Subgroups — #72323180

Order 2048.

561 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72321743	C₂	1	1	yes	maximal, characteristic, complemented
2	#72321570	C₂	1	2	yes	maximal
3	#72286415	C₂	1	2	yes	maximal, complemented
4	#72234379	C₂	1	1	yes	maximal, characteristic, complemented
5	#72231861	C₂	1	1	yes	maximal, characteristic
6	#512731	C₄	1	1	yes	characteristic
7	#512545	C₂²	1	1	yes	characteristic
8	#479917	C₄	1	1	yes	characteristic
9	#153432	C₄	1	4	yes	—
10	C₂×#7511	C₂²	1	2	yes	—
11	C₂×#7511	C₂²	1	2	yes	—
12	#149509	—	2	2	no	—
13	#149508	—	2	2	no	—
14	#149502	C₄	1	1	yes	characteristic
15	#149501	C₄	1	1	yes	characteristic
16	#149496	C₂²	1	1	yes	characteristic
17	C₂×#7330	C₂²	1	1	yes	characteristic
18	#94759	C₄	1	4	yes	—
19	#94509	—	2	2	no	—
20	#94506	—	2	2	no	—
21	C₂×#2258	C₂×C₄	1	2	yes	—
22	#33420	C₈	1	1	yes	characteristic
23	C₄×#504	C₂×C₄	1	1	yes	characteristic
24	#17335	C₈	1	1	yes	characteristic
25	C₂×M₁₂₈	C₂×C₄	1	4	yes	—
26	C₂×M₁₂₈	C₂×C₄	1	4	yes	—
27	#13641	—	2	2	no	—
28	#13640	—	2	2	no	—
29	#13628	C₈	1	2	yes	—
30	#13554	C₈	1	2	yes	—
31	#13549	—	2	2	no	—
32	#13549	C₂×C₄	1	2	yes	—
33	C₄×M₆₄	—	2	1	no	—
34	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
35	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
36	C₂×#1397	—	2	1	no	—
37	C₂×#1397	C₂×C₄	1	1	yes	characteristic
38	C₂×#1397	C₂³	1	1	yes	characteristic
39	#11409	—	2	2	no	—
40	#11408	—	2	2	no	—
41	#11402	C₈	1	1	yes	characteristic
42	#11401	C₈	1	1	yes	characteristic
43	#11399	C₂×C₄	1	1	yes	characteristic
44	C₂×#1311	C₂×C₄	1	1	yes	characteristic
45	#7514	—	2	8	no	—
46	#7511	—	2	2	no	—
47	#7511	—	2	2	no	—
48	#7511	—	2	2	no	—
49	#7511	—	2	2	no	—
50	#7332	—	2	2	no	—
51	#7332	—	2	2	no	—
52	#7330	—	2	1	no	—
53	#7330	—	2	1	no	—
54	#7330	D₄	1	2	yes	—
55	#7330	D₄	1	2	yes	—
56	C₂×#504	C₄²	1	1	yes	characteristic
57	C₂×#504	C₂×C₈	1	1	yes	characteristic
58	#2965	—	2	1	no	—
59	C₄×#131	C₂×C₈	1	1	yes	characteristic
60	#2258	—	2	4	no	—
61	#2258	—	2	4	no	—
62	C₂×M₆₄	—	2	2	no	—
63	C₂×M₆₄	C₄²	1	2	yes	—
64	C₂×M₆₄	C₂×C₈	1	2	yes	—
65	C₂²×C₃₂	—	2	2	no	—
66	C₂²×C₃₂	C₂×C₈	1	2	yes	—
67	C₂²×C₃₂	C₂²×C₄	1	2	yes	—
68	#2164	—	2	2	no	—
69	#2162	—	2	2	no	—
70	#2151	—	2	1	no	—
71	#2113	—	2	1	no	—
72	#2108	—	2	1	no	—
73	#2108	—	2	1	no	—
74	#2108	C₄²	1	1	yes	characteristic
75	#2108	C₂×C₈	1	1	yes	characteristic
76	C₄×M₃₂	—	2	1	no	—
77	C₄×M₃₂	C₂×C₈	1	1	yes	characteristic
78	C₄×M₃₂	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₆	—	2	1	no	—
80	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
82	#1455	—	2	1	no	—
83	#1453	—	2	1	no	—
84	#1451	—	2	1	no	—
85	#1450	C₂×C₈	1	1	yes	characteristic
86	C₈×M₁₆	—	2	2	no	—
87	C₂×C₈²	C₂×C₈	1	1	yes	characteristic
88	M₁₂₈	—	2	4	no	—
89	M₁₂₈	—	2	4	no	—
90	M₁₂₈	—	2	4	no	—
91	M₁₂₈	M₁₆	1	8	yes	complemented
92	C₂×C₆₄	—	2	4	no	—
93	C₂×C₆₄	—	2	4	no	—
94	C₄⋊C₃₂	—	2	4	no	—
95	C₂²⋊C₃₂	—	2	4	no	—
96	#1397	—	2	2	no	—
97	#1397	—	2	1	no	—
98	#1397	—	2	1	no	—
99	#1397	—	2	1	no	—
100	#1397	C₂²⋊C₄	1	2	yes	—
101	#1397	C₂×D₄	1	1	yes	characteristic
102	#1397	C₂×D₄	1	1	yes	characteristic
103	#1397	#41	1	2	yes	—
104	C₄×C₃₂	—	2	2	no	—
105	C₄×C₃₂	—	2	1	no	—
106	C₄×C₃₂	M₁₆	1	2	yes	—
107	#1312	—	2	2	no	—
108	#1312	—	2	2	no	—
109	#1311	—	2	1	no	—
110	#1311	—	2	1	no	—
111	#1311	C₂²⋊C₄	1	2	yes	—
112	C₈×C₁₆	M₁₆	1	2	yes	—
113	C₂×#131	C₄×C₈	1	1	yes	characteristic
114	C₂×#131	M₃₂	1	1	yes	characteristic
115	C₄×#41	M₃₂	1	1	yes	characteristic
116	#504	—	2	2	no	—
117	#504	—	2	2	no	—
118	#504	—	2	4	no	—
119	C₂×M₃₂	—	2	1	no	—
120	C₂×M₃₂	—	2	1	no	—
121	C₂×M₃₂	—	2	1	no	—
122	C₂×M₃₂	C₄×C₈	1	1	yes	characteristic
123	C₂×M₃₂	C₄×C₈	1	1	yes	characteristic
124	C₂²×C₁₆	—	2	1	no	—
125	C₂²×C₁₆	—	2	1	no	—
126	C₂²×C₁₆	—	2	1	no	—
127	C₂²×C₁₆	C₂×C₄²	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
129	#446	—	4	1	no	—
130	#436	—	4	1	no	—
131	#435	—	4	2	no	—
132	#431	—	2	2	no	—
133	#431	—	2	1	no	—
134	#405	—	2	1	no	—
135	#405	—	2	1	no	—
136	#405	C₄×C₈	1	1	yes	characteristic
137	#405	M₃₂	1	1	yes	characteristic
138	C₄×M₁₆	—	2	2	no	—
139	C₄×M₁₆	—	2	1	no	—
140	C₄×M₁₆	M₃₂	1	1	yes	characteristic
141	C₂×C₄×C₈	—	2	1	no	—
142	C₂×C₄×C₈	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₈	1	1	yes	characteristic
144	M₆₄	—	2	4	no	—
145	M₆₄	—	2	2	no	—
146	M₆₄	—	2	2	no	—
147	M₆₄	—	4	4	no	—
148	C₂×C₃₂	—	2	2	no	—
149	C₂×C₃₂	—	2	2	no	—
150	C₂×C₃₂	—	2	4	no	—
151	C₂×C₃₂	—	4	4	no	—
152	C₂×C₃₂	C₂²⋊C₈	1	4	yes	—
153	C₂×C₃₂	C₄×D₄	1	2	yes	—
154	C₂×C₃₂	C₄×D₄	1	2	yes	—
155	C₂×C₃₂	C₂×M₁₆	1	4	yes	—
156	C₂×C₃₂	#131	1	4	yes	—
157	C₄⋊C₁₆	—	2	4	no	—
158	C₂²⋊C₁₆	—	2	4	no	—
159	#346	—	2	1	no	—
160	#346	—	2	1	no	—
161	#346	—	2	1	no	—
162	#346	—	2	1	no	—
163	C₄×C₁₆	—	2	1	no	—
164	C₄×C₁₆	—	2	1	no	—
165	C₄×C₁₆	—	2	1	no	—
166	C₄×C₁₆	—	2	1	no	—
167	C₄×C₁₆	C₂²⋊C₈	1	2	yes	—
168	C₄×C₁₆	M₃₂	1	2	yes	—
169	C₄×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₄×C₁₆	#117	1	1	yes	characteristic
171	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
172	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
173	#322	—	2	2	no	—
174	#322	—	2	2	no	—
175	#322	—	2	1	no	—
176	C₈²	—	2	1	no	—
177	C₈²	C₂²⋊C₈	1	2	yes	—
178	C₈²	M₃₂	1	1	yes	characteristic
179	C₈²	M₃₂	1	1	yes	characteristic
180	C₆₄	—	2	8	no	—
181	C₆₄	—	4	4	no	—
182	C₂×#41	#346	1	1	yes	characteristic
183	#131	—	2	2	no	—
184	#131	—	2	2	no	—
185	#131	—	4	2	no	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	4	2	no	—
189	C₂×M₁₆	#346	1	1	yes	characteristic
190	C₂×M₁₆	#346	1	1	yes	characteristic
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	4	1	no	—
194	C₂²×C₈	C₂×C₄×C₈	1	1	yes	characteristic
195	C₂²×C₈	C₂×M₃₂	1	1	yes	characteristic
196	C₄×Q₈	—	2	1	no	—
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	—	2	2	no	—
199	#117	—	2	1	no	—
200	#117	#346	1	1	yes	characteristic
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	C₂×M₃₂	1	1	yes	characteristic
203	M₃₂	—	2	2	no	—
204	M₃₂	—	2	2	no	—
205	M₃₂	—	4	2	no	—
206	M₃₂	—	4	2	no	—
207	M₃₂	—	4	1	no	—
208	M₃₂	—	4	1	no	—
209	C₂×C₁₆	—	2	2	no	—
210	C₂×C₁₆	—	2	2	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	4	2	no	—
214	C₂×C₁₆	—	4	2	no	—
215	C₂×C₁₆	—	4	2	no	—
216	C₂×C₁₆	C₄×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₁₆	C₄×M₁₆	1	1	yes	characteristic
218	C₂×C₁₆	C₄×M₁₆	1	2	yes	—
219	C₂×C₁₆	#405	1	2	yes	—
220	C₂×C₁₆	C₂×C₂²⋊C₈	1	1	yes	characteristic
221	C₂×C₁₆	#431	1	1	yes	characteristic
222	C₂×C₁₆	#504	1	4	yes	—
223	C₄⋊C₈	—	4	2	no	—
224	C₄⋊C₈	—	4	1	no	—
225	C₂²⋊C₈	—	4	1	no	—
226	C₂²⋊C₈	—	4	2	no	—
227	#97	—	2	1	no	—
228	#97	—	2	1	no	—
229	#97	—	2	1	no	—
230	#97	—	4	2	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	—	4	1	no	—
238	C₄×C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
239	C₄×C₈	#431	1	1	yes	characteristic
240	C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
241	C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
242	C₃₂	—	2	4	no	—
243	C₃₂	—	4	2	no	—
244	C₃₂	—	4	2	no	—
245	C₃₂	#435	1	4	yes	—
246	C₃₂	#436	1	4	yes	—
247	#41	—	2	2	no	—
248	#41	—	2	2	no	—
249	C₂×Q₈	#1398	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	#1398	1	1	yes	characteristic
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	C₂×#346	1	1	yes	characteristic
256	M₁₆	—	2	2	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	2	no	—
259	M₁₆	—	4	2	no	—
260	M₁₆	—	16	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	—	16	1	no	—
271	C₂×C₈	C₈×M₁₆	1	2	yes	—
272	C₂×C₈	C₄×C₂²⋊C₈	1	1	yes	characteristic
273	C₂×C₈	C₄×M₃₂	1	1	yes	characteristic
274	C₂×C₈	#2108	1	2	yes	—
275	C₂×C₈	#2112	1	1	yes	characteristic
276	C₂×C₈	#2163	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	#1398	1	2	yes	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	#2112	1	1	yes	characteristic
285	C₄²	#2163	1	1	yes	characteristic
286	C₁₆	—	4	2	no	—
287	C₁₆	—	4	2	no	—
288	C₁₆	—	4	1	no	—
289	C₁₆	—	4	1	no	—
290	C₁₆	#1453	1	1	yes	characteristic
291	C₁₆	#1455	1	1	yes	characteristic
292	C₁₆	#1753	1	2	yes	—
293	C₁₆	#2113	1	2	yes	—
294	C₁₆	#2164	1	2	yes	—
295	C₂³	—	2	1	no	—
296	C₂³	C₂×#1398	1	1	yes	characteristic
297	Q₈	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	#11408	1	2	yes	—
309	C₂×C₄	#12599	1	1	yes	characteristic
310	C₂×C₄	C₂×#1398	1	1	yes	characteristic
311	C₂×C₄	#13551	1	2	yes	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	4	1	no	—
315	C₈	—	4	1	no	—
316	C₈	—	8	1	no	—
317	C₈	—	8	1	no	—
318	C₈	—	8	2	no	—
319	C₈	#8146	1	1	yes	characteristic
320	C₈	#11409	1	2	yes	—
321	C₈	#12600	1	1	yes	characteristic
322	C₂²	—	4	1	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#149509	1	2	yes	—
326	C₂²	#151891	1	1	yes	characteristic
327	C₄	—	4	1	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	#128272	1	1	yes	characteristic
332	C₄	#149508	1	2	yes	—
333	C₄	#151892	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	2	no	—
336	C₂	—	8	1	no	—
337	C₂	#72322700	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#8146
commutator	C₂²×C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72322700
prime radical	C₂×#41	#346
LCS	G ▷ C₂²×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72321743	C₂	semidirect
3. #72286415	C₂	semidirect
4. #72234379	C₂	semidirect
91. M₁₂₈	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72321743	2
#72321570	2
#72286415	2
#72234379	2
#72231861	2