Subgroups — #72227239

Order 4096.

742 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72227235	C₂	1	2	yes	maximal
2	#72227183	C₂	1	1	yes	maximal, characteristic
3	#72227181	C₄	1	2	yes	—
4	#72227180	C₄	1	2	yes	—
5	#72225376	C₂	1	1	yes	maximal, characteristic
6	#72225373	C₄	1	2	yes	—
7	#72225362	C₄	1	1	yes	characteristic
8	#72225352	C₄	1	1	yes	characteristic
9	#72225206	C₂	1	1	yes	maximal, characteristic
10	#72225138	C₂²	1	1	yes	characteristic
11	C₄×#7333	C₄	1	2	yes	—
12	C₄×#7333	C₄	1	2	yes	—
13	#72216255	C₂	1	2	yes	maximal
14	#72216250	—	2	2	no	—
15	#72216248	—	2	2	no	—
16	#72216247	C₂²	1	2	yes	—
17	#72216243	C₂²	1	2	yes	—
18	C₈×#1398	C₂²	1	1	yes	characteristic
19	#72216161	C₂²	1	1	yes	characteristic
20	#151981	—	2	2	no	—
21	#151981	—	2	2	no	—
22	C₄×#1398	C₂×C₄	1	2	yes	—
23	C₄×#1398	C₂×C₄	1	1	yes	characteristic
24	C₄×#1398	C₂×C₄	1	1	yes	characteristic
25	C₂×#7459	C₂×C₄	1	2	yes	—
26	C₂×#7459	C₂×C₄	1	2	yes	—
27	C₂×#7456	C₂×C₄	1	4	yes	—
28	C₂×#7333	—	2	4	no	—
29	C₂×#7333	—	2	2	no	—
30	C₂×#7333	—	2	2	no	—
31	C₂×#7333	—	4	2	no	—
32	#149502	—	2	1	no	—
33	#149499	—	2	1	no	—
34	#149495	C₂×C₄	1	1	yes	characteristic
35	#149495	C₂×C₄	1	1	yes	characteristic
36	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
37	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
38	C₄×#1313	C₂×C₄	1	2	yes	—
39	C₄×#1313	D₄	1	2	yes	—
40	#126522	Q₈	1	1	yes	characteristic
41	C₈×#346	C₂³	1	1	yes	characteristic
42	#94857	Q₈	1	1	yes	characteristic
43	#94019	—	2	2	no	—
44	#94018	—	2	2	no	—
45	#93997	—	4	2	no	—
46	#93885	C₈	1	4	yes	complemented
47	#93882	C₈	1	4	yes	complemented
48	#93878	—	2	4	no	—
49	C₂×#1422	—	2	2	no	—
50	C₂×#1422	—	2	2	no	—
51	C₂×#1422	—	2	4	no	—
52	C₂×#1398	—	2	2	no	—
53	C₂×#1398	—	2	2	no	—
54	C₂×#1398	—	4	2	no	—
55	C₂×#1398	—	4	2	no	—
56	C₂×#1398	M₁₆	1	2	yes	—
57	C₂×#1398	M₁₆	1	2	yes	—
58	C₂×#1398	M₁₆	1	4	yes	—
59	C₄×M₆₄	—	2	2	no	—
60	C₄×M₆₄	C₄²	1	1	yes	characteristic
61	C₄×M₆₄	C₄²	1	1	yes	characteristic
62	C₄×M₆₄	C₄²	1	1	yes	characteristic
63	C₄×M₆₄	C₄²	1	1	yes	characteristic
64	C₄×#346	C₄⋊C₄	1	2	yes	—
65	C₄×#346	C₂²×C₄	1	1	yes	characteristic
66	C₄×#346	C₂×D₄	1	1	yes	characteristic
67	C₂×#1313	—	2	2	no	—
68	C₂×#1313	—	2	2	no	—
69	C₂×#1313	—	2	2	no	—
70	C₂×#1313	—	2	2	no	—
71	C₂×#1312	—	4	1	no	—
72	C₂×#1311	C₂²×C₄	1	1	yes	characteristic
73	C₂×#1311	#41	1	1	yes	characteristic
74	C₂×C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₆	#41	1	1	yes	characteristic
76	C₄×#322	—	4	1	no	—
77	C₄×#322	C₄⋊C₄	1	2	yes	—
78	C₄×C₈²	C₂×Q₈	1	1	yes	characteristic
79	#7468	—	4	2	no	—
80	#7468	—	4	2	no	—
81	#7459	—	2	2	no	—
82	#7459	—	2	2	no	—
83	#7459	C₂×C₈	1	4	yes	—
84	#7459	C₂×C₈	1	4	yes	—
85	#7456	—	2	4	no	—
86	#7456	—	2	4	no	—
87	#7333	—	4	8	no	—
88	#7333	—	4	4	no	—
89	#7333	—	4	4	no	—
90	#7333	—	8	4	no	—
91	#7332	—	4	1	no	—
92	#7332	—	4	1	no	—
93	#7331	—	4	2	no	—
94	#7330	—	2	2	no	—
95	#7330	M₁₆	1	4	yes	—
96	#7329	—	2	1	no	—
97	#7329	C₂×C₈	1	2	yes	—
98	C₈×C₃₂	—	2	1	no	—
99	C₈×C₃₂	C₂×C₈	1	2	yes	—
100	C₂×M₆₄	—	2	1	no	—
101	C₂×M₆₄	—	2	1	no	—
102	C₂×M₆₄	—	2	4	no	—
103	C₂×M₆₄	—	4	2	no	—
104	C₂×M₆₄	—	4	2	no	—
105	C₂×M₆₄	—	4	4	no	—
106	C₂×M₆₄	#97	1	2	yes	—
107	C₂×M₆₄	#97	1	1	yes	characteristic
108	C₂×M₆₄	#97	1	1	yes	characteristic
109	C₂×#346	—	2	2	no	—
110	C₂×#346	—	2	2	no	—
111	C₂×#346	C₂×M₁₆	1	2	yes	—
112	C₂×#346	C₂×M₁₆	1	2	yes	—
113	C₂×#346	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×#346	#137	1	2	yes	—
115	C₂×C₄×C₁₆	—	2	2	no	—
116	C₂×C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₆	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
120	C₄²×C₈	—	2	1	no	—
121	C₄²×C₈	—	2	2	no	—
122	C₄²×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×#322	—	2	2	no	—
124	C₂×#322	—	2	2	no	—
125	C₂×#322	—	8	2	no	—
126	C₂×C₈²	—	2	1	no	—
127	C₂×C₈²	C₄×Q₈	1	1	yes	characteristic
128	C₂×C₈²	C₄×Q₈	1	1	yes	characteristic
129	#1422	—	2	4	no	—
130	#1422	—	2	4	no	—
131	#1422	—	4	4	no	—
132	#1422	—	4	2	no	—
133	#1422	—	4	2	no	—
134	#1422	—	4	4	no	—
135	#1398	—	2	8	no	—
136	#1398	—	2	4	no	—
137	#1398	—	2	4	no	—
138	#1398	—	4	4	no	—
139	#1398	—	4	2	no	—
140	#1398	—	4	2	no	—
141	#1398	—	8	2	no	—
142	#1398	—	8	2	no	—
143	#1398	—	8	2	no	—
144	#1398	—	8	2	no	—
145	#1397	—	2	1	no	—
146	#1397	—	2	1	no	—
147	#1397	—	4	2	no	—
148	#1397	#97	1	2	yes	—
149	#1397	#97	1	2	yes	—
150	C₄×C₃₂	—	2	1	no	—
151	C₄×C₃₂	—	2	1	no	—
152	C₄×C₃₂	—	4	2	no	—
153	C₄×C₃₂	C₄×C₈	1	2	yes	—
154	C₄×C₃₂	C₄×C₈	1	2	yes	—
155	#1313	—	4	2	no	—
156	#1313	—	4	2	no	—
157	#1313	—	4	2	no	—
158	#1313	—	4	4	no	—
159	#1313	—	4	2	no	—
160	#1313	—	8	2	no	—
161	#1312	—	8	2	no	—
162	#1311	—	2	1	no	—
163	#1311	—	2	1	no	—
164	#1311	C₂×M₁₆	1	2	yes	—
165	#1311	#131	1	2	yes	—
166	C₈×C₁₆	—	2	2	no	—
167	C₈×C₁₆	C₂²×C₈	1	1	yes	characteristic
168	C₈×C₁₆	C₂²×C₈	1	1	yes	characteristic
169	C₈×C₁₆	#131	1	1	yes	characteristic
170	C₈×C₁₆	#131	1	1	yes	characteristic
171	C₂²×C₁₆	—	4	2	no	—
172	C₂²×C₁₆	C₂×#97	1	1	yes	characteristic
173	C₂²×C₁₆	C₄×M₁₆	1	1	yes	characteristic
174	C₂²×C₁₆	#432	1	2	yes	—
175	C₂²×C₁₆	#440	1	2	yes	—
176	C₂²×C₁₆	#441	1	1	yes	characteristic
177	C₂²×C₁₆	#442	1	1	yes	characteristic
178	C₂×C₄×C₈	—	2	1	no	—
179	C₂×C₄×C₈	—	2	1	no	—
180	C₂×C₄×C₈	—	4	2	no	—
181	C₂×C₄×C₈	—	4	2	no	—
182	C₂×C₄×C₈	—	8	2	no	—
183	C₂×C₄×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄×C₈	#423	1	2	yes	—
185	C₂×C₄×C₈	#428	1	2	yes	—
186	C₂×C₄×C₈	#430	1	1	yes	characteristic
187	C₄³	#430	1	1	yes	characteristic
188	M₆₄	—	2	4	no	—
189	M₆₄	—	2	4	no	—
190	M₆₄	—	2	2	no	—
191	M₆₄	—	2	2	no	—
192	M₆₄	—	4	4	no	—
193	M₆₄	—	4	4	no	—
194	M₆₄	—	4	4	no	—
195	M₆₄	—	4	4	no	—
196	M₆₄	—	8	2	no	—
197	M₆₄	—	8	2	no	—
198	M₆₄	—	8	4	no	—
199	M₆₄	—	8	4	no	—
200	M₆₄	#347	1	4	yes	—
201	M₆₄	#347	1	4	yes	—
202	C₂×C₃₂	—	2	2	no	—
203	C₂×C₃₂	—	2	2	no	—
204	C₂×C₃₂	—	2	2	no	—
205	C₂×C₃₂	—	4	4	no	—
206	C₂×C₃₂	—	8	2	no	—
207	C₂×C₃₂	—	8	2	no	—
208	C₂×C₃₂	—	8	4	no	—
209	C₂×C₃₂	#322	1	2	yes	—
210	C₂×C₃₂	#322	1	2	yes	—
211	#346	—	2	2	no	—
212	#346	—	2	2	no	—
213	#346	—	2	4	no	—
214	#346	—	2	2	no	—
215	#346	—	2	2	no	—
216	#346	—	4	2	no	—
217	#346	—	4	2	no	—
218	#346	—	4	2	no	—
219	#346	—	4	2	no	—
220	#346	—	4	2	no	—
221	C₄×C₁₆	—	2	1	no	—
222	C₄×C₁₆	—	2	1	no	—
223	C₄×C₁₆	—	8	2	no	—
224	C₄×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
225	C₄×C₁₆	C₂×#97	1	1	yes	characteristic
226	C₄×C₁₆	#405	1	1	yes	characteristic
227	C₄×C₁₆	#405	1	1	yes	characteristic
228	C₄×C₁₆	C₈×D₄	1	1	yes	characteristic
229	C₄×C₁₆	C₈×D₄	1	1	yes	characteristic
230	C₄×C₁₆	#436	1	2	yes	—
231	C₄×C₁₆	#444	1	1	yes	characteristic
232	C₄×C₁₆	#444	1	1	yes	characteristic
233	C₄×C₁₆	#444	1	2	yes	—
234	#322	—	4	2	no	—
235	#322	—	4	2	no	—
236	#322	—	4	2	no	—
237	#322	—	4	2	no	—
238	#322	—	16	4	no	—
239	C₈²	—	2	1	no	—
240	C₈²	—	2	1	no	—
241	C₈²	—	4	2	no	—
242	C₈²	C₈×Q₈	1	1	yes	characteristic
243	C₈²	C₈×Q₈	1	1	yes	characteristic
244	C₈²	#446	1	2	yes	—
245	C₂²×C₈	—	2	2	no	—
246	C₂²×C₈	—	4	1	no	—
247	C₂²×C₈	—	8	2	no	—
248	C₂²×C₈	#1569	1	2	yes	—
249	C₂²×C₈	#1767	1	1	yes	characteristic
250	C₂²×C₈	#1776	1	1	yes	characteristic
251	C₂×C₄²	—	4	1	no	—
252	C₂×C₄²	#1569	1	2	yes	—
253	C₂×C₄²	#1778	1	1	yes	characteristic
254	C₂×C₁₆	—	2	2	no	—
255	C₂×C₁₆	—	2	2	no	—
256	C₂×C₁₆	—	2	2	no	—
257	C₂×C₁₆	—	2	2	no	—
258	C₂×C₁₆	—	2	2	no	—
259	C₂×C₁₆	—	2	2	no	—
260	C₂×C₁₆	—	8	2	no	—
261	C₂×C₁₆	—	8	2	no	—
262	C₂×C₁₆	—	8	2	no	—
263	C₂×C₁₆	C₂×#322	1	1	yes	characteristic
264	C₂×C₁₆	C₈×M₁₆	1	1	yes	characteristic
265	C₂×C₁₆	#1451	1	1	yes	characteristic
266	C₂×C₁₆	#1452	1	1	yes	characteristic
267	C₂×C₁₆	#1579	1	1	yes	characteristic
268	C₂×C₁₆	#1579	1	1	yes	characteristic
269	C₂×C₁₆	#1580	1	1	yes	characteristic
270	C₂×C₁₆	#1580	1	1	yes	characteristic
271	C₂×C₁₆	#1592	1	4	yes	—
272	C₂×C₁₆	C₂×#347	1	2	yes	—
273	C₂×C₁₆	C₂×#347	1	2	yes	—
274	C₂×C₁₆	#2165	1	4	yes	—
275	C₂×C₁₆	#2180	1	2	yes	—
276	C₂×C₁₆	AGL(1,ℤ/16)	1	2	yes	—
277	C₄×C₈	—	2	2	no	—
278	C₄×C₈	—	2	2	no	—
279	C₄×C₈	—	2	2	no	—
280	C₄×C₈	—	2	2	no	—
281	C₄×C₈	—	2	2	no	—
282	C₄×C₈	—	2	1	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	2	2	no	—
285	C₄×C₈	—	4	1	no	—
286	C₄×C₈	—	4	1	no	—
287	C₄×C₈	—	8	2	no	—
288	C₄×C₈	—	8	2	no	—
289	C₄×C₈	—	8	4	no	—
290	C₄×C₈	—	16	4	no	—
291	C₄×C₈	C₈×C₄⋊C₄	1	1	yes	characteristic
292	C₄×C₈	#1770	1	1	yes	characteristic
293	C₄×C₈	#1773	1	1	yes	characteristic
294	C₄×C₈	#1773	1	1	yes	characteristic
295	C₃₂	—	2	4	no	—
296	C₃₂	—	2	4	no	—
297	C₃₂	—	2	2	no	—
298	C₃₂	—	2	4	no	—
299	C₃₂	—	8	4	no	—
300	C₃₂	—	8	4	no	—
301	C₃₂	—	8	4	no	—
302	C₃₂	—	16	4	no	—
303	C₃₂	#1313	1	4	yes	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	#8276	1	1	yes	characteristic
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	4	4	no	—
313	C₂×C₈	—	4	2	no	—
314	C₂×C₈	—	8	2	no	—
315	C₂×C₈	—	16	2	no	—
316	C₂×C₈	—	16	4	no	—
317	C₂×C₈	#8261	1	1	yes	characteristic
318	C₂×C₈	#8269	1	1	yes	characteristic
319	C₂×C₈	#8284	1	1	yes	characteristic
320	C₂×C₈	#8286	1	1	yes	characteristic
321	C₂×C₈	#12658	1	2	yes	—
322	C₂×C₈	#12659	1	2	yes	—
323	C₂×C₈	#12677	1	2	yes	—
324	C₂×C₈	#12678	1	2	yes	—
325	C₄²	—	2	1	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	8	2	no	—
329	C₄²	#8294	1	1	yes	characteristic
330	C₄²	#8295	1	1	yes	characteristic
331	C₁₆	—	2	1	no	—
332	C₁₆	—	2	1	no	—
333	C₁₆	—	2	4	no	—
334	C₁₆	—	2	1	no	—
335	C₁₆	—	2	1	no	—
336	C₁₆	—	2	4	no	—
337	C₁₆	—	16	2	no	—
338	C₁₆	C₂×#1313	1	1	yes	characteristic
339	C₁₆	#11404	1	1	yes	characteristic
340	C₁₆	#11415	1	2	yes	—
341	C₁₆	#12065	1	2	yes	—
342	C₁₆	#12066	1	1	yes	characteristic
343	C₁₆	#12067	1	1	yes	characteristic
344	C₂³	#129125	1	1	yes	characteristic
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	#96894	1	1	yes	characteristic
351	C₂×C₄	#129123	1	2	yes	—
352	C₂×C₄	#129124	1	2	yes	—
353	C₂×C₄	#129125	1	1	yes	characteristic
354	C₈	—	2	4	no	—
355	C₈	—	4	2	no	—
356	C₈	—	4	2	no	—
357	C₈	—	8	2	no	—
358	C₈	—	32	2	no	—
359	C₈	#129054	1	1	yes	characteristic
360	C₈	#129055	1	1	yes	characteristic
361	C₈	#129084	1	2	yes	—
362	C₈	#129165	1	1	yes	characteristic
363	C₈	#129166	1	1	yes	characteristic
364	C₈	#129174	1	2	yes	—
365	C₂²	—	4	1	no	—
366	C₂²	#72216225	1	1	yes	characteristic
367	C₂²	#72227236	1	2	yes	—
368	C₄	—	2	2	no	—
369	C₄	—	4	1	no	—
370	C₄	—	8	2	no	—
371	C₄	#72216252	1	1	yes	characteristic
372	C₄	#72225332	1	1	yes	characteristic
373	C₄	#72227236	1	2	yes	—
374	C₂	—	2	1	no	—
375	C₂	—	2	2	no	—
376	C₂	#72227238	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#72225332
commutator	C₄×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₈×#346	C₂³
socle	C₂	#72227238
prime radical	C₂³	#129125
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ #72216161 ◁ 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
46. #93885	C₈	semidirect
47. #93882	C₈	semidirect

Maximal subgroups

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

subgroup	index
#72227235	2
#72227183	2
#72225376	2
#72225206	2
#72216255	2