Subgroups — #72227185

Order 2048.

710 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72227182	C₂	1	1	yes	maximal, characteristic
2	#72225535	C₂	1	2	yes	maximal
3	#72225325	C₂	1	1	yes	maximal, characteristic
4	#72216542	C₂	1	2	yes	maximal
5	#72216182	C₂	1	1	yes	maximal, characteristic
6	C₄×#1368	C₄	1	2	yes	—
7	#129086	C₂²	1	2	yes	—
8	#129049	—	2	2	no	—
9	C₈×#346	C₂²	1	1	yes	characteristic
10	C₈×#346	C₂²	1	1	yes	characteristic
11	C₈×#346	C₂²	1	1	yes	characteristic
12	C₂×#7099	C₄	1	2	yes	—
13	C₂×#7052	C₄	1	1	yes	characteristic
14	C₂×#7052	C₄	1	1	yes	characteristic
15	C₂×#7052	C₄	1	1	yes	characteristic
16	C₂×C₁₆²	C₄	1	1	yes	characteristic
17	#96889	C₂²	1	2	yes	—
18	#93269	C₄	1	2	yes	—
19	#93261	—	2	2	no	—
20	#92913	C₄	1	2	yes	—
21	#12653	—	2	2	no	—
22	C₄×#346	—	2	1	no	—
23	C₄×#346	C₂×C₄	1	1	yes	characteristic
24	C₄×#346	C₂×C₄	1	1	yes	characteristic
25	C₄×#346	D₄	1	1	yes	characteristic
26	C₄×#346	D₄	1	1	yes	characteristic
27	C₂×#1368	—	2	2	no	—
28	C₂×#1367	C₂×C₄	1	4	yes	—
29	C₂×#1313	—	4	2	no	—
30	C₂×#1311	—	2	1	no	—
31	C₂×#1311	C₂×C₄	1	1	yes	characteristic
32	C₂×#1311	C₂×C₄	1	1	yes	characteristic
33	C₂×#1311	Q₈	1	1	yes	characteristic
34	C₂×#1311	Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
41	C₄×#334	C₂×C₄	1	4	yes	—
42	#8260	—	2	2	no	—
43	C₄×C₈²	C₂³	1	1	yes	characteristic
44	#7156	—	2	2	no	—
45	#7151	—	2	2	no	—
46	#7137	—	4	2	no	—
47	#7099	—	2	4	no	—
48	#7052	—	2	2	no	—
49	#7052	—	2	2	no	—
50	#7052	C₈	1	4	yes	complemented
51	C₁₆²	C₈	1	4	yes	complemented
52	#7015	—	2	2	no	—
53	C₂×C₄⋊C₁₆	—	2	4	no	—
54	C₂×#346	—	2	4	no	—
55	C₂×#346	—	2	2	no	—
56	C₂×#346	—	2	2	no	—
57	C₂×#346	—	4	2	no	—
58	C₂×#346	M₁₆	1	2	yes	—
59	C₂×#346	M₁₆	1	2	yes	—
60	C₂×#346	SD₁₆	1	2	yes	—
61	C₂×#346	SD₁₆	1	2	yes	—
62	C₂×C₄×C₁₆	—	2	1	no	—
63	C₂×C₄×C₁₆	—	2	1	no	—
64	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
65	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
66	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
67	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
68	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₄⋊C₈	—	2	4	no	—
73	C₄²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₈	#41	1	1	yes	characteristic
76	C₂×#334	—	2	4	no	—
77	C₂×#322	—	4	2	no	—
78	C₂×C₈²	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈²	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈²	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈²	#41	1	1	yes	characteristic
82	#1374	—	4	2	no	—
83	#1368	—	4	4	no	—
84	#1367	—	2	4	no	—
85	#1367	—	2	4	no	—
86	#1313	—	8	4	no	—
87	#1311	—	2	4	no	—
88	#1311	—	2	2	no	—
89	#1311	—	2	2	no	—
90	#1311	—	2	2	no	—
91	#1311	M₁₆	1	4	yes	—
92	C₈×C₁₆	—	2	1	no	—
93	C₈×C₁₆	—	2	1	no	—
94	C₈×C₁₆	—	2	1	no	—
95	C₈×C₁₆	—	2	1	no	—
96	C₈×C₁₆	C₂×C₈	1	2	yes	—
97	C₈×C₁₆	C₂×C₈	1	2	yes	—
98	C₈×C₁₆	C₂×C₈	1	2	yes	—
99	C₈×C₁₆	C₂×C₈	1	2	yes	—
100	C₈×C₁₆	C₂×C₈	1	2	yes	—
101	C₈×C₁₆	C₂×C₈	1	2	yes	—
102	C₈×C₁₆	M₁₆	1	2	yes	—
103	C₈×C₁₆	M₁₆	1	2	yes	—
104	#1277	—	2	4	no	—
105	#1273	—	4	2	no	—
106	C₂²×C₁₆	—	2	2	no	—
107	C₂²×C₁₆	—	2	1	no	—
108	C₂²×C₁₆	—	2	1	no	—
109	C₂²×C₁₆	—	2	2	no	—
110	C₂²×C₁₆	—	2	1	no	—
111	C₂²×C₁₆	—	2	1	no	—
112	C₂²×C₁₆	—	4	4	no	—
113	C₂²×C₁₆	#97	1	2	yes	—
114	C₂²×C₁₆	#97	1	1	yes	characteristic
115	C₂²×C₁₆	#97	1	1	yes	characteristic
116	C₂²×C₁₆	#106	1	2	yes	—
117	C₂²×C₁₆	#106	1	1	yes	characteristic
118	C₂²×C₁₆	#106	1	1	yes	characteristic
119	C₂×C₄⋊C₈	—	4	4	no	—
120	C₂×C₄×C₈	—	2	2	no	—
121	C₂×C₄×C₈	—	4	4	no	—
122	C₂×C₄×C₈	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈	#117	1	1	yes	characteristic
125	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈	C₄×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₈	C₄×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₈	#131	1	2	yes	—
130	C₂×C₄×C₈	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₈	#135	1	2	yes	—
132	C₄³	C₄×D₄	1	1	yes	characteristic
133	C₄⋊C₁₆	—	4	4	no	—
134	C₄⋊C₁₆	—	4	4	no	—
135	#346	—	2	4	no	—
136	#346	—	2	4	no	—
137	#346	—	4	2	no	—
138	#346	—	4	2	no	—
139	#346	—	4	4	no	—
140	#346	—	4	4	no	—
141	#346	—	4	8	no	—
142	#346	—	8	2	no	—
143	#346	—	8	2	no	—
144	C₄×C₁₆	—	2	2	no	—
145	C₄×C₁₆	—	2	2	no	—
146	C₄×C₁₆	—	2	1	no	—
147	C₄×C₁₆	—	2	1	no	—
148	C₄×C₁₆	—	2	1	no	—
149	C₄×C₁₆	—	2	1	no	—
150	C₄×C₁₆	—	2	1	no	—
151	C₄×C₁₆	—	2	1	no	—
152	C₄×C₁₆	—	2	2	no	—
153	C₄×C₁₆	—	2	2	no	—
154	C₄×C₁₆	C₄×C₈	1	2	yes	—
155	C₄×C₁₆	C₄×C₈	1	2	yes	—
156	C₄×C₁₆	#97	1	2	yes	—
157	C₄×C₁₆	#97	1	2	yes	—
158	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
159	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
160	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
161	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
162	C₄×C₁₆	#108	1	2	yes	—
163	C₄×C₁₆	#108	1	2	yes	—
164	#336	—	4	4	no	—
165	#334	—	4	4	no	—
166	#334	—	4	4	no	—
167	#322	—	8	2	no	—
168	#322	—	8	2	no	—
169	C₈²	—	2	1	no	—
170	C₈²	—	2	1	no	—
171	C₈²	—	2	2	no	—
172	C₈²	C₂²×C₈	1	1	yes	characteristic
173	C₈²	C₂²×C₈	1	1	yes	characteristic
174	C₈²	C₂×M₁₆	1	2	yes	—
175	C₈²	C₂×M₁₆	1	1	yes	characteristic
176	C₈²	C₂×M₁₆	1	1	yes	characteristic
177	C₈²	#131	1	2	yes	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	8	4	no	—
181	C₂²×C₈	C₂×#97	1	1	yes	characteristic
182	C₂²×C₈	#405	1	1	yes	characteristic
183	C₂²×C₈	C₂×#106	1	1	yes	characteristic
184	C₂²×C₈	#427	1	1	yes	characteristic
185	C₂²×C₈	C₄×SD₁₆	1	2	yes	—
186	C₂²×C₈	#446	1	2	yes	—
187	C₂×C₄²	C₄×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₄²	#436	1	2	yes	—
189	C₂×C₄²	C₄×SD₁₆	1	2	yes	—
190	C₂×C₄²	#443	1	2	yes	—
191	C₂×C₁₆	—	2	4	no	—
192	C₂×C₁₆	—	2	8	no	—
193	C₂×C₁₆	—	2	4	no	—
194	C₂×C₁₆	—	2	4	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	—	2	2	no	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	4	4	no	—
208	C₂×C₁₆	—	4	2	no	—
209	C₂×C₁₆	—	4	2	no	—
210	C₂×C₁₆	—	8	4	no	—
211	C₂×C₁₆	—	8	4	no	—
212	C₂×C₁₆	—	8	4	no	—
213	C₂×C₁₆	#322	1	2	yes	—
214	C₂×C₁₆	#322	1	2	yes	—
215	C₂×C₁₆	#334	1	2	yes	—
216	C₂×C₁₆	#334	1	4	yes	—
217	C₂×C₁₆	#334	1	2	yes	—
218	C₂×C₁₆	#347	1	4	yes	—
219	C₂×C₁₆	#365	1	4	yes	—
220	C₄⋊C₈	—	8	4	no	—
221	C₄⋊C₈	—	8	4	no	—
222	C₄×C₈	—	2	2	no	—
223	C₄×C₈	—	2	2	no	—
224	C₄×C₈	—	2	2	no	—
225	C₄×C₈	—	2	2	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	4	2	no	—
231	C₄×C₈	—	4	2	no	—
232	C₄×C₈	—	4	4	no	—
233	C₄×C₈	—	8	4	no	—
234	C₄×C₈	—	8	4	no	—
235	C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
236	C₄×C₈	C₂×#97	1	1	yes	characteristic
237	C₄×C₈	C₄×M₁₆	1	1	yes	characteristic
238	C₄×C₈	#405	1	1	yes	characteristic
239	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
240	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
241	C₄×C₈	C₂×#108	1	1	yes	characteristic
242	C₄×C₈	C₂×#108	1	1	yes	characteristic
243	C₄×C₈	#431	1	1	yes	characteristic
244	C₄×C₈	#431	1	1	yes	characteristic
245	C₄×C₈	#436	1	2	yes	—
246	C₄×C₈	#443	1	2	yes	—
247	C₄×C₈	#446	1	2	yes	—
248	C₂²×C₄	#1590	1	4	yes	—
249	C₂²×C₄	#1770	1	1	yes	characteristic
250	C₂²×C₄	#1772	1	1	yes	characteristic
251	C₂²×C₄	C₄×#106	1	1	yes	characteristic
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	16	4	no	—
265	C₂×C₈	—	16	4	no	—
266	C₂×C₈	—	16	4	no	—
267	C₂×C₈	C₂×#322	1	1	yes	characteristic
268	C₂×C₈	#1450	1	1	yes	characteristic
269	C₂×C₈	#1451	1	1	yes	characteristic
270	C₂×C₈	#1451	1	1	yes	characteristic
271	C₂×C₈	C₂×#334	1	1	yes	characteristic
272	C₂×C₈	C₂×#334	1	1	yes	characteristic
273	C₂×C₈	#1564	1	1	yes	characteristic
274	C₂×C₈	#1564	1	1	yes	characteristic
275	C₂×C₈	#1590	1	4	yes	—
276	C₂×C₈	C₂×#347	1	2	yes	—
277	C₂×C₈	#2110	1	2	yes	—
278	C₂×C₈	C₂×#365	1	2	yes	—
279	C₂×C₈	#2155	1	2	yes	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	C₄×C₄⋊C₈	1	1	yes	characteristic
286	C₄²	#1770	1	1	yes	characteristic
287	C₄²	#1772	1	1	yes	characteristic
288	C₄²	C₄×#108	1	1	yes	characteristic
289	C₁₆	—	2	8	no	—
290	C₁₆	—	2	4	no	—
291	C₁₆	—	2	4	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	—	2	2	no	—
294	C₁₆	—	2	4	no	—
295	C₁₆	—	4	4	no	—
296	C₁₆	—	4	2	no	—
297	C₁₆	—	4	2	no	—
298	C₁₆	—	4	2	no	—
299	C₁₆	—	16	4	no	—
300	C₁₆	#1313	1	4	yes	complemented
301	C₁₆	#1368	1	4	yes	complemented
302	C₂³	#8283	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	#8260	1	1	yes	characteristic
308	C₂×C₄	#8266	1	1	yes	characteristic
309	C₂×C₄	C₄×#334	1	1	yes	characteristic
310	C₂×C₄	#8283	1	1	yes	characteristic
311	C₂×C₄	#8293	1	1	yes	characteristic
312	C₂×C₄	#8293	1	1	yes	characteristic
313	C₂×C₄	#12666	1	2	yes	—
314	C₂×C₄	#12667	1	2	yes	—
315	C₂×C₄	#12668	1	2	yes	—
316	C₂×C₄	C₄×#365	1	2	yes	—
317	C₈	—	2	2	no	—
318	C₈	—	2	2	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	4	no	—
324	C₈	—	4	2	no	—
325	C₈	—	4	2	no	—
326	C₈	—	32	4	no	—
327	C₈	C₂×#1313	1	1	yes	characteristic
328	C₈	#11405	1	1	yes	characteristic
329	C₈	#11414	1	2	yes	—
330	C₈	C₂×#1368	1	1	yes	characteristic
331	C₈	#11999	1	1	yes	characteristic
332	C₈	#12000	1	2	yes	—
333	C₂²	#96889	1	1	yes	characteristic
334	C₂²	#129158	1	2	yes	—
335	C₂²	#129159	1	2	yes	—
336	C₂²	#129160	1	2	yes	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	4	no	—
340	C₄	#129049	1	1	yes	characteristic
341	C₄	#129051	1	1	yes	characteristic
342	C₄	#129088	1	1	yes	characteristic
343	C₄	C₄×#1368	1	1	yes	characteristic
344	C₄	#129160	1	2	yes	—
345	C₄	#129195	1	2	yes	—
346	C₂	#72216215	1	1	yes	characteristic
347	C₂	#72225325	1	1	yes	characteristic
348	C₂	#72225535	1	1	yes	characteristic
349	C₂	#72227184	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₄×#106
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×C₈²	C₂³
socle, prime radical	C₂³	#8283
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
50. #7052	C₈	semidirect
51. C₁₆²	C₈	semidirect
300. C₁₆	#1313	semidirect
301. C₁₆	#1368	semidirect

Maximal subgroups

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

subgroup	index
#72227182	2
#72225535	2
#72225325	2
#72216542	2
#72216182	2