Subgroups — #72234392

Order 4096.

917 conjugacy classes of subgroups.

The subgroup lattice has 277 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234377	C₂	1	4	yes	maximal
2	#72234371	C₂	1	2	yes	maximal
3	#72234367	C₄	1	4	yes	—
4	C₈×M₁₂₈	C₄	1	4	yes	—
5	#72234363	C₂²	1	4	yes	—
6	#72234241	C₂	1	1	yes	maximal, characteristic
7	#72234237	C₄	1	2	yes	—
8	#72234237	C₂²	1	2	yes	—
9	C₈×#1397	C₄	1	1	yes	characteristic
10	C₈×#1397	C₂²	1	1	yes	characteristic
11	#72234224	C₄	1	1	yes	characteristic
12	C₄×M₁₂₈	C₂×C₄	1	8	yes	—
13	C₄×#1397	C₂×C₄	1	1	yes	characteristic
14	C₄×#1397	C₂×C₄	1	2	yes	—
15	C₄×#1397	C₂³	1	1	yes	characteristic
16	C₂×#7329	C₈	1	2	yes	—
17	C₂×#7329	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₃₂	C₈	1	2	yes	—
19	C₂×C₈×C₃₂	C₂×C₄	1	2	yes	—
20	C₄×#1311	C₈	1	1	yes	characteristic
21	C₄×#1311	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₆	C₈	1	2	yes	—
23	C₄×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
24	#94852	C₈	1	1	yes	characteristic
25	#94756	—	2	8	no	—
26	#94508	—	2	8	no	—
27	#94504	C₈	1	8	yes	complemented
28	C₈×C₆₄	C₈	1	8	yes	complemented
29	#93990	—	2	4	no	—
30	#93990	—	2	4	no	—
31	C₂×M₁₂₈	—	2	8	no	—
32	C₂×#1397	—	2	2	no	—
33	C₂×#1397	—	2	2	no	—
34	C₂×#1397	M₁₆	1	4	yes	—
35	C₂×#1397	#41	1	4	yes	—
36	C₂×C₄×C₃₂	C₄²	1	2	yes	—
37	C₂×C₄×C₃₂	C₂×C₈	1	4	yes	—
38	C₂×C₄×C₃₂	C₂²×C₄	1	2	yes	—
39	C₄²×C₁₆	C₂×C₈	1	2	yes	—
40	C₄²×C₁₆	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₁₆	C₂²×C₄	1	1	yes	characteristic
42	C₂×#1311	—	2	2	no	—
43	C₂×#1311	—	2	2	no	—
44	C₂×#1311	—	2	2	no	—
45	C₂×C₈×C₁₆	C₁₆	1	4	yes	—
46	C₂×C₈×C₁₆	C₁₆	1	4	yes	—
47	C₂×C₈×C₁₆	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₁₆	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
51	#8131	—	2	2	no	—
52	C₄×C₈²	C₂×C₈	1	2	yes	—
53	C₄×C₈²	C₂×C₈	1	1	yes	characteristic
54	#7510	—	2	8	no	—
55	C₄×C₆₄	C₂×C₈	1	16	yes	—
56	#7329	—	2	4	no	—
57	#7329	M₁₆	1	8	yes	—
58	C₈×C₃₂	C₁₆	1	4	yes	—
59	C₈×C₃₂	C₁₆	1	4	yes	—
60	C₈×C₃₂	C₂×C₈	1	4	yes	—
61	C₈×C₃₂	C₂×C₈	1	4	yes	—
62	C₂²×C₃₂	—	2	4	no	—
63	C₂²×C₃₂	#97	1	4	yes	—
64	C₂²×C₃₂	#117	1	4	yes	—
65	C₂×C₄×C₁₆	—	2	2	no	—
66	C₂×C₄×C₁₆	—	2	2	no	—
67	C₂×C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₆	C₂×C₁₆	1	4	yes	—
70	C₂×C₄×C₁₆	C₂×C₁₆	1	2	yes	—
71	C₂×C₄×C₁₆	C₂×C₁₆	1	2	yes	—
72	C₂×C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₁₆	#117	1	2	yes	—
74	C₂×C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₆	C₂×M₁₆	1	2	yes	—
76	C₄²×C₈	—	2	2	no	—
77	C₄²×C₈	M₃₂	1	2	yes	—
78	C₄²×C₈	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₈²	—	2	2	no	—
80	C₂×C₈²	C₄×C₈	1	1	yes	characteristic
81	C₂×C₈²	C₄×C₈	1	1	yes	characteristic
82	C₂×C₈²	C₂×C₁₆	1	4	yes	—
83	C₂×C₈²	M₃₂	1	2	yes	—
84	M₁₂₈	—	4	16	no	—
85	C₂×C₆₄	—	2	16	no	—
86	#1397	—	4	4	no	—
87	#1397	—	4	4	no	—
88	#1397	—	4	4	no	—
89	#1397	—	4	4	no	—
90	C₄×C₃₂	—	2	4	no	—
91	C₄×C₃₂	C₄×C₈	1	4	yes	—
92	C₄×C₃₂	#97	1	4	yes	—
93	C₄×C₃₂	C₂×C₁₆	1	8	yes	—
94	C₄×C₃₂	C₂²×C₈	1	4	yes	—
95	C₄×C₃₂	C₂×M₁₆	1	4	yes	—
96	#1311	—	4	4	no	—
97	#1311	—	4	4	no	—
98	#1311	—	4	4	no	—
99	C₈×C₁₆	—	2	8	no	—
100	C₈×C₁₆	C₃₂	1	16	yes	—
101	C₈×C₁₆	C₄×C₈	1	2	yes	—
102	C₈×C₁₆	C₄×C₈	1	2	yes	—
103	C₈×C₁₆	#97	1	4	yes	—
104	C₈×C₁₆	C₂×C₁₆	1	2	yes	—
105	C₈×C₁₆	C₂×C₁₆	1	2	yes	—
106	C₈×C₁₆	M₃₂	1	4	yes	—
107	#1273	—	4	4	no	—
108	C₂²×C₁₆	—	2	4	no	—
109	C₂²×C₁₆	C₄×C₁₆	1	1	yes	characteristic
110	C₂²×C₁₆	C₄×C₁₆	1	1	yes	characteristic
111	C₂²×C₁₆	C₄×C₁₆	1	2	yes	—
112	C₂²×C₁₆	#376	1	1	yes	characteristic
113	C₂²×C₁₆	C₂×#97	1	1	yes	characteristic
114	C₂²×C₁₆	#431	1	2	yes	—
115	C₂×C₄×C₈	—	2	4	no	—
116	C₂×C₄×C₈	—	2	4	no	—
117	C₂×C₄×C₈	—	4	4	no	—
118	C₂×C₄×C₈	#346	1	2	yes	—
119	C₂×C₄×C₈	#346	1	2	yes	—
120	C₂×C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈	#431	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₁₆	1	2	yes	—
123	C₂×C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
124	C₄³	C₂×M₃₂	1	1	yes	characteristic
125	C₂×C₃₂	—	2	8	no	—
126	C₂×C₃₂	—	2	8	no	—
127	C₂×C₃₂	—	2	8	no	—
128	C₂×C₃₂	—	4	4	no	—
129	C₂×C₃₂	—	4	4	no	—
130	C₂×C₃₂	#322	1	8	yes	—
131	C₂×C₃₂	#431	1	8	yes	—
132	C₄×C₁₆	—	2	4	no	—
133	C₄×C₁₆	—	2	4	no	—
134	C₄×C₁₆	—	2	4	no	—
135	C₄×C₁₆	—	4	2	no	—
136	C₄×C₁₆	—	4	2	no	—
137	C₄×C₁₆	—	4	4	no	—
138	C₄×C₁₆	C₈²	1	1	yes	characteristic
139	C₄×C₁₆	C₈²	1	1	yes	characteristic
140	C₄×C₁₆	#322	1	2	yes	—
141	C₄×C₁₆	C₄×C₁₆	1	1	yes	characteristic
142	C₄×C₁₆	C₄×C₁₆	1	1	yes	characteristic
143	C₄×C₁₆	#346	1	2	yes	—
144	C₄×C₁₆	C₂×C₃₂	1	8	yes	—
145	C₄×C₁₆	C₂×C₃₂	1	8	yes	—
146	C₄×C₁₆	M₆₄	1	8	yes	—
147	C₄×C₁₆	C₂×C₄×C₈	1	2	yes	—
148	C₄×C₁₆	C₂×#97	1	2	yes	—
149	C₄×C₁₆	C₂²×C₁₆	1	2	yes	—
150	C₄×C₁₆	C₂×M₃₂	1	2	yes	—
151	C₈²	—	2	4	no	—
152	C₈²	—	2	4	no	—
153	C₈²	—	4	2	no	—
154	C₈²	—	4	2	no	—
155	C₈²	C₈²	1	2	yes	—
156	C₈²	#322	1	2	yes	—
157	C₈²	C₄×C₁₆	1	2	yes	—
158	C₈²	#346	1	2	yes	—
159	C₈²	C₂×C₃₂	1	8	yes	—
160	C₆₄	—	4	16	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	4	4	no	—
163	C₂²×C₈	#1398	1	4	yes	—
164	C₂²×C₈	#1744	1	1	yes	characteristic
165	C₂²×C₈	C₂×C₄×C₁₆	1	1	yes	characteristic
166	C₂²×C₈	#2162	1	2	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂×#346	1	1	yes	characteristic
169	C₂×C₄²	#2162	1	2	yes	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	8	no	—
173	C₂×C₁₆	—	4	4	no	—
174	C₂×C₁₆	—	4	4	no	—
175	C₂×C₁₆	C₈×C₁₆	1	1	yes	characteristic
176	C₂×C₁₆	C₈×C₁₆	1	1	yes	characteristic
177	C₂×C₁₆	C₈×C₁₆	1	1	yes	characteristic
178	C₂×C₁₆	C₈×C₁₆	1	1	yes	characteristic
179	C₂×C₁₆	#1312	1	4	yes	—
180	C₂×C₁₆	C₄×C₃₂	1	4	yes	—
181	C₂×C₁₆	C₄×C₃₂	1	4	yes	—
182	C₂×C₁₆	#1397	1	4	yes	—
183	C₂×C₁₆	#1397	1	4	yes	—
184	C₂×C₁₆	C₂×#322	1	2	yes	—
185	C₂×C₁₆	#1744	1	2	yes	—
186	C₂×C₁₆	#1747	1	4	yes	—
187	C₂×C₁₆	#2110	1	4	yes	—
188	C₂×C₁₆	#2162	1	4	yes	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	—	2	8	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	—	4	4	no	—
195	C₄×C₈	—	8	4	no	—
196	C₄×C₈	—	8	8	no	—
197	C₄×C₈	#1311	1	4	yes	—
198	C₄×C₈	#1312	1	4	yes	—
199	C₄×C₈	#1398	1	4	yes	—
200	C₄×C₈	C₂×C₈²	1	1	yes	characteristic
201	C₄×C₈	C₂×#322	1	1	yes	characteristic
202	C₄×C₈	C₂×C₄×C₁₆	1	1	yes	characteristic
203	C₄×C₈	C₂×#346	1	1	yes	characteristic
204	C₄×C₈	C₂²×C₃₂	1	4	yes	—
205	C₄×C₈	C₂×M₆₄	1	4	yes	—
206	C₃₂	—	2	8	no	—
207	C₃₂	—	2	8	no	—
208	C₃₂	—	4	8	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	#12590	1	1	yes	characteristic
211	C₂²×C₄	C₂×#1398	1	2	yes	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	—	4	8	no	—
218	C₂×C₈	—	8	8	no	—
219	C₂×C₈	#7330	1	8	yes	—
220	C₂×C₈	#8131	1	1	yes	characteristic
221	C₂×C₈	#8137	1	2	yes	—
222	C₂×C₈	C₂×C₈×C₁₆	1	1	yes	characteristic
223	C₂×C₈	C₂×#1312	1	1	yes	characteristic
224	C₂×C₈	#12590	1	1	yes	characteristic
225	C₂×C₈	#12593	1	2	yes	—
226	C₂×C₈	C₂×C₄×C₃₂	1	2	yes	—
227	C₂×C₈	C₂×#1397	1	2	yes	—
228	C₂×C₈	#13639	1	4	yes	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₂×#1311	1	1	yes	characteristic
232	C₄²	C₂×#1312	1	1	yes	characteristic
233	C₄²	C₂×#1398	1	2	yes	—
234	C₄²	#13639	1	4	yes	—
235	C₁₆	—	2	4	no	—
236	C₁₆	—	4	8	no	—
237	C₁₆	C₈×C₃₂	1	2	yes	—
238	C₁₆	C₈×C₃₂	1	2	yes	—
239	C₁₆	#7329	1	2	yes	—
240	C₁₆	#7329	1	2	yes	—
241	C₁₆	#7331	1	8	yes	—
242	C₁₆	#8137	1	4	yes	—
243	C₁₆	#11405	1	4	yes	—
244	C₂³	#151879	1	1	yes	characteristic
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	#128203	1	1	yes	characteristic
249	C₂×C₄	#128225	1	2	yes	—
250	C₂×C₄	C₂×#7330	1	2	yes	—
251	C₂×C₄	#149502	1	2	yes	—
252	C₂×C₄	#151876	1	2	yes	—
253	C₂×C₄	#151879	1	1	yes	characteristic
254	C₈	—	2	4	no	—
255	C₈	—	2	8	no	—
256	C₈	—	4	4	no	—
257	C₈	—	8	8	no	—
258	C₈	#96658	1	1	yes	characteristic
259	C₈	#128206	1	1	yes	characteristic
260	C₈	#128225	1	2	yes	—
261	C₈	C₂×C₈×C₃₂	1	1	yes	characteristic
262	C₈	C₂×#7329	1	1	yes	characteristic
263	C₈	C₂×#7331	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	#72234237	1	1	yes	characteristic
266	C₂²	#72234277	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	4	4	no	—
269	C₄	#72210645	1	1	yes	characteristic
270	C₄	#72234277	1	2	yes	—
271	C₄	#72234287	1	1	yes	characteristic
272	C₄	#72234379	1	4	yes	—
273	C₂	—	2	2	no	—
274	C₂	#72234289	1	1	yes	characteristic
275	C₂	#72234385	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₈²
commutator	C₈	C₂×C₈×C₃₂
Frattini, cosocle	C₄×#1397	C₂³
socle	C₂²	#72234237
prime radical	C₂³	#151879
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×#1397 ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
27. #94504	C₈	semidirect
28. C₈×C₆₄	C₈	semidirect

Maximal subgroups

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

subgroup	index
#72234377	2
#72234371	2
#72234241	2