Subgroups — C₂×#10589126 #11796891

Order 1152.

557 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#63639	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic, complemented
4	#10589126	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4563	—	3	1	no	maximal
6	C₂×#4495	—	3	1	no	maximal
7	C₂²×C₃×D₁₂	—	2	1	no	—
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×#208	—	2	1	no	—
10	C₂×C₃×D₂₄	—	2	1	no	—
11	C₂×C₃×D₂₄	—	2	1	no	—
12	C₂×C₃×D₂₄	—	2	1	no	—
13	C₂×C₃×D₂₄	C₄	1	2	yes	—
14	C₂×C₃×D₂₄	C₄	1	2	yes	—
15	C₂×C₃×D₂₄	C₂²	1	2	yes	—
16	#63639	—	2	1	no	—
17	#63639	C₂²	1	2	yes	complemented
18	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
19	C₃×#819	—	2	1	no	—
20	C₃×#819	C₂²	1	2	yes	complemented
21	C₂²×C₃×D₈	—	3	1	no	—
22	C₂²×D₂₄	D₃	1	1	yes	characteristic
23	C₂×C₃×#108	—	3	1	no	—
24	C₂×#819	D₃	1	1	yes	characteristic, complemented
25	C₂×#812	—	3	1	no	—
26	C₂×#812	—	3	1	no	—
27	C₂×#812	—	6	1	no	—
28	#4563	—	3	4	no	—
29	#4495	—	3	4	no	—
30	C₂³×C₃×D₃	—	4	1	no	—
31	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃×D₁₂	—	2	2	no	—
33	C₂×C₃×D₁₂	—	2	2	no	—
34	C₂×C₃×D₁₂	—	2	2	no	—
35	C₂×C₃×D₁₂	—	4	1	no	—
36	C₂×C₃×D₁₂	—	4	1	no	—
37	C₂×C₃×D₁₂	—	4	1	no	—
38	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
42	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
44	C₃×#208	—	2	2	no	—
45	C₃×#208	—	4	1	no	—
46	C₂×C₃×C₃⋊C₈	—	4	1	no	—
47	C₃×D₂₄	—	2	2	no	—
48	C₃×D₂₄	—	2	2	no	—
49	C₃×D₂₄	—	2	2	no	—
50	C₃×D₂₄	C₂×C₄	1	4	yes	—
51	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
52	C₂×#361	—	9	1	no	Sylow
53	C₂²×C₃×D₄	—	6	1	no	—
54	C₂²×D₁₂	—	2	1	no	—
55	C₂×C₃×D₈	—	3	2	no	—
56	C₂×C₃×D₈	—	3	2	no	—
57	C₂×C₃×D₈	—	3	2	no	—
58	C₂×C₃×D₈	—	6	1	no	—
59	C₂×C₃×D₈	—	6	1	no	—
60	C₂×C₃×D₈	—	6	1	no	—
61	C₂×C₃×M₁₆	—	6	1	no	—
62	C₂²×C₃×C₈	—	2	1	no	—
63	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂×#208	—	2	1	no	—
66	C₂×D₂₄	—	2	1	no	—
67	C₂×D₂₄	—	2	1	no	—
68	C₂×D₂₄	—	2	1	no	—
69	C₂×D₂₄	Dic₃	1	2	yes	—
70	C₂×D₂₄	Dic₃	1	2	yes	—
71	C₂×D₂₄	C₂×D₃	1	2	yes	—
72	C₃×#108	—	3	2	no	—
73	C₃×#108	—	6	1	no	—
74	#819	—	2	1	no	—
75	#819	C₂×D₃	1	2	yes	complemented
76	#812	—	3	2	no	—
77	#812	—	3	2	no	—
78	#812	—	6	2	no	—
79	#812	—	6	2	no	—
80	#812	—	6	1	no	—
81	#812	—	6	1	no	—
82	C₂×C₃⋊C₁₆	—	6	2	no	—
83	C₂×C₃⋊C₁₆	—	6	1	no	—
84	C₂×C₃⋊C₁₆	—	6	1	no	—
85	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
86	C₂²×C₃×D₃	—	4	2	no	—
87	C₂²×C₃×D₃	—	4	2	no	—
88	C₂²×C₃×D₃	—	4	2	no	—
89	C₂²×C₃×D₃	—	8	4	no	—
90	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
93	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
94	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₃×D₁₂	—	2	4	no	—
97	C₃×D₁₂	—	4	2	no	—
98	C₃×D₁₂	—	4	2	no	—
99	C₃×D₁₂	—	4	2	no	—
100	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
104	C₃×C₃⋊C₈	—	4	2	no	—
105	C₂²×D₈	—	3	1	no	—
106	C₂×M₃₂	—	9	1	no	—
107	C₂×#108	—	3	1	no	—
108	#361	—	9	4	no	—
109	C₂⁴×C₃	—	12	1	no	—
110	C₂³×D₃	—	4	1	no	—
111	C₂×C₃×D₄	—	6	2	no	—
112	C₂×C₃×D₄	—	6	2	no	—
113	C₂×C₃×D₄	—	6	2	no	—
114	C₂×C₃×D₄	—	12	1	no	—
115	C₂×C₃×D₄	—	12	1	no	—
116	C₂×C₃×D₄	—	12	1	no	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	#67	1	1	yes	characteristic
119	C₂²×C₃×C₄	#67	1	1	yes	characteristic
120	C₂×D₁₂	—	2	2	no	—
121	C₂×D₁₂	—	2	2	no	—
122	C₂×D₁₂	—	2	2	no	—
123	C₂×D₁₂	—	4	1	no	—
124	C₂×D₁₂	—	4	1	no	—
125	C₂×D₁₂	—	4	1	no	—
126	C₃×D₈	—	3	4	no	—
127	C₃×D₈	—	6	2	no	—
128	C₃×D₈	—	6	2	no	—
129	C₃×D₈	—	6	2	no	—
130	C₃×M₁₆	—	6	2	no	—
131	C₃×M₁₆	—	12	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	—	2	1	no	—
138	C₂×C₃×C₈	—	12	1	no	—
139	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
140	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
141	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
142	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
145	C₂×C₃×C₈	#67	1	1	yes	characteristic
146	C₂×C₃×C₈	#67	1	1	yes	characteristic
147	C₂×C₃×C₈	#67	1	1	yes	characteristic
148	C₂×C₃×C₈	#67	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
151	#208	—	2	2	no	—
152	#208	—	4	1	no	—
153	C₂×C₃⋊C₈	—	4	1	no	—
154	D₂₄	—	2	2	no	—
155	D₂₄	—	2	2	no	—
156	D₂₄	—	2	2	no	—
157	D₂₄	C₂×Dic₃	1	4	yes	—
158	C₃⋊C₁₆	—	6	4	no	—
159	C₃⋊C₁₆	—	6	2	no	—
160	C₃⋊C₁₆	—	6	2	no	—
161	C₂²×C₃²	—	2	2	no	—
162	C₂²×C₃²	#102	1	1	yes	characteristic
163	C₂²×C₃²	#102	1	1	yes	characteristic
164	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
165	C₂×C₃×D₃	—	4	4	no	—
166	C₂×C₃×D₃	—	8	4	no	—
167	C₂×C₃×D₃	—	8	4	no	—
168	C₂×C₃×D₃	—	8	4	no	—
169	C₃²×C₄	#102	1	1	yes	characteristic
170	C₃²×C₄	#102	1	1	yes	characteristic
171	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
173	C₂²×D₄	—	6	1	no	—
174	C₂×D₈	—	3	2	no	—
175	C₂×D₈	—	3	2	no	—
176	C₂×D₈	—	3	2	no	—
177	C₂×D₈	—	6	1	no	—
178	C₂×D₈	—	6	1	no	—
179	C₂×D₈	—	6	1	no	—
180	C₂×M₁₆	—	6	1	no	—
181	C₂²×C₈	D₃²	1	1	yes	characteristic
182	M₃₂	—	9	2	no	—
183	M₃₂	—	18	1	no	—
184	C₂×C₁₆	—	18	1	no	—
185	#108	—	3	2	no	—
186	#108	—	6	1	no	—
187	C₂³×C₃	—	2	1	no	—
188	C₂³×C₃	—	12	2	no	—
189	C₂³×C₃	—	12	2	no	—
190	C₂³×C₃	—	12	2	no	—
191	C₂³×C₃	—	24	4	no	—
192	C₂³×C₃	#214	1	1	yes	characteristic
193	C₂³×C₃	#215	1	1	yes	characteristic
194	C₂²×D₃	—	4	2	no	—
195	C₂²×D₃	—	4	2	no	—
196	C₂²×D₃	—	4	2	no	—
197	C₂²×D₃	—	8	4	no	—
198	C₃×D₄	—	6	4	no	—
199	C₃×D₄	—	12	2	no	—
200	C₃×D₄	—	12	2	no	—
201	C₃×D₄	—	12	2	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
209	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
210	C₂×C₃×C₄	#213	1	1	yes	characteristic
211	C₂×C₃×C₄	#213	1	1	yes	characteristic
212	C₂×C₃×C₄	#213	1	1	yes	characteristic
213	C₂×C₃×C₄	#213	1	1	yes	characteristic
214	C₂×C₃×C₄	#214	1	1	yes	characteristic
215	C₂×C₃×C₄	#215	1	1	yes	characteristic
216	C₂×C₃×C₄	#217	1	1	yes	characteristic
217	C₂×C₃×C₄	#217	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
220	D₁₂	—	2	4	no	—
221	D₁₂	—	4	2	no	—
222	D₁₂	—	4	2	no	—
223	D₁₂	—	4	2	no	—
224	C₃×C₈	—	2	1	no	—
225	C₃×C₈	—	2	1	no	—
226	C₃×C₈	—	2	1	no	—
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	12	2	no	—
229	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
230	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
231	C₃×C₈	#217	1	1	yes	characteristic
232	C₃×C₈	#217	1	1	yes	characteristic
233	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
234	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
235	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
236	C₃×C₈	C₂×#67	1	1	yes	characteristic
237	C₃⋊C₈	—	4	2	no	—
238	C₂×C₃²	—	2	1	no	—
239	C₂×C₃²	—	2	1	no	—
240	C₂×C₃²	#361	1	2	yes	complemented
241	C₂×C₃²	C₂×#102	1	1	yes	characteristic
242	C₃×D₃	—	8	4	no	—
243	C₂⁴	—	12	1	no	—
244	C₂×D₄	—	6	2	no	—
245	C₂×D₄	—	6	2	no	—
246	C₂×D₄	—	6	2	no	—
247	C₂×D₄	—	12	1	no	—
248	C₂×D₄	—	12	1	no	—
249	C₂×D₄	—	12	1	no	—
250	C₂²×C₄	#625	1	1	yes	characteristic
251	D₈	—	3	4	no	—
252	D₈	—	6	2	no	—
253	D₈	—	6	2	no	—
254	D₈	—	6	2	no	—
255	M₁₆	—	6	2	no	—
256	M₁₆	—	12	1	no	—
257	C₂×C₈	—	12	1	no	—
258	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
259	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
260	C₂×C₈	#625	1	1	yes	characteristic
261	C₂×C₈	#626	1	1	yes	characteristic
262	C₂×C₈	#626	1	1	yes	characteristic
263	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
264	C₁₆	—	18	2	no	—
265	C₂²×C₃	—	2	2	no	—
266	C₂²×C₃	—	2	2	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	4	2	no	—
271	C₂²×C₃	—	12	4	no	—
272	C₂²×C₃	—	24	4	no	—
273	C₂²×C₃	—	24	4	no	—
274	C₂²×C₃	—	24	4	no	—
275	C₂²×C₃	#809	1	1	yes	characteristic
276	C₂²×C₃	#809	1	1	yes	characteristic
277	C₂²×C₃	#832	1	1	yes	characteristic
278	C₂²×C₃	#832	1	1	yes	characteristic
279	C₂²×C₃	C₂×#214	1	1	yes	characteristic
280	C₂²×C₃	C₂×#215	1	1	yes	characteristic
281	C₂×D₃	—	4	4	no	—
282	C₂×D₃	—	8	4	no	—
283	C₂×D₃	—	8	4	no	—
284	C₂×D₃	—	8	4	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	#809	1	1	yes	characteristic
290	C₃×C₄	#809	1	1	yes	characteristic
291	C₃×C₄	#832	1	1	yes	characteristic
292	C₃×C₄	#832	1	1	yes	characteristic
293	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
294	C₃×C₄	C₂×#213	1	1	yes	characteristic
295	C₃×C₄	C₂×#213	1	1	yes	characteristic
296	C₃×C₄	C₂×#217	1	1	yes	characteristic
297	C₃²	C₂×#361	1	1	yes	characteristic, complemented, Sylow
298	C₂³	—	12	2	no	—
299	C₂³	—	12	2	no	—
300	C₂³	—	12	2	no	—
301	C₂³	—	24	4	no	—
302	C₂³	#3715	1	1	yes	characteristic
303	D₄	—	6	4	no	—
304	D₄	—	12	2	no	—
305	D₄	—	12	2	no	—
306	D₄	—	12	2	no	—
307	C₂×C₄	#3713	1	1	yes	characteristic
308	C₂×C₄	#3713	1	1	yes	characteristic
309	C₂×C₄	#3715	1	1	yes	characteristic
310	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
311	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
312	C₂×C₄	C₂×#625	1	1	yes	characteristic
313	C₈	—	12	2	no	—
314	C₈	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
315	C₈	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
316	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
317	C₈	C₂×#626	1	1	yes	characteristic
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	—	4	1	no	—
325	C₂×C₃	—	4	1	no	—
326	C₂×C₃	—	24	4	no	—
327	C₂×C₃	#4495	1	2	yes	complemented
328	C₂×C₃	#4563	1	2	yes	complemented
329	C₂×C₃	C₂×#809	1	1	yes	characteristic
330	C₂×C₃	C₂×#832	1	1	yes	characteristic
331	D₃	—	8	4	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	12	4	no	—
334	C₂²	—	24	4	no	—
335	C₂²	—	24	4	no	—
336	C₂²	—	24	4	no	—
337	C₂²	#63562	1	1	yes	characteristic
338	C₂²	#63562	1	1	yes	characteristic
339	C₂²	C₂×#3715	1	1	yes	characteristic
340	C₄	#63562	1	1	yes	characteristic
341	C₄	#63562	1	1	yes	characteristic
342	C₄	C₂×#3713	1	1	yes	characteristic
343	C₄	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
344	C₃	—	2	1	no	—
345	C₃	C₂×#4495	1	1	yes	characteristic, complemented
346	C₃	C₂×#4563	1	1	yes	characteristic, complemented
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	24	4	no	—
350	C₂	#10589126	1	2	yes	direct product factor
351	C₂	C₂×#63562	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63562
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	#102
prime radical	C₂²×C₃×D₂₄	C₂
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	C₂×#361
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#63639	C₂	semidirect
3. C₂×C₃×#819	C₂	semidirect
4. #10589126	C₂	direct (N × H)
17. #63639	C₂²	semidirect
20. C₃×#819	C₂²	semidirect
24. C₂×#819	D₃	semidirect
75. #819	C₂×D₃	semidirect
240. C₂×C₃²	#361	semidirect
297. C₃²	C₂×#361	semidirect
327. C₂×C₃	#4495	semidirect
328. C₂×C₃	#4563	semidirect
345. C₃	C₂×#4495	semidirect
346. C₃	C₂×#4563	semidirect
350. C₂	#10589126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₂₄	2
C₂×#63639	2
C₂×C₃×#819	2
#10589126	2
C₂×#4563	3
C₂×#4495	3

Subgroups — C2×#10589126 #11796891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589126 #11796891