes	complemented
193	C₂×#379	#99	1	2	yes	complemented
194	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂⁶	C₂×#99	1	1	yes	characteristic
203	C₂⁶	#410	1	1	yes	characteristic
204	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
205	C₂⁶	#518	1	1	yes	characteristic
206	C₂⁶	C₂×#120	1	1	yes	characteristic
207	C₂⁶	C₂×#120	1	1	yes	characteristic
208	C₂⁶	#534	1	2	yes	—
209	C₂⁶	#534	1	2	yes	—
210	C₂⁶	#535	1	1	yes	characteristic
211	C₂⁶	#535	1	1	yes	characteristic
212	C₂⁶	#535	1	1	yes	characteristic
213	C₂⁶	#535	1	1	yes	characteristic
214	C₂⁶	#535	1	1	yes	characteristic
215	C₂³×D₄	#379	1	1	yes	characteristic
216	C₂³×D₄	#379	1	1	yes	characteristic
217	C₂³×D₄	#379	1	2	yes	—
218	C₂³×D₄	C₂×#99	1	1	yes	characteristic
219	C₂³×D₄	C₂×#99	1	1	yes	characteristic
220	C₂³×D₄	#410	1	2	yes	—
221	C₂³×D₄	C₂²×C₂²⋊C₄	1	2	yes	—
222	C₂³×D₄	#518	1	2	yes	—
223	C₂³×D₄	C₂×#120	1	2	yes	—
224	C₂³×D₄	#534	1	2	yes	—
225	C₂³×D₄	#534	1	2	yes	—
226	C₂³×D₄	#534	1	2	yes	—
227	C₂³×D₄	#534	1	2	yes	—
228	C₂³×D₄	#535	1	2	yes	—
229	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
230	C₂⁴×C₄	#410	1	1	yes	characteristic
231	C₂⁴×C₄	#518	1	1	yes	characteristic
232	C₂⁴×C₄	#534	1	2	yes	—
233	C₂⁴×C₄	#534	1	2	yes	—
234	C₂⁴×C₄	#535	1	1	yes	characteristic
235	C₂×#120	B(4,𝔽₂)	1	2	yes	—
236	C₂×#120	B(4,𝔽₂)	1	2	yes	—
237	C₂×#120	#458	1	2	yes	—
238	C₂×#120	#458	1	2	yes	—
239	C₂²×C₂²⋊C₄	#379	1	2	yes	—
240	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
241	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
242	#379	C₂×#99	1	2	yes	—
243	#379	C₂×#99	1	2	yes	—
244	#379	C₂×#99	1	2	yes	—
245	#379	C₂×#99	1	2	yes	—
246	C₂⁵	#1781	1	1	yes	characteristic
247	C₂⁵	#1781	1	1	yes	characteristic
248	C₂⁵	#1782	1	2	yes	—
249	C₂⁵	C₂×#379	1	1	yes	characteristic
250	C₂⁵	#2308	1	2	yes	—
251	C₂⁵	#2309	1	1	yes	characteristic
252	C₂⁵	#2309	1	1	yes	characteristic
253	C₂⁵	#2613	1	2	yes	—
254	C₂⁵	#2613	1	1	yes	characteristic
255	C₂⁵	#2613	1	1	yes	characteristic
256	C₂⁵	#2613	1	1	yes	characteristic
257	C₂⁵	#2613	1	1	yes	characteristic
258	C₂⁵	C₂²×#99	1	1	yes	characteristic
259	C₂⁵	C₂²×#99	1	1	yes	characteristic
260	C₂⁵	C₂×#410	1	1	yes	characteristic
261	C₂⁵	C₂×#410	1	1	yes	characteristic
262	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
263	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
264	C₂⁵	C₂×#458	1	1	yes	characteristic
265	C₂⁵	C₂×#458	1	1	yes	characteristic
266	C₂⁵	#3026	1	1	yes	characteristic
267	C₂⁵	#3026	1	1	yes	characteristic
268	C₂⁵	#3026	1	1	yes	characteristic
269	C₂⁵	#3026	1	1	yes	characteristic
270	C₂²×D₄	#1790	1	2	yes	—
271	C₂²×D₄	#1790	1	2	yes	—
272	C₂³×C₄	#1781	1	1	yes	characteristic
273	C₂³×C₄	#1781	1	1	yes	characteristic
274	C₂³×C₄	#1782	1	2	yes	—
275	C₂³×C₄	#2308	1	2	yes	—
276	C₂³×C₄	#2613	1	2	yes	—
277	C₂×C₂²⋊C₄	#1790	1	2	yes	—
278	C₂×C₂²⋊C₄	#1790	1	2	yes	—
279	C₂⁴	#8534	1	2	yes	—
280	C₂⁴	#14335	1	1	yes	characteristic
281	C₂⁴	C₂×#1781	1	1	yes	characteristic
282	C₂⁴	C₂×#1782	1	1	yes	characteristic
283	C₂⁴	C₂×#1790	1	1	yes	characteristic
284	C₂⁴	C₂×#1790	1	1	yes	characteristic
285	C₂⁴	#20609	1	1	yes	characteristic
286	C₂⁴	#20609	1	1	yes	characteristic
287	C₂⁴	#21776	1	4	yes	—
288	C₂⁴	#21793	1	1	yes	characteristic
289	C₂⁴	#21793	1	1	yes	characteristic
290	C₂⁴	#21793	1	1	yes	characteristic
291	C₂⁴	#21793	1	1	yes	characteristic
292	C₂⁴	#29634	1	1	yes	characteristic
293	C₂⁴	#29634	1	1	yes	characteristic
294	C₂⁴	#29634	1	1	yes	characteristic
295	C₂⁴	#29634	1	1	yes	characteristic
296	C₂⁴	#29635	1	1	yes	characteristic
297	C₂⁴	#29635	1	1	yes	characteristic
298	C₂⁴	#29635	1	1	yes	characteristic
299	C₂⁴	#29635	1	1	yes	characteristic
300	C₂²×C₄	#8534	1	2	yes	—
301	C₂²×C₄	#21776	1	4	yes	—
302	C₂³	C₂×#8534	1	1	yes	characteristic
303	C₂³	#187929	1	1	yes	characteristic
304	C₂³	#187929	1	1	yes	characteristic
305	C₂³	#187944	1	1	yes	characteristic
306	C₂³	#382387	1	1	yes	characteristic
307	C₂³	#382387	1	1	yes	characteristic
308	C₂³	#382390	1	1	yes	characteristic
309	C₂³	#382434	1	2	yes	—
310	C₂³	#382434	1	2	yes	—
311	C₂²	#72233032	1	1	yes	characteristic
312	C₂²	#72233032	1	2	yes	—
313	C₂²	#72233032	1	1	yes	characteristic
314	C₂²	#72234620	1	2	yes	—
315	C₂²	#72265753	1	1	yes	characteristic
316	C₂	#72266251	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266251
commutator	C₂⁴×D₄	C₂³×C₄
Frattini, cosocle	C₂⁵×D₄	C₂⁴
prime radical	#72266812	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72317133	C₂	semidirect
2. #72314482	C₂	semidirect
3. #72312430	C₂	semidirect
4. #72310388	C₂	semidirect
5. #72310380	C₂	semidirect
6. #72309591	C₂	semidirect
7. #72309574	C₂	semidirect
8. #72309379	C₂	semidirect
10. #72266811	C₄	semidirect
11. #72266808	C₄	semidirect
12. #72266808	C₄	semidirect
13. #72266808	C₄	semidirect
14. #72266073	C₂²	semidirect
15. #72266028	C₂²	semidirect
16. #72264815	C₂	semidirect
17. #72251565	C₂	semidirect
18. #72251103	C₄	semidirect
19. #72240522	C₄	semidirect
25. #72219418	C₂²	semidirect
26. #72219415	C₂²	semidirect
27. #72219414	C₂²	semidirect
28. #72219243	C₂²	semidirect
29. #72219237	C₂²	semidirect
30. #72219229	C₂²	semidirect
31. #72219228	C₂²	semidirect
33. #72218455	C₂²	semidirect
35. #72218180	C₂²	semidirect
36. #72218069	C₂²	semidirect
37. #72218069	C₂²	semidirect
38. #72218057	C₂²	semidirect
39. #72218057	C₂²	semidirect
190. #2309	#99	semidirect
191. #2308	#99	semidirect
192. #2280	#99	semidirect
193. C₂×#379	#99	semidirect

Maximal subgroups

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

subgroup	index
#72317133	2
#72314482	2
#72312430	2
#72310388	2
#72310380	2
#72309591	2
#72309574	2
#72309379	2
#72266812	2
#72264815	2
#72251565	2