Subgroups — C₂×#72224659 #72245339

Order 2048.

488 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#93878	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#94509	C₂	1	1	yes	maximal, characteristic, complemented
3	#72224659	C₂	1	4	yes	maximal, direct product factor
4	C₂×#151085	C₂	1	1	yes	maximal, characteristic
5	C₂×#13654	C₄	1	1	yes	characteristic
6	C₂×#12045	—	2	1	no	—
7	C₂×#7511	—	2	1	no	—
8	#151085	—	2	1	no	—
9	#151085	—	2	1	no	—
10	#151085	C₄	1	2	yes	—
11	#151085	C₂²	1	2	yes	—
12	C₂×#7463	C₄	1	1	yes	characteristic
13	C₂×#7456	—	2	1	no	—
14	C₂×#7385	—	2	1	no	—
15	C₂×#7330	C₂²	1	1	yes	characteristic
16	#94509	—	2	1	no	—
17	#94509	C₂²	1	2	yes	complemented
18	#93878	—	2	1	no	—
19	#93878	C₂²	1	2	yes	complemented
20	C₂²×D₃₂	C₈	1	1	yes	characteristic
21	C₂×C₄×D₁₆	—	2	1	no	—
22	C₂×C₈×D₈	—	4	1	no	—
23	C₂×M₁₂₈	—	2	2	no	—
24	#13654	—	2	1	no	—
25	#13654	—	2	1	no	—
26	#13654	C₈	1	2	yes	—
27	#13654	C₂×C₄	1	2	yes	—
28	C₂×#1424	C₈	1	1	yes	characteristic
29	C₂×#1422	—	4	1	no	—
30	C₂×#1415	—	2	1	no	—
31	C₂×#1397	C₂×C₄	1	1	yes	characteristic
32	C₂×#1397	D₄	1	1	yes	characteristic
33	#12045	—	2	2	no	—
34	#12045	—	2	2	no	—
35	#12045	—	4	1	no	—
36	#12045	—	4	1	no	—
37	C₂×#1372	—	2	1	no	—
38	C₂×#1331	—	4	1	no	—
39	C₂×#1311	D₄	1	1	yes	characteristic
40	#7511	—	2	2	no	—
41	#7511	—	2	2	no	—
42	#7463	—	2	1	no	—
43	#7463	C₂×C₄	1	2	yes	—
44	#7456	—	2	2	no	—
45	#7456	—	4	1	no	—
46	#7385	—	2	2	no	—
47	#7385	—	2	2	no	—
48	#7330	C₂×C₄	1	1	yes	characteristic
49	#7330	D₄	1	1	yes	characteristic
50	#7330	D₄	1	1	yes	characteristic
51	#7330	C₂³	1	1	yes	characteristic
52	C₂²×D₁₆	—	2	1	no	—
53	C₂×C₄×D₈	—	4	1	no	—
54	C₂×C₈×D₄	—	8	1	no	—
55	C₂×D₃₂	—	2	1	no	—
56	C₂×D₃₂	—	2	1	no	—
57	C₂×D₃₂	—	2	1	no	—
58	C₂×D₃₂	C₁₆	1	2	yes	—
59	C₂×D₃₂	C₁₆	1	2	yes	—
60	C₂×D₃₂	C₂×C₈	1	2	yes	—
61	C₂×M₆₄	—	8	1	no	—
62	C₂²×C₃₂	—	2	1	no	—
63	C₂²×C₃₂	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₃₂	M₁₆	1	1	yes	characteristic
65	C₄×D₁₆	—	2	2	no	—
66	C₄×D₁₆	—	2	2	no	—
67	C₄×D₁₆	—	4	1	no	—
68	C₄×D₁₆	—	4	1	no	—
69	C₂×#366	—	2	1	no	—
70	C₂×#357	—	4	1	no	—
71	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₆	D₈	1	1	yes	characteristic
73	C₈×D₈	—	4	2	no	—
74	C₈×D₈	—	4	2	no	—
75	C₈×D₈	—	8	1	no	—
76	C₈×D₈	—	8	1	no	—
77	C₂×#335	—	4	1	no	—
78	C₂×D₄⋊C₈	—	8	1	no	—
79	C₂×C₈²	SD₁₆	1	1	yes	characteristic
80	M₁₂₈	—	2	4	no	—
81	M₁₂₈	—	4	2	no	—
82	C₂×C₆₄	—	4	2	no	—
83	#1424	—	2	1	no	—
84	#1424	C₂×C₈	1	2	yes	—
85	#1422	—	4	2	no	—
86	#1422	—	8	1	no	—
87	#1415	—	2	2	no	—
88	#1415	—	2	2	no	—
89	#1397	—	2	1	no	—
90	#1397	C₂²⋊C₄	1	1	yes	characteristic
91	#1397	C₂²⋊C₄	1	1	yes	characteristic
92	#1397	C₂×C₈	1	1	yes	characteristic
93	#1397	M₁₆	1	1	yes	characteristic
94	#1397	C₂²×C₄	1	1	yes	characteristic
95	#1397	C₂×D₄	1	1	yes	characteristic
96	#1372	—	2	2	no	—
97	#1372	—	4	1	no	—
98	#1331	—	4	2	no	—
99	#1331	—	4	2	no	—
100	#1311	C₂²⋊C₄	1	1	yes	characteristic
101	#1311	D₈	1	1	yes	characteristic
102	#1311	SD₁₆	1	1	yes	characteristic
103	#1311	C₂×D₄	1	1	yes	characteristic
104	C₂²×D₈	—	4	1	no	—
105	C₂³×C₈	—	16	1	no	—
106	C₂×C₄×D₄	—	8	1	no	—
107	C₂×D₁₆	—	2	2	no	—
108	C₂×D₁₆	—	2	2	no	—
109	C₂×D₁₆	—	2	2	no	—
110	C₂×D₁₆	—	4	1	no	—
111	C₂×D₁₆	—	4	1	no	—
112	C₂×D₁₆	—	4	1	no	—
113	C₂²×C₁₆	—	2	1	no	—
114	C₂²×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₁₆	#104	1	1	yes	characteristic
116	C₄×D₈	—	4	2	no	—
117	C₄×D₈	—	4	2	no	—
118	C₄×D₈	—	8	1	no	—
119	C₄×D₈	—	8	1	no	—
120	C₈×D₄	—	8	2	no	—
121	C₈×D₄	—	8	2	no	—
122	C₈×D₄	—	16	1	no	—
123	C₈×D₄	—	16	1	no	—
124	C₂×#107	—	4	1	no	—
125	C₂×C₄⋊C₈	—	8	1	no	—
126	C₂×#102	—	8	1	no	—
127	C₂×C₂²⋊C₈	—	16	1	no	—
128	C₂×C₄×C₈	—	2	1	no	—
129	C₂×C₄×C₈	#102	1	1	yes	characteristic
130	D₃₂	—	2	2	no	—
131	D₃₂	—	2	2	no	—
132	D₃₂	—	2	2	no	—
133	D₃₂	C₂×C₁₆	1	4	yes	—
134	M₆₄	—	8	2	no	—
135	M₆₄	—	16	1	no	—
136	C₂×C₃₂	—	2	2	no	—
137	C₂×C₃₂	—	2	2	no	—
138	C₂×C₃₂	—	2	1	no	—
139	C₂×C₃₂	—	2	1	no	—
140	C₂×C₃₂	—	2	1	no	—
141	C₂×C₃₂	—	2	1	no	—
142	C₂×C₃₂	—	16	1	no	—
143	C₂×C₃₂	C₂²⋊C₈	1	1	yes	characteristic
144	C₂×C₃₂	C₂²⋊C₈	1	1	yes	characteristic
145	C₂×C₃₂	C₂×C₁₆	1	1	yes	characteristic
146	C₂×C₃₂	C₂×C₁₆	1	1	yes	characteristic
147	C₂×C₃₂	M₃₂	1	1	yes	characteristic
148	C₂×C₃₂	M₃₂	1	1	yes	characteristic
149	C₂×C₃₂	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₃₂	C₂×M₁₆	1	1	yes	characteristic
151	#366	—	2	2	no	—
152	#366	—	4	1	no	—
153	#357	—	4	2	no	—
154	#357	—	4	2	no	—
155	C₄×C₁₆	—	2	1	no	—
156	C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
157	C₄×C₁₆	#102	1	1	yes	characteristic
158	C₄×C₁₆	#102	1	1	yes	characteristic
159	C₄×C₁₆	#104	1	1	yes	characteristic
160	C₄×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₄×C₁₆	C₂×D₈	1	1	yes	characteristic
162	#335	—	4	2	no	—
163	#335	—	8	1	no	—
164	D₄⋊C₈	—	8	2	no	—
165	D₄⋊C₈	—	8	2	no	—
166	C₈²	—	2	1	no	—
167	C₈²	#102	1	1	yes	characteristic
168	C₈²	C₂×SD₁₆	1	1	yes	characteristic
169	C₆₄	—	4	4	no	—
170	C₂²×D₄	—	8	1	no	—
171	C₂³×C₄	—	16	1	no	—
172	C₂×D₈	—	4	2	no	—
173	C₂×D₈	—	4	2	no	—
174	C₂×D₈	—	4	2	no	—
175	C₂×D₈	—	8	1	no	—
176	C₂×D₈	—	8	1	no	—
177	C₂×D₈	—	8	1	no	—
178	C₂²×C₈	—	4	1	no	—
179	C₂²×C₈	—	16	2	no	—
180	C₂²×C₈	—	16	2	no	—
181	C₂²×C₈	—	16	1	no	—
182	C₂²×C₈	—	32	4	no	—
183	C₂²×C₈	D₄⋊C₈	1	1	yes	characteristic
184	C₂²×C₈	#360	1	1	yes	characteristic
185	C₄×D₄	—	8	2	no	—
186	C₄×D₄	—	8	2	no	—
187	C₄×D₄	—	16	1	no	—
188	C₄×D₄	—	16	1	no	—
189	C₂×C₄⋊C₄	—	8	1	no	—
190	C₂×C₂²⋊C₄	—	16	1	no	—
191	C₂×C₄²	#361	1	1	yes	characteristic
192	D₁₆	—	2	4	no	—
193	D₁₆	—	4	2	no	—
194	D₁₆	—	4	2	no	—
195	D₁₆	—	4	2	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	4	1	no	—
201	C₂×C₁₆	—	4	1	no	—
202	C₂×C₁₆	D₄⋊C₈	1	1	yes	characteristic
203	C₂×C₁₆	D₄⋊C₈	1	1	yes	characteristic
204	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
205	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
206	C₂×C₁₆	#350	1	1	yes	characteristic
207	C₂×C₁₆	#350	1	1	yes	characteristic
208	C₂×C₁₆	C₂×C₂²⋊C₈	1	1	yes	characteristic
209	C₂×C₁₆	C₂×#104	1	1	yes	characteristic
210	#107	—	4	2	no	—
211	#107	—	8	1	no	—
212	C₄⋊C₈	—	8	2	no	—
213	C₄⋊C₈	—	16	1	no	—
214	#102	—	8	2	no	—
215	#102	—	8	2	no	—
216	C₂²⋊C₈	—	16	2	no	—
217	C₂²⋊C₈	—	16	2	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	4	1	no	—
221	C₄×C₈	D₄⋊C₈	1	1	yes	characteristic
222	C₄×C₈	#360	1	1	yes	characteristic
223	C₄×C₈	#361	1	1	yes	characteristic
224	C₄×C₈	C₂×#102	1	1	yes	characteristic
225	C₃₂	—	2	2	no	—
226	C₃₂	—	2	2	no	—
227	C₃₂	—	2	1	no	—
228	C₃₂	—	2	1	no	—
229	C₃₂	—	16	2	no	—
230	C₃₂	C₂²⋊C₁₆	1	1	yes	characteristic
231	C₃₂	C₂²⋊C₁₆	1	1	yes	characteristic
232	C₃₂	C₂²×C₁₆	1	1	yes	characteristic
233	C₃₂	C₂×M₃₂	1	1	yes	characteristic
234	C₂⁴	—	16	1	no	—
235	C₂×D₄	—	8	2	no	—
236	C₂×D₄	—	8	2	no	—
237	C₂×D₄	—	8	2	no	—
238	C₂×D₄	—	16	1	no	—
239	C₂×D₄	—	16	1	no	—
240	C₂×D₄	—	16	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	16	2	no	—
243	C₂²×C₄	—	16	2	no	—
244	C₂²×C₄	—	16	1	no	—
245	C₂²×C₄	—	32	2	no	—
246	C₂²×C₄	—	32	2	no	—
247	C₂²×C₄	#1333	1	1	yes	characteristic
248	D₈	—	4	4	no	—
249	D₈	—	8	2	no	—
250	D₈	—	8	2	no	—
251	D₈	—	8	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	8	2	no	—
257	C₂×C₈	—	16	2	no	—
258	C₂×C₈	—	32	4	no	—
259	C₂×C₈	—	32	4	no	—
260	C₂×C₈	—	32	2	no	—
261	C₂×C₈	D₄⋊C₁₆	1	1	yes	characteristic
262	C₂×C₈	D₄⋊C₁₆	1	1	yes	characteristic
263	C₂×C₈	#1333	1	1	yes	characteristic
264	C₂×C₈	#1333	1	1	yes	characteristic
265	C₂×C₈	#1336	1	1	yes	characteristic
266	C₂×C₈	#1336	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄⋊C₈	1	1	yes	characteristic
268	C₂×C₈	C₂×#360	1	1	yes	characteristic
269	C₄⋊C₄	—	8	2	no	—
270	C₄⋊C₄	—	16	1	no	—
271	C₂²⋊C₄	—	16	2	no	—
272	C₂²⋊C₄	—	16	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	#1333	1	1	yes	characteristic
275	C₄²	C₂×#361	1	1	yes	characteristic
276	C₁₆	—	2	1	no	—
277	C₁₆	—	2	1	no	—
278	C₁₆	—	4	1	no	—
279	C₁₆	—	4	1	no	—
280	C₁₆	D₄⋊C₁₆	1	1	yes	characteristic
281	C₁₆	D₄⋊C₁₆	1	1	yes	characteristic
282	C₁₆	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
283	C₁₆	C₂×#350	1	1	yes	characteristic
284	C₂³	—	16	2	no	—
285	C₂³	—	16	2	no	—
286	C₂³	—	16	2	no	—
287	C₂³	—	32	2	no	—
288	C₂³	—	32	2	no	—
289	C₂³	#7390	1	1	yes	characteristic
290	D₄	—	8	4	no	—
291	D₄	—	16	2	no	—
292	D₄	—	16	2	no	—
293	D₄	—	16	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	—	32	2	no	—
299	C₂×C₄	—	32	2	no	—
300	C₂×C₄	—	32	4	no	—
301	C₂×C₄	—	32	2	no	—
302	C₂×C₄	#7070	1	1	yes	characteristic
303	C₂×C₄	#7070	1	1	yes	characteristic
304	C₂×C₄	#7389	1	1	yes	characteristic
305	C₂×C₄	#7389	1	1	yes	characteristic
306	C₂×C₄	#7390	1	1	yes	characteristic
307	C₂×C₄	C₂×#1333	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	8	1	no	—
311	C₈	—	8	1	no	—
312	C₈	—	32	2	no	—
313	C₈	#7070	1	1	yes	characteristic
314	C₈	#7070	1	1	yes	characteristic
315	C₈	C₂×D₄⋊C₁₆	1	1	yes	characteristic
316	C₈	C₂×#1336	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	16	4	no	—
319	C₂²	—	32	2	no	—
320	C₂²	—	32	2	no	—
321	C₂²	—	32	4	no	—
322	C₂²	—	32	4	no	—
323	C₂²	#93757	1	1	yes	characteristic
324	C₂²	#93757	1	1	yes	characteristic
325	C₂²	C₂×#7390	1	1	yes	characteristic
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	32	2	no	—
329	C₄	#93757	1	1	yes	characteristic
330	C₄	#93757	1	1	yes	characteristic
331	C₄	C₂×#7070	1	1	yes	characteristic
332	C₄	C₂×#7389	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	32	4	no	—
336	C₂	#72224659	1	2	yes	direct product factor
337	C₂	C₂×#93757	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#93757
commutator	C₃₂	C₂²×C₁₆
Frattini, cosocle	#7330	C₂³
prime radical	C₂²×D₃₂	C₈
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
1. C₂×#93878	C₂	semidirect
2. C₂×#94509	C₂	semidirect
3. #72224659	C₂	direct (N × H)
17. #94509	C₂²	semidirect
19. #93878	C₂²	semidirect
336. C₂	#72224659	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×#93878	2
C₂×#94509	2
#72224659	2
C₂×#151085	2

Subgroups — C2×#72224659 #72245339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72224659 #72245339