Subgroups — #72225351

Order 2048.

475 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	#72225350	C₂	1	2	yes	maximal
2	#72225349	C₂	1	2	yes	maximal
3	#72225126	C₂	1	1	yes	maximal, characteristic
4	#72216243	C₂	1	1	yes	maximal, characteristic
5	#72216215	C₂	1	1	yes	maximal, characteristic
6	#151981	C₂²	1	2	yes	—
7	C₂×#7459	C₄	1	1	yes	characteristic
8	C₂×#7459	C₄	1	1	yes	characteristic
9	#151024	—	2	2	no	—
10	#135718	—	2	1	no	—
11	#135718	C₂²	1	2	yes	—
12	#135713	—	2	1	no	—
13	#129090	C₂²	1	1	yes	characteristic
14	#129088	C₂²	1	1	yes	characteristic
15	#129051	—	2	1	no	—
16	C₈×#346	C₂²	1	1	yes	characteristic
17	#94031	—	2	2	no	—
18	#94019	C₄	1	2	yes	—
19	#93264	—	2	1	no	—
20	C₂×#1422	C₂×C₄	1	2	yes	—
21	C₂×#1422	C₂×C₄	1	2	yes	—
22	C₂×#1398	—	2	2	no	—
23	#12900	—	2	2	no	—
24	C₄×#364	—	2	1	no	—
25	C₄×#346	D₄	1	1	yes	characteristic
26	C₄×#346	C₂³	1	1	yes	characteristic
27	C₂×#1369	—	2	2	no	—
28	C₂×#1369	—	2	1	no	—
29	C₂×#1369	—	2	2	no	—
30	C₂×#1369	—	4	1	no	—
31	C₂×#1369	—	4	1	no	—
32	#11999	D₄	1	2	yes	—
33	#11996	—	4	2	no	—
34	#11405	—	4	1	no	—
35	C₂×#1311	C₂×C₄	1	1	yes	characteristic
36	C₂×#1311	Q₈	1	1	yes	characteristic
37	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₆	Q₈	1	1	yes	characteristic
39	#8266	—	2	1	no	—
40	C₄×C₈²	D₄	1	1	yes	characteristic
41	#7468	—	2	2	no	—
42	#7459	—	2	1	no	—
43	#7459	—	2	1	no	—
44	#7459	C₈	1	2	yes	complemented
45	#7459	C₈	1	2	yes	complemented
46	#7457	—	4	4	no	—
47	#7161	Q₈	1	2	yes	—
48	#7157	—	2	1	no	—
49	#7140	—	4	1	no	—
50	C₂×M₆₄	—	2	2	no	—
51	C₂×M₆₄	—	2	2	no	—
52	C₂×#364	—	2	2	no	—
53	C₂×#364	—	4	2	no	—
54	C₂×#364	—	4	2	no	—
55	#2150	—	4	2	no	—
56	C₄×M₃₂	—	4	1	no	—
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×#346	SD₁₆	1	1	yes	characteristic
61	C₂×#346	C₂×D₄	1	1	yes	characteristic
62	C₂×#346	C₂×Q₈	1	1	yes	characteristic
63	C₂×#346	#41	1	1	yes	characteristic
64	C₂×#346	#41	1	1	yes	characteristic
65	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₆	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₈	—	2	1	no	—
70	C₄²×C₈	C₂×D₄	1	1	yes	characteristic
71	#1564	—	2	2	no	—
72	#1450	—	4	1	no	—
73	C₂×C₈²	—	2	1	no	—
74	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
76	#1422	—	2	2	no	—
77	#1422	—	2	2	no	—
78	#1422	—	2	2	no	—
79	#1422	C₂×C₈	1	4	yes	—
80	#1398	—	4	2	no	—
81	#1398	—	4	2	no	—
82	#1376	—	4	1	no	—
83	#1369	—	4	1	no	—
84	#1369	—	4	1	no	—
85	#1369	—	4	2	no	—
86	#1369	—	4	2	no	—
87	#1369	—	8	2	no	—
88	#1369	—	8	2	no	—
89	#1369	—	8	2	no	—
90	#1368	—	2	2	no	—
91	#1368	—	2	2	no	—
92	#1367	—	8	4	no	—
93	#1313	—	8	2	no	—
94	#1311	—	2	1	no	—
95	#1311	—	2	1	no	—
96	#1311	—	2	1	no	—
97	#1311	—	2	1	no	—
98	C₈×C₁₆	—	4	1	no	—
99	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
100	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
101	C₈×C₁₆	M₁₆	1	1	yes	characteristic
102	C₈×C₁₆	M₁₆	1	1	yes	characteristic
103	#1276	—	4	1	no	—
104	C₂×M₃₂	—	4	2	no	—
105	C₂×M₃₂	—	8	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	C₂²×C₁₆	C₄×D₄	1	1	yes	characteristic
111	C₂²×C₁₆	C₄×Q₈	1	1	yes	characteristic
112	C₂²×C₁₆	#125	1	1	yes	characteristic
113	C₂²×C₁₆	C₄⋊Q₈	1	1	yes	characteristic
114	#431	—	4	2	no	—
115	C₂×C₄×C₈	—	2	1	no	—
116	C₂×C₄×C₈	—	2	1	no	—
117	C₂×C₄×C₈	—	4	2	no	—
118	C₂×C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₈	#121	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₄³	#137	1	1	yes	characteristic
125	M₆₄	—	2	4	no	—
126	M₆₄	—	4	2	no	—
127	M₆₄	—	4	2	no	—
128	M₆₄	—	4	2	no	—
129	C₂×C₃₂	—	4	2	no	—
130	C₂×C₃₂	—	4	2	no	—
131	#364	—	4	2	no	—
132	#364	—	4	2	no	—
133	#364	—	8	2	no	—
134	#364	—	8	2	no	—
135	#364	—	8	2	no	—
136	#364	—	8	2	no	—
137	C₄⋊C₁₆	—	8	4	no	—
138	#346	—	2	2	no	—
139	#346	—	2	2	no	—
140	#346	—	2	2	no	—
141	#346	—	2	1	no	—
142	#346	—	2	1	no	—
143	#346	—	2	2	no	—
144	#346	—	4	1	no	—
145	#346	—	4	1	no	—
146	#346	—	8	1	no	—
147	#346	AGL(1,ℤ/8)	1	2	yes	—
148	#346	#137	1	2	yes	—
149	C₄×C₁₆	—	2	1	no	—
150	C₄×C₁₆	—	2	1	no	—
151	C₄×C₁₆	—	2	1	no	—
152	C₄×C₁₆	—	2	1	no	—
153	C₄×C₁₆	—	8	1	no	—
154	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
155	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
156	C₄×C₁₆	#108	1	1	yes	characteristic
157	C₄×C₁₆	#108	1	1	yes	characteristic
158	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
159	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
160	C₄×C₁₆	#131	1	1	yes	characteristic
161	C₄×C₁₆	#131	1	1	yes	characteristic
162	#335	—	4	2	no	—
163	#334	—	4	2	no	—
164	#322	—	8	1	no	—
165	C₈²	—	2	4	no	—
166	C₈²	—	2	1	no	—
167	C₈²	—	2	1	no	—
168	C₈²	—	2	1	no	—
169	C₈²	—	8	1	no	—
170	C₈²	C₄⋊C₈	1	1	yes	characteristic
171	C₈²	C₄⋊C₈	1	1	yes	characteristic
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	4	1	no	—
174	C₂²×C₈	#389	1	1	yes	characteristic
175	C₂²×C₈	C₂×#106	1	1	yes	characteristic
176	C₂²×C₈	#460	1	1	yes	characteristic
177	C₂²×C₈	#476	1	1	yes	characteristic
178	C₂×C₄²	—	4	1	no	—
179	C₂×C₄²	#428	1	1	yes	characteristic
180	C₂×C₄²	#461	1	1	yes	characteristic
181	C₂×C₄²	#475	1	1	yes	characteristic
182	M₃₂	—	8	2	no	—
183	M₃₂	—	8	2	no	—
184	M₃₂	—	16	2	no	—
185	M₃₂	—	16	2	no	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	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	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	8	2	no	—
201	C₂×C₁₆	—	16	2	no	—
202	C₂×C₁₆	#334	1	1	yes	characteristic
203	C₂×C₁₆	#334	1	1	yes	characteristic
204	C₂×C₁₆	#334	1	1	yes	characteristic
205	C₂×C₁₆	#334	1	1	yes	characteristic
206	C₂×C₁₆	#365	1	2	yes	—
207	C₂×C₁₆	#365	1	2	yes	—
208	C₂×C₁₆	#365	1	2	yes	—
209	C₂×C₁₆	C₈×D₄	1	1	yes	characteristic
210	C₂×C₁₆	#436	1	1	yes	characteristic
211	C₂×C₁₆	#441	1	1	yes	characteristic
212	C₂×C₁₆	#442	1	1	yes	characteristic
213	C₂×C₁₆	#444	1	1	yes	characteristic
214	C₂×C₁₆	#444	1	1	yes	characteristic
215	C₂×C₁₆	C₈×Q₈	1	1	yes	characteristic
216	C₂×C₁₆	#446	1	1	yes	characteristic
217	C₂×C₁₆	#501	1	2	yes	—
218	C₄⋊C₈	—	8	2	no	—
219	C₂²⋊C₈	—	8	2	no	—
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	2	no	—
228	C₄×C₈	—	2	2	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	4	1	no	—
232	C₄×C₈	—	8	2	no	—
233	C₄×C₈	—	8	4	no	—
234	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
235	C₄×C₈	#424	1	1	yes	characteristic
236	C₄×C₈	C₂×#108	1	1	yes	characteristic
237	C₄×C₈	#430	1	1	yes	characteristic
238	C₄×C₈	#468	1	2	yes	—
239	C₄×C₈	#471	1	2	yes	—
240	C₃₂	—	4	4	no	—
241	C₃₂	—	8	2	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	#1948	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	4	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	8	2	no	—
255	C₂×C₈	—	16	2	no	—
256	C₂×C₈	C₂×#334	1	1	yes	characteristic
257	C₂×C₈	#1570	1	1	yes	characteristic
258	C₂×C₈	#1686	1	2	yes	—
259	C₂×C₈	#1939	1	1	yes	characteristic
260	C₂×C₈	#1943	1	1	yes	characteristic
261	C₂×C₈	#1946	1	1	yes	characteristic
262	C₂×C₈	#1950	1	1	yes	characteristic
263	C₂×C₈	#1951	1	1	yes	characteristic
264	C₂×C₈	#1953	1	1	yes	characteristic
265	C₂×C₈	C₂×#365	1	1	yes	characteristic
266	C₂×C₈	C₂×#365	1	1	yes	characteristic
267	C₂×C₈	C₂×#365	1	1	yes	characteristic
268	C₂×C₈	#2155	1	1	yes	characteristic
269	C₂×C₈	#2230	1	2	yes	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	8	2	no	—
275	C₄²	#1565	1	1	yes	characteristic
276	C₄²	#1574	1	1	yes	characteristic
277	C₄²	#1691	1	2	yes	—
278	C₁₆	—	2	1	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	—	2	2	no	—
281	C₁₆	—	2	1	no	—
282	C₁₆	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	4	1	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	16	2	no	—
287	C₁₆	—	32	2	no	—
288	C₁₆	#1368	1	2	yes	—
289	C₁₆	#1368	1	2	yes	—
290	C₁₆	#1579	1	1	yes	characteristic
291	C₁₆	#1580	1	1	yes	characteristic
292	C₁₆	#1593	1	1	yes	characteristic
293	C₁₆	#1594	1	1	yes	characteristic
294	C₂³	#13188	1	1	yes	characteristic
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	#9602	1	1	yes	characteristic
303	C₂×C₄	#9618	1	1	yes	characteristic
304	C₂×C₄	#9631	1	1	yes	characteristic
305	C₂×C₄	#13183	1	1	yes	characteristic
306	C₂×C₄	#13185	1	1	yes	characteristic
307	C₂×C₄	#13186	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	8	2	no	—
313	C₈	—	32	2	no	—
314	C₈	#9595	1	1	yes	characteristic
315	C₈	#9607	1	1	yes	characteristic
316	C₈	#9625	1	1	yes	characteristic
317	C₈	#9635	1	1	yes	characteristic
318	C₈	C₂×#1368	1	1	yes	characteristic
319	C₈	#12000	1	1	yes	characteristic
320	C₈	#12029	1	1	yes	characteristic
321	C₈	#12030	1	1	yes	characteristic
322	C₂²	—	4	1	no	—
323	C₂²	#140390	1	1	yes	characteristic
324	C₂²	#140465	1	1	yes	characteristic
325	C₂²	#140503	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	2	no	—
329	C₄	—	8	1	no	—
330	C₄	#101934	1	1	yes	characteristic
331	C₄	#140388	1	1	yes	characteristic
332	C₄	#140463	1	1	yes	characteristic
333	C₄	#140505	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	#72216306	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#140505
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72216306
prime radical	C₂³	#13188
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
44. #7459	C₈	semidirect
45. #7459	C₈	semidirect

Maximal subgroups

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

subgroup	index
#72225350	2
#72225349	2
#72225126	2
#72216243	2
#72216215	2