Subgroups — #72227235

Order 2048.

417 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225138	C₂	1	1	yes	maximal, characteristic
2	#72216247	C₂	1	1	yes	maximal, characteristic
3	#72216243	C₂	1	1	yes	maximal, characteristic
4	#151981	—	2	1	no	—
5	#151981	—	2	1	no	—
6	C₄×#1398	C₄	1	1	yes	characteristic
7	C₄×#1398	C₄	1	1	yes	characteristic
8	C₂×#7459	C₄	1	1	yes	characteristic
9	C₂×#7459	C₄	1	1	yes	characteristic
10	C₂×#7456	C₄	1	2	yes	—
11	#149502	—	2	1	no	—
12	#149499	—	2	1	no	—
13	C₈×#346	C₂²	1	1	yes	characteristic
14	#94019	—	2	1	no	—
15	#94018	—	2	1	no	—
16	C₂×#1422	—	2	2	no	—
17	C₂×#1422	—	2	1	no	—
18	C₂×#1422	—	2	1	no	—
19	C₂×#1398	—	2	1	no	—
20	C₂×#1398	—	2	1	no	—
21	C₂×#1398	—	4	1	no	—
22	C₂×#1398	—	4	1	no	—
23	C₂×#1398	C₈	1	2	yes	—
24	C₂×#1398	C₈	1	2	yes	—
25	C₄×M₆₄	—	2	1	no	—
26	C₄×M₆₄	—	2	1	no	—
27	C₄×#346	C₂×C₄	1	1	yes	characteristic
28	C₄×#346	D₄	1	1	yes	characteristic
29	C₂×#1311	C₂×C₄	1	1	yes	characteristic
30	C₂×#1311	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₆	D₄	1	1	yes	characteristic
33	C₄×C₈²	Q₈	1	1	yes	characteristic
34	#7468	—	4	1	no	—
35	#7468	—	4	1	no	—
36	#7459	—	2	1	no	—
37	#7459	—	2	1	no	—
38	#7459	C₈	1	2	yes	—
39	#7459	C₈	1	2	yes	—
40	#7456	—	2	2	no	—
41	#7456	C₈	1	4	yes	—
42	#7332	—	2	2	no	—
43	#7332	—	2	2	no	—
44	#7331	—	2	4	no	—
45	C₂×M₆₄	—	2	2	no	—
46	C₂×M₆₄	—	2	2	no	—
47	C₂×M₆₄	—	4	2	no	—
48	C₂×M₆₄	—	4	1	no	—
49	C₂×M₆₄	—	4	1	no	—
50	C₂×#346	C₂×C₈	1	1	yes	characteristic
51	C₂×#346	C₂×C₈	1	1	yes	characteristic
52	C₂×#346	M₁₆	1	1	yes	characteristic
53	C₂×#346	M₁₆	1	1	yes	characteristic
54	C₂×#346	D₈	1	1	yes	characteristic
55	C₂×#346	SD₁₆	1	1	yes	characteristic
56	C₂×#346	SD₁₆	1	1	yes	characteristic
57	C₂×#346	Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
59	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₈	—	2	1	no	—
63	C₄²×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈²	—	2	1	no	—
65	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
67	#1422	—	2	4	no	—
68	#1422	—	2	2	no	—
69	#1422	—	2	2	no	—
70	#1422	—	4	2	no	—
71	#1422	—	4	1	no	—
72	#1422	—	4	1	no	—
73	#1398	—	4	2	no	—
74	#1398	—	4	2	no	—
75	#1398	—	4	2	no	—
76	#1398	—	4	2	no	—
77	#1398	—	8	1	no	—
78	#1398	—	8	1	no	—
79	#1398	—	8	2	no	—
80	#1397	—	2	2	no	—
81	#1397	—	2	2	no	—
82	C₄×C₃₂	—	2	2	no	—
83	C₄×C₃₂	—	2	2	no	—
84	#1311	—	2	1	no	—
85	#1311	—	2	1	no	—
86	#1311	C₂×C₈	1	2	yes	—
87	#1311	M₁₆	1	2	yes	—
88	C₈×C₁₆	—	2	2	no	—
89	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
90	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
91	C₈×C₁₆	M₁₆	1	1	yes	characteristic
92	C₈×C₁₆	M₁₆	1	1	yes	characteristic
93	C₂²×C₁₆	C₄×C₈	1	1	yes	characteristic
94	C₂²×C₁₆	#97	1	1	yes	characteristic
95	C₂²×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
96	C₂²×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
97	C₂²×C₁₆	#102	1	1	yes	characteristic
98	C₂²×C₁₆	#102	1	1	yes	characteristic
99	C₂²×C₁₆	#103	1	1	yes	characteristic
100	C₂²×C₁₆	#103	1	1	yes	characteristic
101	C₂×C₄×C₈	—	2	1	no	—
102	C₂×C₄×C₈	—	2	1	no	—
103	C₂×C₄×C₈	—	4	2	no	—
104	C₂×C₄×C₈	#95	1	1	yes	characteristic
105	C₂×C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₈	#106	1	1	yes	characteristic
108	C₂×C₄×C₈	#107	1	1	yes	characteristic
109	C₂×C₄×C₈	#108	1	1	yes	characteristic
110	C₄³	#108	1	1	yes	characteristic
111	M₆₄	—	4	2	no	—
112	M₆₄	—	4	2	no	—
113	M₆₄	—	4	2	no	—
114	M₆₄	—	4	2	no	—
115	M₆₄	—	4	4	no	—
116	M₆₄	—	4	2	no	—
117	M₆₄	—	4	2	no	—
118	M₆₄	—	8	2	no	—
119	M₆₄	—	8	1	no	—
120	M₆₄	—	8	1	no	—
121	C₂×C₃₂	—	2	4	no	—
122	C₂×C₃₂	—	2	4	no	—
123	C₂×C₃₂	—	8	2	no	—
124	C₂×C₃₂	—	8	1	no	—
125	C₂×C₃₂	—	8	1	no	—
126	#346	—	2	2	no	—
127	#346	—	2	2	no	—
128	#346	—	2	2	no	—
129	#346	—	2	2	no	—
130	#346	—	2	1	no	—
131	#346	—	2	1	no	—
132	#346	—	2	1	no	—
133	#346	—	2	1	no	—
134	#346	—	4	1	no	—
135	#346	—	4	1	no	—
136	C₄×C₁₆	—	2	1	no	—
137	C₄×C₁₆	—	2	1	no	—
138	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
139	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
140	C₄×C₁₆	#97	1	1	yes	characteristic
141	C₄×C₁₆	#97	1	1	yes	characteristic
142	C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
144	C₄×C₁₆	C₂²⋊C₈	1	2	yes	—
145	C₄×C₁₆	#104	1	2	yes	—
146	C₄×C₁₆	#104	1	1	yes	characteristic
147	C₄×C₁₆	#104	1	1	yes	characteristic
148	C₈²	—	2	1	no	—
149	C₈²	—	2	1	no	—
150	C₈²	—	4	2	no	—
151	C₈²	C₄⋊C₈	1	1	yes	characteristic
152	C₈²	C₄⋊C₈	1	1	yes	characteristic
153	C₈²	C₄⋊C₈	1	2	yes	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	4	1	no	—
156	C₂²×C₈	#334	1	1	yes	characteristic
157	C₂²×C₈	#335	1	1	yes	characteristic
158	C₂²×C₈	#336	1	1	yes	characteristic
159	C₂²×C₈	#340	1	1	yes	characteristic
160	C₂×C₄²	—	4	1	no	—
161	C₂×C₄²	#334	1	1	yes	characteristic
162	C₂×C₄²	#335	1	1	yes	characteristic
163	C₂×C₄²	#341	1	1	yes	characteristic
164	C₂×C₁₆	—	2	1	no	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	—	2	1	no	—
167	C₂×C₁₆	—	2	1	no	—
168	C₂×C₁₆	—	2	1	no	—
169	C₂×C₁₆	—	2	1	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	C₈²	1	1	yes	characteristic
177	C₂×C₁₆	#322	1	1	yes	characteristic
178	C₂×C₁₆	#322	1	1	yes	characteristic
179	C₂×C₁₆	#322	1	1	yes	characteristic
180	C₂×C₁₆	D₄⋊C₈	1	2	yes	—
181	C₂×C₁₆	D₄⋊C₈	1	1	yes	characteristic
182	C₂×C₁₆	D₄⋊C₈	1	1	yes	characteristic
183	C₂×C₁₆	Q₈⋊C₈	1	2	yes	—
184	C₂×C₁₆	Q₈⋊C₈	1	1	yes	characteristic
185	C₂×C₁₆	Q₈⋊C₈	1	1	yes	characteristic
186	C₂×C₁₆	#347	1	2	yes	—
187	C₂×C₁₆	#347	1	2	yes	—
188	C₂×C₁₆	#349	1	2	yes	—
189	C₂×C₁₆	#349	1	2	yes	—
190	C₂×C₁₆	#360	1	1	yes	characteristic
191	C₂×C₁₆	#360	1	1	yes	characteristic
192	C₂×C₁₆	#361	1	1	yes	characteristic
193	C₂×C₁₆	#362	1	1	yes	characteristic
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	—	4	1	no	—
207	C₄×C₈	—	4	1	no	—
208	C₄×C₈	—	8	4	no	—
209	C₄×C₈	#336	1	1	yes	characteristic
210	C₄×C₈	#336	1	1	yes	characteristic
211	C₄×C₈	#339	1	1	yes	characteristic
212	C₄×C₈	#339	1	1	yes	characteristic
213	C₃₂	—	4	4	no	—
214	C₃₂	—	4	4	no	—
215	C₃₂	—	8	4	no	—
216	C₃₂	—	8	2	no	—
217	C₃₂	—	8	2	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	#1277	1	1	yes	characteristic
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	8	2	no	—
233	C₂×C₈	#1273	1	1	yes	characteristic
234	C₂×C₈	#1276	1	1	yes	characteristic
235	C₂×C₈	#1279	1	1	yes	characteristic
236	C₂×C₈	#1279	1	1	yes	characteristic
237	C₂×C₈	#1375	1	1	yes	characteristic
238	C₂×C₈	#1375	1	1	yes	characteristic
239	C₂×C₈	#1377	1	1	yes	characteristic
240	C₂×C₈	#1377	1	1	yes	characteristic
241	C₂×C₈	#1386	1	1	yes	characteristic
242	C₂×C₈	#1387	1	1	yes	characteristic
243	C₂×C₈	#1388	1	1	yes	characteristic
244	C₂×C₈	#1389	1	1	yes	characteristic
245	C₄²	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	8	2	no	—
251	C₄²	#1278	1	1	yes	characteristic
252	C₄²	#1278	1	1	yes	characteristic
253	C₁₆	—	2	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	2	1	no	—
256	C₁₆	—	2	1	no	—
257	C₁₆	—	2	1	no	—
258	C₁₆	—	2	1	no	—
259	C₁₆	—	2	2	no	—
260	C₁₆	—	2	2	no	—
261	C₁₆	#1313	1	2	yes	—
262	C₁₆	#1313	1	2	yes	—
263	C₁₆	#1333	1	1	yes	characteristic
264	C₁₆	#1333	1	1	yes	characteristic
265	C₁₆	#1334	1	1	yes	characteristic
266	C₁₆	#1334	1	1	yes	characteristic
267	C₂³	#7162	1	1	yes	characteristic
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	#7015	1	1	yes	characteristic
276	C₂×C₄	#7152	1	1	yes	characteristic
277	C₂×C₄	#7152	1	1	yes	characteristic
278	C₂×C₄	#7158	1	1	yes	characteristic
279	C₂×C₄	#7159	1	1	yes	characteristic
280	C₂×C₄	#7162	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	1	no	—
285	C₈	—	8	2	no	—
286	C₈	#7137	1	1	yes	characteristic
287	C₈	#7139	1	1	yes	characteristic
288	C₈	#7149	1	1	yes	characteristic
289	C₈	#7149	1	1	yes	characteristic
290	C₈	#7175	1	1	yes	characteristic
291	C₈	#7176	1	1	yes	characteristic
292	C₈	#7177	1	1	yes	characteristic
293	C₈	#7178	1	1	yes	characteristic
294	C₂²	—	4	1	no	—
295	C₂²	#92921	1	1	yes	characteristic
296	C₂²	#92922	1	1	yes	characteristic
297	C₂²	#93273	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	8	1	no	—
301	C₄	—	8	1	no	—
302	C₄	#92921	1	1	yes	characteristic
303	C₄	#92922	1	1	yes	characteristic
304	C₄	#93258	1	1	yes	characteristic
305	C₄	#93270	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	#72227234	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#7175
commutator	C₂×C₁₆	C₈²
Frattini, cosocle	C₈×#346	C₂²
socle	C₂	#72227234
prime radical	C₂³	#7162
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#72225138	2
#72216247	2
#72216243	2