Subgroups — #72310618

Order 4096.

At least 17644 conjugacy classes of subgroups; only the 533 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	#72310595	C₂	1	2	yes	maximal, complemented
2	#72310535	C₂	1	2	yes	maximal, complemented
3	#72237969	C₂	1	2	yes	maximal, complemented
4	#72237965	C₂	1	4	yes	maximal, complemented
5	#72237875	C₂²	1	4	yes	complemented
6	#72237770	C₂	1	2	yes	maximal, complemented
7	#72237747	C₂²	1	4	yes	complemented
8	#72237630	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237628	C₂	1	1	yes	maximal, characteristic
10	#72237625	C₂²	1	2	yes	—
11	#72237624	C₂²	1	2	yes	—
12	#72237620	C₄	1	2	yes	complemented
13	#72237618	C₂²	1	1	yes	characteristic
14	#72237612	C₄	1	2	yes	complemented
15	#72237594	C₂²	1	4	yes	complemented
16	#72237478	C₂²	1	4	yes	complemented
17	#72234668	C₄	1	1	yes	characteristic, complemented
18	#72234625	C₄	1	1	yes	characteristic, complemented
19	#72232634	C₂²	1	2	yes	complemented
20	#72232592	C₂²	1	1	yes	characteristic
21	#72225873	C₂²	1	2	yes	complemented
22	#72225781	C₂	1	1	yes	maximal, characteristic, complemented
23	#72225777	C₂²	1	1	yes	characteristic, complemented
24	#72225776	C₂²	1	1	yes	characteristic, complemented
25	#72225733	C₂²	1	2	yes	—
26	#72219340	C₂²	1	2	yes	—
27	#72219340	C₂²	1	2	yes	complemented
28	C₂×#7670203	C₂²	1	1	yes	characteristic
29	#72217789	C₄	1	1	yes	characteristic, complemented
30	#72217511	C₄	1	1	yes	characteristic, complemented
31	C₂×#62817	C₂×C₄	1	1	yes	characteristic, complemented
32	C₂²×#3484	D₄	1	1	yes	characteristic
33	C₂²×#3484	D₄	1	1	yes	characteristic
34	C₂⁴×#120	C₂×C₄	1	1	yes	characteristic
35	C₂⁴×#120	D₄	1	1	yes	characteristic
36	#7642273	C₂×C₄	1	2	yes	—
37	C₂²×#2846	D₄	1	1	yes	characteristic
38	#7630944	C₂×C₄	1	2	yes	complemented
39	C₂×#35390	D₄	1	1	yes	characteristic
40	C₂×#35348	C₂×C₄	1	1	yes	characteristic
41	C₂×#35348	D₄	1	1	yes	characteristic
42	C₂×#35346	C₂³	1	1	yes	characteristic
43	C₂×#34981	C₂³	1	1	yes	characteristic
44	C₂×#34981	C₂³	1	1	yes	characteristic
45	C₂×#34980	C₂×C₄	1	1	yes	characteristic
46	C₂×#34980	D₄	1	1	yes	characteristic
47	#7626202	C₂×C₄	1	1	yes	characteristic, complemented
48	C₂×#34198	C₂×C₄	1	1	yes	characteristic
49	C₂×#34198	D₄	1	1	yes	characteristic
50	#6453039	C₂×C₄	1	2	yes	complemented
51	#6451874	C₂×C₄	1	2	yes	—
52	#6451874	C₂×C₄	1	2	yes	complemented
53	#6451874	C₂³	1	2	yes	—
54	C₂×#21746	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂×#21746	C₂×C₄	1	1	yes	characteristic, complemented
56	#480250	C₂³	1	4	yes	—
57	#417896	D₄	1	2	yes	—
58	#417896	D₄	1	2	yes	—
59	#417896	D₄	1	2	yes	—
60	#417896	D₄	1	2	yes	—
61	#417824	D₄	1	2	yes	—
62	#417824	D₄	1	2	yes	—
63	#417824	D₄	1	2	yes	—
64	#417824	D₄	1	2	yes	—
65	#392207	C₂×C₄	1	2	yes	complemented
66	#382422	C₂×C₄	1	2	yes	—
67	#359921	C₂×C₄	1	2	yes	complemented
68	#359705	C₂×C₄	1	2	yes	—
69	#359705	C₂×C₄	1	2	yes	complemented
70	#359705	C₂³	1	2	yes	—
71	#190147	C₂³	1	2	yes	—
72	#190116	C₂³	1	2	yes	—
73	#133916	D₄	1	8	yes	complemented
74	C₂⁸	C₂²⋊C₄	1	1	yes	characteristic
75	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
76	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
77	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×#3490	C₂²×C₄	1	2	yes	—
79	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×#534	C₂×D₄	1	1	yes	characteristic
83	C₂²×#534	C₂×D₄	1	1	yes	characteristic
84	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
85	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
86	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
87	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
88	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
89	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
90	C₂³×#120	C₂×D₄	1	1	yes	characteristic
91	C₂³×#120	C₂×D₄	1	1	yes	characteristic
92	C₂³×#120	C₂×D₄	1	1	yes	characteristic
93	C₂³×#120	C₂×D₄	1	1	yes	characteristic
94	C₂³×#120	C₂×D₄	1	1	yes	characteristic
95	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic
97	C₂⁴×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
98	C₂⁴×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
99	#35390	C₂²⋊C₄	1	2	yes	—
100	#35390	C₂²⋊C₄	1	2	yes	—
101	#35367	C₂²⋊C₄	1	2	yes	—
102	#35367	C₂²⋊C₄	1	2	yes	—
103	#35346	C₂×D₄	1	2	yes	—
104	#35346	C₂×D₄	1	2	yes	—
105	#35346	C₂×D₄	1	2	yes	—
106	#35346	C₂×D₄	1	2	yes	—
107	#34981	C₂×D₄	1	4	yes	—
108	#34981	#41	1	4	yes	—
109	#34176	C₂²×C₄	1	2	yes	—
110	C₂×#2315	C₂²×C₄	1	1	yes	characteristic
111	C₂×#2315	C₂×D₄	1	1	yes	characteristic
112	C₂×#2308	C₂²×C₄	1	1	yes	characteristic
113	C₂×#2308	C₂²×C₄	1	1	yes	characteristic
114	C₂×#2308	C₂×D₄	1	1	yes	characteristic
115	C₂×#2308	C₂×D₄	1	1	yes	characteristic
116	C₂×#2308	C₂⁴	1	1	yes	characteristic
117	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
120	C₂×#2273	C₂²⋊C₄	1	1	yes	characteristic
121	#23468	C₂²⋊C₄	1	2	yes	—
122	#23468	C₂²⋊C₄	1	2	yes	—
123	#23468	C₂×D₄	1	2	yes	—
124	#23468	C₂×D₄	1	2	yes	—
125	#23444	C₂²⋊C₄	1	2	yes	—
126	#23444	C₂²⋊C₄	1	2	yes	—
127	#23444	C₂×D₄	1	2	yes	—
128	#23444	C₂×D₄	1	2	yes	—
129	C₂×#1819	C₂²×C₄	1	2	yes	—
130	C₂×#1819	C₂²×C₄	1	2	yes	—
131	#10965	C₂²⋊C₄	1	2	yes	—
132	#10965	C₂²⋊C₄	1	2	yes	—
133	#10965	C₂²⋊C₄	1	2	yes	—
134	#10965	C₂²⋊C₄	1	2	yes	—
135	#10938	C₂²⋊C₄	1	2	yes	—
136	#10938	C₂²⋊C₄	1	2	yes	—
137	#10938	C₂²⋊C₄	1	2	yes	—
138	#10938	C₂²⋊C₄	1	2	yes	—
139	#10938	C₂×D₄	1	2	yes	—
140	#10938	C₂×D₄	1	2	yes	—
141	#10938	C₂×D₄	1	2	yes	—
142	#10938	C₂×D₄	1	2	yes	—
143	#8752	C₂×D₄	1	4	yes	complemented
144	#8747	C₂×D₄	1	4	yes	complemented
145	C₂⁷	#99	1	1	yes	characteristic
146	C₂⁷	#99	1	1	yes	characteristic
147	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
151	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
152	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
153	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
154	C₂⁴×D₄	#99	1	1	yes	characteristic
155	C₂⁴×D₄	#99	1	1	yes	characteristic
156	C₂⁴×D₄	#99	1	1	yes	characteristic
157	C₂⁴×D₄	#99	1	1	yes	characteristic
158	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
163	C₂⁵×C₄	#99	1	1	yes	characteristic
164	C₂⁵×C₄	#99	1	1	yes	characteristic
165	C₂×#543	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂×#534	#99	1	2	yes	—
167	C₂×#534	#99	1	2	yes	—
168	C₂×#534	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂×#534	#120	1	2	yes	—
170	C₂×#534	#123	1	2	yes	—
171	C₂²×#120	#99	1	2	yes	—
172	C₂²×#120	#99	1	2	yes	—
173	C₂²×#120	C₂×C₂²⋊C₄	1	2	yes	—
174	C₂²×#120	C₂×C₂²⋊C₄	1	2	yes	—
175	C₂²×#120	C₄×D₄	1	4	yes	—
176	C₂²×#120	#120	1	2	yes	—
177	C₂²×#120	#120	1	2	yes	—
178	C₂²×#120	#120	1	2	yes	—
179	C₂²×#120	#120	1	2	yes	—
180	C₂²×#120	#120	1	2	yes	—
181	C₂²×#120	#123	1	2	yes	—
182	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
184	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₂³×C₂²⋊C₄	#121	1	4	yes	—
188	C₂³×C₂²⋊C₄	C₂³×C₄	1	1	yes	characteristic
189	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
190	#2315	C₄×D₄	1	4	yes	—
191	#2308	C₂×C₂²⋊C₄	1	2	yes	—
192	#2308	C₂×C₂²⋊C₄	1	2	yes	—
193	#2308	#120	1	2	yes	—
194	#2308	#120	1	2	yes	—
195	#2308	#121	1	4	yes	—
196	#2308	C₂²×D₄	1	2	yes	—
197	#2308	C₂²×D₄	1	2	yes	—
198	#2308	C₂²×D₄	1	2	yes	—
199	#2308	C₂×#41	1	2	yes	—
200	C₂×#381	H(5,𝔽₂)	1	2	yes	—
201	C₂×#379	H(5,𝔽₂)	1	2	yes	—
202	#2273	C₂×C₂²⋊C₄	1	2	yes	—
203	#2273	C₂×C₂²⋊C₄	1	2	yes	—
204	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
205	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
206	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
207	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂⁶	#379	1	2	yes	—
209	C₂⁶	#386	1	2	yes	—
210	C₂⁶	C₂×#99	1	1	yes	characteristic
211	C₂⁶	C₂×#99	1	1	yes	characteristic
212	C₂⁶	C₂×#99	1	1	yes	characteristic
213	C₂⁶	C₂×#99	1	1	yes	characteristic
214	C₂⁶	C₂×#99	1	2	yes	—
215	C₂⁶	C₂×#99	1	1	yes	characteristic
216	C₂⁶	C₂×#99	1	1	yes	characteristic
217	C₂⁶	C₂×#99	1	1	yes	characteristic
218	C₂⁶	C₂×#99	1	1	yes	characteristic
219	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
220	C₂⁶	#534	1	1	yes	characteristic
221	C₂³×D₄	#379	1	1	yes	characteristic
222	C₂³×D₄	#379	1	1	yes	characteristic
223	C₂³×D₄	#379	1	1	yes	characteristic
224	C₂³×D₄	#379	1	1	yes	characteristic
225	C₂³×D₄	#386	1	2	yes	—
226	C₂³×D₄	C₂×#99	1	2	yes	—
227	C₂³×D₄	C₂×#99	1	1	yes	characteristic
228	C₂³×D₄	C₂×#99	1	1	yes	characteristic
229	C₂³×D₄	C₂×#99	1	1	yes	characteristic
230	C₂³×D₄	C₂×#99	1	1	yes	characteristic
231	C₂³×D₄	C₂×#120	1	1	yes	characteristic
232	C₂³×D₄	C₂×#120	1	1	yes	characteristic
233	C₂³×D₄	C₂×#120	1	1	yes	characteristic
234	C₂³×D₄	C₂×#123	1	1	yes	characteristic
235	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
236	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
237	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
238	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
239	C₂⁴×C₄	#534	1	1	yes	characteristic
240	C₂×#120	AΣL(1,𝔽₁₆)	1	4	yes	—
241	C₂×#120	AΣL(1,𝔽₁₆)	1	4	yes	—
242	C₂²×C₂²⋊C₄	#379	1	2	yes	—
243	C₂²×C₂²⋊C₄	#386	1	2	yes	—
244	C₂²×C₂²⋊C₄	#386	1	2	yes	—
245	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
246	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
247	C₂²×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	2	yes	—
248	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	2	yes	—
249	C₂²×C₂²⋊C₄	#518	1	2	yes	—
250	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
251	C₂²×C₂²⋊C₄	C₂×#121	1	2	yes	—
252	C₂²×C₂²⋊C₄	#534	1	2	yes	—
253	#381	D₄²	1	4	yes	—
254	#381	#546	1	4	yes	—
255	#379	D₄²	1	4	yes	—
256	#379	#546	1	4	yes	—
257	C₂⁵	#1781	1	1	yes	characteristic
258	C₂⁵	#1781	1	1	yes	characteristic
259	C₂⁵	#1781	1	1	yes	characteristic
260	C₂⁵	#1781	1	1	yes	characteristic
261	C₂⁵	#1898	1	2	yes	—
262	C₂⁵	#1898	1	2	yes	—
263	C₂⁵	#1898	1	2	yes	—
264	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
265	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
266	C₂⁵	#2121	1	2	yes	—
267	C₂⁵	#2121	1	2	yes	—
268	C₂⁵	C₂×#379	1	1	yes	characteristic
269	C₂⁵	C₂×#386	1	1	yes	characteristic
270	C₂⁵	#2308	1	1	yes	characteristic
271	C₂⁵	#2403	1	1	yes	characteristic
272	C₂⁵	#2405	1	1	yes	characteristic
273	C₂⁵	C₂²×#99	1	1	yes	characteristic
274	C₂⁵	C₂²×#99	1	1	yes	characteristic
275	C₂⁵	C₂²×#99	1	1	yes	characteristic
276	C₂⁵	C₂²×#99	1	1	yes	characteristic
277	C₂²×D₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
278	C₂³×C₄	#1781	1	1	yes	characteristic
279	C₂³×C₄	#1781	1	1	yes	characteristic
280	C₂³×C₄	#1781	1	1	yes	characteristic
281	C₂³×C₄	#1781	1	1	yes	characteristic
282	C₂³×C₄	#1898	1	2	yes	—
283	C₂³×C₄	#2403	1	1	yes	characteristic
284	C₂³×C₄	#2405	1	1	yes	characteristic
285	C₂×C₂²⋊C₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
286	C₂×C₂²⋊C₄	D₄×C₂²⋊C₄	1	4	yes	—
287	C₂×C₂²⋊C₄	#2410	1	4	yes	—
288	C₂⁴	#12687	1	2	yes	—
289	C₂⁴	#12707	1	2	yes	—
290	C₂⁴	#13086	1	2	yes	—
291	C₂⁴	#13086	1	2	yes	—
292	C₂⁴	#13086	1	2	yes	—
293	C₂⁴	#13086	1	2	yes	—
294	C₂⁴	#13867	1	1	yes	characteristic
295	C₂⁴	C₂×#1781	1	1	yes	characteristic
296	C₂⁴	C₂×#1781	1	1	yes	characteristic
297	C₂⁴	C₂×#1898	1	1	yes	characteristic
298	C₂⁴	C₂×#1898	1	1	yes	characteristic
299	C₂⁴	#21746	1	1	yes	characteristic
300	C₂⁴	#21746	1	1	yes	characteristic
301	C₂⁴	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
302	C₂⁴	C₂×#2121	1	2	yes	—
303	C₂²×C₄	#12687	1	2	yes	—
304	C₂²×C₄	#12707	1	2	yes	—
305	C₂³	#359428	1	1	yes	characteristic
306	C₂³	#359534	1	2	yes	—
307	C₂³	C₂×#12687	1	1	yes	characteristic
308	C₂³	#493357	1	1	yes	characteristic
309	C₂³	C₂×#12707	1	1	yes	characteristic
310	C₂³	C₂×#12707	1	1	yes	characteristic
311	C₂³	C₂×#13086	1	2	yes	—
312	C₂³	#496225	1	2	yes	—
313	C₂²	#72218057	1	1	yes	characteristic
314	C₂²	#72219294	1	1	yes	characteristic
315	C₂²	#72256085	1	1	yes	characteristic
316	C₂	#72266172	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266172
commutator	C₂³×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂×#2308	C₂⁴
prime radical	#72237628	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. #72310595	C₂	semidirect
2. #72310535	C₂	semidirect
3. #72237969	C₂	semidirect
4. #72237965	C₂	semidirect
5. #72237875	C₂²	semidirect
6. #72237770	C₂	semidirect
7. #72237747	C₂²	semidirect
8. #72237630	C₂	semidirect
12. #72237620	C₄	semidirect
14. #72237612	C₄	semidirect
15. #72237594	C₂²	semidirect
16. #72237478	C₂²	semidirect
17. #72234668	C₄	semidirect
18. #72234625	C₄	semidirect
19. #72232634	C₂²	semidirect
21. #72225873	C₂²	semidirect
22. #72225781	C₂	semidirect
23. #72225777	C₂²	semidirect
24. #72225776	C₂²	semidirect
27. #72219340	C₂²	semidirect
29. #72217789	C₄	semidirect
30. #72217511	C₄	semidirect
31. C₂×#62817	C₂×C₄	semidirect
38. #7630944	C₂×C₄	semidirect
47. #7626202	C₂×C₄	semidirect
50. #6453039	C₂×C₄	semidirect
52. #6451874	C₂×C₄	semidirect
54. C₂×#21746	C₂×C₄	semidirect
55. C₂×#21746	C₂×C₄	semidirect
65. #392207	C₂×C₄	semidirect
67. #359921	C₂×C₄	semidirect
69. #359705	C₂×C₄	semidirect
73. #133916	D₄	semidirect
143. #8752	C₂×D₄	semidirect
144. #8747	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72310595	2
#72310535	2
#72237969	2
#72237965	2
#72237770	2
#72237630	2
#72237628	2
#72225781	2