Subgroups — #72225417

Order 4096.

606 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225410	C₂	1	2	yes	maximal
2	#72225408	C₂	1	2	yes	maximal
3	#72225405	—	2	2	no	—
4	#72225404	—	2	2	no	—
5	#72225402	C₂	1	2	yes	maximal
6	#72225398	C₄	1	4	yes	complemented
7	#72225397	C₄	1	4	yes	complemented
8	#72225395	C₂²	1	4	yes	—
9	#72216231	C₂	1	1	yes	maximal, characteristic
10	#72216226	C₄	1	1	yes	characteristic
11	#72216226	C₂²	1	1	yes	characteristic
12	C₈×#1398	C₄	1	1	yes	characteristic
13	C₈×#1398	C₂²	1	1	yes	characteristic
14	#72216220	C₄	1	1	yes	characteristic
15	C₁₆×M₆₄	C₂²	1	1	yes	characteristic
16	#72216182	C₄	1	1	yes	characteristic
17	#153582	—	2	4	no	—
18	C₄×#1398	C₈	1	1	yes	characteristic
19	C₄×#1398	C₈	1	1	yes	characteristic
20	C₄×#1398	D₄	1	1	yes	characteristic
21	C₄×#1398	Q₈	1	1	yes	characteristic
22	#149495	C₈	1	1	yes	characteristic
23	#149495	C₈	1	1	yes	characteristic
24	#149495	D₄	1	1	yes	characteristic
25	#149495	Q₈	1	1	yes	characteristic
26	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
27	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
28	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
29	C₈×M₆₄	C₂³	1	1	yes	characteristic
30	C₈×#346	—	2	1	no	—
31	C₈×#346	C₂×C₄	1	1	yes	characteristic
32	C₂×#7052	—	2	1	no	—
33	C₂×#7052	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆²	C₂×C₄	1	1	yes	characteristic
35	#94748	C₂×C₄	1	8	yes	—
36	#94746	—	2	4	no	—
37	#94509	—	4	2	no	—
38	#94506	—	4	2	no	—
39	#93666	—	2	2	no	—
40	#93666	C₈	1	4	yes	—
41	#93665	—	2	2	no	—
42	#93664	C₂×C₄	1	2	yes	—
43	C₁₆×C₃₂	C₂×C₄	1	2	yes	—
44	C₂×#1398	—	2	2	no	—
45	C₂×#1398	—	2	2	no	—
46	C₂×#1398	—	2	2	no	—
47	C₂×#1398	—	2	2	no	—
48	C₄×M₆₄	C₄⋊C₄	1	1	yes	characteristic
49	C₄×M₆₄	C₄⋊C₄	1	1	yes	characteristic
50	C₄×M₆₄	C₂×C₈	1	1	yes	characteristic
51	C₄×M₆₄	C₂×C₈	1	1	yes	characteristic
52	C₄×M₆₄	C₂×C₈	1	1	yes	characteristic
53	C₄×M₆₄	C₂×C₈	1	1	yes	characteristic
54	C₄×M₆₄	C₂×D₄	1	1	yes	characteristic
55	C₄×M₆₄	C₂×Q₈	1	1	yes	characteristic
56	C₄×#346	—	2	2	no	—
57	C₄×#346	C₄⋊C₄	1	1	yes	characteristic
58	C₄×#346	C₂×C₈	1	1	yes	characteristic
59	C₂×#1311	—	2	2	no	—
60	C₂×#1311	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#1311	C₂×C₈	1	1	yes	characteristic
62	C₂×C₈×C₁₆	—	2	1	no	—
63	C₂×C₈×C₁₆	—	2	1	no	—
64	C₂×C₈×C₁₆	M₁₆	1	1	yes	characteristic
65	C₂×C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
66	C₄×C₈²	M₁₆	1	1	yes	characteristic
67	#7544	—	2	8	no	—
68	#7513	—	4	4	no	—
69	#7511	—	4	4	no	—
70	#7330	—	2	2	no	—
71	#7330	—	2	1	no	—
72	#7330	—	2	1	no	—
73	#7330	—	2	2	no	—
74	#7330	D₈	1	2	yes	—
75	#7330	Q₁₆	1	2	yes	—
76	#7329	—	2	1	no	—
77	#7329	C₂×C₈	1	2	yes	—
78	#7329	C₂×C₈	1	2	yes	—
79	#7329	#41	1	2	yes	—
80	C₈×C₃₂	—	2	1	no	—
81	C₈×C₃₂	C₄²	1	2	yes	—
82	C₈×C₃₂	M₁₆	1	2	yes	—
83	C₈×C₃₂	C₂²×C₄	1	2	yes	—
84	#7052	—	2	4	no	—
85	#7052	—	2	1	no	—
86	#7052	C₂×C₈	1	2	yes	—
87	C₁₆²	—	2	1	no	—
88	C₁₆²	C₄²	1	1	yes	characteristic
89	C₁₆²	C₄²	1	1	yes	characteristic
90	C₂×M₆₄	—	2	1	no	—
91	C₂×M₆₄	—	2	1	no	—
92	C₂×M₆₄	—	2	1	no	—
93	C₂×M₆₄	—	2	1	no	—
94	C₂×M₆₄	#108	1	2	yes	—
95	C₂×M₆₄	M₃₂	1	2	yes	—
96	C₂×M₆₄	M₃₂	1	2	yes	—
97	C₂×M₆₄	#135	1	2	yes	—
98	C₂×#346	—	2	1	no	—
99	C₂×#346	—	2	1	no	—
100	C₂×#346	—	2	1	no	—
101	C₂×#346	—	2	1	no	—
102	C₂×#346	—	4	4	no	—
103	C₂×C₄×C₁₆	—	2	2	no	—
104	C₂×C₄×C₁₆	—	2	2	no	—
105	C₂×C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
109	C₄²×C₈	—	2	1	no	—
110	C₄²×C₈	C₄⋊C₈	1	1	yes	characteristic
111	C₂×C₈²	—	2	1	no	—
112	C₂×C₈²	C₄⋊C₈	1	1	yes	characteristic
113	C₂×C₈²	C₂×M₁₆	1	1	yes	characteristic
114	M₁₂₈	—	4	8	no	—
115	#1398	—	4	4	no	—
116	#1398	—	4	4	no	—
117	#1398	—	4	4	no	—
118	#1398	—	4	4	no	—
119	#1397	—	2	1	no	—
120	#1397	—	2	1	no	—
121	#1397	—	2	1	no	—
122	#1397	—	2	1	no	—
123	#1397	#107	1	2	yes	—
124	#1397	M₃₂	1	2	yes	—
125	#1397	M₃₂	1	1	yes	characteristic
126	#1397	M₃₂	1	1	yes	characteristic
127	#1397	C₂×D₈	1	1	yes	characteristic
128	#1397	C₂×Q₁₆	1	1	yes	characteristic
129	C₄×C₃₂	—	2	1	no	—
130	C₄×C₃₂	—	2	1	no	—
131	C₄×C₃₂	C₄×C₈	1	2	yes	—
132	C₄×C₃₂	C₄×C₈	1	2	yes	—
133	C₄×C₃₂	C₄⋊C₈	1	2	yes	—
134	C₄×C₃₂	C₄⋊C₈	1	2	yes	—
135	C₄×C₃₂	C₄×D₄	1	2	yes	—
136	C₄×C₃₂	C₄×Q₈	1	2	yes	—
137	#1311	—	2	1	no	—
138	#1311	—	2	1	no	—
139	#1311	—	4	4	no	—
140	#1311	#107	1	2	yes	—
141	#1311	#108	1	2	yes	—
142	C₈×C₁₆	—	2	2	no	—
143	C₈×C₁₆	—	2	2	no	—
144	C₈×C₁₆	—	2	2	no	—
145	C₈×C₁₆	—	2	1	no	—
146	C₈×C₁₆	—	2	1	no	—
147	C₈×C₁₆	—	2	2	no	—
148	C₈×C₁₆	C₂×C₄²	1	1	yes	characteristic
149	C₈×C₁₆	#117	1	1	yes	characteristic
150	C₈×C₁₆	C₂²×C₈	1	1	yes	characteristic
151	C₈×C₁₆	C₂×M₁₆	1	1	yes	characteristic
152	C₂²×C₁₆	—	2	4	no	—
153	C₂²×C₁₆	—	2	1	no	—
154	C₂²×C₁₆	—	2	1	no	—
155	C₂²×C₁₆	—	4	2	no	—
156	C₂²×C₁₆	#427	1	1	yes	characteristic
157	C₂²×C₁₆	C₂×#108	1	1	yes	characteristic
158	C₂²×C₁₆	C₂×M₃₂	1	1	yes	characteristic
159	C₂²×C₁₆	C₂×M₃₂	1	1	yes	characteristic
160	C₂×C₄×C₈	—	2	1	no	—
161	C₂×C₄×C₈	—	2	1	no	—
162	C₂×C₄×C₈	—	2	1	no	—
163	C₂×C₄×C₈	—	2	1	no	—
164	C₂×C₄×C₈	—	4	2	no	—
165	C₂×C₄×C₈	#364	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
167	C₄³	#364	1	1	yes	characteristic
168	M₆₄	—	2	4	no	—
169	M₆₄	—	2	4	no	—
170	M₆₄	—	2	4	no	—
171	M₆₄	—	2	4	no	—
172	M₆₄	—	4	2	no	—
173	M₆₄	—	4	2	no	—
174	M₆₄	—	4	2	no	—
175	M₆₄	—	4	2	no	—
176	C₂×C₃₂	—	2	2	no	—
177	C₂×C₃₂	—	2	2	no	—
178	C₂×C₃₂	—	2	2	no	—
179	C₂×C₃₂	—	2	2	no	—
180	C₂×C₃₂	—	2	2	no	—
181	C₂×C₃₂	#335	1	4	yes	—
182	C₂×C₃₂	#346	1	2	yes	—
183	C₂×C₃₂	#346	1	2	yes	—
184	C₂×C₃₂	C₄×D₈	1	2	yes	—
185	C₂×C₃₂	C₄×Q₁₆	1	2	yes	—
186	#346	—	4	2	no	—
187	#346	—	4	2	no	—
188	#346	—	4	2	no	—
189	#346	—	4	2	no	—
190	#346	—	8	8	no	—
191	C₄×C₁₆	—	2	2	no	—
192	C₄×C₁₆	—	2	2	no	—
193	C₄×C₁₆	—	2	4	no	—
194	C₄×C₁₆	—	2	4	no	—
195	C₄×C₁₆	—	2	1	no	—
196	C₄×C₁₆	—	2	1	no	—
197	C₄×C₁₆	—	4	2	no	—
198	C₄×C₁₆	#335	1	2	yes	—
199	C₄×C₁₆	#364	1	2	yes	—
200	C₄×C₁₆	C₄×C₄⋊C₄	1	1	yes	characteristic
201	C₄×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
202	C₄×C₁₆	C₂×C₄⋊C₈	1	1	yes	characteristic
203	C₄×C₁₆	C₂×#107	1	1	yes	characteristic
204	C₄×C₁₆	C₂×#108	1	1	yes	characteristic
205	C₄×C₁₆	#431	1	1	yes	characteristic
206	C₄×C₁₆	C₂×M₃₂	1	1	yes	characteristic
207	C₄×C₁₆	C₂×M₃₂	1	1	yes	characteristic
208	C₈²	—	2	1	no	—
209	C₈²	—	2	1	no	—
210	C₈²	—	4	2	no	—
211	C₈²	#335	1	2	yes	—
212	C₈²	C₄×M₁₆	1	1	yes	characteristic
213	C₈²	#431	1	1	yes	characteristic
214	C₆₄	—	8	8	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	4	1	no	—
218	C₂²×C₈	#1564	1	1	yes	characteristic
219	C₂²×C₈	#2150	1	1	yes	characteristic
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	—	4	1	no	—
222	C₂×C₄²	C₂×#364	1	1	yes	characteristic
223	C₂×C₁₆	—	2	4	no	—
224	C₂×C₁₆	—	2	2	no	—
225	C₂×C₁₆	—	2	2	no	—
226	C₂×C₁₆	—	2	2	no	—
227	C₂×C₁₆	—	2	2	no	—
228	C₂×C₁₆	—	2	4	no	—
229	C₂×C₁₆	—	2	2	no	—
230	C₂×C₁₆	—	2	2	no	—
231	C₂×C₁₆	—	4	8	no	—
232	C₂×C₁₆	—	4	1	no	—
233	C₂×C₁₆	—	4	1	no	—
234	C₂×C₁₆	—	4	1	no	—
235	C₂×C₁₆	—	4	1	no	—
236	C₂×C₁₆	—	4	4	no	—
237	C₂×C₁₆	—	8	4	no	—
238	C₂×C₁₆	C₂×#335	1	1	yes	characteristic
239	C₂×C₁₆	#1564	1	1	yes	characteristic
240	C₂×C₁₆	C₄×#107	1	1	yes	characteristic
241	C₂×C₁₆	C₄×#108	1	1	yes	characteristic
242	C₂×C₁₆	C₂×#346	1	1	yes	characteristic
243	C₂×C₁₆	C₄×M₃₂	1	1	yes	characteristic
244	C₂×C₁₆	#2163	1	1	yes	characteristic
245	C₂×C₁₆	#2163	1	1	yes	characteristic
246	C₄×C₈	—	2	2	no	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	2	2	no	—
250	C₄×C₈	—	4	1	no	—
251	C₄×C₈	—	4	1	no	—
252	C₄×C₈	—	4	1	no	—
253	C₄×C₈	—	4	1	no	—
254	C₄×C₈	—	4	1	no	—
255	C₄×C₈	—	4	1	no	—
256	C₄×C₈	—	8	4	no	—
257	C₄×C₈	C₂×#335	1	1	yes	characteristic
258	C₄×C₈	C₄×C₄⋊C₈	1	1	yes	characteristic
259	C₄×C₈	#2150	1	1	yes	characteristic
260	C₄×C₈	C₂×#364	1	1	yes	characteristic
261	C₃₂	—	2	4	no	—
262	C₃₂	—	2	4	no	—
263	C₃₂	—	2	2	no	—
264	C₃₂	—	2	4	no	—
265	C₃₂	—	2	2	no	—
266	C₃₂	—	4	2	no	—
267	C₃₂	—	4	2	no	—
268	C₃₂	—	4	2	no	—
269	C₃₂	#1398	1	4	yes	—
270	C₃₂	#2179	1	4	yes	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	#12013	1	1	yes	characteristic
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	4	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	8	2	no	—
286	C₂×C₈	C₄×#335	1	1	yes	characteristic
287	C₂×C₈	#12004	1	1	yes	characteristic
288	C₂×C₈	#12013	1	1	yes	characteristic
289	C₂×C₈	#12653	1	1	yes	characteristic
290	C₄²	—	2	1	no	—
291	C₄²	—	4	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	8	2	no	—
294	C₄²	#12004	1	1	yes	characteristic
295	C₄²	C₄×#364	1	1	yes	characteristic
296	C₁₆	—	2	8	no	—
297	C₁₆	—	2	1	no	—
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	1	no	—
300	C₁₆	—	2	1	no	—
301	C₁₆	—	4	4	no	—
302	C₁₆	—	4	2	no	—
303	C₁₆	—	4	2	no	—
304	C₁₆	—	8	4	no	—
305	C₁₆	#12011	1	2	yes	—
306	C₁₆	#12671	1	2	yes	—
307	C₁₆	C₂×#1398	1	2	yes	—
308	C₁₆	#13642	1	2	yes	—
309	C₂³	#151045	1	1	yes	characteristic
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	#129093	1	1	yes	characteristic
318	C₂×C₄	#151045	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	4	2	no	—
323	C₈	—	8	1	no	—
324	C₈	—	8	2	no	—
325	C₈	#129096	1	1	yes	characteristic
326	C₈	#129101	1	1	yes	characteristic
327	C₈	#151979	1	1	yes	characteristic
328	C₈	#151981	1	1	yes	characteristic
329	C₂²	—	2	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	#72216243	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	16	1	no	—
336	C₄	#72225412	1	1	yes	characteristic
337	C₄	#72225413	1	1	yes	characteristic
338	C₂	—	2	1	no	—
339	C₂	—	4	1	no	—
340	C₂	#72225415	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#129096
commutator	C₄×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₈×M₆₄	C₂³
socle	C₂	#72225415
prime radical	C₂³	#151045
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #1397 ◁ #149495 ◁ C₈×#1398 ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
6. #72225398	C₄	semidirect
7. #72225397	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72225410	2
#72225408	2
#72225402	2
#72216231	2