Subgroups — #72216544

Order 2048.

563 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72216247	C₂	1	1	yes	maximal, characteristic
2	#72216215	C₂	1	1	yes	maximal, characteristic
3	#72216182	C₂	1	1	yes	maximal, characteristic
4	#151981	—	2	1	no	—
5	C₂×#7456	C₄	1	2	yes	—
6	#129090	C₄	1	1	yes	characteristic
7	#129088	C₄	1	1	yes	characteristic
8	#129051	—	2	1	no	—
9	C₈×#346	—	2	1	no	—
10	C₈×#346	C₂²	1	1	yes	characteristic
11	C₂×#7052	—	2	1	no	—
12	C₂×#7052	C₄	1	1	yes	characteristic
13	C₂×C₁₆²	C₄	1	1	yes	characteristic
14	#94018	—	2	1	no	—
15	#93264	—	2	1	no	—
16	C₂×#1422	—	2	2	no	—
17	C₂×#1398	—	4	1	no	—
18	C₄×#364	—	2	1	no	—
19	C₄×#346	—	2	2	no	—
20	C₄×#346	C₂×C₄	1	1	yes	characteristic
21	C₄×#346	D₄	1	1	yes	characteristic
22	C₂×#1369	—	2	1	no	—
23	#11999	—	2	1	no	—
24	#11405	—	4	1	no	—
25	C₂×#1311	—	2	2	no	—
26	C₂×#1311	C₂×C₄	1	1	yes	characteristic
27	C₂×#1311	Q₈	1	1	yes	characteristic
28	C₂×C₈×C₁₆	—	2	1	no	—
29	C₂×C₈×C₁₆	—	2	1	no	—
30	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₆	D₄	1	1	yes	characteristic
32	#8266	—	2	1	no	—
33	C₄×C₈²	D₄	1	1	yes	characteristic
34	#7468	—	4	1	no	—
35	#7456	—	2	2	no	—
36	#7456	C₈	1	4	yes	complemented
37	#7161	C₈	1	2	yes	—
38	#7157	C₈	1	2	yes	—
39	#7140	—	4	1	no	—
40	#7052	—	2	4	no	—
41	#7052	—	2	1	no	—
42	#7052	C₈	1	2	yes	complemented
43	C₁₆²	—	2	1	no	—
44	C₁₆²	C₈	1	2	yes	complemented
45	C₂×M₆₄	—	4	2	no	—
46	C₂×#364	—	2	2	no	—
47	C₄×M₃₂	—	4	1	no	—
48	C₂×#346	—	2	8	no	—
49	C₂×#346	C₂×C₈	1	1	yes	characteristic
50	C₂×#346	C₂×C₈	1	1	yes	characteristic
51	C₂×#346	M₁₆	1	1	yes	characteristic
52	C₂×#346	M₁₆	1	1	yes	characteristic
53	C₂×#346	D₈	1	1	yes	characteristic
54	C₂×#346	SD₁₆	1	1	yes	characteristic
55	C₂×#346	SD₁₆	1	1	yes	characteristic
56	C₂×#346	Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₆	—	2	2	no	—
58	C₂×C₄×C₁₆	—	2	2	no	—
59	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
60	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
63	C₄²×C₈	—	2	1	no	—
64	C₄²×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	#1564	—	2	2	no	—
66	#1450	—	4	1	no	—
67	C₂×C₈²	—	2	1	no	—
68	C₂×C₈²	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
70	#1422	—	2	4	no	—
71	#1422	—	4	2	no	—
72	#1398	—	8	2	no	—
73	#1376	—	4	1	no	—
74	#1369	—	4	2	no	—
75	#1368	—	4	2	no	—
76	#1313	—	8	2	no	—
77	#1311	—	2	8	no	—
78	#1311	—	2	1	no	—
79	#1311	—	2	1	no	—
80	#1311	C₂×C₈	1	2	yes	—
81	#1311	M₁₆	1	2	yes	—
82	C₈×C₁₆	—	2	2	no	—
83	C₈×C₁₆	—	2	2	no	—
84	C₈×C₁₆	—	2	2	no	—
85	C₈×C₁₆	—	2	2	no	—
86	C₈×C₁₆	—	2	2	no	—
87	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
88	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
89	C₈×C₁₆	M₁₆	1	1	yes	characteristic
90	C₈×C₁₆	M₁₆	1	1	yes	characteristic
91	#1276	—	4	1	no	—
92	C₂×M₃₂	—	4	2	no	—
93	C₂²×C₁₆	—	2	4	no	—
94	C₂²×C₁₆	—	2	4	no	—
95	C₂²×C₁₆	C₄×C₈	1	1	yes	characteristic
96	C₂²×C₁₆	#97	1	1	yes	characteristic
97	C₂²×C₁₆	#102	1	1	yes	characteristic
98	C₂²×C₁₆	#103	1	1	yes	characteristic
99	C₂²×C₁₆	C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₁₆	C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₁₆	#106	1	1	yes	characteristic
102	C₂²×C₁₆	#107	1	1	yes	characteristic
103	#431	—	4	2	no	—
104	C₂×C₄×C₈	—	2	1	no	—
105	C₂×C₄×C₈	—	2	1	no	—
106	C₂×C₄×C₈	—	2	4	no	—
107	C₂×C₄×C₈	#95	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₈	#102	1	1	yes	characteristic
111	C₂×C₄×C₈	#103	1	1	yes	characteristic
112	C₂×C₄×C₈	#104	1	1	yes	characteristic
113	C₄³	#104	1	1	yes	characteristic
114	M₆₄	—	4	4	no	—
115	M₆₄	—	8	2	no	—
116	C₂×C₃₂	—	8	2	no	—
117	#364	—	4	2	no	—
118	#364	—	4	2	no	—
119	#346	—	2	2	no	—
120	#346	—	2	2	no	—
121	#346	—	2	2	no	—
122	#346	—	2	2	no	—
123	#346	—	4	1	no	—
124	#346	—	4	1	no	—
125	#346	—	4	1	no	—
126	#346	—	4	1	no	—
127	#346	—	4	16	no	—
128	#346	—	8	1	no	—
129	C₄×C₁₆	—	2	4	no	—
130	C₄×C₁₆	—	2	4	no	—
131	C₄×C₁₆	—	2	4	no	—
132	C₄×C₁₆	—	2	4	no	—
133	C₄×C₁₆	—	2	1	no	—
134	C₄×C₁₆	—	2	1	no	—
135	C₄×C₁₆	—	8	1	no	—
136	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
137	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
138	C₄×C₁₆	#97	1	1	yes	characteristic
139	C₄×C₁₆	#97	1	1	yes	characteristic
140	C₄×C₁₆	C₂²⋊C₈	1	2	yes	—
141	C₄×C₁₆	#104	1	2	yes	—
142	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
143	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
144	C₄×C₁₆	#108	1	1	yes	characteristic
145	C₄×C₁₆	#108	1	1	yes	characteristic
146	#335	—	4	2	no	—
147	#334	—	4	2	no	—
148	#322	—	8	1	no	—
149	C₈²	—	2	4	no	—
150	C₈²	—	2	1	no	—
151	C₈²	—	2	1	no	—
152	C₈²	—	8	1	no	—
153	C₈²	C₂²⋊C₈	1	1	yes	characteristic
154	C₈²	C₂²⋊C₈	1	1	yes	characteristic
155	C₈²	C₄⋊C₈	1	2	yes	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	D₄⋊C₈	1	1	yes	characteristic
159	C₂²×C₈	Q₈⋊C₈	1	1	yes	characteristic
160	C₂²×C₈	#336	1	1	yes	characteristic
161	C₂²×C₈	#340	1	1	yes	characteristic
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	D₄⋊C₈	1	1	yes	characteristic
164	C₂×C₄²	Q₈⋊C₈	1	1	yes	characteristic
165	C₂×C₄²	#339	1	1	yes	characteristic
166	M₃₂	—	8	2	no	—
167	M₃₂	—	8	2	no	—
168	C₂×C₁₆	—	2	8	no	—
169	C₂×C₁₆	—	2	16	no	—
170	C₂×C₁₆	—	2	1	no	—
171	C₂×C₁₆	—	2	1	no	—
172	C₂×C₁₆	—	2	1	no	—
173	C₂×C₁₆	—	2	1	no	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	8	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	4	8	no	—
186	C₂×C₁₆	—	8	2	no	—
187	C₂×C₁₆	C₈²	1	1	yes	characteristic
188	C₂×C₁₆	#322	1	1	yes	characteristic
189	C₂×C₁₆	#322	1	1	yes	characteristic
190	C₂×C₁₆	#322	1	1	yes	characteristic
191	C₂×C₁₆	D₄⋊C₈	1	2	yes	—
192	C₂×C₁₆	Q₈⋊C₈	1	2	yes	—
193	C₂×C₁₆	#334	1	1	yes	characteristic
194	C₂×C₁₆	#334	1	1	yes	characteristic
195	C₂×C₁₆	#335	1	1	yes	characteristic
196	C₂×C₁₆	#335	1	1	yes	characteristic
197	C₂×C₁₆	#347	1	2	yes	—
198	C₂×C₁₆	#347	1	2	yes	—
199	C₂×C₁₆	#365	1	2	yes	—
200	C₂×C₁₆	#365	1	2	yes	—
201	C₄⋊C₈	—	8	2	no	—
202	C₂²⋊C₈	—	8	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	2	2	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	—	4	8	no	—
218	C₄×C₈	—	8	2	no	—
219	C₄×C₈	#336	1	1	yes	characteristic
220	C₄×C₈	#336	1	1	yes	characteristic
221	C₄×C₈	#339	1	1	yes	characteristic
222	C₄×C₈	#341	1	1	yes	characteristic
223	C₃₂	—	8	4	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	#1279	1	1	yes	characteristic
227	C₂²×C₄	#1335	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	4	4	no	—
238	C₂×C₈	—	16	2	no	—
239	C₂×C₈	#1273	1	1	yes	characteristic
240	C₂×C₈	#1277	1	1	yes	characteristic
241	C₂×C₈	#1278	1	1	yes	characteristic
242	C₂×C₈	#1279	1	1	yes	characteristic
243	C₂×C₈	#1335	1	2	yes	—
244	C₂×C₈	#1375	1	1	yes	characteristic
245	C₂×C₈	#1375	1	1	yes	characteristic
246	C₂×C₈	#1378	1	1	yes	characteristic
247	C₂×C₈	#1378	1	1	yes	characteristic
248	C₂×C₈	#1383	1	1	yes	characteristic
249	C₂×C₈	#1383	1	1	yes	characteristic
250	C₂×C₈	#1384	1	1	yes	characteristic
251	C₂×C₈	#1385	1	1	yes	characteristic
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	4	4	no	—
258	C₄²	#1276	1	1	yes	characteristic
259	C₄²	#1278	1	1	yes	characteristic
260	C₁₆	—	2	16	no	—
261	C₁₆	—	2	2	no	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	2	1	no	—
264	C₁₆	—	2	1	no	—
265	C₁₆	—	2	1	no	—
266	C₁₆	—	2	1	no	—
267	C₁₆	—	4	8	no	—
268	C₁₆	—	4	1	no	—
269	C₁₆	—	4	1	no	—
270	C₁₆	—	16	2	no	—
271	C₁₆	#1313	1	2	yes	—
272	C₁₆	#1313	1	2	yes	—
273	C₁₆	#1368	1	2	yes	—
274	C₁₆	#1368	1	2	yes	—
275	C₂³	#7179	1	1	yes	characteristic
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	#7015	1	1	yes	characteristic
284	C₂×C₄	#7168	1	1	yes	characteristic
285	C₂×C₄	#7168	1	1	yes	characteristic
286	C₂×C₄	#7173	1	1	yes	characteristic
287	C₂×C₄	#7174	1	1	yes	characteristic
288	C₂×C₄	#7179	1	1	yes	characteristic
289	C₈	—	2	4	no	—
290	C₈	—	2	4	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	4	no	—
293	C₈	—	4	1	no	—
294	C₈	—	32	2	no	—
295	C₈	#7137	1	1	yes	characteristic
296	C₈	#7140	1	1	yes	characteristic
297	C₈	#7156	1	1	yes	characteristic
298	C₈	#7157	1	1	yes	characteristic
299	C₈	#7164	1	1	yes	characteristic
300	C₈	#7164	1	1	yes	characteristic
301	C₈	#7173	1	1	yes	characteristic
302	C₈	#7174	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	#92937	1	1	yes	characteristic
305	C₂²	#92938	1	1	yes	characteristic
306	C₂²	#93264	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	1	no	—
311	C₄	#92937	1	1	yes	characteristic
312	C₄	#92938	1	1	yes	characteristic
313	C₄	#93261	1	1	yes	characteristic
314	C₄	#93269	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	#72214361	1	2	yes	—
318	C₂	#72216542	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#1384
commutator	C₂×C₁₆	C₈²
Frattini, cosocle	C₈×#346	C₂²
socle	C₂²	#92938
prime radical	C₂³	#7179
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
36. #7456	C₈	semidirect
42. #7052	C₈	semidirect
44. C₁₆²	C₈	semidirect

Maximal subgroups

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

subgroup	index
#72216247	2
#72216215	2
#72216182	2