Subgroups — C₂×#72225141 #72231963

Order 2048.

1292 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#149502	C₂	1	1	yes	maximal, characteristic
2	C₂²×#7053	C₂	1	1	yes	maximal, characteristic
3	C₂×#93668	C₂	1	4	yes	maximal, complemented
4	#72225141	C₂	1	8	yes	maximal, direct product factor
5	C₂×#149495	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×M₆₄	C₄	1	2	yes	—
7	C₂×C₄×M₆₄	C₄	1	1	yes	characteristic
8	C₂×C₄×M₆₄	C₄	1	1	yes	characteristic
9	C₂²×#1312	C₄	1	2	yes	—
10	C₂²×#1311	C₂²	1	1	yes	characteristic
11	#149502	C₄	1	4	yes	—
12	#149502	C₂²	1	4	yes	—
13	C₂×#7332	C₄	1	2	yes	—
14	C₂×#7332	C₂²	1	2	yes	—
15	#149495	C₄	1	4	yes	—
16	#149495	C₂²	1	4	yes	—
17	C₂×#7330	C₄	1	2	yes	—
18	C₂×#7330	C₂²	1	2	yes	—
19	C₂×#7053	C₄	1	4	yes	—
20	C₂×#7053	C₄	1	2	yes	—
21	C₂×#7053	C₂²	1	4	yes	—
22	C₂×#7053	C₂²	1	2	yes	—
23	#93668	C₂²	1	16	yes	complemented
24	C₂²×M₆₄	—	2	2	no	—
25	C₂²×M₆₄	C₈	1	2	yes	—
26	C₂²×M₆₄	C₈	1	2	yes	—
27	C₂²×C₄×C₁₆	—	2	2	no	—
28	C₂²×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈²	C₂×C₄	1	1	yes	characteristic
31	C₄×M₆₄	—	2	4	no	—
32	C₄×M₆₄	C₈	1	4	yes	—
33	C₄×M₆₄	C₈	1	4	yes	—
34	C₄×M₆₄	C₂×C₄	1	8	yes	—
35	C₄×M₆₄	C₂×C₄	1	4	yes	—
36	C₄×M₆₄	C₂×C₄	1	4	yes	—
37	C₂×#1397	C₂×C₄	1	4	yes	—
38	C₂×#1397	C₂×C₄	1	2	yes	—
39	C₂×#1397	C₂×C₄	1	2	yes	—
40	C₂×C₄×C₃₂	—	2	2	no	—
41	C₂×C₄×C₃₂	C₈	1	2	yes	—
42	C₂×C₄×C₃₂	C₈	1	2	yes	—
43	C₂×#1312	—	2	2	no	—
44	C₂×#1312	—	2	4	no	—
45	C₂×#1312	C₂×C₄	1	8	yes	—
46	C₂×#1312	C₂×C₄	1	4	yes	—
47	C₂×#1311	C₂×C₄	1	2	yes	—
48	C₂×#1311	C₂×C₄	1	1	yes	characteristic
49	C₂×#1311	C₂×C₄	1	1	yes	characteristic
50	C₂×#1311	C₂×C₄	1	2	yes	—
51	C₂×#1311	C₂×C₄	1	1	yes	characteristic
52	C₂×#1311	C₂×C₄	1	2	yes	—
53	C₂×#1311	C₂³	1	1	yes	characteristic
54	C₂×#1311	C₂³	1	2	yes	—
55	#7332	C₂×C₄	1	8	yes	—
56	#7332	C₂×C₄	1	4	yes	—
57	#7332	C₂³	1	4	yes	—
58	#7330	C₂×C₄	1	8	yes	—
59	#7330	C₂×C₄	1	4	yes	—
60	#7330	C₂³	1	4	yes	—
61	#7053	C₂×C₄	1	8	yes	—
62	#7053	C₂×C₄	1	4	yes	—
63	#7053	C₂³	1	4	yes	—
64	C₂³×C₁₆	—	4	2	no	—
65	C₂³×C₁₆	C₂×C₈	1	1	yes	characteristic
66	C₂³×C₁₆	C₂×C₈	1	1	yes	characteristic
67	C₂³×C₁₆	M₁₆	1	2	yes	—
68	C₂²×C₄×C₈	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₈	M₁₆	1	2	yes	—
70	C₂×M₆₄	—	2	8	no	—
71	C₂×M₆₄	—	2	8	no	—
72	C₂×M₆₄	—	4	8	no	—
73	C₂×M₆₄	C₁₆	1	16	yes	complemented
74	C₂×M₆₄	C₂×C₈	1	8	yes	—
75	C₂×M₆₄	C₂×C₈	1	8	yes	—
76	C₂²×C₃₂	—	2	4	no	—
77	C₂²×C₃₂	C₂×C₈	1	4	yes	—
78	C₂²×C₃₂	C₂×C₈	1	4	yes	—
79	C₂×C₄×C₁₆	—	2	8	no	—
80	C₂×C₄×C₁₆	—	2	4	no	—
81	C₂×C₄×C₁₆	—	4	2	no	—
82	C₂×C₄×C₁₆	—	4	4	no	—
83	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₆	C₄²	1	2	yes	—
85	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₆	C₄²	1	2	yes	—
87	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₆	C₂×C₈	1	2	yes	—
90	C₂×C₄×C₁₆	C₂×C₈	1	2	yes	—
91	C₂×C₄×C₁₆	M₁₆	1	2	yes	—
92	C₂×C₄×C₁₆	M₁₆	1	4	yes	—
93	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₆	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₆	C₂²×C₄	1	2	yes	—
97	C₂×C₈²	C₄²	1	1	yes	characteristic
98	C₂×C₈²	C₄²	1	2	yes	—
99	C₂×C₈²	M₁₆	1	4	yes	—
100	C₂×C₈²	M₁₆	1	2	yes	—
101	C₂×C₈²	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈²	C₂²×C₄	1	2	yes	—
103	#1397	—	2	4	no	—
104	#1397	C₂×C₈	1	4	yes	—
105	#1397	C₂×C₈	1	4	yes	—
106	#1397	C₂²×C₄	1	8	yes	—
107	#1397	C₂²×C₄	1	4	yes	—
108	#1397	C₂²×C₄	1	4	yes	—
109	C₄×C₃₂	—	2	8	no	—
110	C₄×C₃₂	C₂×C₈	1	8	yes	—
111	C₄×C₃₂	C₂×C₈	1	8	yes	—
112	#1312	—	2	8	no	—
113	#1312	—	2	4	no	—
114	#1312	C₂²×C₄	1	8	yes	—
115	#1311	C₄²	1	2	yes	—
116	#1311	C₄²	1	2	yes	—
117	#1311	C₄²	1	2	yes	—
118	#1311	C₂²×C₄	1	2	yes	—
119	#1311	C₂²×C₄	1	1	yes	characteristic
120	#1311	C₂²×C₄	1	2	yes	—
121	#1311	C₂²×C₄	1	1	yes	characteristic
122	#1311	C₂²×C₄	1	1	yes	characteristic
123	#1311	C₂²×C₄	1	2	yes	—
124	#1311	C₂⁴	1	1	yes	characteristic
125	C₂³×C₈	—	2	2	no	—
126	C₂³×C₈	C₄×C₈	1	1	yes	characteristic
127	C₂³×C₈	#97	1	1	yes	characteristic
128	C₂²×C₄²	#97	1	1	yes	characteristic
129	C₂²×C₁₆	—	2	8	no	—
130	C₂²×C₁₆	—	4	4	no	—
131	C₂²×C₁₆	—	4	8	no	—
132	C₂²×C₁₆	—	8	8	no	—
133	C₂²×C₁₆	C₄×C₈	1	1	yes	characteristic
134	C₂²×C₁₆	C₄×C₈	1	2	yes	—
135	C₂²×C₁₆	C₄×C₈	1	1	yes	characteristic
136	C₂²×C₁₆	C₄×C₈	1	2	yes	—
137	C₂²×C₁₆	#97	1	2	yes	—
138	C₂²×C₁₆	#97	1	4	yes	—
139	C₂²×C₁₆	C₂×C₁₆	1	8	yes	—
140	C₂²×C₁₆	M₃₂	1	8	yes	—
141	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
142	C₂²×C₁₆	C₂²×C₈	1	2	yes	—
143	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
144	C₂²×C₁₆	C₂²×C₈	1	2	yes	—
145	C₂²×C₁₆	C₂×M₁₆	1	2	yes	—
146	C₂²×C₁₆	C₂×M₁₆	1	4	yes	—
147	C₂×C₄×C₈	—	2	2	no	—
148	C₂×C₄×C₈	—	2	4	no	—
149	C₂×C₄×C₈	C₄×C₈	1	2	yes	—
150	C₂×C₄×C₈	C₄×C₈	1	1	yes	characteristic
151	C₂×C₄×C₈	#97	1	2	yes	—
152	C₂×C₄×C₈	#97	1	4	yes	—
153	C₂×C₄×C₈	#97	1	2	yes	—
154	C₂×C₄×C₈	#97	1	1	yes	characteristic
155	C₂×C₄×C₈	#97	1	1	yes	characteristic
156	C₂×C₄×C₈	#97	1	2	yes	—
157	C₂×C₄×C₈	C₂×C₄²	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×C₄²	1	2	yes	—
159	C₂×C₄×C₈	C₂×M₁₆	1	2	yes	—
160	C₂×C₄×C₈	C₂×M₁₆	1	4	yes	—
161	M₆₄	—	2	16	no	—
162	M₆₄	—	4	16	no	—
163	M₆₄	C₂×C₁₆	1	32	yes	complemented
164	C₂×C₃₂	—	2	8	no	—
165	C₂×C₃₂	—	2	8	no	—
166	C₂×C₃₂	—	4	8	no	—
167	C₂×C₃₂	C₂×C₁₆	1	16	yes	complemented
168	C₂×C₃₂	C₂²×C₈	1	8	yes	—
169	C₂×C₃₂	C₂²×C₈	1	8	yes	—
170	C₄×C₁₆	—	2	8	no	—
171	C₄×C₁₆	—	4	4	no	—
172	C₄×C₁₆	—	4	8	no	—
173	C₄×C₁₆	C₄×C₈	1	2	yes	—
174	C₄×C₁₆	C₄×C₈	1	2	yes	—
175	C₄×C₁₆	C₄×C₈	1	2	yes	—
176	C₄×C₁₆	#97	1	4	yes	—
177	C₄×C₁₆	#97	1	2	yes	—
178	C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
179	C₄×C₁₆	C₂×C₄²	1	2	yes	—
180	C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
181	C₄×C₁₆	C₂×C₄²	1	2	yes	—
182	C₄×C₁₆	C₂²×C₈	1	2	yes	—
183	C₄×C₁₆	C₂²×C₈	1	2	yes	—
184	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
185	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
186	C₄×C₁₆	C₂×M₁₆	1	4	yes	—
187	C₄×C₁₆	C₂×M₁₆	1	2	yes	—
188	C₄×C₁₆	C₂³×C₄	1	1	yes	characteristic
189	C₄×C₁₆	C₂³×C₄	1	1	yes	characteristic
190	C₈²	#97	1	2	yes	—
191	C₈²	#97	1	4	yes	—
192	C₈²	C₂×C₄²	1	2	yes	—
193	C₈²	C₂×C₄²	1	1	yes	characteristic
194	C₈²	C₂×M₁₆	1	2	yes	—
195	C₈²	C₂×M₁₆	1	4	yes	—
196	C₈²	C₂³×C₄	1	1	yes	characteristic
197	C₂³×C₄	#322	1	1	yes	characteristic
198	C₂³×C₄	#347	1	2	yes	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	2	4	no	—
203	C₂²×C₈	—	4	8	no	—
204	C₂²×C₈	#322	1	2	yes	—
205	C₂²×C₈	#322	1	1	yes	characteristic
206	C₂²×C₈	#322	1	2	yes	—
207	C₂²×C₈	#322	1	1	yes	characteristic
208	C₂²×C₈	C₄×C₁₆	1	4	yes	—
209	C₂²×C₈	#346	1	4	yes	—
210	C₂²×C₈	#347	1	8	yes	—
211	C₂²×C₈	C₂×C₄×C₈	1	2	yes	—
212	C₂²×C₈	C₂×C₄×C₈	1	1	yes	characteristic
213	C₂²×C₈	C₂×#97	1	2	yes	—
214	C₂²×C₈	C₂×#97	1	1	yes	characteristic
215	C₂×C₄²	#322	1	2	yes	—
216	C₂×C₄²	#322	1	1	yes	characteristic
217	C₂×C₄²	#347	1	2	yes	—
218	C₂×C₄²	#347	1	4	yes	—
219	C₂×C₄²	C₂×#97	1	1	yes	characteristic
220	C₂×C₄²	C₂×#97	1	2	yes	—
221	C₂×C₁₆	—	2	4	no	—
222	C₂×C₁₆	—	2	8	no	—
223	C₂×C₁₆	—	2	8	no	—
224	C₂×C₁₆	—	2	4	no	—
225	C₂×C₁₆	—	4	8	no	—
226	C₂×C₁₆	—	8	8	no	—
227	C₂×C₁₆	—	8	16	no	—
228	C₂×C₁₆	C₄×C₁₆	1	8	yes	—
229	C₂×C₁₆	C₄×C₁₆	1	4	yes	—
230	C₂×C₁₆	#346	1	8	yes	—
231	C₂×C₁₆	#346	1	4	yes	—
232	C₂×C₁₆	C₂×C₄×C₈	1	2	yes	—
233	C₂×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
234	C₂×C₁₆	C₂×C₄×C₈	1	2	yes	—
235	C₂×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
236	C₂×C₁₆	C₂×#97	1	4	yes	—
237	C₂×C₁₆	C₂×#97	1	2	yes	—
238	C₂×C₁₆	C₂²×C₁₆	1	4	yes	—
239	C₂×C₁₆	C₂²×C₁₆	1	8	yes	—
240	C₂×C₁₆	C₂×M₃₂	1	4	yes	—
241	C₂×C₁₆	C₂×M₃₂	1	8	yes	—
242	C₂×C₁₆	C₂³×C₈	1	1	yes	characteristic
243	C₂×C₁₆	C₂³×C₈	1	1	yes	characteristic
244	C₂×C₁₆	C₂²×M₁₆	1	2	yes	—
245	C₄×C₈	—	2	2	no	—
246	C₄×C₈	—	2	4	no	—
247	C₄×C₈	—	2	2	no	—
248	C₄×C₈	—	2	4	no	—
249	C₄×C₈	#322	1	2	yes	—
250	C₄×C₈	#322	1	2	yes	—
251	C₄×C₈	#322	1	2	yes	—
252	C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
253	C₄×C₈	C₂×C₄×C₈	1	2	yes	—
254	C₄×C₈	C₂×#97	1	4	yes	—
255	C₄×C₈	C₂×#97	1	2	yes	—
256	C₄×C₈	C₂×#97	1	1	yes	characteristic
257	C₄×C₈	C₂×#97	1	2	yes	—
258	C₄×C₈	C₂×#97	1	1	yes	characteristic
259	C₄×C₈	C₂×#97	1	2	yes	—
260	C₄×C₈	C₂²×C₄²	1	1	yes	characteristic
261	C₄×C₈	C₂²×M₁₆	1	2	yes	—
262	C₃₂	—	2	8	no	—
263	C₃₂	—	4	8	no	—
264	C₃₂	C₂²×C₁₆	1	16	yes	complemented
265	C₂⁴	#1313	1	1	yes	characteristic
266	C₂²×C₄	—	2	8	no	—
267	C₂²×C₄	#1312	1	4	yes	—
268	C₂²×C₄	#1313	1	4	yes	—
269	C₂²×C₄	#1313	1	4	yes	—
270	C₂²×C₄	#1313	1	2	yes	—
271	C₂²×C₄	#1313	1	1	yes	characteristic
272	C₂²×C₄	#1313	1	2	yes	—
273	C₂²×C₄	C₂×#322	1	2	yes	—
274	C₂²×C₄	C₂×#322	1	1	yes	characteristic
275	C₂²×C₄	C₂×#347	1	2	yes	—
276	C₂²×C₄	C₂×#347	1	4	yes	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	4	4	no	—
281	C₂×C₈	—	4	8	no	—
282	C₂×C₈	—	4	4	no	—
283	C₂×C₈	—	4	8	no	—
284	C₂×C₈	#1312	1	8	yes	—
285	C₂×C₈	#1312	1	4	yes	—
286	C₂×C₈	#1313	1	8	yes	—
287	C₂×C₈	#1313	1	4	yes	—
288	C₂×C₈	C₂×#322	1	2	yes	—
289	C₂×C₈	C₂×#322	1	1	yes	characteristic
290	C₂×C₈	C₂×#322	1	2	yes	—
291	C₂×C₈	C₂×#322	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×C₁₆	1	2	yes	—
293	C₂×C₈	C₂×C₄×C₁₆	1	4	yes	—
294	C₂×C₈	C₂×#346	1	2	yes	—
295	C₂×C₈	C₂×#346	1	4	yes	—
296	C₂×C₈	C₂×#347	1	4	yes	—
297	C₂×C₈	C₂×#347	1	8	yes	—
298	C₂×C₈	C₂²×C₄×C₈	1	1	yes	characteristic
299	C₂×C₈	C₂²×#97	1	1	yes	characteristic
300	C₄²	#1313	1	4	yes	—
301	C₄²	#1313	1	2	yes	—
302	C₄²	C₂×#322	1	1	yes	characteristic
303	C₄²	C₂×#322	1	2	yes	—
304	C₄²	C₂×#347	1	2	yes	—
305	C₄²	C₂×#347	1	4	yes	—
306	C₄²	C₂²×#97	1	1	yes	characteristic
307	C₁₆	—	2	2	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	2	4	no	—
310	C₁₆	—	8	8	no	—
311	C₁₆	C₂×C₄×C₁₆	1	2	yes	—
312	C₁₆	C₂×C₄×C₁₆	1	4	yes	—
313	C₁₆	C₂×#346	1	2	yes	—
314	C₁₆	C₂×#346	1	4	yes	—
315	C₁₆	C₂³×C₁₆	1	2	yes	—
316	C₁₆	C₂²×M₃₂	1	2	yes	—
317	C₂³	—	2	4	no	—
318	C₂³	#7053	1	4	yes	—
319	C₂³	C₂×#1313	1	2	yes	—
320	C₂³	C₂×#1313	1	1	yes	characteristic
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	8	no	—
323	C₂×C₄	—	2	8	no	—
324	C₂×C₄	#7053	1	8	yes	—
325	C₂×C₄	#7053	1	4	yes	—
326	C₂×C₄	#7333	1	8	yes	—
327	C₂×C₄	C₂×#1312	1	4	yes	—
328	C₂×C₄	C₂×#1312	1	2	yes	—
329	C₂×C₄	C₂×#1313	1	2	yes	—
330	C₂×C₄	C₂×#1313	1	2	yes	—
331	C₂×C₄	C₂×#1313	1	1	yes	characteristic
332	C₂×C₄	C₂×#1313	1	4	yes	—
333	C₂×C₄	C₂×#1313	1	2	yes	—
334	C₂×C₄	C₂×#1313	1	4	yes	—
335	C₂×C₄	C₂²×#322	1	1	yes	characteristic
336	C₂×C₄	C₂²×#347	1	2	yes	—
337	C₈	—	4	2	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	4	no	—
340	C₈	C₂×#1312	1	2	yes	—
341	C₈	C₂×#1312	1	4	yes	—
342	C₈	C₂×#1313	1	2	yes	—
343	C₈	C₂×#1313	1	4	yes	—
344	C₈	C₂²×C₄×C₁₆	1	1	yes	characteristic
345	C₈	C₂²×#346	1	1	yes	characteristic
346	C₈	C₂²×#347	1	2	yes	—
347	C₂²	—	2	8	no	—
348	C₂²	—	2	4	no	—
349	C₂²	C₂×#7053	1	4	yes	—
350	C₂²	C₂×#7053	1	2	yes	—
351	C₂²	C₂×#7333	1	4	yes	—
352	C₂²	C₂²×#1313	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	C₂×#7053	1	4	yes	—
356	C₄	C₂×#7053	1	2	yes	—
357	C₄	C₂×#7333	1	8	yes	—
358	C₄	C₂²×#1312	1	1	yes	characteristic
359	C₄	C₂²×#1313	1	1	yes	characteristic
360	C₂	—	2	4	no	—
361	C₂	C₂²×#7333	1	2	yes	—
362	C₂	#72225141	1	4	yes	direct product factor
363	C₂	C₂²×#7053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	#322
commutator	C₈	C₂²×C₄×C₁₆
Frattini, cosocle	#1311	C₂⁴
socle	C₂³	C₂×#1313
prime radical	C₂⁴	#1313
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#93668	C₂	semidirect
4. #72225141	C₂	direct (N × H)
23. #93668	C₂²	semidirect
73. C₂×M₆₄	C₁₆	semidirect
163. M₆₄	C₂×C₁₆	semidirect
167. C₂×C₃₂	C₂×C₁₆	semidirect
264. C₃₂	C₂²×C₁₆	semidirect
362. C₂	#72225141	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#149502	2
C₂²×#7053	2
C₂×#93668	2
#72225141	2
C₂×#149495	2

Subgroups — C2×#72225141 #72231963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72225141 #72231963