Subgroups — #72227014

Order 4096.

418 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225402	C₂	1	1	yes	maximal, characteristic
2	#72225398	—	2	1	no	—
3	#72225397	C₄	1	2	yes	complemented
4	#72225395	—	2	1	no	—
5	#72225366	C₂	1	1	yes	maximal, characteristic
6	#72225363	—	2	1	no	—
7	#72225362	C₄	1	1	yes	characteristic
8	#72225313	C₄	1	1	yes	characteristic
9	#72225076	C₂	1	1	yes	maximal, characteristic
10	#72225071	—	2	1	no	—
11	#72225071	C₄	1	2	yes	complemented
12	#72216534	—	2	1	no	—
13	#72216220	—	2	1	no	—
14	C₁₆×M₆₄	C₂²	1	1	yes	characteristic
15	#153582	—	4	1	no	—
16	#151998	C₈	1	1	yes	characteristic
17	#151989	—	4	1	no	—
18	#151040	C₈	1	1	yes	characteristic
19	C₂×#7459	—	2	1	no	—
20	C₈×M₆₄	—	2	2	no	—
21	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
22	C₈×M₆₄	D₄	1	1	yes	characteristic
23	#129142	—	2	1	no	—
24	C₂×C₁₆²	D₄	1	1	yes	characteristic
25	#94748	—	2	2	no	—
26	#94746	—	4	1	no	—
27	#94053	—	2	1	no	—
28	#94041	—	4	1	no	—
29	#93885	—	2	1	no	—
30	#93885	C₈	1	2	yes	—
31	#93665	—	2	2	no	—
32	#93665	—	2	2	no	—
33	#93664	C₂×C₄	1	1	yes	characteristic
34	#93664	D₄	1	1	yes	characteristic
35	C₁₆×C₃₂	C₂×C₄	1	1	yes	characteristic
36	C₁₆×C₃₂	Q₈	1	1	yes	characteristic
37	#13638	—	2	1	no	—
38	C₂×#1422	—	4	1	no	—
39	#13551	—	8	1	no	—
40	C₄×M₆₄	—	2	4	no	—
41	C₄×M₆₄	C₂×C₈	1	1	yes	characteristic
42	C₄×M₆₄	M₁₆	1	1	yes	characteristic
43	C₄×M₆₄	D₈	1	1	yes	characteristic
44	C₄×M₆₄	SD₁₆	1	1	yes	characteristic
45	#12670	—	2	1	no	—
46	#12664	—	4	1	no	—
47	C₂×#1373	—	2	1	no	—
48	C₂×C₈×C₁₆	—	2	1	no	—
49	C₂×C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
50	#7544	—	4	2	no	—
51	#7513	—	8	1	no	—
52	#7511	—	8	1	no	—
53	#7476	C₁₆	1	1	yes	characteristic
54	#7475	C₁₆	1	1	yes	characteristic
55	#7469	—	8	1	no	—
56	#7464	—	2	1	no	—
57	#7464	C₁₆	1	2	yes	—
58	#7459	—	2	2	no	—
59	#7459	—	4	1	no	—
60	#7329	—	2	4	no	—
61	#7329	C₂×C₈	1	1	yes	characteristic
62	#7329	M₁₆	1	1	yes	characteristic
63	#7329	SD₁₆	1	1	yes	characteristic
64	#7329	Q₁₆	1	1	yes	characteristic
65	C₈×C₃₂	—	2	4	no	—
66	C₈×C₃₂	C₄²	1	1	yes	characteristic
67	C₈×C₃₂	C₂²⋊C₄	1	1	yes	characteristic
68	C₈×C₃₂	C₄⋊C₄	1	1	yes	characteristic
69	C₈×C₃₂	C₄⋊C₄	1	1	yes	characteristic
70	#7179	—	4	1	no	—
71	C₁₆²	—	2	1	no	—
72	C₁₆²	C₂²⋊C₄	1	1	yes	characteristic
73	C₁₆²	C₄⋊C₄	1	1	yes	characteristic
74	C₂×M₆₄	—	2	1	no	—
75	C₂×M₆₄	—	2	1	no	—
76	C₂×M₆₄	—	4	4	no	—
77	C₂×M₆₄	—	8	1	no	—
78	C₂×M₆₄	C₂×C₁₆	1	1	yes	characteristic
79	C₂×M₆₄	M₃₂	1	1	yes	characteristic
80	C₂×M₆₄	D₁₆	1	1	yes	characteristic
81	C₂×M₆₄	SD₃₂	1	1	yes	characteristic
82	C₂×#368	—	2	1	no	—
83	#2158	—	2	1	no	—
84	C₂×#364	—	4	1	no	—
85	#2108	—	8	1	no	—
86	C₂×C₄×C₁₆	—	2	2	no	—
87	C₂×C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₆	#102	1	1	yes	characteristic
89	#1772	—	4	1	no	—
90	C₂×C₈²	#104	1	1	yes	characteristic
91	M₁₂₈	—	8	2	no	—
92	#1426	—	4	2	no	—
93	#1422	—	4	2	no	—
94	#1422	—	8	1	no	—
95	#1398	—	16	2	no	—
96	#1397	—	2	1	no	—
97	#1397	—	2	1	no	—
98	#1397	—	4	4	no	—
99	#1397	C₂×C₁₆	1	1	yes	characteristic
100	#1397	M₃₂	1	1	yes	characteristic
101	#1397	SD₃₂	1	1	yes	characteristic
102	#1397	Q₃₂	1	1	yes	characteristic
103	C₄×C₃₂	—	2	8	no	—
104	C₄×C₃₂	C₄×C₈	1	1	yes	characteristic
105	C₄×C₃₂	#97	1	1	yes	characteristic
106	C₄×C₃₂	#102	1	1	yes	characteristic
107	C₄×C₃₂	#103	1	1	yes	characteristic
108	C₄×C₃₂	C₄⋊C₈	1	1	yes	characteristic
109	C₄×C₃₂	C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₃₂	#106	1	1	yes	characteristic
111	C₄×C₃₂	#107	1	1	yes	characteristic
112	#1382	—	4	1	no	—
113	#1379	—	8	1	no	—
114	#1373	—	2	2	no	—
115	#1373	—	4	1	no	—
116	C₈×C₁₆	—	2	2	no	—
117	C₈×C₁₆	—	2	1	no	—
118	C₈×C₁₆	—	2	1	no	—
119	C₈×C₁₆	#95	1	1	yes	characteristic
120	C₈×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
121	C₈×C₁₆	#103	1	1	yes	characteristic
122	C₈×C₁₆	#104	1	1	yes	characteristic
123	C₂×M₃₂	—	8	1	no	—
124	C₂²×C₁₆	—	2	4	no	—
125	C₂²×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
126	C₂²×C₁₆	#350	1	1	yes	characteristic
127	C₂²×C₁₆	#357	1	1	yes	characteristic
128	C₂²×C₁₆	#359	1	1	yes	characteristic
129	C₂×#108	—	4	1	no	—
130	#427	—	4	1	no	—
131	#405	—	8	1	no	—
132	C₂×C₄×C₈	—	2	1	no	—
133	C₂×C₄×C₈	D₄⋊C₈	1	1	yes	characteristic
134	M₆₄	—	2	2	no	—
135	M₆₄	—	2	2	no	—
136	M₆₄	—	4	1	no	—
137	M₆₄	—	4	1	no	—
138	M₆₄	—	4	2	no	—
139	M₆₄	—	4	2	no	—
140	M₆₄	—	8	8	no	—
141	M₆₄	—	8	2	no	—
142	M₆₄	—	16	1	no	—
143	C₂×C₃₂	—	2	1	no	—
144	C₂×C₃₂	—	2	1	no	—
145	C₂×C₃₂	—	2	1	no	—
146	C₂×C₃₂	—	2	1	no	—
147	C₂×C₃₂	—	4	8	no	—
148	C₂×C₃₂	—	16	1	no	—
149	C₂×C₃₂	C₄×C₁₆	1	1	yes	characteristic
150	C₂×C₃₂	#346	1	1	yes	characteristic
151	C₂×C₃₂	#357	1	1	yes	characteristic
152	C₂×C₃₂	#358	1	1	yes	characteristic
153	C₂×C₃₂	C₄⋊C₁₆	1	1	yes	characteristic
154	C₂×C₃₂	C₄⋊C₁₆	1	1	yes	characteristic
155	C₂×C₃₂	#366	1	1	yes	characteristic
156	C₂×C₃₂	#367	1	1	yes	characteristic
157	#368	—	4	1	no	—
158	#368	—	4	1	no	—
159	#367	—	4	1	no	—
160	#366	—	4	1	no	—
161	#364	—	4	2	no	—
162	#364	—	8	1	no	—
163	#346	—	16	1	no	—
164	C₄×C₁₆	—	2	2	no	—
165	C₄×C₁₆	—	2	2	no	—
166	C₄×C₁₆	—	2	4	no	—
167	C₄×C₁₆	—	16	1	no	—
168	C₄×C₁₆	D₄⋊C₈	1	1	yes	characteristic
169	C₄×C₁₆	Q₈⋊C₈	1	1	yes	characteristic
170	C₄×C₁₆	#336	1	1	yes	characteristic
171	C₄×C₁₆	#340	1	1	yes	characteristic
172	C₄×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
173	C₄×C₁₆	#350	1	1	yes	characteristic
174	C₄×C₁₆	#358	1	1	yes	characteristic
175	C₄×C₁₆	#359	1	1	yes	characteristic
176	#339	—	8	1	no	—
177	C₈²	—	2	1	no	—
178	C₈²	Q₈⋊C₈	1	1	yes	characteristic
179	C₈²	#339	1	1	yes	characteristic
180	C₆₄	—	16	2	no	—
181	C₂×M₁₆	—	8	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	D₄⋊C₁₆	1	1	yes	characteristic
184	C₂²×C₈	#1331	1	1	yes	characteristic
185	#117	—	8	1	no	—
186	C₂×C₄²	#1335	1	1	yes	characteristic
187	M₃₂	—	8	2	no	—
188	M₃₂	—	16	1	no	—
189	C₂×C₁₆	—	2	4	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	4	no	—
195	C₂×C₁₆	—	4	1	no	—
196	C₂×C₁₆	—	4	1	no	—
197	C₂×C₁₆	—	4	8	no	—
198	C₂×C₁₆	—	16	1	no	—
199	C₂×C₁₆	#1331	1	1	yes	characteristic
200	C₂×C₁₆	#1332	1	1	yes	characteristic
201	C₂×C₁₆	Q₈⋊C₁₆	1	1	yes	characteristic
202	C₂×C₁₆	Q₈⋊C₁₆	1	1	yes	characteristic
203	C₂×C₁₆	#1374	1	1	yes	characteristic
204	C₂×C₁₆	#1379	1	1	yes	characteristic
205	C₂×C₁₆	#1380	1	1	yes	characteristic
206	C₂×C₁₆	#1382	1	1	yes	characteristic
207	#108	—	8	1	no	—
208	#108	—	8	1	no	—
209	#107	—	8	1	no	—
210	#106	—	8	1	no	—
211	#97	—	16	1	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	16	1	no	—
216	C₄×C₈	#1279	1	1	yes	characteristic
217	C₄×C₈	D₄⋊C₁₆	1	1	yes	characteristic
218	C₄×C₈	#1332	1	1	yes	characteristic
219	C₄×C₈	#1335	1	1	yes	characteristic
220	C₃₂	—	2	2	no	—
221	C₃₂	—	2	2	no	—
222	C₃₂	—	2	1	no	—
223	C₃₂	—	2	1	no	—
224	C₃₂	—	2	1	no	—
225	C₃₂	—	2	1	no	—
226	C₃₂	—	2	1	no	—
227	C₃₂	—	2	1	no	—
228	C₃₂	—	4	1	no	—
229	C₃₂	—	4	1	no	—
230	C₃₂	—	8	8	no	—
231	C₃₂	—	16	2	no	—
232	C₃₂	#1398	1	2	yes	—
233	C₃₂	#1418	1	2	yes	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	#7068	1	1	yes	characteristic
236	M₁₆	—	16	1	no	—
237	M₁₆	—	16	1	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	—	16	1	no	—
246	C₂×C₈	#7068	1	1	yes	characteristic
247	C₂×C₈	#7069	1	1	yes	characteristic
248	C₂×C₈	#7167	1	1	yes	characteristic
249	C₂×C₈	#7171	1	1	yes	characteristic
250	C₄⋊C₄	—	16	1	no	—
251	C₂²⋊C₄	—	16	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	16	1	no	—
254	C₄²	#7069	1	1	yes	characteristic
255	C₄²	#7179	1	1	yes	characteristic
256	C₁₆	—	2	8	no	—
257	C₁₆	—	2	1	no	—
258	C₁₆	—	2	1	no	—
259	C₁₆	—	2	1	no	—
260	C₁₆	—	2	1	no	—
261	C₁₆	—	2	1	no	—
262	C₁₆	—	2	1	no	—
263	C₁₆	—	4	4	no	—
264	C₁₆	—	16	2	no	—
265	C₁₆	#7466	1	1	yes	characteristic
266	C₁₆	#7469	1	1	yes	characteristic
267	C₁₆	#7477	1	1	yes	characteristic
268	C₁₆	#7480	1	1	yes	characteristic
269	C₂³	#93752	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	32	1	no	—
276	C₂×C₄	#92931	1	1	yes	characteristic
277	C₂×C₄	#93752	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	4	2	no	—
282	C₈	—	32	1	no	—
283	C₈	#94037	1	1	yes	characteristic
284	C₈	#94041	1	1	yes	characteristic
285	C₈	#94048	1	1	yes	characteristic
286	C₈	#94054	1	1	yes	characteristic
287	C₂²	—	2	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	#72216535	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	64	1	no	—
294	C₄	#72227004	1	1	yes	characteristic
295	C₄	#72227011	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	4	1	no	—
298	C₂	#72227013	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#94048
commutator	C₂×C₃₂	C₄×C₁₆
Frattini, cosocle	C₁₆×M₆₄	C₂²
socle	C₂	#72227013
prime radical	C₂³	#93752
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₃₂ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₁₆×M₆₄ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #72225397	C₄	semidirect
11. #72225071	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72225402	2
#72225366	2
#72225076	2