Subgroups — #72234406

Order 4096.

575 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234385	C₂	1	2	yes	maximal
2	#72234375	—	2	2	no	—
3	#72234375	—	2	2	no	—
4	C₄×#7511	C₄	1	4	yes	—
5	#72234267	C₂	1	1	yes	maximal, characteristic
6	#72234264	—	2	2	no	—
7	#72234239	C₂²	1	1	yes	characteristic
8	#72210680	C₄	1	1	yes	characteristic
9	#72210619	C₄	1	1	yes	characteristic
10	C₂×#7511	—	2	4	no	—
11	C₂×#7511	—	2	4	no	—
12	C₄×M₁₂₈	C₈	1	8	yes	—
13	#151879	—	2	1	no	—
14	C₄×#1397	C₂×C₄	1	2	yes	—
15	#149502	—	2	1	no	—
16	C₂×#7330	D₄	1	2	yes	—
17	C₂×#7330	D₄	1	1	yes	characteristic
18	C₂×#7330	Q₈	1	1	yes	characteristic
19	C₈×M₆₄	C₈	1	2	yes	—
20	#128225	—	2	1	no	—
21	#126519	C₂×C₄	1	1	yes	characteristic
22	#122214	—	2	1	no	—
23	#122206	C₈	1	2	yes	—
24	C₂×M₁₂₈	—	2	8	no	—
25	C₄×M₆₄	—	2	2	no	—
26	C₂×#1397	C₂²⋊C₄	1	2	yes	—
27	C₂×#1397	C₄⋊C₄	1	2	yes	—
28	C₂×#1397	M₁₆	1	4	yes	—
29	C₂×C₄×C₃₂	C₂×C₈	1	4	yes	—
30	C₄×#346	M₁₆	1	1	yes	characteristic
31	C₄²×C₁₆	C₄²	1	1	yes	characteristic
32	#11410	—	2	2	no	—
33	C₂×#1311	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×#1311	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
37	#7841	M₁₆	1	1	yes	characteristic
38	#7511	—	4	4	no	—
39	#7511	—	4	4	no	—
40	#7511	—	4	8	no	—
41	#7510	—	2	8	no	—
42	C₄×C₆₄	C₁₆	1	16	yes	complemented
43	#7466	—	4	2	no	—
44	#7332	—	4	2	no	—
45	#7330	—	4	2	no	—
46	#7330	—	4	1	no	—
47	#7330	—	4	1	no	—
48	#7329	—	2	2	no	—
49	C₈×C₃₂	C₁₆	1	4	yes	—
50	#7145	—	4	1	no	—
51	#7142	—	4	1	no	—
52	#7053	—	4	2	no	—
53	#7052	—	2	2	no	—
54	C₁₆²	C₁₆	1	4	yes	—
55	C₂×M₆₄	—	4	2	no	—
56	C₂²×C₃₂	C₂²⋊C₈	1	4	yes	—
57	C₂²×C₃₂	C₄⋊C₈	1	4	yes	—
58	#2162	—	4	2	no	—
59	C₂×#346	—	2	2	no	—
60	C₂×C₄×C₁₆	#95	1	1	yes	characteristic
61	C₂×C₄×C₁₆	C₄×C₈	1	2	yes	—
62	C₂×C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₆	#97	1	2	yes	—
64	C₂×C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
66	#1744	—	2	1	no	—
67	C₄²×C₈	#97	1	1	yes	characteristic
68	C₂×#322	—	2	1	no	—
69	C₂×C₈²	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₈²	C₄⋊C₈	1	1	yes	characteristic
71	M₁₂₈	—	4	8	no	—
72	M₁₂₈	—	4	8	no	—
73	C₂×C₆₄	—	2	16	no	—
74	#1397	—	2	2	no	—
75	#1397	—	2	2	no	—
76	#1397	—	2	2	no	—
77	#1397	—	2	2	no	—
78	#1397	—	2	8	no	—
79	#1397	—	4	2	no	—
80	C₄×C₃₂	—	2	4	no	—
81	C₄×C₃₂	C₂×C₁₆	1	8	yes	—
82	C₄×C₃₂	M₃₂	1	8	yes	—
83	#1312	—	4	2	no	—
84	#1311	—	2	2	no	—
85	#1311	—	2	1	no	—
86	#1311	—	2	1	no	—
87	C₈×C₁₆	—	2	4	no	—
88	C₈×C₁₆	C₂×C₁₆	1	2	yes	—
89	C₈×C₁₆	C₂×C₁₆	1	2	yes	—
90	C₈×C₁₆	M₃₂	1	2	yes	—
91	C₈×C₁₆	M₃₂	1	2	yes	—
92	C₂²×C₁₆	—	2	1	no	—
93	C₂²×C₁₆	—	2	1	no	—
94	C₂²×C₁₆	C₈²	1	1	yes	characteristic
95	C₂²×C₁₆	#322	1	1	yes	characteristic
96	C₂²×C₁₆	#336	1	2	yes	—
97	C₂×C₄×C₈	—	2	1	no	—
98	C₂×C₄×C₈	—	2	1	no	—
99	C₂×C₄×C₈	#322	1	2	yes	—
100	C₂×C₄×C₈	#336	1	1	yes	characteristic
101	C₂×C₄×C₈	#347	1	2	yes	—
102	C₂×C₄×C₈	#347	1	1	yes	characteristic
103	C₄³	#347	1	1	yes	characteristic
104	M₆₄	—	8	4	no	—
105	C₂×C₃₂	—	2	4	no	—
106	C₂×C₃₂	—	2	4	no	—
107	C₂×C₃₂	—	2	4	no	—
108	C₂×C₃₂	—	2	4	no	—
109	C₂×C₃₂	—	4	4	no	—
110	C₂×C₃₂	C₂²⋊C₁₆	1	8	yes	—
111	C₂×C₃₂	C₄⋊C₁₆	1	8	yes	—
112	C₄⋊C₁₆	—	8	2	no	—
113	C₂²⋊C₁₆	—	8	2	no	—
114	#346	—	8	2	no	—
115	C₄×C₁₆	—	2	1	no	—
116	C₄×C₁₆	—	2	1	no	—
117	C₄×C₁₆	—	4	4	no	—
118	C₄×C₁₆	H(3,ℤ/4)	1	2	yes	—
119	C₄×C₁₆	H(3,ℤ/4)	1	1	yes	characteristic
120	C₄×C₁₆	#338	1	1	yes	characteristic
121	C₄×C₁₆	C₄×C₁₆	1	4	yes	—
122	C₄×C₁₆	C₄×C₁₆	1	1	yes	characteristic
123	C₄×C₁₆	C₄×C₁₆	1	1	yes	characteristic
124	C₄×C₁₆	#346	1	4	yes	—
125	C₄×C₁₆	#346	1	1	yes	characteristic
126	C₄×C₁₆	#346	1	1	yes	characteristic
127	C₄×C₁₆	#347	1	8	yes	—
128	C₄×C₁₆	C₂²⋊C₁₆	1	2	yes	—
129	C₄×C₁₆	C₄⋊C₁₆	1	2	yes	—
130	#336	—	8	1	no	—
131	#322	—	8	1	no	—
132	C₈²	—	2	1	no	—
133	C₈²	—	2	1	no	—
134	C₈²	C₂²⋊C₁₆	1	2	yes	—
135	C₈²	C₄⋊C₁₆	1	2	yes	—
136	C₆₄	—	4	16	no	—
137	C₂²×C₈	—	4	1	no	—
138	C₂²×C₈	—	4	1	no	—
139	C₂²×C₈	#1273	1	1	yes	characteristic
140	C₂²×C₈	#1313	1	2	yes	—
141	C₂²×C₈	#1375	1	1	yes	characteristic
142	C₂×C₄²	—	4	1	no	—
143	C₂×C₄²	#1313	1	2	yes	—
144	C₂×C₄²	#1375	1	1	yes	characteristic
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	4	2	no	—
148	C₂×C₁₆	—	4	2	no	—
149	C₂×C₁₆	—	8	4	no	—
150	C₂×C₁₆	#1274	1	2	yes	—
151	C₂×C₁₆	#1275	1	2	yes	—
152	C₂×C₁₆	C₈×C₁₆	1	2	yes	—
153	C₂×C₁₆	C₈×C₁₆	1	1	yes	characteristic
154	C₂×C₁₆	#1311	1	2	yes	—
155	C₂×C₁₆	#1311	1	1	yes	characteristic
156	C₂×C₁₆	#1312	1	1	yes	characteristic
157	C₂×C₁₆	#1312	1	1	yes	characteristic
158	C₂×C₁₆	#1313	1	4	yes	—
159	C₂×C₁₆	#1374	1	4	yes	—
160	C₂×C₁₆	#1375	1	4	yes	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	4	1	no	—
164	C₄×C₈	—	4	2	no	—
165	C₄×C₈	—	4	1	no	—
166	C₄×C₈	—	4	1	no	—
167	C₄×C₈	#1274	1	1	yes	characteristic
168	C₄×C₈	#1275	1	1	yes	characteristic
169	C₄×C₈	#1312	1	4	yes	—
170	C₄×C₈	#1313	1	4	yes	—
171	C₄×C₈	#1374	1	1	yes	characteristic
172	C₄×C₈	#1374	1	1	yes	characteristic
173	C₃₂	—	2	8	no	—
174	C₃₂	—	2	8	no	—
175	C₃₂	—	8	4	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	#7137	1	1	yes	characteristic
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	16	1	no	—
185	C₂×C₈	—	16	1	no	—
186	C₂×C₈	#7014	1	1	yes	characteristic
187	C₂×C₈	#7053	1	4	yes	—
188	C₂×C₈	#7136	1	2	yes	—
189	C₂×C₈	#7136	1	1	yes	characteristic
190	C₂×C₈	#7137	1	2	yes	—
191	C₂×C₈	#7142	1	1	yes	characteristic
192	C₂×C₈	#7145	1	1	yes	characteristic
193	C₄²	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	16	1	no	—
196	C₄²	#7053	1	4	yes	—
197	C₄²	#7142	1	1	yes	characteristic
198	C₄²	#7145	1	1	yes	characteristic
199	C₁₆	—	4	2	no	—
200	C₁₆	—	4	2	no	—
201	C₁₆	—	16	4	no	—
202	C₁₆	C₁₆²	1	1	yes	characteristic
203	C₁₆	#7052	1	2	yes	—
204	C₁₆	#7052	1	1	yes	characteristic
205	C₁₆	#7053	1	2	yes	—
206	C₁₆	#7053	1	2	yes	—
207	C₁₆	#7142	1	4	yes	—
208	C₁₆	#7145	1	4	yes	—
209	C₂³	#93995	1	1	yes	characteristic
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	#92896	1	2	yes	—
216	C₂×C₄	#93249	1	1	yes	characteristic
217	C₂×C₄	#93995	1	1	yes	characteristic
218	C₂×C₄	#93995	1	2	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	8	2	no	—
223	C₈	—	8	2	no	—
224	C₈	#92895	1	1	yes	characteristic
225	C₈	#92896	1	1	yes	characteristic
226	C₈	#93249	1	2	yes	—
227	C₂²	—	4	1	no	—
228	C₂²	#72210682	1	1	yes	characteristic
229	C₂²	#72234264	1	2	yes	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	4	no	—
233	C₄	#72210682	1	1	yes	characteristic
234	C₄	#72210714	1	1	yes	characteristic
235	C₄	#72234264	1	2	yes	—
236	C₂	—	2	2	no	—
237	C₂	—	2	1	no	—
238	C₂	#72234265	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₁₆	C₁₆²
Frattini, cosocle	#72234239	C₂²
socle	C₂	#72234265
prime radical	C₂³	#93995
LCS	G ▷ C₁₆ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ G
LECS	G ▷ #72234239 ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
42. C₄×C₆₄	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72234385	2
#72234267	2