Subgroups — #72225326

Order 2048.

565 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225323	C₂	1	1	yes	maximal, characteristic
2	#72225322	C₂	1	1	yes	maximal, characteristic
3	#72223681	C₂	1	2	yes	maximal
4	C₈×#1398	C₂	1	1	yes	maximal, characteristic
5	#72216215	C₂	1	2	yes	maximal
6	C₄×#1398	C₂²	1	1	yes	characteristic
7	C₄×#1398	C₂²	1	1	yes	characteristic
8	#151024	C₄	1	4	yes	—
9	#149495	C₄	1	1	yes	characteristic
10	#149495	C₄	1	1	yes	characteristic
11	C₈×M₆₄	C₄	1	1	yes	characteristic
12	C₈×M₆₄	C₄	1	1	yes	characteristic
13	#129090	C₂²	1	2	yes	—
14	#129088	C₂²	1	2	yes	—
15	#129051	—	2	2	no	—
16	C₈×#346	C₂²	1	1	yes	characteristic
17	#94031	C₄	1	4	yes	—
18	#93264	—	2	2	no	—
19	C₂×#1398	—	2	2	no	—
20	C₂×#1398	C₂×C₄	1	2	yes	—
21	C₂×#1398	C₂×C₄	1	2	yes	—
22	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
23	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
24	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
25	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
26	C₄×#364	—	2	2	no	—
27	C₄×#346	D₄	1	1	yes	characteristic
28	C₄×#346	C₂³	1	1	yes	characteristic
29	C₂×#1369	C₂×C₄	1	4	yes	—
30	#11999	C₂×C₄	1	4	yes	—
31	#11405	—	4	2	no	—
32	C₂×#1311	C₂×C₄	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	Q₈	1	1	yes	characteristic
36	#8266	—	2	2	no	—
37	C₄×C₈²	D₄	1	1	yes	characteristic
38	#7457	—	2	8	no	—
39	#7330	—	2	2	no	—
40	#7330	C₈	1	4	yes	complemented
41	#7329	—	2	1	no	—
42	#7329	C₈	1	2	yes	complemented
43	C₈×C₃₂	—	2	1	no	—
44	C₈×C₃₂	C₈	1	2	yes	complemented
45	#7161	—	2	2	no	—
46	#7157	—	2	2	no	—
47	#7140	—	4	2	no	—
48	C₂×M₆₄	—	2	1	no	—
49	C₂×M₆₄	—	2	1	no	—
50	C₂×M₆₄	C₄²	1	2	yes	—
51	C₂×M₆₄	C₄²	1	1	yes	characteristic
52	C₂×M₆₄	C₄²	1	1	yes	characteristic
53	C₂×#364	—	2	4	no	—
54	C₄×M₃₂	—	4	2	no	—
55	C₂×#346	SD₁₆	1	2	yes	—
56	C₂×#346	SD₁₆	1	2	yes	—
57	C₂×#346	C₂²×C₄	1	2	yes	—
58	C₂×#346	#41	1	2	yes	—
59	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₆	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₈	—	2	1	no	—
64	C₄²×C₈	C₂×D₄	1	1	yes	characteristic
65	#1564	—	2	4	no	—
66	#1450	—	4	2	no	—
67	C₂×C₈²	—	2	1	no	—
68	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
70	#1398	—	2	8	no	—
71	#1398	—	2	4	no	—
72	#1398	M₁₆	1	8	yes	—
73	#1397	—	2	1	no	—
74	#1397	—	2	1	no	—
75	#1397	C₂×C₈	1	2	yes	—
76	#1397	C₂×C₈	1	2	yes	—
77	C₄×C₃₂	—	2	1	no	—
78	C₄×C₃₂	—	2	1	no	—
79	C₄×C₃₂	C₂×C₈	1	2	yes	—
80	C₄×C₃₂	C₂×C₈	1	2	yes	—
81	#1376	—	4	2	no	—
82	#1369	—	2	4	no	—
83	#1369	—	2	4	no	—
84	#1368	—	2	4	no	—
85	#1368	—	2	4	no	—
86	#1313	—	8	4	no	—
87	#1311	—	2	1	no	—
88	#1311	—	2	1	no	—
89	#1311	C₂×C₈	1	2	yes	—
90	#1311	M₁₆	1	2	yes	—
91	C₈×C₁₆	—	2	2	no	—
92	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
93	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
94	C₈×C₁₆	M₁₆	1	1	yes	characteristic
95	C₈×C₁₆	M₁₆	1	1	yes	characteristic
96	#1276	—	4	2	no	—
97	C₂×M₃₂	—	4	4	no	—
98	C₂²×C₁₆	#106	1	2	yes	—
99	C₂²×C₁₆	#106	1	1	yes	characteristic
100	C₂²×C₁₆	#106	1	1	yes	characteristic
101	C₂²×C₁₆	C₂×C₄²	1	1	yes	characteristic
102	C₂²×C₁₆	#117	1	1	yes	characteristic
103	C₂²×C₁₆	C₄×Q₈	1	2	yes	—
104	#431	—	4	4	no	—
105	C₂×C₄×C₈	—	2	1	no	—
106	C₂×C₄×C₈	—	2	1	no	—
107	C₂×C₄×C₈	—	4	2	no	—
108	C₂×C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈	C₂×SD₁₆	1	2	yes	—
111	C₂×C₄×C₈	#135	1	1	yes	characteristic
112	C₄³	#135	1	1	yes	characteristic
113	M₆₄	—	2	4	no	—
114	M₆₄	—	2	4	no	—
115	M₆₄	—	2	2	no	—
116	M₆₄	—	2	2	no	—
117	M₆₄	#97	1	4	yes	—
118	M₆₄	#97	1	4	yes	—
119	C₂×C₃₂	—	2	2	no	—
120	C₂×C₃₂	—	2	2	no	—
121	C₂×C₃₂	—	2	2	no	—
122	C₂×C₃₂	C₄×C₈	1	2	yes	—
123	C₂×C₃₂	C₄×C₈	1	2	yes	—
124	#364	—	4	4	no	—
125	#364	—	4	4	no	—
126	#346	—	2	4	no	—
127	#346	—	2	4	no	—
128	#346	—	4	2	no	—
129	#346	—	8	2	no	—
130	#346	C₂×M₁₆	1	4	yes	—
131	#346	#131	1	4	yes	—
132	C₄×C₁₆	—	2	1	no	—
133	C₄×C₁₆	—	2	1	no	—
134	C₄×C₁₆	—	8	2	no	—
135	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
136	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
137	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
138	C₄×C₁₆	#108	1	2	yes	—
139	C₄×C₁₆	#108	1	1	yes	characteristic
140	C₄×C₁₆	#108	1	1	yes	characteristic
141	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
142	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
143	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
144	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
145	#335	—	4	4	no	—
146	#334	—	4	4	no	—
147	#322	—	8	2	no	—
148	C₈²	—	2	1	no	—
149	C₈²	—	2	1	no	—
150	C₈²	—	4	2	no	—
151	C₈²	—	8	2	no	—
152	C₈²	C₄⋊C₈	1	1	yes	characteristic
153	C₈²	C₄⋊C₈	1	1	yes	characteristic
154	C₈²	C₄⋊C₈	1	2	yes	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	4	1	no	—
157	C₂²×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₈	C₂×#106	1	1	yes	characteristic
159	C₂²×C₈	C₄×SD₁₆	1	2	yes	—
160	C₂×C₄²	—	4	1	no	—
161	C₂×C₄²	#427	1	1	yes	characteristic
162	C₂×C₄²	C₄×SD₁₆	1	2	yes	—
163	M₃₂	—	8	4	no	—
164	M₃₂	—	8	4	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	8	4	no	—
172	C₂×C₁₆	#334	1	1	yes	characteristic
173	C₂×C₁₆	#334	1	1	yes	characteristic
174	C₂×C₁₆	#334	1	4	yes	—
175	C₂×C₁₆	#334	1	1	yes	characteristic
176	C₂×C₁₆	#334	1	1	yes	characteristic
177	C₂×C₁₆	#365	1	4	yes	—
178	C₂×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
179	C₂×C₁₆	C₂×#97	1	2	yes	—
180	C₂×C₁₆	C₄×M₁₆	1	1	yes	characteristic
181	C₂×C₁₆	#405	1	2	yes	—
182	C₂×C₁₆	#431	1	1	yes	characteristic
183	C₂×C₁₆	#431	1	1	yes	characteristic
184	C₂×C₁₆	#446	1	4	yes	—
185	C₄⋊C₈	—	8	4	no	—
186	C₂²⋊C₈	—	8	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	4	1	no	—
194	C₄×C₈	—	4	1	no	—
195	C₄×C₈	—	8	4	no	—
196	C₄×C₈	—	8	4	no	—
197	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
198	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
199	C₄×C₈	C₂×#108	1	1	yes	characteristic
200	C₄×C₈	C₂×#108	1	1	yes	characteristic
201	C₄×C₈	#436	1	4	yes	—
202	C₄×C₈	#443	1	4	yes	—
203	C₃₂	—	2	4	no	—
204	C₃₂	—	2	4	no	—
205	C₃₂	—	2	2	no	—
206	C₃₂	—	2	4	no	—
207	C₃₂	#322	1	4	yes	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	C₄×#106	1	1	yes	characteristic
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	8	2	no	—
218	C₂×C₈	—	16	4	no	—
219	C₂×C₈	C₂×#334	1	1	yes	characteristic
220	C₂×C₈	C₂×#334	1	1	yes	characteristic
221	C₂×C₈	#1590	1	4	yes	—
222	C₂×C₈	C₄×C₄⋊C₈	1	1	yes	characteristic
223	C₂×C₈	#1770	1	2	yes	—
224	C₂×C₈	#1772	1	2	yes	—
225	C₂×C₈	C₄×#108	1	1	yes	characteristic
226	C₂×C₈	C₂×#365	1	2	yes	—
227	C₂×C₈	#2155	1	2	yes	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	8	2	no	—
231	C₄²	#1564	1	1	yes	characteristic
232	C₄²	#1564	1	1	yes	characteristic
233	C₄²	#1590	1	4	yes	—
234	C₁₆	—	2	1	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	2	4	no	—
237	C₁₆	—	2	1	no	—
238	C₁₆	—	2	1	no	—
239	C₁₆	—	2	4	no	—
240	C₁₆	—	16	4	no	—
241	C₁₆	#1368	1	2	yes	—
242	C₁₆	#1368	1	2	yes	—
243	C₁₆	C₂×#322	1	1	yes	characteristic
244	C₁₆	#1450	1	1	yes	characteristic
245	C₁₆	#1451	1	2	yes	—
246	C₂³	#12668	1	1	yes	characteristic
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	C₄×#334	1	1	yes	characteristic
253	C₂×C₄	#8283	1	2	yes	—
254	C₂×C₄	#12668	1	1	yes	characteristic
255	C₂×C₄	C₄×#365	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	2	4	no	—
258	C₈	—	4	2	no	—
259	C₈	—	8	2	no	—
260	C₈	—	32	4	no	—
261	C₈	#8260	1	1	yes	characteristic
262	C₈	#8266	1	1	yes	characteristic
263	C₈	#8293	1	2	yes	—
264	C₈	C₂×#1368	1	1	yes	characteristic
265	C₈	#11999	1	1	yes	characteristic
266	C₈	#12000	1	2	yes	—
267	C₂²	—	4	1	no	—
268	C₂²	#129088	1	1	yes	characteristic
269	C₂²	#129160	1	2	yes	—
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	4	no	—
273	C₄	#96889	1	1	yes	characteristic
274	C₄	C₄×#1368	1	1	yes	characteristic
275	C₄	#129160	1	2	yes	—
276	C₂	—	2	1	no	—
277	C₂	—	2	2	no	—
278	C₂	#72225325	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#11999
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72225325
prime radical	C₂³	#12668
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ 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
40. #7330	C₈	semidirect
42. #7329	C₈	semidirect
44. 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
#72225323	2
#72225322	2
#72223681	2
C₈×#1398	2
#72216215	2