Subgroups — C₂×#10589095 #11776167

Order 1152.

581 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589095	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4546	—	3	1	no	maximal
6	C₂×#4481	—	3	1	no	maximal
7	C₂²×C₃×Dic₆	C₄	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊C₈)	—	2	1	no	—
10	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
11	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
12	C₂²×C₃×C₃⋊C₈	—	2	1	no	—
13	C₃×#869	—	2	1	no	—
14	C₃×#869	—	2	1	no	—
15	C₃×#869	C₄	1	2	yes	—
16	C₃×#869	C₂²	1	2	yes	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
18	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
19	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
22	C₂×#869	D₃	1	1	yes	characteristic
23	C₂×C₃×C₄⋊C₈	—	3	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
25	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
26	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
27	C₂×C₄⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
28	#4546	—	3	4	no	—
29	#4481	—	3	4	no	—
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	—	2	2	no	—
34	C₂×C₃×Dic₆	—	2	2	no	—
35	C₂×C₃×Dic₆	—	4	4	no	—
36	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
37	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₃²×C₄²	D₄	1	1	yes	characteristic
39	C₃²×C₄²	D₄	1	1	yes	characteristic
40	C₃²×C₄²	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	4	2	no	—
43	C₃×C₄⋊Dic₃	—	2	2	no	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄⋊Dic₃	—	2	2	no	—
46	C₃×C₄⋊Dic₃	—	2	2	no	—
47	C₂×C₃×C₃⋊C₈	—	2	2	no	—
48	C₂×C₃×C₃⋊C₈	—	4	2	no	—
49	C₂×#333	—	9	1	no	Sylow
50	C₂²×C₃×Q₈	—	3	2	no	—
51	C₂²×Dic₆	Dic₃	1	2	yes	—
52	C₂²×C₃×C₈	—	6	1	no	—
53	C₃×C₄⋊Q₈	—	3	2	no	—
54	C₃×C₄⋊Q₈	—	3	2	no	—
55	C₃×C₄⋊Q₈	—	6	1	no	—
56	C₃×C₄⋊Q₈	—	6	1	no	—
57	C₂×C₃×C₄⋊C₄	—	6	1	no	—
58	C₂×C₃×C₄⋊C₄	—	6	1	no	—
59	C₂×C₃×C₄²	—	2	1	no	—
60	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₂×C₄⋊Dic₃	—	2	1	no	—
63	C₂×C₄⋊Dic₃	—	2	1	no	—
64	C₂²×C₃⋊C₈	—	2	1	no	—
65	C₂²×C₃⋊C₈	—	6	2	no	—
66	C₂²×C₃⋊C₈	—	6	1	no	—
67	C₂²×C₃⋊C₈	—	6	1	no	—
68	#869	—	2	1	no	—
69	#869	—	2	1	no	—
70	#869	Dic₃	1	2	yes	—
71	#869	C₂×D₃	1	2	yes	—
72	C₃×C₄⋊C₈	—	3	2	no	—
73	C₃×C₄⋊C₈	—	6	1	no	—
74	C₄⋊(C₃⋊C₈)	—	2	1	no	—
75	C₄⋊(C₃⋊C₈)	—	3	2	no	—
76	C₄⋊(C₃⋊C₈)	—	3	2	no	—
77	C₄⋊(C₃⋊C₈)	—	6	2	no	—
78	C₄⋊(C₃⋊C₈)	—	6	2	no	—
79	C₄⋊(C₃⋊C₈)	—	6	1	no	—
80	C₄⋊(C₃⋊C₈)	—	6	1	no	—
81	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
82	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
88	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
89	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
90	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
91	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×Dic₃	—	2	4	no	—
95	C₂×C₃×Dic₃	—	4	4	no	—
96	C₃×Dic₆	—	4	4	no	—
97	C₃×Dic₆	—	4	4	no	—
98	C₃⋊(C₃⋊C₈)	—	4	2	no	—
99	C₃×C₃⋊C₈	—	4	2	no	—
100	C₂×C₄⋊Q₈	—	3	1	no	—
101	C₂×C₄⋊C₈	—	3	1	no	—
102	C₂×C₄⋊C₈	—	9	1	no	—
103	#333	—	9	4	no	—
104	C₂×C₃×Q₈	—	3	4	no	—
105	C₂×C₃×Q₈	—	6	2	no	—
106	C₂×C₃×Q₈	—	12	4	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	6	2	no	—
111	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	#67	1	1	yes	characteristic
115	C₂²×C₃×C₄	#67	1	1	yes	characteristic
116	C₂²×C₃×C₄	#67	1	1	yes	characteristic
117	C₂²×Dic₃	—	2	2	no	—
118	C₂×Dic₆	—	2	2	no	—
119	C₂×Dic₆	—	4	4	no	—
120	C₂×Dic₆	C₂×Dic₃	1	4	yes	—
121	C₂×C₃×C₈	—	6	2	no	—
122	C₂×C₃×C₈	—	12	2	no	—
123	C₃×C₄⋊C₄	—	6	2	no	—
124	C₃×C₄⋊C₄	—	6	2	no	—
125	C₃×C₄⋊C₄	—	6	2	no	—
126	C₃×C₄⋊C₄	—	6	2	no	—
127	C₃×C₄²	—	2	1	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
132	C₃×C₄²	D₁₂	1	1	yes	characteristic
133	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
134	C₃×C₄²	#67	1	1	yes	characteristic
135	C₃×C₄²	#67	1	1	yes	characteristic
136	C₃×C₄²	#67	1	1	yes	characteristic
137	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
138	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
139	C₄⋊Dic₃	—	2	2	no	—
140	C₄⋊Dic₃	—	2	2	no	—
141	C₄⋊Dic₃	—	2	2	no	—
142	C₄⋊Dic₃	—	2	2	no	—
143	C₂×C₃⋊C₈	—	2	2	no	—
144	C₂×C₃⋊C₈	—	4	2	no	—
145	C₂×C₃⋊C₈	—	6	4	no	—
146	C₂×C₃⋊C₈	—	6	2	no	—
147	C₂×C₃⋊C₈	—	6	2	no	—
148	C₂×C₃⋊C₈	—	12	4	no	—
149	C₂×C₃⋊C₈	—	12	2	no	—
150	C₂×C₃⋊C₈	—	12	2	no	—
151	C₂²×C₃²	#100	1	2	yes	—
152	C₂²×C₃²	#103	1	2	yes	—
153	C₂²×C₃²	#103	1	2	yes	—
154	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₃²×C₄	—	2	1	no	—
156	C₃²×C₄	—	2	1	no	—
157	C₃²×C₄	#103	1	2	yes	—
158	C₃²×C₄	#103	1	2	yes	—
159	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
160	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
161	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
162	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
163	C₃×Dic₃	—	4	4	no	—
164	C₂²×Q₈	—	3	2	no	—
165	C₂²×C₈	—	6	1	no	—
166	C₂²×C₈	—	18	1	no	—
167	C₄⋊Q₈	—	3	2	no	—
168	C₄⋊Q₈	—	3	2	no	—
169	C₄⋊Q₈	—	6	1	no	—
170	C₄⋊Q₈	—	6	1	no	—
171	C₂×C₄⋊C₄	—	6	1	no	—
172	C₂×C₄⋊C₄	—	6	1	no	—
173	C₂×C₄²	D₃²	1	1	yes	characteristic
174	C₄⋊C₈	—	3	2	no	—
175	C₄⋊C₈	—	6	1	no	—
176	C₄⋊C₈	—	9	2	no	—
177	C₄⋊C₈	—	18	1	no	—
178	C₂³×C₃	—	2	1	no	—
179	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂³×C₃	#217	1	1	yes	characteristic
181	C₃×Q₈	—	12	4	no	—
182	C₃×Q₈	—	12	4	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	4	2	no	—
196	C₂×C₃×C₄	—	6	4	no	—
197	C₂×C₃×C₄	—	12	4	no	—
198	C₂×C₃×C₄	#204	1	2	yes	—
199	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
200	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
201	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₄	#214	1	2	yes	—
204	C₂×C₃×C₄	#215	1	2	yes	—
205	C₂×C₃×C₄	#215	1	2	yes	—
206	C₂×C₃×C₄	#216	1	2	yes	—
207	C₂×C₃×C₄	#216	1	2	yes	—
208	C₂×C₃×C₄	#216	1	2	yes	—
209	C₂×C₃×C₄	#217	1	1	yes	characteristic
210	C₂×C₃×C₄	#217	1	1	yes	characteristic
211	C₂×C₃×C₄	#217	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
218	C₂×Dic₃	—	2	4	no	—
219	C₂×Dic₃	—	4	4	no	—
220	Dic₆	—	4	4	no	—
221	Dic₆	—	4	4	no	—
222	C₃×C₈	—	12	2	no	—
223	C₃⋊C₈	—	4	2	no	—
224	C₃⋊C₈	—	12	4	no	—
225	C₃⋊C₈	—	12	2	no	—
226	C₃⋊C₈	—	12	2	no	—
227	C₂×C₃²	#333	1	4	yes	complemented
228	C₂×C₃²	C₂×#100	1	1	yes	characteristic
229	C₂×C₃²	C₂×#103	1	1	yes	characteristic
230	C₂×C₃²	C₂×#103	1	1	yes	characteristic
231	C₂×Q₈	—	3	4	no	—
232	C₂×Q₈	—	6	2	no	—
233	C₂×Q₈	—	12	4	no	—
234	C₂²×C₄	—	6	2	no	—
235	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
236	C₂²×C₄	#625	1	1	yes	characteristic
237	C₂²×C₄	#626	1	1	yes	characteristic
238	C₂×C₈	—	6	2	no	—
239	C₂×C₈	—	12	2	no	—
240	C₂×C₈	—	18	2	no	—
241	C₂×C₈	—	36	2	no	—
242	C₄⋊C₄	—	6	2	no	—
243	C₄⋊C₄	—	6	2	no	—
244	C₄⋊C₄	—	6	2	no	—
245	C₄⋊C₄	—	6	2	no	—
246	C₄²	D₃×Dic₃	1	1	yes	characteristic
247	C₄²	#625	1	1	yes	characteristic
248	C₄²	#626	1	1	yes	characteristic
249	C₄²	C₂×D₃²	1	1	yes	characteristic
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	#810	1	2	yes	—
255	C₂²×C₃	#816	1	2	yes	—
256	C₂²×C₃	#823	1	2	yes	—
257	C₂²×C₃	#833	1	2	yes	—
258	C₂²×C₃	#835	1	2	yes	—
259	C₂²×C₃	#835	1	2	yes	—
260	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
261	C₂²×C₃	C₂×#217	1	1	yes	characteristic
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	4	1	no	—
273	C₃×C₄	—	4	1	no	—
274	C₃×C₄	—	12	4	no	—
275	C₃×C₄	#810	1	2	yes	—
276	C₃×C₄	#816	1	2	yes	—
277	C₃×C₄	#835	1	2	yes	—
278	C₃×C₄	#835	1	2	yes	—
279	C₃×C₄	C₂×#204	1	1	yes	characteristic
280	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
281	C₃×C₄	C₂×#214	1	1	yes	characteristic
282	C₃×C₄	C₂×#215	1	1	yes	characteristic
283	C₃×C₄	C₂×#215	1	1	yes	characteristic
284	C₃×C₄	C₂×#216	1	1	yes	characteristic
285	C₃×C₄	C₂×#216	1	1	yes	characteristic
286	C₃×C₄	C₂×#216	1	1	yes	characteristic
287	Dic₃	—	4	4	no	—
288	C₃²	C₂×#333	1	1	yes	characteristic, complemented, Sylow
289	C₂³	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
290	Q₈	—	12	4	no	—
291	Q₈	—	12	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	6	4	no	—
294	C₂×C₄	—	12	4	no	—
295	C₂×C₄	#3715	1	2	yes	—
296	C₂×C₄	#3717	1	2	yes	—
297	C₂×C₄	#3718	1	2	yes	—
298	C₂×C₄	#3719	1	2	yes	—
299	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
300	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
301	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
302	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
303	C₂×C₄	C₂×#625	1	1	yes	characteristic
304	C₂×C₄	C₂×#626	1	1	yes	characteristic
305	C₈	—	12	2	no	—
306	C₈	—	36	2	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	4	no	—
311	C₂×C₃	#4481	1	4	yes	complemented
312	C₂×C₃	#4546	1	4	yes	complemented
313	C₂×C₃	C₂×#810	1	1	yes	characteristic
314	C₂×C₃	C₂×#816	1	1	yes	characteristic
315	C₂×C₃	C₂×#823	1	1	yes	characteristic
316	C₂×C₃	C₂×#833	1	1	yes	characteristic
317	C₂×C₃	C₂×#835	1	1	yes	characteristic
318	C₂×C₃	C₂×#835	1	1	yes	characteristic
319	C₂²	#63558	1	2	yes	—
320	C₂²	#63565	1	2	yes	—
321	C₂²	#63566	1	2	yes	—
322	C₂²	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	12	4	no	—
326	C₄	#63565	1	2	yes	—
327	C₄	#63566	1	2	yes	—
328	C₄	C₂×#3715	1	1	yes	characteristic
329	C₄	C₂×#3717	1	1	yes	characteristic
330	C₄	C₂×#3718	1	1	yes	characteristic
331	C₄	C₂×#3719	1	1	yes	characteristic
332	C₃	—	2	1	no	—
333	C₃	C₂×#4481	1	1	yes	characteristic, complemented
334	C₃	C₂×#4546	1	1	yes	characteristic, complemented
335	C₂	#10589095	1	4	yes	direct product factor
336	C₂	C₂×#63558	1	1	yes	characteristic
337	C₂	C₂×#63565	1	1	yes	characteristic
338	C₂	C₂×#63566	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×D₃)⋊Dic₃
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂²⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	C₂×#333
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #10589095	C₂	direct (N × H)
227. C₂×C₃²	#333	semidirect
288. C₃²	C₂×#333	semidirect
311. C₂×C₃	#4481	semidirect
312. C₂×C₃	#4546	semidirect
333. C₃	C₂×#4481	semidirect
334. C₃	C₂×#4546	semidirect
335. C₂	#10589095	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×C₃×#869	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
#10589095	2
C₂×#4546	3
C₂×#4481	3

Subgroups — C2×#10589095 #11776167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589095 #11776167