Subgroups — C₂×#10590303 #11874892

Order 1152.

993 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64078	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#63573	C₂	1	4	yes	maximal
5	#10590303	C₂	1	8	yes	maximal, direct product factor
6	C₂×#5080	—	3	1	no	maximal
7	C₂×#4680	—	3	1	no	maximal
8	C₂²×#634	—	2	1	no	—
9	C₂²×C₃×Dic₆	—	2	1	no	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
12	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
13	#64078	—	2	2	no	—
14	#64078	C₂²	1	4	yes	—
15	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
16	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃×#869	—	2	2	no	—
19	C₃×#869	C₂²	1	4	yes	—
20	C₃×C₄×C₃⋊C₈	C₂²	1	4	yes	—
21	#63573	C₂²	1	16	yes	—
22	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
23	C₂×#869	—	3	1	no	—
24	C₂×#869	—	3	1	no	—
25	C₂×#869	—	6	1	no	—
26	C₂×#869	D₃	1	1	yes	characteristic
27	C₂×C₃×C₄×C₈	—	3	1	no	—
28	C₂×#817	—	3	4	no	—
29	#5080	—	3	8	no	—
30	C₂×#808	—	3	4	no	—
31	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
32	#4680	—	3	8	no	—
33	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
34	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
35	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₃⋊Dic₃	—	2	2	no	—
37	C₂×#634	—	2	4	no	—
38	C₂×#634	—	4	4	no	—
39	C₂²×C₃×Dic₃	—	2	2	no	—
40	C₂×C₃×Dic₆	—	2	4	no	—
41	C₂×C₃×Dic₆	—	4	4	no	—
42	C₃²×C₄²	D₄	1	2	yes	—
43	C₃²×C₄²	C₂³	1	2	yes	—
44	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
45	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
46	C₄⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
47	C₃×C₄⋊Dic₃	—	2	4	no	—
48	C₃×C₄⋊Dic₃	—	2	2	no	—
49	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
50	C₂×C₃×C₃⋊C₈	Q₈	1	8	yes	complemented
51	C₂×C₃×C₃⋊C₈	C₂³	1	4	yes	—
52	C₂×#498	—	9	1	no	Sylow
53	C₂²×C₃×Q₈	—	6	1	no	—
54	C₂²×Dic₆	—	2	1	no	—
55	C₂²×Dic₆	—	6	2	no	—
56	C₂²×Dic₆	—	6	1	no	—
57	C₂²×Dic₆	—	6	1	no	—
58	C₂²×C₃×C₈	—	3	2	no	—
59	C₃×C₄⋊Q₈	—	3	4	no	—
60	C₃×C₄⋊Q₈	—	6	2	no	—
61	C₂×C₃×C₄⋊C₄	—	3	2	no	—
62	C₂×C₃×C₄⋊C₄	—	6	1	no	—
63	C₂×C₃×C₄²	—	2	1	no	—
64	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
66	C₂×C₄⋊Dic₃	—	2	1	no	—
67	C₂×C₄⋊Dic₃	—	3	2	no	—
68	C₂×C₄⋊Dic₃	—	3	2	no	—
69	C₂×C₄⋊Dic₃	—	6	2	no	—
70	C₂×C₄⋊Dic₃	—	6	2	no	—
71	C₂×C₄⋊Dic₃	—	6	1	no	—
72	C₂×C₄⋊Dic₃	—	6	1	no	—
73	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
74	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
75	#869	—	2	2	no	—
76	#869	—	3	4	no	—
77	#869	—	3	4	no	—
78	#869	—	6	4	no	—
79	#869	—	6	4	no	—
80	#869	—	6	2	no	—
81	#869	—	6	2	no	—
82	#869	C₂×D₃	1	4	yes	—
83	C₃×C₄×C₈	—	3	4	no	—
84	#817	—	3	16	no	—
85	#808	—	3	16	no	—
86	C₄×C₃⋊C₈	C₂×D₃	1	4	yes	—
87	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
89	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
90	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
92	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
93	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
94	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
95	C₂×C₃⋊Dic₃	—	2	4	no	—
96	C₂×C₃⋊Dic₃	—	4	4	no	—
97	#634	—	4	8	no	—
98	C₂×C₃×Dic₃	—	2	4	no	—
99	C₂×C₃×Dic₃	—	4	4	no	—
100	C₃×Dic₆	—	4	8	no	—
101	C₃×C₃⋊C₈	C₂×Q₈	1	8	yes	complemented
102	C₂×C₄⋊Q₈	—	3	1	no	—
103	C₂×C₄⋊Q₈	—	9	1	no	—
104	#498	—	9	8	no	—
105	C₂×#106	—	9	4	no	—
106	C₂×C₄×C₈	—	3	1	no	—
107	C₂×C₃×Q₈	—	6	4	no	—
108	C₂×C₃×Q₈	—	12	4	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	—	6	2	no	—
113	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
114	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
115	C₂²×C₃×C₄	#67	1	1	yes	characteristic
116	C₂²×C₃×C₄	#67	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂²×Dic₃	—	2	2	no	—
120	C₂²×Dic₃	—	6	4	no	—
121	C₂²×Dic₃	—	6	2	no	—
122	C₂²×Dic₃	—	6	2	no	—
123	C₂×Dic₆	—	2	4	no	—
124	C₂×Dic₆	—	4	4	no	—
125	C₂×Dic₆	—	6	8	no	—
126	C₂×Dic₆	—	6	4	no	—
127	C₂×Dic₆	—	6	4	no	—
128	C₂×Dic₆	—	12	8	no	—
129	C₂×Dic₆	—	12	4	no	—
130	C₂×Dic₆	—	12	4	no	—
131	C₂×C₃×C₈	—	3	8	no	—
132	C₂×C₃×C₈	—	3	4	no	—
133	C₃×C₄⋊C₄	—	3	4	no	—
134	C₃×C₄⋊C₄	—	6	2	no	—
135	C₃×C₄⋊C₄	—	6	4	no	—
136	C₃×C₄²	—	2	2	no	—
137	C₃×C₄²	—	2	2	no	—
138	C₃×C₄²	D₁₂	1	2	yes	—
139	C₃×C₄²	#67	1	2	yes	—
140	C₃×C₄²	C₂²×D₃	1	2	yes	—
141	C₃×C₄²	C₂²×D₃	1	2	yes	—
142	C₄⋊Dic₃	—	2	2	no	—
143	C₄⋊Dic₃	—	2	4	no	—
144	C₄⋊Dic₃	—	3	4	no	—
145	C₄⋊Dic₃	—	3	4	no	—
146	C₄⋊Dic₃	—	6	4	no	—
147	C₄⋊Dic₃	—	6	4	no	—
148	C₄⋊Dic₃	—	6	8	no	—
149	C₄⋊Dic₃	—	6	4	no	—
150	C₄⋊Dic₃	—	6	2	no	—
151	C₄⋊Dic₃	—	6	4	no	—
152	C₄⋊Dic₃	—	6	2	no	—
153	C₄⋊Dic₃	C₂²×D₃	1	4	yes	—
154	C₂×C₃⋊C₈	Dic₆	1	8	yes	complemented
155	C₂×C₃⋊C₈	C₂²×D₃	1	4	yes	—
156	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
157	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
158	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
159	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
160	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
161	C₃²×C₄	C₂×SD₁₆	1	4	yes	—
162	C₃²×C₄	C₂×SD₁₆	1	4	yes	—
163	C₃²×C₄	C₂²×Q₈	1	2	yes	—
164	C₃⋊Dic₃	—	4	4	no	—
165	C₃×Dic₃	—	4	4	no	—
166	C₂²×Q₈	—	6	1	no	—
167	C₂²×Q₈	—	18	1	no	—
168	C₂²×C₈	—	3	2	no	—
169	C₄⋊Q₈	—	3	4	no	—
170	C₄⋊Q₈	—	6	2	no	—
171	C₄⋊Q₈	—	9	4	no	—
172	C₄⋊Q₈	—	18	2	no	—
173	C₂×C₄⋊C₄	—	3	2	no	—
174	C₂×C₄⋊C₄	—	6	1	no	—
175	C₂×C₄⋊C₄	—	9	2	no	—
176	C₂×C₄⋊C₄	—	18	1	no	—
177	C₂×C₄²	D₃²	1	1	yes	characteristic
178	#106	—	9	16	no	—
179	C₄×C₈	—	3	4	no	—
180	C₂³×C₃	—	2	1	no	—
181	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
182	C₂³×C₃	C₂×#67	1	1	yes	characteristic
183	C₃×Q₈	—	12	8	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	6	4	no	—
192	C₂×C₃×C₄	—	12	4	no	—
193	C₂×C₃×C₄	#204	1	4	yes	—
194	C₂×C₃×C₄	#204	1	4	yes	—
195	C₂×C₃×C₄	#214	1	4	yes	—
196	C₂×C₃×C₄	#215	1	4	yes	—
197	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
198	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
200	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
201	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
202	C₂×C₃×C₄	C₂×#67	1	2	yes	—
203	C₂×C₃×C₄	C₂×#67	1	2	yes	—
204	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
207	C₂×Dic₃	—	2	4	no	—
208	C₂×Dic₃	—	4	4	no	—
209	C₂×Dic₃	—	6	8	no	—
210	C₂×Dic₃	—	6	4	no	—
211	C₂×Dic₃	—	6	4	no	—
212	C₂×Dic₃	—	12	8	no	—
213	C₂×Dic₃	—	12	4	no	—
214	C₂×Dic₃	—	12	4	no	—
215	Dic₆	—	4	8	no	—
216	Dic₆	—	12	16	no	—
217	Dic₆	—	12	8	no	—
218	Dic₆	—	12	8	no	—
219	C₃×C₈	—	3	8	no	—
220	C₃⋊C₈	C₂×Dic₆	1	8	yes	complemented
221	C₂×C₃²	#498	1	4	yes	complemented
222	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
223	C₂×C₃²	C₂²×SD₁₆	1	1	yes	characteristic
224	C₂×C₃²	C₂²×SD₁₆	1	1	yes	characteristic
225	C₂×Q₈	—	6	4	no	—
226	C₂×Q₈	—	12	4	no	—
227	C₂×Q₈	—	18	4	no	—
228	C₂×Q₈	—	36	4	no	—
229	C₂²×C₄	—	6	2	no	—
230	C₂²×C₄	—	18	2	no	—
231	C₂²×C₄	#626	1	1	yes	characteristic
232	C₂²×C₄	#626	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
234	C₂×C₈	—	3	4	no	—
235	C₂×C₈	—	3	8	no	—
236	C₄⋊C₄	—	3	4	no	—
237	C₄⋊C₄	—	6	2	no	—
238	C₄⋊C₄	—	6	4	no	—
239	C₄⋊C₄	—	9	4	no	—
240	C₄⋊C₄	—	18	2	no	—
241	C₄⋊C₄	—	18	4	no	—
242	C₄²	#626	1	2	yes	—
243	C₄²	C₂×D₃²	1	2	yes	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	#869	1	2	yes	—
249	C₂²×C₃	C₂×#204	1	2	yes	—
250	C₂²×C₃	C₂×#204	1	2	yes	—
251	C₂²×C₃	C₂×#214	1	2	yes	—
252	C₂²×C₃	C₂×#215	1	2	yes	—
253	C₂²×C₃	#944	1	2	yes	—
254	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
255	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	4	no	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	12	4	no	—
261	C₃×C₄	#869	1	2	yes	—
262	C₃×C₄	C₂×#204	1	4	yes	—
263	C₃×C₄	C₂×#204	1	4	yes	—
264	C₃×C₄	C₂×#214	1	4	yes	—
265	C₃×C₄	C₂×#215	1	4	yes	—
266	C₃×C₄	#944	1	2	yes	—
267	C₃×C₄	C₂²×Dic₆	1	2	yes	—
268	C₃×C₄	C₂×D₃×Q₈	1	2	yes	—
269	Dic₃	—	4	4	no	—
270	Dic₃	—	12	8	no	—
271	Dic₃	—	12	4	no	—
272	Dic₃	—	12	4	no	—
273	C₃²	C₂×#498	1	1	yes	characteristic, complemented, Sylow
274	C₂³	C₂×#626	1	1	yes	characteristic
275	Q₈	—	12	8	no	—
276	Q₈	—	36	8	no	—
277	C₂×C₄	—	6	4	no	—
278	C₂×C₄	—	12	4	no	—
279	C₂×C₄	—	18	4	no	—
280	C₂×C₄	—	36	4	no	—
281	C₂×C₄	#3716	1	4	yes	—
282	C₂×C₄	#3717	1	4	yes	—
283	C₂×C₄	D₃×Dic₆	1	4	yes	—
284	C₂×C₄	C₂×#626	1	2	yes	—
285	C₂×C₄	C₂×#626	1	2	yes	—
286	C₂×C₄	C₂×#626	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₃²	1	1	yes	characteristic
288	C₈	—	3	8	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	4	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	#4680	1	4	yes	complemented
294	C₂×C₃	#5080	1	4	yes	complemented
295	C₂×C₃	C₂×#869	1	1	yes	characteristic
296	C₂×C₃	C₂²×#204	1	1	yes	characteristic
297	C₂×C₃	C₂²×#204	1	1	yes	characteristic
298	C₂×C₃	C₂²×#214	1	1	yes	characteristic
299	C₂×C₃	C₂²×#215	1	1	yes	characteristic
300	C₂×C₃	C₂×#944	1	1	yes	characteristic
301	C₂²	C₂×#3716	1	2	yes	—
302	C₂²	C₂×#3717	1	2	yes	—
303	C₂²	#63866	1	2	yes	—
304	C₂²	C₂²×#626	1	1	yes	characteristic
305	C₄	—	12	4	no	—
306	C₄	—	36	4	no	—
307	C₄	C₂×#3716	1	4	yes	—
308	C₄	C₂×#3717	1	4	yes	—
309	C₄	#63866	1	2	yes	—
310	C₄	C₂×D₃×Dic₆	1	2	yes	—
311	C₃	—	2	1	no	—
312	C₃	C₂×#4680	1	1	yes	characteristic, complemented
313	C₃	C₂×#5080	1	1	yes	characteristic, complemented
314	C₂	#10590303	1	4	yes	direct product factor
315	C₂	C₂²×#3716	1	1	yes	characteristic
316	C₂	C₂²×#3717	1	1	yes	characteristic
317	C₂	C₂×#63866	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#626
commutator, cosocle	C₂×C₃²×C₄	C₂⁴
Frattini	C₂×C₄	C₂²×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	C₂×#498
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. #10590303	C₂	direct (N × H)
50. C₂×C₃×C₃⋊C₈	Q₈	semidirect
101. C₃×C₃⋊C₈	C₂×Q₈	semidirect
154. C₂×C₃⋊C₈	Dic₆	semidirect
220. C₃⋊C₈	C₂×Dic₆	semidirect
221. C₂×C₃²	#498	semidirect
273. C₃²	C₂×#498	semidirect
293. C₂×C₃	#4680	semidirect
294. C₂×C₃	#5080	semidirect
312. C₃	C₂×#4680	semidirect
313. C₃	C₂×#5080	semidirect
314. C₂	#10590303	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×#64078	2
C₂×C₃×#869	2
C₂×C₃×C₄×C₃⋊C₈	2
C₂×#63573	2
#10590303	2
C₂×#5080	3
C₂×#4680	3

Subgroups — C2×#10590303 #11874892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10590303 #11874892