Subgroups — #72227278

Order 2048.

578 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225535	C₂	1	1	yes	maximal, characteristic
2	#72225375	C₂	1	1	yes	maximal, characteristic
3	#72216557	C₂	1	2	yes	maximal
4	#72216306	C₂	1	2	yes	maximal
5	#72216168	C₂	1	1	yes	maximal, characteristic
6	#140388	—	2	2	no	—
7	#135715	—	2	1	no	—
8	#135715	—	2	1	no	—
9	#129093	C₂²	1	2	yes	—
10	#129086	C₂²	1	1	yes	characteristic
11	#129049	—	2	1	no	—
12	C₈×#346	C₂²	1	1	yes	characteristic
13	C₂×#7101	C₄	1	1	yes	characteristic
14	C₂×#7100	C₄	1	1	yes	characteristic
15	#99295	C₂²	1	2	yes	—
16	#96889	C₂²	1	1	yes	characteristic
17	#93452	—	2	2	no	—
18	#93261	—	2	1	no	—
19	#92911	C₄	1	2	yes	—
20	#12653	—	2	2	no	—
21	#12653	—	2	1	no	—
22	C₄×#346	D₄	1	1	yes	characteristic
23	C₄×#346	D₄	1	1	yes	characteristic
24	C₂×#1370	C₂×C₄	1	2	yes	—
25	C₂×#1370	C₂×C₄	1	2	yes	—
26	C₂×#1368	—	4	2	no	—
27	C₂×#1368	—	4	2	no	—
28	C₂×#1368	—	4	1	no	—
29	C₂×#1368	—	4	1	no	—
30	C₂×#1367	—	2	1	no	—
31	C₂×#1313	—	4	1	no	—
32	C₂×#1311	Q₈	1	1	yes	characteristic
33	C₂×#1311	Q₈	1	1	yes	characteristic
34	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
36	#8805	—	2	2	no	—
37	#8805	—	2	2	no	—
38	#8805	—	2	2	no	—
39	C₄×#334	D₄	1	2	yes	—
40	#8260	—	2	1	no	—
41	C₄×C₈²	C₂³	1	1	yes	characteristic
42	#7156	—	4	2	no	—
43	#7151	—	2	2	no	—
44	#7151	Q₈	1	2	yes	—
45	#7137	—	4	1	no	—
46	#7101	—	2	2	no	—
47	#7100	C₈	1	4	yes	complemented
48	#7015	—	2	1	no	—
49	C₂×C₄⋊C₁₆	—	2	2	no	—
50	C₂×C₄⋊C₁₆	—	2	2	no	—
51	C₂×C₄⋊C₁₆	—	2	2	no	—
52	C₂×#346	—	2	1	no	—
53	C₂×#346	—	2	1	no	—
54	C₂×#346	—	4	1	no	—
55	C₂×#346	—	4	2	no	—
56	C₂×#346	SD₁₆	1	1	yes	characteristic
57	C₂×#346	SD₁₆	1	1	yes	characteristic
58	C₂×#346	SD₁₆	1	1	yes	characteristic
59	C₂×#346	SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
64	#1842	—	4	2	no	—
65	C₄×C₄⋊C₈	—	2	2	no	—
66	C₄²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₈	#41	1	1	yes	characteristic
69	C₂×#334	—	2	2	no	—
70	C₂×#334	—	4	2	no	—
71	C₂×#334	—	4	2	no	—
72	C₂×#334	—	4	2	no	—
73	C₂×#334	—	4	2	no	—
74	C₂×#334	—	8	2	no	—
75	C₂×#322	—	4	1	no	—
76	C₂×C₈²	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈²	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈²	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈²	#41	1	1	yes	characteristic
80	#1374	—	4	1	no	—
81	#1374	—	4	2	no	—
82	#1370	—	2	2	no	—
83	#1370	—	2	2	no	—
84	#1370	—	2	2	no	—
85	#1370	C₂×C₈	1	4	yes	—
86	#1368	—	8	4	no	—
87	#1368	—	8	4	no	—
88	#1368	—	8	2	no	—
89	#1368	—	8	2	no	—
90	#1367	—	4	1	no	—
91	#1367	—	4	1	no	—
92	#1313	—	8	2	no	—
93	#1311	—	2	2	no	—
94	#1311	—	4	1	no	—
95	C₈×C₁₆	—	2	1	no	—
96	C₈×C₁₆	—	2	1	no	—
97	C₈×C₁₆	C₂×C₈	1	2	yes	—
98	C₈×C₁₆	M₁₆	1	2	yes	—
99	#1277	—	2	2	no	—
100	#1273	—	4	1	no	—
101	C₂²×C₁₆	—	2	1	no	—
102	C₂²×C₁₆	—	2	1	no	—
103	C₂²×C₁₆	—	4	2	no	—
104	C₂²×C₁₆	—	4	2	no	—
105	C₂²×C₁₆	—	4	2	no	—
106	C₂²×C₁₆	#106	1	1	yes	characteristic
107	C₂²×C₁₆	#106	1	1	yes	characteristic
108	C₂²×C₁₆	#106	1	1	yes	characteristic
109	C₂²×C₁₆	#106	1	1	yes	characteristic
110	C₂×C₄⋊C₈	—	4	2	no	—
111	C₂×C₄⋊C₈	—	8	2	no	—
112	C₂×C₄⋊C₈	—	8	2	no	—
113	C₂×C₄⋊C₈	—	8	2	no	—
114	C₂×C₄×C₈	—	2	2	no	—
115	C₂×C₄×C₈	—	4	2	no	—
116	C₂×C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈	C₄×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₈	#125	1	1	yes	characteristic
121	C₂×C₄×C₈	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₈	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₈	#137	1	1	yes	characteristic
127	C₂×C₄×C₈	#137	1	2	yes	—
128	C₄³	#121	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	4	no	—
130	C₄⋊C₁₆	—	4	2	no	—
131	C₄⋊C₁₆	—	4	2	no	—
132	C₄⋊C₁₆	—	4	2	no	—
133	C₄⋊C₁₆	—	4	2	no	—
134	C₄⋊C₁₆	—	4	2	no	—
135	#346	—	2	4	no	—
136	#346	—	2	2	no	—
137	#346	—	2	2	no	—
138	#346	—	2	2	no	—
139	#346	—	4	1	no	—
140	#346	—	4	2	no	—
141	#346	—	8	1	no	—
142	#346	—	8	1	no	—
143	#346	—	8	2	no	—
144	#346	—	8	2	no	—
145	C₄×C₁₆	—	2	1	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	C₄⋊C₈	1	2	yes	—
151	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
152	C₄×C₁₆	#108	1	2	yes	—
153	#336	—	4	2	no	—
154	#334	—	4	2	no	—
155	#334	—	4	2	no	—
156	#334	—	8	2	no	—
157	#334	—	8	2	no	—
158	#334	—	8	2	no	—
159	#334	—	8	2	no	—
160	#334	—	8	4	no	—
161	#334	—	16	2	no	—
162	#334	—	16	4	no	—
163	#322	—	8	1	no	—
164	#322	—	8	1	no	—
165	C₈²	—	2	1	no	—
166	C₈²	—	2	1	no	—
167	C₈²	—	2	1	no	—
168	C₈²	—	2	1	no	—
169	C₈²	—	4	1	no	—
170	C₈²	C₂²×C₈	1	1	yes	characteristic
171	C₈²	C₂×M₁₆	1	1	yes	characteristic
172	C₈²	#131	1	1	yes	characteristic
173	C₈²	#131	1	1	yes	characteristic
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	8	2	no	—
177	C₂²×C₈	—	16	2	no	—
178	C₂²×C₈	C₂×#106	1	1	yes	characteristic
179	C₂²×C₈	#428	1	1	yes	characteristic
180	C₂²×C₈	#441	1	1	yes	characteristic
181	C₂²×C₈	#442	1	1	yes	characteristic
182	C₂²×C₈	#475	1	1	yes	characteristic
183	C₂²×C₈	#476	1	1	yes	characteristic
184	C₂²×C₈	#501	1	2	yes	—
185	C₂×C₄²	#389	1	1	yes	characteristic
186	C₂×C₄²	#460	1	1	yes	characteristic
187	C₂×C₄²	#461	1	1	yes	characteristic
188	C₂×C₄²	#468	1	2	yes	—
189	C₂×C₄²	#471	1	2	yes	—
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	2	2	no	—
192	C₂×C₁₆	—	2	2	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	2	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	4	1	no	—
202	C₂×C₁₆	—	4	1	no	—
203	C₂×C₁₆	—	4	4	no	—
204	C₂×C₁₆	—	8	2	no	—
205	C₂×C₁₆	—	8	2	no	—
206	C₂×C₁₆	—	8	2	no	—
207	C₂×C₁₆	—	8	2	no	—
208	C₂×C₁₆	—	8	2	no	—
209	C₂×C₁₆	—	8	2	no	—
210	C₂×C₁₆	—	8	2	no	—
211	C₂×C₁₆	—	8	2	no	—
212	C₂×C₁₆	#334	1	2	yes	—
213	C₂×C₁₆	#334	1	2	yes	—
214	C₂×C₁₆	#365	1	2	yes	—
215	C₂×C₁₆	#365	1	2	yes	—
216	C₂×C₁₆	#365	1	2	yes	—
217	C₄⋊C₈	—	8	2	no	—
218	C₄⋊C₈	—	8	2	no	—
219	C₄⋊C₈	—	16	2	no	—
220	C₄⋊C₈	—	16	2	no	—
221	C₄⋊C₈	—	16	2	no	—
222	C₄⋊C₈	—	16	2	no	—
223	C₄⋊C₈	—	16	4	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	2	no	—
229	C₄×C₈	—	2	2	no	—
230	C₄×C₈	—	2	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	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	4	2	no	—
239	C₄×C₈	—	4	1	no	—
240	C₄×C₈	—	4	4	no	—
241	C₄×C₈	—	8	2	no	—
242	C₄×C₈	—	8	2	no	—
243	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
244	C₄×C₈	#424	1	1	yes	characteristic
245	C₄×C₈	C₂×#108	1	1	yes	characteristic
246	C₄×C₈	#430	1	1	yes	characteristic
247	C₄×C₈	C₈×D₄	1	1	yes	characteristic
248	C₄×C₈	#436	1	1	yes	characteristic
249	C₄×C₈	#444	1	1	yes	characteristic
250	C₄×C₈	#444	1	1	yes	characteristic
251	C₄×C₈	C₈×Q₈	1	1	yes	characteristic
252	C₄×C₈	#446	1	1	yes	characteristic
253	C₂²×C₄	#1686	1	2	yes	—
254	C₂²×C₄	#1691	1	2	yes	—
255	C₂²×C₄	#1943	1	1	yes	characteristic
256	C₂²×C₄	#1948	1	1	yes	characteristic
257	C₂²×C₄	#1951	1	1	yes	characteristic
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	16	2	no	—
275	C₂×C₈	—	16	2	no	—
276	C₂×C₈	—	16	2	no	—
277	C₂×C₈	—	32	2	no	—
278	C₂×C₈	—	32	2	no	—
279	C₂×C₈	—	32	2	no	—
280	C₂×C₈	C₂×#334	1	1	yes	characteristic
281	C₂×C₈	#1565	1	1	yes	characteristic
282	C₂×C₈	#1570	1	1	yes	characteristic
283	C₂×C₈	#1574	1	1	yes	characteristic
284	C₂×C₈	#1579	1	1	yes	characteristic
285	C₂×C₈	#1580	1	1	yes	characteristic
286	C₂×C₈	#1593	1	1	yes	characteristic
287	C₂×C₈	#1594	1	1	yes	characteristic
288	C₂×C₈	C₂×#365	1	1	yes	characteristic
289	C₂×C₈	C₂×#365	1	1	yes	characteristic
290	C₂×C₈	C₂×#365	1	1	yes	characteristic
291	C₂×C₈	#2155	1	1	yes	characteristic
292	C₂×C₈	#2180	1	2	yes	—
293	C₂×C₈	AGL(1,ℤ/16)	1	2	yes	—
294	C₂×C₈	#2230	1	2	yes	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	4	2	no	—
302	C₄²	#1939	1	1	yes	characteristic
303	C₄²	#1946	1	1	yes	characteristic
304	C₄²	#1950	1	1	yes	characteristic
305	C₄²	#1953	1	1	yes	characteristic
306	C₁₆	—	2	4	no	—
307	C₁₆	—	2	2	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	4	2	no	—
310	C₁₆	—	4	1	no	—
311	C₁₆	—	8	4	no	—
312	C₁₆	—	16	2	no	—
313	C₁₆	—	16	2	no	—
314	C₁₆	#1368	1	4	yes	complemented
315	C₂³	#9618	1	1	yes	characteristic
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	#9595	1	1	yes	characteristic
324	C₂×C₄	#9602	1	1	yes	characteristic
325	C₂×C₄	#9607	1	1	yes	characteristic
326	C₂×C₄	#9625	1	1	yes	characteristic
327	C₂×C₄	#9631	1	1	yes	characteristic
328	C₂×C₄	#9635	1	1	yes	characteristic
329	C₂×C₄	#13174	1	2	yes	—
330	C₂×C₄	#13175	1	2	yes	—
331	C₂×C₄	#13183	1	1	yes	characteristic
332	C₂×C₄	#13185	1	1	yes	characteristic
333	C₂×C₄	#13186	1	1	yes	characteristic
334	C₂×C₄	#13188	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	2	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	2	no	—
342	C₈	—	4	2	no	—
343	C₈	—	4	1	no	—
344	C₈	—	8	1	no	—
345	C₈	—	32	2	no	—
346	C₈	—	64	2	no	—
347	C₈	C₂×#1368	1	1	yes	characteristic
348	C₈	#12000	1	1	yes	characteristic
349	C₈	#12029	1	1	yes	characteristic
350	C₈	#12030	1	1	yes	characteristic
351	C₈	#12066	1	1	yes	characteristic
352	C₈	#12067	1	1	yes	characteristic
353	C₈	#12087	1	2	yes	—
354	C₂²	#101934	1	1	yes	characteristic
355	C₂²	#140463	1	1	yes	characteristic
356	C₂²	#140465	1	1	yes	characteristic
357	C₂²	#140466	1	2	yes	—
358	C₂²	#140467	1	2	yes	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	2	no	—
363	C₄	#140354	1	1	yes	characteristic
364	C₄	#140355	1	1	yes	characteristic
365	C₄	#140388	1	1	yes	characteristic
366	C₄	#140390	1	1	yes	characteristic
367	C₄	#140487	1	2	yes	—
368	C₄	#140503	1	1	yes	characteristic
369	C₄	#140505	1	1	yes	characteristic
370	C₂	#72216306	1	1	yes	characteristic
371	C₂	#72216390	1	1	yes	characteristic
372	C₂	#72226984	1	1	yes	characteristic
373	C₂	#72227276	1	2	yes	—
374	C₂	#72227277	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	#9618
commutator	C₈²	C₂²×C₈
Frattini, cosocle	C₄×C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₈² ◁ 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
47. #7100	C₈	semidirect
314. C₁₆	#1368	semidirect

Maximal subgroups

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

subgroup	index
#72225535	2
#72225375	2
#72216557	2
#72216306	2
#72216168	2