Subgroups — #11791189

Order 1152.

681 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#904	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63568	C₂	1	2	yes	maximal, complemented
5	#10591071	C₂	1	2	yes	maximal, complemented
6	#72084	—	3	1	no	maximal
7	#70696	—	3	1	no	maximal
8	C₂×C₃×#235	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₃×#208	C₂²	1	2	yes	—
13	C₂×C₃×Dic₁₂	C₄	1	1	yes	characteristic, complemented
14	C₃×#904	—	2	2	no	—
15	C₃×#904	—	2	2	no	—
16	C₂×C₃×D₂₄	C₄	1	1	yes	characteristic, complemented
17	C₂×C₃×#204	C₄	1	2	yes	complemented
18	C₈×C₃⋊Dic₃	—	2	2	no	—
19	C₃×#819	—	2	2	no	—
20	#63568	—	2	4	no	—
21	(C₂×D₃)⋊(C₃⋊C₈)	—	2	4	no	—
22	C₂×C₃×#135	—	3	1	no	—
23	C₂×#904	D₃	1	1	yes	characteristic
24	C₂×C₃×#108	—	3	1	no	—
25	C₂×#837	—	3	2	no	—
26	#5234	—	3	2	no	—
27	C₂×#825	—	3	2	no	—
28	#5129	—	3	2	no	—
29	C₂×#819	D₃	1	1	yes	characteristic, complemented
30	C₂×C₈×Dic₃	—	3	1	no	—
31	C₂×C₈×Dic₃	—	3	1	no	—
32	C₂×C₈×Dic₃	—	6	1	no	—
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃⋊Dic₃	—	2	1	no	—
35	C₂×C₃×#67	—	2	2	no	—
36	C₃×#235	—	2	2	no	—
37	C₃×#235	—	2	2	no	—
38	C₃×#235	—	2	2	no	—
39	C₃×#235	—	2	2	no	—
40	C₂×C₃×D₁₂	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₄×D₃	—	2	2	no	—
42	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	D₄	1	2	yes	—
46	C₂×C₃²×C₈	D₄	1	2	yes	—
47	C₄×C₃⋊Dic₃	—	2	1	no	—
48	C₄×C₃⋊Dic₃	D₄	1	2	yes	—
49	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
51	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
52	C₃×#208	—	2	2	no	—
53	C₃×#208	—	2	2	no	—
54	C₂×C₃×C₃⋊C₈	—	2	2	no	—
55	C₃×Dic₁₂	—	2	2	no	—
56	C₃×D₂₄	—	2	2	no	—
57	C₃×#204	—	2	4	no	—
58	#1974	—	9	1	no	Sylow
59	C₂×C₃×#41	—	3	2	no	—
60	C₂×#235	C₂×D₃	1	2	yes	—
61	C₃×#135	—	6	2	no	—
62	C₃×#135	—	6	2	no	—
63	C₂×C₃×Q₁₆	—	3	1	no	—
64	C₂×C₃×SD₁₆	—	3	2	no	—
65	C₂×C₃×D₈	—	3	1	no	—
66	C₂×C₃×M₁₆	—	3	2	no	—
67	C₂²×C₃×C₈	—	2	1	no	—
68	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
70	C₂×C₄×Dic₃	—	3	1	no	—
71	C₂×C₄×Dic₃	—	3	1	no	—
72	C₂×C₄×Dic₃	—	6	1	no	—
73	C₂×#208	C₂×D₃	1	2	yes	—
74	C₂²×C₃⋊C₈	—	3	1	no	—
75	C₂²×C₃⋊C₈	—	3	1	no	—
76	C₂²×C₃⋊C₈	—	6	1	no	—
77	C₂×Dic₁₂	Dic₃	1	1	yes	characteristic, complemented
78	#904	—	2	2	no	—
79	#904	—	2	2	no	—
80	C₂×D₂₄	Dic₃	1	1	yes	characteristic, complemented
81	C₂×#204	Dic₃	1	2	yes	complemented
82	C₃×#108	—	6	2	no	—
83	#837	—	6	4	no	—
84	#830	—	6	4	no	—
85	#825	—	6	4	no	—
86	(C₂×D₃)⋊C₈	—	6	4	no	—
87	#819	—	2	2	no	—
88	C₈×Dic₃	—	6	4	no	—
89	C₈×Dic₃	—	6	2	no	—
90	C₈×Dic₃	—	6	2	no	—
91	C₂³×C₃²	#41	1	1	yes	characteristic
92	C₂²×C₃×D₃	—	2	2	no	—
93	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃²×C₄	#41	1	1	yes	characteristic
99	C₂×C₃⋊Dic₃	—	2	4	no	—
100	C₂×C₃⋊Dic₃	—	2	2	no	—
101	C₃×#67	—	4	2	no	—
102	C₃×#67	—	4	2	no	—
103	C₂×C₃×Dic₃	—	2	2	no	—
104	C₃×D₁₂	—	2	2	no	—
105	C₃×D₁₂	—	2	2	no	—
106	C₃×C₄×D₃	—	4	2	no	—
107	C₃×C₄×D₃	—	4	2	no	—
108	C₃×Dic₆	—	2	2	no	—
109	C₃×Dic₆	—	2	2	no	—
110	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
111	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
112	C₃⋊(C₃⋊C₈)	—	2	2	no	—
113	C₃×C₃⋊C₈	—	4	2	no	—
114	C₂×#135	—	3	1	no	—
115	C₂×#108	—	3	1	no	—
116	C₂×#104	—	9	2	no	—
117	#408	—	9	2	no	—
118	C₂×C₄×C₈	—	9	1	no	—
119	C₃×#41	—	6	2	no	—
120	C₃×#41	—	6	2	no	—
121	C₃×#41	—	6	2	no	—
122	C₃×#41	—	6	2	no	—
123	C₂×C₃×Q₈	—	3	2	no	—
124	C₂×C₃×D₄	—	3	2	no	—
125	C₂×C₃×D₄	—	6	2	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	6	2	no	—
128	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
130	C₂×#67	—	2	2	no	—
131	C₂²×Dic₃	—	6	2	no	—
132	C₂²×Dic₃	—	6	1	no	—
133	C₂²×Dic₃	—	6	1	no	—
134	#235	—	2	2	no	—
135	#235	—	2	2	no	—
136	#235	—	2	2	no	—
137	#235	—	2	2	no	—
138	C₂×D₁₂	C₂×Dic₃	1	2	yes	—
139	C₂×C₄×D₃	—	2	2	no	—
140	C₂×Dic₆	C₂×Dic₃	1	2	yes	—
141	C₃×Q₁₆	—	6	2	no	—
142	C₃×SD₁₆	—	6	4	no	—
143	C₃×D₈	—	6	2	no	—
144	C₃×M₁₆	—	6	2	no	—
145	C₃×M₁₆	—	6	2	no	—
146	C₂×C₃×C₈	—	2	1	no	—
147	C₂×C₃×C₈	—	2	1	no	—
148	C₂×C₃×C₈	—	2	2	no	—
149	C₂×C₃×C₈	—	2	2	no	—
150	C₂×C₃×C₈	—	6	2	no	—
151	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
152	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
153	C₂×C₃×C₈	D₁₂	1	2	yes	—
154	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
155	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
156	C₂×C₃×C₈	#67	1	2	yes	—
157	C₂×C₃×C₈	#67	1	2	yes	—
158	C₂×C₃×C₈	#67	1	2	yes	—
159	C₄×Dic₃	—	3	2	no	—
160	C₄×Dic₃	—	3	2	no	—
161	C₄×Dic₃	—	6	2	no	—
162	C₄×Dic₃	—	6	2	no	—
163	C₄×Dic₃	—	6	1	no	—
164	C₄×Dic₃	—	6	1	no	—
165	#208	—	2	2	no	—
166	#208	—	2	2	no	—
167	C₂×C₃⋊C₈	—	2	2	no	—
168	C₂×C₃⋊C₈	—	3	2	no	—
169	C₂×C₃⋊C₈	—	3	2	no	—
170	C₂×C₃⋊C₈	—	6	2	no	—
171	C₂×C₃⋊C₈	—	6	2	no	—
172	C₂×C₃⋊C₈	—	6	2	no	—
173	C₂×C₃⋊C₈	—	6	1	no	—
174	C₂×C₃⋊C₈	—	6	1	no	—
175	C₂×C₃⋊C₈	—	6	1	no	—
176	C₂×C₃⋊C₈	—	6	1	no	—
177	Dic₁₂	—	2	2	no	—
178	D₂₄	—	2	2	no	—
179	#204	—	2	4	no	—
180	C₂²×C₃²	—	2	2	no	—
181	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
182	C₂²×C₃²	#121	1	1	yes	characteristic
183	C₂²×C₃²	#124	1	1	yes	characteristic
184	C₂×C₃×D₃	—	4	2	no	—
185	C₂×C₃×D₃	—	4	4	no	—
186	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
188	C₃²×C₄	#121	1	1	yes	characteristic
189	C₃²×C₄	#127	1	1	yes	characteristic
190	C₃⋊Dic₃	—	2	4	no	—
191	C₃×Dic₃	—	4	2	no	—
192	C₂×#41	—	3	2	no	—
193	#135	—	6	2	no	—
194	#135	—	6	2	no	—
195	C₂×Q₁₆	—	3	1	no	—
196	C₂×SD₁₆	—	3	2	no	—
197	C₂×D₈	—	3	1	no	—
198	C₂×M₁₆	—	3	2	no	—
199	C₂²×C₈	—	9	1	no	—
200	C₂²×C₈	D₃²	1	1	yes	characteristic
201	C₂×C₄²	—	9	1	no	—
202	#108	—	6	2	no	—
203	#104	—	18	4	no	—
204	C₂²⋊C₈	—	18	4	no	—
205	C₄×C₈	—	18	2	no	—
206	C₂³×C₃	—	2	1	no	—
207	C₂³×C₃	—	6	2	no	—
208	C₂³×C₃	#237	1	1	yes	characteristic
209	C₂³×C₃	#239	1	1	yes	characteristic
210	C₂²×D₃	—	2	2	no	—
211	C₃×Q₈	—	6	2	no	—
212	C₃×Q₈	—	6	2	no	—
213	C₃×D₄	—	6	2	no	—
214	C₃×D₄	—	6	2	no	—
215	C₃×D₄	—	12	2	no	—
216	C₃×D₄	—	12	2	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	6	2	no	—
224	C₂×C₃×C₄	—	12	2	no	—
225	C₂×C₃×C₄	—	12	2	no	—
226	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
228	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
230	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	#237	1	1	yes	characteristic
233	C₂×C₃×C₄	#239	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
238	#67	—	4	2	no	—
239	#67	—	4	2	no	—
240	C₂×Dic₃	—	2	2	no	—
241	C₂×Dic₃	—	6	4	no	—
242	C₂×Dic₃	—	6	8	no	—
243	C₂×Dic₃	—	6	4	no	—
244	C₂×Dic₃	—	6	4	no	—
245	C₂×Dic₃	—	6	2	no	—
246	C₂×Dic₃	—	6	2	no	—
247	D₁₂	—	2	2	no	—
248	D₁₂	—	2	2	no	—
249	C₄×D₃	—	4	2	no	—
250	C₄×D₃	—	4	2	no	—
251	Dic₆	—	2	2	no	—
252	Dic₆	—	2	2	no	—
253	C₃×C₈	—	2	2	no	—
254	C₃×C₈	—	2	2	no	—
255	C₃×C₈	—	12	2	no	—
256	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
257	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
258	C₃×C₈	#217	1	2	yes	—
259	C₃×C₈	#217	1	2	yes	—
260	C₃⋊C₈	—	4	2	no	—
261	C₃⋊C₈	—	6	4	no	—
262	C₃⋊C₈	—	6	2	no	—
263	C₃⋊C₈	—	6	2	no	—
264	C₂×C₃²	—	2	1	no	—
265	C₂×C₃²	—	2	1	no	—
266	C₂×C₃²	#390	1	1	yes	characteristic
267	C₂×C₃²	#443	1	2	yes	—
268	C₃×D₃	—	8	2	no	—
269	#41	—	6	2	no	—
270	#41	—	6	2	no	—
271	#41	—	6	2	no	—
272	#41	—	6	2	no	—
273	C₂×Q₈	—	3	2	no	—
274	C₂×D₄	—	3	2	no	—
275	C₂×D₄	—	6	2	no	—
276	C₂²×C₄	—	6	2	no	—
277	C₂²×C₄	—	18	1	no	—
278	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
279	Q₁₆	—	6	2	no	—
280	SD₁₆	—	6	4	no	—
281	D₈	—	6	2	no	—
282	M₁₆	—	6	2	no	—
283	M₁₆	—	6	2	no	—
284	C₂×C₈	—	6	2	no	—
285	C₂×C₈	—	9	2	no	—
286	C₂×C₈	—	18	1	no	—
287	C₂×C₈	—	18	1	no	—
288	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
289	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
290	C₂×C₈	#625	1	2	yes	—
291	C₂×C₈	#626	1	2	yes	—
292	C₄²	—	9	2	no	—
293	C₄²	—	18	1	no	—
294	C₂²×C₃	—	2	4	no	—
295	C₂²×C₃	—	2	2	no	—
296	C₂²×C₃	—	2	2	no	—
297	C₂²×C₃	—	2	1	no	—
298	C₂²×C₃	—	2	1	no	—
299	C₂²×C₃	—	2	1	no	—
300	C₂²×C₃	—	12	4	no	—
301	C₂²×C₃	—	12	2	no	—
302	C₂²×C₃	#893	1	1	yes	characteristic
303	C₂²×C₃	#895	1	1	yes	characteristic
304	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
305	C₂²×C₃	#936	1	1	yes	characteristic
306	C₂²×C₃	#938	1	1	yes	characteristic
307	C₂²×C₃	#947	1	1	yes	characteristic
308	C₂×D₃	—	4	4	no	—
309	C₂×D₃	—	4	2	no	—
310	C₃×C₄	—	2	1	no	—
311	C₃×C₄	—	2	1	no	—
312	C₃×C₄	—	2	1	no	—
313	C₃×C₄	—	2	1	no	—
314	C₃×C₄	—	12	2	no	—
315	C₃×C₄	#893	1	1	yes	characteristic
316	C₃×C₄	#895	1	1	yes	characteristic
317	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
318	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
319	C₃×C₄	#938	1	1	yes	characteristic
320	C₃×C₄	#940	1	1	yes	characteristic
321	C₃×C₄	#940	1	1	yes	characteristic
322	C₃×C₄	C₂×#217	1	1	yes	characteristic
323	Dic₃	—	4	2	no	—
324	Dic₃	—	6	8	no	—
325	Dic₃	—	6	4	no	—
326	Dic₃	—	6	4	no	—
327	C₃²	#1974	1	1	yes	characteristic, complemented, Sylow
328	C₂³	—	6	2	no	—
329	C₂³	#3796	1	1	yes	characteristic
330	Q₈	—	6	2	no	—
331	Q₈	—	6	2	no	—
332	D₄	—	6	2	no	—
333	D₄	—	6	2	no	—
334	D₄	—	12	2	no	—
335	D₄	—	12	2	no	—
336	C₂×C₄	—	6	2	no	—
337	C₂×C₄	—	12	2	no	—
338	C₂×C₄	—	12	2	no	—
339	C₂×C₄	—	18	4	no	—
340	C₂×C₄	—	18	2	no	—
341	C₂×C₄	#3796	1	1	yes	characteristic
342	C₂×C₄	#3802	1	1	yes	characteristic
343	C₂×C₄	#3802	1	1	yes	characteristic
344	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
345	C₂×C₄	C₂×#625	1	1	yes	characteristic
346	C₂×C₄	C₂×#626	1	1	yes	characteristic
347	C₈	—	12	2	no	—
348	C₈	—	18	2	no	—
349	C₈	(C₂×D₃)⋊Dic₃	1	2	yes	—
350	C₈	(C₂×D₃)⋊Dic₃	1	2	yes	—
351	C₂×C₃	—	2	2	no	—
352	C₂×C₃	—	2	2	no	—
353	C₂×C₃	—	2	1	no	—
354	C₂×C₃	—	2	1	no	—
355	C₂×C₃	—	2	1	no	—
356	C₂×C₃	—	2	1	no	—
357	C₂×C₃	—	2	1	no	—
358	C₂×C₃	—	2	2	no	—
359	C₂×C₃	—	24	2	no	—
360	C₂×C₃	#4895	1	2	yes	—
361	C₂×C₃	#4988	1	1	yes	characteristic
362	C₂×C₃	#5196	1	2	yes	—
363	C₂×C₃	#5221	1	1	yes	characteristic
364	D₃	—	8	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	12	2	no	—
367	C₂²	—	12	4	no	—
368	C₂²	#63863	1	1	yes	characteristic
369	C₂²	#63898	1	1	yes	characteristic
370	C₂²	#63916	1	1	yes	characteristic
371	C₄	—	12	2	no	—
372	C₄	—	18	4	no	—
373	C₄	#63898	1	1	yes	characteristic
374	C₄	#63914	1	1	yes	characteristic
375	C₄	#63916	1	1	yes	characteristic
376	C₄	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
377	C₃	—	2	1	no	—
378	C₃	#70696	1	1	yes	characteristic, complemented
379	C₃	#72084	1	1	yes	characteristic, complemented
380	C₂	—	2	1	no	—
381	C₂	—	2	1	no	—
382	C₂	—	24	2	no	—
383	C₂	#10590919	1	2	yes	—
384	C₂	#10591323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3802
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×#904	C₂
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1974
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×C₃⋊Dic₃	C₂	semidirect
3. C₂×C₃×#819	C₂	semidirect
4. C₂×#63568	C₂	semidirect
5. #10591071	C₂	semidirect
13. C₂×C₃×Dic₁₂	C₄	semidirect
16. C₂×C₃×D₂₄	C₄	semidirect
17. C₂×C₃×#204	C₄	semidirect
29. C₂×#819	D₃	semidirect
77. C₂×Dic₁₂	Dic₃	semidirect
80. C₂×D₂₄	Dic₃	semidirect
81. C₂×#204	Dic₃	semidirect
327. C₃²	#1974	semidirect
378. C₃	#70696	semidirect
379. C₃	#72084	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#904	2
C₂×C₈×C₃⋊Dic₃	2
C₂×C₃×#819	2
C₂×#63568	2
#10591071	2
#72084	3
#70696	3