Subgroups — C₂×C₈×#6122 #16792570

Order 3328.

1046 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6122	C₂	1	1	yes	maximal, characteristic
3	C₈×#6122	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1057⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#1057	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#6122	C₄	1	2	yes	—
8	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊#1057	C₄	1	2	yes	—
12	#11719125	C₄	1	4	yes	—
13	#11719125	C₂²	1	4	yes	—
14	C₂×#88112	C₄	1	2	yes	—
15	C₄×#6122	C₄	1	4	yes	—
16	C₄×#6122	C₂²	1	4	yes	—
17	C₂×#88110	C₄	1	2	yes	—
18	C₂×C₄×#1057	C₂²	1	2	yes	—
19	C₂×C₄×#1057	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₃	C₄	1	4	yes	—
21	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
23	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
26	#1057⋊C₈	C₂²	1	8	yes	complemented
27	C₈×#1057	C₂²	1	8	yes	complemented
28	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#6122	C₈	1	16	yes	direct product factor
32	C₂×#6122	C₂×C₄	1	8	yes	—
33	C₂²×#1057	C₂×C₄	1	4	yes	—
34	C₂²×#1057	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₁₃	—	2	4	no	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
47	C₂²×#1055	—	2	4	no	—
48	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
49	C₂²×#1055	C₂×C₄	1	4	yes	—
50	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊#1057	C₂×C₄	1	8	yes	—
52	#88112	C₂×C₄	1	8	yes	—
53	#88110	C₂×C₄	1	8	yes	—
54	C₄×#1057	C₂×C₄	1	4	yes	—
55	C₄×#1057	C₂×C₄	1	4	yes	—
56	C₄×#1057	C₂³	1	4	yes	—
57	C₄×#1057	C₂³	1	4	yes	—
58	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
60	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
61	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
62	C₈×Dic₁₃	C₂³	1	2	yes	—
63	C₈×Dic₁₃	C₂³	1	2	yes	—
64	C₂×C₈×M₁₆	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
74	C₂²×Dic₁₃	C₄²	1	2	yes	—
75	C₂²×Dic₁₃	C₄²	1	2	yes	—
76	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
77	C₂²×Dic₁₃	C₂×C₈	1	8	yes	—
78	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
82	#6122	C₂×C₈	1	32	yes	direct product factor
83	C₂×#1057	—	2	8	no	—
84	C₂×#1057	C₂×C₈	1	16	yes	complemented
85	C₂×#1057	C₂²×C₄	1	8	yes	—
86	C₂×#1057	C₂²×C₄	1	8	yes	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	8	no	—
89	C₂×C₈×C₁₃	M₁₆	1	4	yes	complemented
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
92	C₄×Dic₁₃	C₄²	1	2	yes	—
93	C₄×Dic₁₃	C₄²	1	2	yes	—
94	C₄×Dic₁₃	C₄²	1	2	yes	—
95	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
96	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
101	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	8	no	—
104	C₂×#1055	M₁₆	1	4	yes	complemented
105	C₂×#1055	C₂²×C₄	1	2	yes	—
106	C₂×#1055	C₂²×C₄	1	2	yes	—
107	C₂×#431	—	13	1	no	—
108	C₂×C₄×M₁₆	—	13	1	no	—
109	C₂²×C₄×C₈	—	13	1	no	—
110	C₈×M₁₆	—	13	8	no	—
111	C₂×#322	—	13	2	no	—
112	C₂×C₈²	—	13	2	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	C₄×C₈	1	4	yes	—
116	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
117	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	2	no	—
128	C₂×Dic₁₃	—	2	4	no	—
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	C₄×C₈	1	8	yes	—
131	C₂×Dic₁₃	C₄×C₈	1	4	yes	—
132	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
133	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
134	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
135	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
137	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
138	C₂×Dic₁₃	#131	1	4	yes	—
139	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
141	#1057	—	2	8	no	—
142	#1057	C₂²×C₈	1	16	yes	complemented
143	C₈×C₁₃	—	2	2	no	—
144	C₈×C₁₃	C₂×M₁₆	1	4	yes	complemented
145	#1055	—	2	2	no	—
146	#1055	C₂×M₁₆	1	4	yes	complemented
147	C₂²×M₁₆	—	13	2	no	—
148	C₂³×C₈	—	13	1	no	—
149	C₂³×C₈	#261	1	1	yes	characteristic
150	C₂²×C₄²	—	13	1	no	—
151	#431	—	13	4	no	—
152	#431	—	13	4	no	—
153	C₂×C₄⋊C₈	—	13	2	no	—
154	C₂×C₂²⋊C₈	—	13	2	no	—
155	C₄×M₁₆	—	13	4	no	—
156	C₄×M₁₆	—	13	4	no	—
157	C₂×#97	—	13	2	no	—
158	C₂×C₄×C₈	—	13	2	no	—
159	C₂×C₄×C₈	—	13	2	no	—
160	C₂×C₄×C₈	—	13	1	no	—
161	C₂×C₄×C₈	—	13	4	no	—
162	C₂×C₄×C₈	—	13	1	no	—
163	C₂×C₄×C₈	—	13	4	no	—
164	C₂×C₄×C₈	—	13	1	no	—
165	C₂×C₄×C₈	—	13	1	no	—
166	#322	—	13	8	no	—
167	C₈²	—	13	8	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	—	2	8	no	—
171	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
172	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
173	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
174	C₂²×C₁₃	#405	1	2	yes	—
175	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
178	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
179	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
180	Dic₁₃	—	2	2	no	—
181	Dic₁₃	C₂×C₄×C₈	1	2	yes	—
182	Dic₁₃	C₂×C₄×C₈	1	4	yes	—
183	Dic₁₃	#405	1	2	yes	—
184	Dic₁₃	C₂³×C₈	1	2	yes	—
185	Dic₁₃	C₂×#131	1	2	yes	—
186	C₂³×C₄	—	13	1	no	—
187	C₂³×C₄	—	13	1	no	—
188	C₂³×C₄	C₂×#261	1	1	yes	characteristic
189	C₂×M₁₆	—	13	8	no	—
190	C₂×M₁₆	—	13	16	no	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	—	13	4	no	—
193	C₂²×C₈	—	13	1	no	—
194	C₂²×C₈	—	13	4	no	—
195	C₂²×C₈	—	13	4	no	—
196	C₂²×C₈	—	13	1	no	—
197	C₂²×C₈	—	26	4	no	—
198	C₂²×C₈	C₂×#261	1	1	yes	characteristic
199	C₂²×C₈	C₂×#261	1	4	yes	—
200	C₂²×C₈	C₂×#261	1	1	yes	characteristic
201	C₂×C₄²	—	13	1	no	—
202	C₂×C₄²	—	13	2	no	—
203	C₂×C₄²	—	13	1	no	—
204	C₂×C₄²	—	13	2	no	—
205	C₂×C₄²	—	13	2	no	—
206	C₂×C₄²	—	13	1	no	—
207	C₂×C₄²	—	13	1	no	—
208	C₂×C₄²	—	13	2	no	—
209	C₄⋊C₈	—	13	8	no	—
210	C₂²⋊C₈	—	13	8	no	—
211	#97	—	13	8	no	—
212	C₄×C₈	—	13	4	no	—
213	C₄×C₈	—	13	4	no	—
214	C₄×C₈	—	13	4	no	—
215	C₄×C₈	—	13	4	no	—
216	C₄×C₈	—	13	2	no	—
217	C₄×C₈	—	13	2	no	—
218	C₄×C₈	—	13	4	no	—
219	C₄×C₈	—	13	2	no	—
220	C₄×C₈	—	13	2	no	—
221	C₄×C₈	—	13	4	no	—
222	C₂×C₁₃	—	2	4	no	—
223	C₂×C₁₃	C₈×M₁₆	1	4	yes	complemented
224	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
225	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
226	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
227	C₂⁴	C₄×#261	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	13	8	no	—
231	C₂²×C₄	—	13	1	no	—
232	C₂²×C₄	—	13	1	no	—
233	C₂²×C₄	—	13	2	no	—
234	C₂²×C₄	—	13	2	no	—
235	C₂²×C₄	—	13	1	no	—
236	C₂²×C₄	—	13	2	no	—
237	C₂²×C₄	—	13	2	no	—
238	C₂²×C₄	—	13	1	no	—
239	C₂²×C₄	—	26	4	no	—
240	C₂²×C₄	C₄×#261	1	1	yes	characteristic
241	C₂²×C₄	C₄×#261	1	2	yes	—
242	C₂²×C₄	C₂²×#261	1	2	yes	—
243	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
244	M₁₆	—	13	32	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	8	no	—
247	C₂×C₈	—	13	8	no	—
248	C₂×C₈	—	13	16	no	—
249	C₂×C₈	—	13	4	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	8	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	26	8	no	—
254	C₂×C₈	—	26	8	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	#6122	1	4	yes	direct product factor
257	C₂×C₈	C₂²×#261	1	2	yes	—
258	C₂×C₈	C₂²×#261	1	2	yes	—
259	C₄²	—	13	1	no	—
260	C₄²	—	13	1	no	—
261	C₄²	—	13	2	no	—
262	C₄²	—	13	2	no	—
263	C₄²	—	13	2	no	—
264	C₄²	—	13	2	no	—
265	C₄²	—	13	2	no	—
266	C₄²	—	13	1	no	—
267	C₄²	—	13	1	no	—
268	C₄²	—	13	2	no	—
269	C₁₃	C₂×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₈×#261	1	4	yes	—
273	C₂³	C₂×C₄×#261	1	2	yes	—
274	C₂³	C₂×C₄×#261	1	1	yes	characteristic
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	13	4	no	—
280	C₂×C₄	—	13	8	no	—
281	C₂×C₄	—	13	8	no	—
282	C₂×C₄	—	13	4	no	—
283	C₂×C₄	—	13	4	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	2	no	—
290	C₂×C₄	—	26	4	no	—
291	C₂×C₄	—	26	4	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
294	C₂×C₄	C₂×C₄×#261	1	2	yes	—
295	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
296	C₂×C₄	C₂×#6122	1	2	yes	—
297	C₂×C₄	C₂³×#261	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	13	16	no	—
300	C₈	—	13	4	no	—
301	C₈	—	26	8	no	—
302	C₈	—	26	2	no	—
303	C₈	C₂×#6122	1	4	yes	direct product factor
304	C₂²	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	C₂×C₈×#261	1	4	yes	—
308	C₂²	C₂×C₈×#261	1	2	yes	—
309	C₂²	#11719089	1	2	yes	—
310	C₂²	C₄×#6122	1	2	yes	—
311	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	13	2	no	—
314	C₄	—	13	2	no	—
315	C₄	—	13	4	no	—
316	C₄	—	13	2	no	—
317	C₄	—	13	2	no	—
318	C₄	—	26	2	no	—
319	C₄	C₄×#6122	1	2	yes	—
320	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
321	C₄	C₂²×#6122	1	1	yes	characteristic
322	C₂	—	2	4	no	—
323	C₂	C₈×#6122	1	4	yes	direct product factor
324	C₂	C₂²×C₈×#261	1	1	yes	characteristic
325	C₂	C₂×C₄×#6122	1	1	yes	characteristic
326	C₂	C₂×#11719089	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂³×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂⁴×C₁₃	C₄²
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂³×C₈	#261
hypocenter	C₁₃	C₂×C₈×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₈×#6122	C₂	direct (N × H)
4. C₂×#1057⋊C₈	C₂	semidirect
5. C₂×C₈×#1057	C₂	semidirect
26. #1057⋊C₈	C₂²	semidirect
27. C₈×#1057	C₂²	semidirect
31. C₂×#6122	C₈	direct (N × H)
82. #6122	C₂×C₈	direct (N × H)
84. C₂×#1057	C₂×C₈	semidirect
89. C₂×C₈×C₁₃	M₁₆	semidirect
104. C₂×#1055	M₁₆	semidirect
142. #1057	C₂²×C₈	semidirect
144. C₈×C₁₃	C₂×M₁₆	semidirect
146. #1055	C₂×M₁₆	semidirect
223. C₂×C₁₃	C₈×M₁₆	semidirect
256. C₂×C₈	#6122	direct (N × H)
269. C₁₃	C₂×C₈×M₁₆	semidirect
303. C₈	C₂×#6122	direct (N × H)
323. C₂	C₈×#6122	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×#11719125	2
C₂×C₄×#6122	2
C₈×#6122	2
C₂×#1057⋊C₈	2
C₂×C₈×#1057	2
C₂²×C₈×Dic₁₃	2
C₂×C₈×M₁₆	13

Subgroups — C2×C8×#6122 #16792570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×#6122 #16792570