Subgroups — #72323155

Order 2048.

438 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#151086	C₂	1	1	yes	maximal, characteristic
2	C₂×#94509	C₂	1	1	yes	maximal, characteristic, complemented
3	#72231924	C₂	1	1	yes	maximal, characteristic, complemented
4	#72224654	C₂	1	4	yes	maximal, complemented
5	C₂×#13656	C₄	1	1	yes	characteristic
6	C₂×#12047	—	2	1	no	—
7	C₂×#7511	—	2	1	no	—
8	#151086	—	2	1	no	—
9	#151086	—	2	1	no	—
10	#151086	C₄	1	2	yes	—
11	#151086	C₂²	1	2	yes	—
12	C₂×#7464	C₄	1	1	yes	characteristic
13	C₂×#7388	—	2	1	no	—
14	C₂×#7330	C₂²	1	1	yes	characteristic
15	#122919	—	2	1	no	—
16	#94509	—	2	1	no	—
17	#94509	C₂²	1	2	yes	complemented
18	#93877	—	2	1	no	—
19	#93877	C₂²	1	2	yes	complemented
20	C₂×#2262	C₈	1	1	yes	characteristic
21	C₂×#2178	—	2	1	no	—
22	C₂×#2170	—	4	1	no	—
23	C₂×M₁₂₈	—	2	2	no	—
24	#13656	—	2	1	no	—
25	#13656	—	2	1	no	—
26	#13656	C₈	1	2	yes	—
27	#13656	C₂×C₄	1	2	yes	—
28	C₂×#1425	C₈	1	1	yes	characteristic
29	C₂×#1417	—	2	1	no	—
30	C₂×#1397	C₂×C₄	1	1	yes	characteristic
31	C₂×#1397	D₄	1	1	yes	characteristic
32	#12047	—	2	2	no	—
33	#12047	—	2	2	no	—
34	#12047	—	4	1	no	—
35	#12047	—	4	1	no	—
36	#12012	—	4	1	no	—
37	C₂×#1373	—	2	1	no	—
38	C₂×#1335	—	4	1	no	—
39	C₂×#1311	D₄	1	1	yes	characteristic
40	#7511	—	2	2	no	—
41	#7511	—	2	2	no	—
42	#7464	—	2	1	no	—
43	#7464	C₂×C₄	1	2	yes	—
44	#7388	—	2	2	no	—
45	#7388	—	2	2	no	—
46	#7330	C₂×C₄	1	1	yes	characteristic
47	#7330	D₄	1	1	yes	characteristic
48	#7330	D₄	1	1	yes	characteristic
49	#7330	C₂³	1	1	yes	characteristic
50	#7099	—	2	2	no	—
51	#7099	—	4	1	no	—
52	C₂×#508	—	2	1	no	—
53	C₂×#504	—	8	1	no	—
54	C₂×#443	—	4	1	no	—
55	#2262	—	2	1	no	—
56	#2262	—	2	1	no	—
57	#2262	C₁₆	1	2	yes	complemented
58	#2262	C₂×C₈	1	2	yes	—
59	C₂×Q₆₄	C₁₆	1	1	yes	characteristic, complemented
60	C₂×SD₆₄	—	2	1	no	—
61	C₂×D₃₂	C₁₆	1	1	yes	characteristic, complemented
62	C₂²×C₃₂	—	2	1	no	—
63	C₂²×C₃₂	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₃₂	M₁₆	1	1	yes	characteristic
65	#2178	—	2	2	no	—
66	#2178	—	2	2	no	—
67	#2178	—	4	1	no	—
68	#2178	—	4	1	no	—
69	#2170	—	4	2	no	—
70	#2170	—	4	2	no	—
71	#2170	—	8	1	no	—
72	#2170	—	8	1	no	—
73	#2162	—	8	1	no	—
74	C₂×#368	—	2	1	no	—
75	C₂×#364	—	4	1	no	—
76	C₂×#359	—	4	1	no	—
77	C₂×#350	—	8	1	no	—
78	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₆	SD₁₆	1	1	yes	characteristic
80	C₂×C₈²	D₈	1	1	yes	characteristic
81	M₁₂₈	—	2	4	no	—
82	M₁₂₈	—	4	2	no	—
83	C₂×C₆₄	—	4	2	no	—
84	#1425	—	2	1	no	—
85	#1425	C₂×C₈	1	2	yes	—
86	#1417	—	2	2	no	—
87	#1417	—	2	2	no	—
88	#1397	—	2	1	no	—
89	#1397	C₂²⋊C₄	1	1	yes	characteristic
90	#1397	C₂²⋊C₄	1	1	yes	characteristic
91	#1397	C₂×C₈	1	1	yes	characteristic
92	#1397	M₁₆	1	1	yes	characteristic
93	#1397	C₂²×C₄	1	1	yes	characteristic
94	#1397	C₂×D₄	1	1	yes	characteristic
95	#1373	—	2	2	no	—
96	#1373	—	4	1	no	—
97	#1370	—	8	1	no	—
98	#1367	—	4	2	no	—
99	#1335	—	4	2	no	—
100	#1335	—	4	2	no	—
101	#1311	C₂²⋊C₄	1	1	yes	characteristic
102	#1311	D₈	1	1	yes	characteristic
103	#1311	SD₁₆	1	1	yes	characteristic
104	#1311	C₂×D₄	1	1	yes	characteristic
105	C₂×#135	—	4	1	no	—
106	C₂×#131	—	8	1	no	—
107	#508	—	2	2	no	—
108	#508	—	2	2	no	—
109	#508	—	4	1	no	—
110	#508	—	4	1	no	—
111	C₂×Q₃₂	—	2	1	no	—
112	C₂×SD₃₂	—	4	1	no	—
113	C₂×D₁₆	—	2	1	no	—
114	#504	—	8	2	no	—
115	#504	—	8	2	no	—
116	#504	—	16	1	no	—
117	#504	—	16	1	no	—
118	C₂×M₃₂	—	8	1	no	—
119	C₂×M₃₂	—	16	1	no	—
120	C₂²×C₁₆	—	2	1	no	—
121	C₂²×C₁₆	—	16	1	no	—
122	C₂²×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₁₆	#104	1	1	yes	characteristic
124	#443	—	4	2	no	—
125	#443	—	4	2	no	—
126	#443	—	8	1	no	—
127	#443	—	8	1	no	—
128	C₂×#108	—	4	1	no	—
129	C₂×#104	—	8	1	no	—
130	C₂×C₄×C₈	—	2	1	no	—
131	C₂×C₄×C₈	#102	1	1	yes	characteristic
132	Q₆₄	—	2	1	no	—
133	Q₆₄	C₂×C₁₆	1	2	yes	—
134	SD₆₄	—	2	2	no	—
135	SD₆₄	—	2	2	no	—
136	D₃₂	—	2	1	no	—
137	D₃₂	C₂×C₁₆	1	2	yes	—
138	C₂×C₃₂	—	2	2	no	—
139	C₂×C₃₂	—	2	2	no	—
140	C₂×C₃₂	—	2	1	no	—
141	C₂×C₃₂	—	2	1	no	—
142	C₂×C₃₂	—	2	1	no	—
143	C₂×C₃₂	—	2	1	no	—
144	C₂×C₃₂	C₂²⋊C₈	1	1	yes	characteristic
145	C₂×C₃₂	C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₃₂	C₂×C₁₆	1	1	yes	characteristic
147	C₂×C₃₂	C₂×C₁₆	1	1	yes	characteristic
148	C₂×C₃₂	M₃₂	1	1	yes	characteristic
149	C₂×C₃₂	M₃₂	1	1	yes	characteristic
150	C₂×C₃₂	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₃₂	C₂×M₁₆	1	1	yes	characteristic
152	#368	—	2	2	no	—
153	#368	—	4	1	no	—
154	#364	—	4	2	no	—
155	#364	—	8	1	no	—
156	C₄⋊C₁₆	—	8	2	no	—
157	#359	—	4	2	no	—
158	#359	—	4	2	no	—
159	#350	—	8	2	no	—
160	#350	—	8	2	no	—
161	C₂²⋊C₁₆	—	16	1	no	—
162	C₄×C₁₆	—	2	1	no	—
163	C₄×C₁₆	—	16	1	no	—
164	C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
165	C₄×C₁₆	#102	1	1	yes	characteristic
166	C₄×C₁₆	#102	1	1	yes	characteristic
167	C₄×C₁₆	#104	1	1	yes	characteristic
168	C₄×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₄×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
170	C₈²	—	2	1	no	—
171	C₈²	#102	1	1	yes	characteristic
172	C₈²	C₂×D₈	1	1	yes	characteristic
173	C₆₄	—	4	4	no	—
174	C₂×#41	—	8	1	no	—
175	#135	—	4	2	no	—
176	#135	—	4	2	no	—
177	#135	—	8	1	no	—
178	#135	—	8	1	no	—
179	C₂×Q₁₆	—	4	1	no	—
180	C₂×SD₁₆	—	8	1	no	—
181	C₂×D₈	—	4	1	no	—
182	#131	—	8	2	no	—
183	#131	—	8	2	no	—
184	#131	—	16	1	no	—
185	#131	—	16	1	no	—
186	C₂×M₁₆	—	8	1	no	—
187	C₂×M₁₆	—	16	1	no	—
188	C₂²×C₈	—	4	1	no	—
189	C₂²×C₈	—	16	1	no	—
190	C₂²×C₈	D₄⋊C₈	1	1	yes	characteristic
191	C₂²×C₈	#361	1	1	yes	characteristic
192	C₂×C₄²	#360	1	1	yes	characteristic
193	Q₃₂	—	2	2	no	—
194	Q₃₂	—	4	1	no	—
195	SD₃₂	—	4	2	no	—
196	SD₃₂	—	4	2	no	—
197	D₁₆	—	2	2	no	—
198	D₁₆	—	4	1	no	—
199	M₃₂	—	8	2	no	—
200	M₃₂	—	16	2	no	—
201	M₃₂	—	16	2	no	—
202	M₃₂	—	16	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	4	1	no	—
208	C₂×C₁₆	—	4	1	no	—
209	C₂×C₁₆	—	16	2	no	—
210	C₂×C₁₆	—	16	2	no	—
211	C₂×C₁₆	—	16	1	no	—
212	C₂×C₁₆	—	16	2	no	—
213	C₂×C₁₆	D₄⋊C₈	1	1	yes	characteristic
214	C₂×C₁₆	D₄⋊C₈	1	1	yes	characteristic
215	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
216	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
217	C₂×C₁₆	#350	1	1	yes	characteristic
218	C₂×C₁₆	#350	1	1	yes	characteristic
219	C₂×C₁₆	C₂×C₂²⋊C₈	1	1	yes	characteristic
220	C₂×C₁₆	C₂×#104	1	1	yes	characteristic
221	#108	—	4	2	no	—
222	#108	—	8	1	no	—
223	#104	—	8	2	no	—
224	#104	—	8	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	4	1	no	—
228	C₄×C₈	D₄⋊C₈	1	1	yes	characteristic
229	C₄×C₈	#360	1	1	yes	characteristic
230	C₄×C₈	#361	1	1	yes	characteristic
231	C₄×C₈	C₂×#102	1	1	yes	characteristic
232	C₃₂	—	2	2	no	—
233	C₃₂	—	2	2	no	—
234	C₃₂	—	2	1	no	—
235	C₃₂	—	2	1	no	—
236	C₃₂	C₂²⋊C₁₆	1	1	yes	characteristic
237	C₃₂	C₂²⋊C₁₆	1	1	yes	characteristic
238	C₃₂	C₂²×C₁₆	1	1	yes	characteristic
239	C₃₂	C₂×M₃₂	1	1	yes	characteristic
240	#41	—	8	2	no	—
241	#41	—	8	2	no	—
242	#41	—	16	1	no	—
243	#41	—	16	1	no	—
244	C₂×Q₈	—	8	1	no	—
245	C₂×D₄	—	8	1	no	—
246	C₂×D₄	—	16	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	16	1	no	—
249	C₂²×C₄	#1333	1	1	yes	characteristic
250	Q₁₆	—	4	2	no	—
251	Q₁₆	—	8	1	no	—
252	SD₁₆	—	8	2	no	—
253	SD₁₆	—	8	2	no	—
254	D₈	—	4	2	no	—
255	D₈	—	8	1	no	—
256	M₁₆	—	8	2	no	—
257	M₁₆	—	16	2	no	—
258	M₁₆	—	16	2	no	—
259	M₁₆	—	16	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	16	2	no	—
265	C₂×C₈	—	16	2	no	—
266	C₂×C₈	—	16	1	no	—
267	C₂×C₈	—	16	1	no	—
268	C₂×C₈	D₄⋊C₁₆	1	1	yes	characteristic
269	C₂×C₈	D₄⋊C₁₆	1	1	yes	characteristic
270	C₂×C₈	#1333	1	1	yes	characteristic
271	C₂×C₈	#1333	1	1	yes	characteristic
272	C₂×C₈	#1336	1	1	yes	characteristic
273	C₂×C₈	#1336	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄⋊C₈	1	1	yes	characteristic
275	C₂×C₈	C₂×#361	1	1	yes	characteristic
276	C₄²	—	2	1	no	—
277	C₄²	#1333	1	1	yes	characteristic
278	C₄²	C₂×#360	1	1	yes	characteristic
279	C₁₆	—	2	1	no	—
280	C₁₆	—	2	1	no	—
281	C₁₆	—	4	1	no	—
282	C₁₆	—	4	1	no	—
283	C₁₆	—	16	2	no	—
284	C₁₆	—	32	2	no	—
285	C₁₆	D₄⋊C₁₆	1	1	yes	characteristic
286	C₁₆	D₄⋊C₁₆	1	1	yes	characteristic
287	C₁₆	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
288	C₁₆	C₂×#350	1	1	yes	characteristic
289	C₂³	—	16	1	no	—
290	C₂³	#7389	1	1	yes	characteristic
291	Q₈	—	8	2	no	—
292	Q₈	—	16	1	no	—
293	D₄	—	8	2	no	—
294	D₄	—	16	2	no	—
295	D₄	—	16	2	no	—
296	D₄	—	16	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	16	1	no	—
301	C₂×C₄	—	16	2	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	#7070	1	1	yes	characteristic
304	C₂×C₄	#7070	1	1	yes	characteristic
305	C₂×C₄	#7389	1	1	yes	characteristic
306	C₂×C₄	#7390	1	1	yes	characteristic
307	C₂×C₄	#7390	1	1	yes	characteristic
308	C₂×C₄	C₂×#1333	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	8	1	no	—
312	C₈	—	8	1	no	—
313	C₈	—	16	2	no	—
314	C₈	#7070	1	1	yes	characteristic
315	C₈	#7070	1	1	yes	characteristic
316	C₈	C₂×D₄⋊C₁₆	1	1	yes	characteristic
317	C₈	C₂×#1336	1	1	yes	characteristic
318	C₂²	—	4	1	no	—
319	C₂²	—	16	2	no	—
320	C₂²	—	32	2	no	—
321	C₂²	#93757	1	1	yes	characteristic
322	C₂²	#93757	1	1	yes	characteristic
323	C₂²	C₂×#7389	1	1	yes	characteristic
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	16	2	no	—
327	C₄	#93757	1	1	yes	characteristic
328	C₄	#93757	1	1	yes	characteristic
329	C₄	C₂×#7070	1	1	yes	characteristic
330	C₄	C₂×#7390	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	32	2	no	—
335	C₂	C₂×#93757	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#93757
commutator	C₃₂	C₂²×C₁₆
Frattini, cosocle	#7330	C₂³
socle	C₂	C₂×#93757
prime radical	C₂×#2262	C₈
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#7330 ◁ G
LECS	G ▷ #7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#94509	C₂	semidirect
3. #72231924	C₂	semidirect
4. #72224654	C₂	semidirect
17. #94509	C₂²	semidirect
19. #93877	C₂²	semidirect
57. #2262	C₁₆	semidirect
59. C₂×Q₆₄	C₁₆	semidirect
61. C₂×D₃₂	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#151086	2
C₂×#94509	2
#72231924	2
#72224654	2