Subgroups — #72225536

Order 2048.

546 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225532	C₂	1	2	yes	maximal
2	#72225526	C₂	1	2	yes	maximal
3	#72216247	C₂	1	1	yes	maximal, characteristic
4	C₈×#1398	C₂	1	1	yes	maximal, characteristic
5	#72216215	C₂	1	1	yes	maximal, characteristic
6	#151981	C₂²	1	2	yes	—
7	C₄×#1398	C₂²	1	1	yes	characteristic
8	C₄×#1398	C₂²	1	1	yes	characteristic
9	C₂×#7456	C₄	1	2	yes	—
10	#149504	—	2	2	no	—
11	#149495	C₄	1	2	yes	—
12	C₈×M₆₄	C₄	1	2	yes	—
13	#129090	C₄	1	1	yes	characteristic
14	#129088	C₄	1	1	yes	characteristic
15	#129051	C₂²	1	2	yes	—
16	C₈×#346	C₂²	1	1	yes	characteristic
17	#94018	C₄	1	2	yes	—
18	#93995	—	2	2	no	—
19	#93264	C₄	1	2	yes	—
20	C₂×#1422	C₂×C₄	1	4	yes	—
21	C₂×#1398	—	2	2	no	—
22	C₂×#1398	—	2	1	no	—
23	C₂×#1398	—	2	1	no	—
24	C₂×#1398	D₄	1	1	yes	characteristic
25	C₂×#1398	D₄	1	1	yes	characteristic
26	C₂×#1398	Q₈	1	2	yes	—
27	C₄×M₆₄	C₂×C₄	1	2	yes	—
28	C₄×M₆₄	C₂×C₄	1	2	yes	—
29	C₄×#364	C₂×C₄	1	2	yes	—
30	C₄×#346	C₂×C₄	1	1	yes	characteristic
31	C₄×#346	C₂³	1	1	yes	characteristic
32	C₂×#1369	—	2	1	no	—
33	#11999	—	2	1	no	—
34	#11405	—	2	2	no	—
35	C₂×#1311	C₂×C₄	1	1	yes	characteristic
36	C₂×#1311	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
39	#8266	C₂×C₄	1	2	yes	—
40	C₄×C₈²	C₂×C₄	1	1	yes	characteristic
41	#7468	—	2	2	no	—
42	#7456	—	2	4	no	—
43	#7333	—	4	4	no	—
44	#7330	—	2	4	no	—
45	#7329	C₈	1	4	yes	complemented
46	C₈×C₃₂	C₈	1	4	yes	complemented
47	#7161	—	2	1	no	—
48	#7157	—	2	1	no	—
49	#7140	—	2	2	no	—
50	C₂×M₆₄	—	2	2	no	—
51	C₂×M₆₄	—	2	4	no	—
52	C₂×M₆₄	C₄⋊C₄	1	2	yes	—
53	C₂×M₆₄	C₄⋊C₄	1	2	yes	—
54	C₂×#364	—	2	2	no	—
55	C₄×M₃₂	—	2	2	no	—
56	C₂×#346	M₁₆	1	1	yes	characteristic
57	C₂×#346	M₁₆	1	1	yes	characteristic
58	C₂×#346	M₁₆	1	1	yes	characteristic
59	C₂×#346	M₁₆	1	1	yes	characteristic
60	C₂×#346	C₂×D₄	1	1	yes	characteristic
61	C₂×#346	C₂×Q₈	1	1	yes	characteristic
62	C₂×#346	#41	1	1	yes	characteristic
63	C₂×#346	#41	1	1	yes	characteristic
64	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
65	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
66	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₈	—	2	1	no	—
69	C₄²×C₈	C₂²×C₄	1	1	yes	characteristic
70	#1564	—	2	2	no	—
71	#1450	—	2	2	no	—
72	C₂×C₈²	—	2	1	no	—
73	C₂×C₈²	C₄²	1	1	yes	characteristic
74	C₂×C₈²	C₄²	1	1	yes	characteristic
75	#1422	—	2	4	no	—
76	#1422	—	2	4	no	—
77	#1398	—	2	4	no	—
78	#1398	—	2	4	no	—
79	#1398	—	2	4	no	—
80	#1398	—	2	2	no	—
81	#1398	—	4	2	no	—
82	#1398	—	4	2	no	—
83	#1398	SD₁₆	1	4	yes	—
84	#1397	—	2	2	no	—
85	#1397	M₁₆	1	4	yes	—
86	C₄×C₃₂	C₂×C₈	1	4	yes	—
87	C₄×C₃₂	C₂×C₈	1	4	yes	—
88	#1376	—	2	2	no	—
89	#1369	—	4	2	no	—
90	#1368	—	4	2	no	—
91	#1313	—	4	2	no	—
92	#1313	—	4	2	no	—
93	#1311	—	2	2	no	—
94	#1311	M₁₆	1	4	yes	—
95	C₈×C₁₆	—	2	2	no	—
96	C₈×C₁₆	C₂×C₈	1	2	yes	—
97	C₈×C₁₆	C₂×C₈	1	2	yes	—
98	#1276	—	2	2	no	—
99	C₂×M₃₂	—	4	2	no	—
100	C₂²×C₁₆	#97	1	1	yes	characteristic
101	C₂²×C₁₆	#97	1	1	yes	characteristic
102	C₂²×C₁₆	#97	1	1	yes	characteristic
103	C₂²×C₁₆	#97	1	1	yes	characteristic
104	C₂²×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₆	#117	1	1	yes	characteristic
106	C₂²×C₁₆	C₄×D₄	1	1	yes	characteristic
107	C₂²×C₁₆	C₄×Q₈	1	1	yes	characteristic
108	#431	—	4	2	no	—
109	C₂×C₄×C₈	—	2	1	no	—
110	C₂×C₄×C₈	—	2	1	no	—
111	C₂×C₄×C₈	—	4	2	no	—
112	C₂×C₄×C₈	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₈	#131	1	1	yes	characteristic
118	C₄³	#131	1	1	yes	characteristic
119	M₆₄	—	2	4	no	—
120	M₆₄	—	2	4	no	—
121	M₆₄	—	2	2	no	—
122	M₆₄	—	2	2	no	—
123	M₆₄	—	4	4	no	—
124	M₆₄	—	4	4	no	—
125	M₆₄	#106	1	4	yes	—
126	M₆₄	#108	1	4	yes	—
127	C₂×C₃₂	—	2	4	no	—
128	C₂×C₃₂	—	4	4	no	—
129	C₂×C₃₂	C₄⋊C₈	1	4	yes	—
130	C₂×C₃₂	C₄⋊C₈	1	4	yes	—
131	#364	—	4	2	no	—
132	#364	—	4	2	no	—
133	#346	—	2	2	no	—
134	#346	—	2	2	no	—
135	#346	—	2	2	no	—
136	#346	—	2	2	no	—
137	#346	—	2	2	no	—
138	#346	—	4	1	no	—
139	#346	—	4	1	no	—
140	#346	—	4	2	no	—
141	#346	C₂×SD₁₆	1	2	yes	—
142	#346	#135	1	2	yes	—
143	C₄×C₁₆	—	2	1	no	—
144	C₄×C₁₆	—	2	1	no	—
145	C₄×C₁₆	—	4	2	no	—
146	C₄×C₁₆	C₄×C₈	1	2	yes	—
147	C₄×C₁₆	#97	1	2	yes	—
148	C₄×C₁₆	C₂²×C₈	1	2	yes	—
149	C₄×C₁₆	C₂×M₁₆	1	2	yes	—
150	C₄×C₁₆	C₂×M₁₆	1	2	yes	—
151	C₄×C₁₆	#131	1	2	yes	—
152	#335	—	4	2	no	—
153	#334	—	4	2	no	—
154	#322	—	4	2	no	—
155	C₈²	—	2	1	no	—
156	C₈²	—	2	1	no	—
157	C₈²	—	4	2	no	—
158	C₈²	—	4	2	no	—
159	C₈²	C₄×C₈	1	1	yes	characteristic
160	C₈²	C₄×C₈	1	1	yes	characteristic
161	C₈²	#97	1	2	yes	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	4	1	no	—
164	C₂²×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₈	C₂×#97	1	1	yes	characteristic
166	C₂²×C₈	#436	1	1	yes	characteristic
167	C₂²×C₈	#446	1	1	yes	characteristic
168	C₂×C₄²	—	4	1	no	—
169	C₂×C₄²	#405	1	1	yes	characteristic
170	C₂×C₄²	#436	1	1	yes	characteristic
171	C₂×C₄²	#446	1	1	yes	characteristic
172	M₃₂	—	8	2	no	—
173	M₃₂	—	8	2	no	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	—	8	2	no	—
187	C₂×C₁₆	#322	1	2	yes	—
188	C₂×C₁₆	#322	1	2	yes	—
189	C₂×C₁₆	#347	1	2	yes	—
190	C₂×C₁₆	#347	1	2	yes	—
191	C₂×C₁₆	C₂×C₄⋊C₈	1	2	yes	—
192	C₂×C₁₆	C₂×#106	1	1	yes	characteristic
193	C₂×C₁₆	#427	1	1	yes	characteristic
194	C₂×C₁₆	C₂×#108	1	2	yes	—
195	C₂×C₁₆	#431	1	2	yes	—
196	C₂×C₁₆	#436	1	2	yes	—
197	C₂×C₁₆	C₄×SD₁₆	1	2	yes	—
198	C₂×C₁₆	#443	1	2	yes	—
199	C₂×C₁₆	#446	1	2	yes	—
200	C₄⋊C₈	—	8	2	no	—
201	C₂²⋊C₈	—	8	2	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	—	4	1	no	—
213	C₄×C₈	—	4	1	no	—
214	C₄×C₈	—	8	4	no	—
215	C₄×C₈	—	8	2	no	—
216	C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
217	C₄×C₈	C₂×#97	1	1	yes	characteristic
218	C₄×C₈	C₄×M₁₆	1	1	yes	characteristic
219	C₄×C₈	#405	1	1	yes	characteristic
220	C₄×C₈	C₄×SD₁₆	1	2	yes	—
221	C₄×C₈	#443	1	2	yes	—
222	C₃₂	—	2	8	no	—
223	C₃₂	—	2	4	no	—
224	C₃₂	—	8	4	no	—
225	C₃₂	#334	1	8	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	#1770	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	4	4	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	8	2	no	—
240	C₂×C₈	—	16	2	no	—
241	C₂×C₈	C₂×#322	1	1	yes	characteristic
242	C₂×C₈	#1451	1	1	yes	characteristic
243	C₂×C₈	#1590	1	2	yes	—
244	C₂×C₈	C₄×C₄⋊C₈	1	1	yes	characteristic
245	C₂×C₈	#1770	1	1	yes	characteristic
246	C₂×C₈	#1772	1	1	yes	characteristic
247	C₂×C₈	#1772	1	1	yes	characteristic
248	C₂×C₈	C₄×#106	1	1	yes	characteristic
249	C₂×C₈	C₄×#108	1	1	yes	characteristic
250	C₂×C₈	C₂×#347	1	1	yes	characteristic
251	C₂×C₈	C₂×#347	1	1	yes	characteristic
252	C₂×C₈	#2110	1	1	yes	characteristic
253	C₂×C₈	#2110	1	1	yes	characteristic
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	8	2	no	—
259	C₄²	#1450	1	1	yes	characteristic
260	C₄²	#1451	1	1	yes	characteristic
261	C₄²	#1590	1	2	yes	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	2	2	no	—
264	C₁₆	—	2	2	no	—
265	C₁₆	—	4	1	no	—
266	C₁₆	—	4	1	no	—
267	C₁₆	—	16	2	no	—
268	C₁₆	#1313	1	4	yes	—
269	C₁₆	C₂×#334	1	2	yes	—
270	C₁₆	#1564	1	2	yes	—
271	C₁₆	#1590	1	4	yes	—
272	C₂³	#12667	1	1	yes	characteristic
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	#8260	1	1	yes	characteristic
281	C₂×C₄	#8283	1	1	yes	characteristic
282	C₂×C₄	#8293	1	1	yes	characteristic
283	C₂×C₄	#12666	1	1	yes	characteristic
284	C₂×C₄	#12666	1	1	yes	characteristic
285	C₂×C₄	#12667	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	—	2	2	no	—
288	C₈	—	4	2	no	—
289	C₈	—	4	1	no	—
290	C₈	—	8	2	no	—
291	C₈	—	32	2	no	—
292	C₈	#8266	1	1	yes	characteristic
293	C₈	C₄×#334	1	1	yes	characteristic
294	C₈	#8283	1	1	yes	characteristic
295	C₈	#8293	1	1	yes	characteristic
296	C₈	C₂×#1313	1	1	yes	characteristic
297	C₈	#11405	1	1	yes	characteristic
298	C₈	#11414	1	1	yes	characteristic
299	C₈	#11414	1	1	yes	characteristic
300	C₂²	—	4	1	no	—
301	C₂²	#129051	1	1	yes	characteristic
302	C₂²	#129158	1	1	yes	characteristic
303	C₂²	#129195	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	1	no	—
308	C₄	#96889	1	1	yes	characteristic
309	C₄	#129049	1	1	yes	characteristic
310	C₄	#129158	1	1	yes	characteristic
311	C₄	#129195	1	1	yes	characteristic
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	#72225535	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#8266
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72225535
prime radical	C₂³	#12667
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#1398 ◁ C₄×#1398 ◁ 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
45. #7329	C₈	semidirect
46. C₈×C₃₂	C₈	semidirect

Maximal subgroups

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

subgroup	index
#72225532	2
#72225526	2
#72216247	2
C₈×#1398	2
#72216215	2