Subgroups — #11790074

Order 1152.

565 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
5	#10589155	C₂	1	2	yes	maximal
6	#69836	—	3	1	no	maximal
7	#69485	—	3	1	no	maximal
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₄	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
15	C₃×C₃⋊#95	C₄	1	1	yes	characteristic
16	C₃×C₃⋊#95	C₄	1	1	yes	characteristic
17	C₃×#803	—	2	2	no	—
18	Dic₃⋊(C₃⋊C₈)	—	2	4	no	—
19	C₂×C₃×#97	—	3	1	no	—
20	C₃×C₄×C₄⋊C₄	—	3	1	no	—
21	C₂×Dic₃⋊C₈	—	3	2	no	—
22	C₄×C₄⋊Dic₃	D₃	1	1	yes	characteristic
23	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
25	C₂×C₄⋊(C₃⋊C₈)	—	3	2	no	—
26	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
27	C₂×#803	D₃	1	1	yes	characteristic
28	#4550	—	3	2	no	—
29	#4524	—	3	2	no	—
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
34	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
35	C₃²×C₄²	D₄	1	2	yes	—
36	C₃²×C₄²	Q₈	1	2	yes	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
40	C₃×C₄⋊Dic₃	—	2	4	no	—
41	C₃×C₄×Dic₃	—	2	2	no	—
42	C₃×C₄×Dic₃	—	2	2	no	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃×C₃⋊C₈	—	2	1	no	—
47	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
49	#1940	—	9	1	no	Sylow
50	C₂²×C₃×C₈	—	3	1	no	—
51	C₂²×C₃×C₈	—	3	1	no	—
52	C₂×C₃×C₄⋊C₄	—	3	2	no	—
53	C₂×C₃×C₄²	—	2	1	no	—
54	C₂×C₃×C₄²	—	3	2	no	—
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄⋊Dic₃	Dic₃	1	2	yes	—
58	C₂×C₄×Dic₃	C₂×D₃	1	2	yes	—
59	C₂²×C₃⋊C₈	—	3	2	no	—
60	C₂²×C₃⋊C₈	—	3	2	no	—
61	C₂²×C₃⋊C₈	—	6	2	no	—
62	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
64	C₃×#97	—	6	2	no	—
65	C₃×#95	—	3	1	no	—
66	C₃×#95	—	3	1	no	—
67	C₃⋊#95	Dic₃	1	1	yes	characteristic
68	C₃⋊#95	Dic₃	1	1	yes	characteristic
69	Dic₃⋊C₈	—	6	4	no	—
70	C₄⋊(C₃⋊C₈)	—	6	4	no	—
71	C₄⋊(C₃⋊C₈)	—	6	2	no	—
72	C₄⋊(C₃⋊C₈)	—	6	2	no	—
73	C₄⋊(C₃⋊C₈)	—	6	4	no	—
74	#803	—	2	2	no	—
75	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
79	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₃²×C₄	#41	1	1	yes	characteristic
85	C₂×C₃²×C₄	#41	1	1	yes	characteristic
86	C₂×C₃×Dic₃	—	2	2	no	—
87	C₂×C₃×Dic₃	—	2	2	no	—
88	C₂×C₃×Dic₃	—	2	2	no	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₂×C₃×Dic₃	—	2	2	no	—
91	C₂×C₃×Dic₃	M₁₆	1	4	yes	—
92	C₃⋊(C₃⋊C₈)	—	4	2	no	—
93	C₃×C₃⋊C₈	—	2	2	no	—
94	C₃×C₃⋊C₈	—	2	2	no	—
95	C₂×C₄⋊C₈	—	9	2	no	—
96	C₂×C₄⋊C₈	—	9	1	no	—
97	C₂×#97	—	3	1	no	—
98	C₄×C₄⋊C₄	—	3	1	no	—
99	#336	—	9	2	no	—
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	—	2	1	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	3	2	no	—
104	C₂²×C₃×C₄	—	3	2	no	—
105	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
112	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
113	C₂×C₃×C₈	—	3	2	no	—
114	C₂×C₃×C₈	—	3	2	no	—
115	C₂×C₃×C₈	—	6	1	no	—
116	C₂×C₃×C₈	—	6	1	no	—
117	C₂×C₃×C₈	—	6	1	no	—
118	C₂×C₃×C₈	—	6	1	no	—
119	C₃×C₄⋊C₄	—	6	4	no	—
120	C₃×C₄²	—	2	2	no	—
121	C₃×C₄²	—	2	2	no	—
122	C₃×C₄²	—	6	2	no	—
123	C₃×C₄²	—	6	2	no	—
124	C₃×C₄²	Dic₆	1	2	yes	—
125	C₃×C₄²	Dic₆	1	2	yes	—
126	C₃×C₄²	D₁₂	1	2	yes	—
127	C₃×C₄²	#67	1	2	yes	—
128	C₄⋊Dic₃	—	2	4	no	—
129	C₄×Dic₃	—	2	2	no	—
130	C₄×Dic₃	—	2	2	no	—
131	C₂×C₃⋊C₈	—	2	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	—	6	4	no	—
136	C₂×C₃⋊C₈	—	6	2	no	—
137	C₂×C₃⋊C₈	—	6	2	no	—
138	C₂×C₃⋊C₈	—	6	2	no	—
139	C₂×C₃⋊C₈	—	6	2	no	—
140	C₂×C₃⋊C₈	—	6	2	no	—
141	C₂×C₃⋊C₈	—	6	2	no	—
142	C₂×C₃⋊C₈	—	12	2	no	—
143	C₂×C₃⋊C₈	—	12	2	no	—
144	C₂×C₃⋊C₈	Dic₆	1	2	yes	—
145	C₂×C₃⋊C₈	D₁₂	1	2	yes	—
146	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
148	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
149	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
150	C₂²×C₃²	#131	1	2	yes	—
151	C₃²×C₄	—	2	2	no	—
152	C₃²×C₄	—	2	2	no	—
153	C₃²×C₄	#121	1	1	yes	characteristic
154	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
155	C₃²×C₄	#125	1	1	yes	characteristic
156	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
157	C₃×Dic₃	—	2	4	no	—
158	C₃×Dic₃	—	4	2	no	—
159	C₂²×C₈	—	3	1	no	—
160	C₂²×C₈	—	3	1	no	—
161	C₂²×C₈	—	9	2	no	—
162	C₂×C₄⋊C₄	—	3	2	no	—
163	C₂×C₄²	—	3	2	no	—
164	C₂×C₄²	D₃²	1	1	yes	characteristic
165	C₄⋊C₈	—	18	4	no	—
166	C₄⋊C₈	—	18	2	no	—
167	#97	—	6	2	no	—
168	#95	—	3	1	no	—
169	#95	—	3	1	no	—
170	C₂³×C₃	—	2	1	no	—
171	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	3	4	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	—	6	2	no	—
189	C₂×C₃×C₄	—	6	2	no	—
190	C₂×C₃×C₄	—	6	2	no	—
191	C₂×C₃×C₄	—	6	2	no	—
192	C₂×C₃×C₄	—	6	2	no	—
193	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
194	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
195	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
196	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
197	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
202	C₂×C₃×C₄	#235	1	1	yes	characteristic
203	C₂×C₃×C₄	#235	1	1	yes	characteristic
204	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	#237	1	1	yes	characteristic
206	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
207	C₂×C₃×C₄	#239	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
209	C₂×Dic₃	—	2	2	no	—
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	2	2	no	—
212	C₂×Dic₃	—	2	2	no	—
213	C₂×Dic₃	—	2	2	no	—
214	C₂×Dic₃	#208	1	4	yes	—
215	C₃×C₈	—	6	2	no	—
216	C₃×C₈	—	6	2	no	—
217	C₃⋊C₈	—	2	2	no	—
218	C₃⋊C₈	—	2	2	no	—
219	C₃⋊C₈	—	12	4	no	—
220	C₃⋊C₈	—	12	2	no	—
221	C₃⋊C₈	—	12	2	no	—
222	C₂×C₃²	#389	1	1	yes	characteristic
223	C₂×C₃²	#423	1	1	yes	characteristic
224	C₂×C₃²	#424	1	1	yes	characteristic
225	C₂×C₃²	#435	1	2	yes	—
226	C₂×C₃²	#446	1	2	yes	—
227	C₂²×C₄	—	3	2	no	—
228	C₂²×C₄	—	3	2	no	—
229	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
230	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
232	C₂×C₈	—	3	2	no	—
233	C₂×C₈	—	3	2	no	—
234	C₂×C₈	—	6	1	no	—
235	C₂×C₈	—	6	1	no	—
236	C₂×C₈	—	6	1	no	—
237	C₂×C₈	—	6	1	no	—
238	C₂×C₈	—	18	2	no	—
239	C₂×C₈	—	18	2	no	—
240	C₂×C₈	—	18	2	no	—
241	C₄⋊C₄	—	6	4	no	—
242	C₄²	—	6	2	no	—
243	C₄²	—	6	2	no	—
244	C₄²	#626	1	2	yes	—
245	C₄²	#627	1	2	yes	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
252	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
253	C₂²×C₃	D₃×M₁₆	1	2	yes	—
254	C₂²×C₃	#907	1	2	yes	—
255	C₂²×C₃	C₂×#208	1	2	yes	—
256	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
257	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
258	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
259	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
260	C₂²×C₃	#948	1	2	yes	—
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	4	2	no	—
271	C₃×C₄	—	6	4	no	—
272	C₃×C₄	—	12	2	no	—
273	C₃×C₄	#869	1	1	yes	characteristic
274	C₃×C₄	#870	1	1	yes	characteristic
275	C₃×C₄	#888	1	1	yes	characteristic
276	C₃×C₄	#890	1	1	yes	characteristic
277	C₃×C₄	#895	1	1	yes	characteristic
278	C₃×C₄	#897	1	1	yes	characteristic
279	C₃×C₄	#924	1	1	yes	characteristic
280	C₃×C₄	#930	1	1	yes	characteristic
281	Dic₃	—	2	4	no	—
282	Dic₃	—	4	2	no	—
283	C₃²	#1940	1	1	yes	characteristic, complemented, Sylow
284	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	3	4	no	—
288	C₂×C₄	—	6	2	no	—
289	C₂×C₄	—	6	2	no	—
290	C₂×C₄	—	6	2	no	—
291	C₂×C₄	—	6	2	no	—
292	C₂×C₄	—	6	2	no	—
293	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
294	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
295	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
296	C₂×C₄	#3795	1	1	yes	characteristic
297	C₂×C₄	#3799	1	1	yes	characteristic
298	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
299	C₂×C₄	C₂×#626	1	1	yes	characteristic
300	C₂×C₄	C₂×#627	1	1	yes	characteristic
301	C₈	—	6	2	no	—
302	C₈	—	6	2	no	—
303	C₈	—	36	2	no	—
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	2	2	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	#4701	1	2	yes	—
310	C₂×C₃	#4706	1	2	yes	—
311	C₂×C₃	#4924	1	1	yes	characteristic
312	C₂×C₃	#4928	1	1	yes	characteristic
313	C₂×C₃	#4984	1	1	yes	characteristic
314	C₂×C₃	#4999	1	1	yes	characteristic
315	C₂×C₃	#5011	1	2	yes	—
316	C₂×C₃	#5024	1	2	yes	—
317	C₂×C₃	#5118	1	1	yes	characteristic
318	C₂×C₃	#5119	1	1	yes	characteristic
319	C₂²	D₃×#208	1	2	yes	—
320	C₂²	#63815	1	2	yes	—
321	C₂²	Dic₃×Dic₆	1	1	yes	characteristic
322	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
323	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	6	4	no	—
327	C₄	—	12	2	no	—
328	C₄	#63857	1	1	yes	characteristic
329	C₄	#63869	1	1	yes	characteristic
330	C₄	#63909	1	1	yes	characteristic
331	C₄	#63918	1	1	yes	characteristic
332	C₃	—	2	1	no	—
333	C₃	#69485	1	1	yes	characteristic, complemented
334	C₃	#69836	1	1	yes	characteristic, complemented
335	C₂	#10590251	1	2	yes	—
336	C₂	#10590326	1	2	yes	—
337	C₂	#10591115	1	1	yes	characteristic
338	C₂	#10591117	1	1	yes	characteristic
339	C₂	#10591352	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃²
commutator, socle, prime radical	C₂³×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄²	C₂²
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1940
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
283. C₃²	#1940	semidirect
333. C₃	#69485	semidirect
334. C₃	#69836	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×C₄×C₄⋊Dic₃	2
C₂×C₃×#803	2
C₂×Dic₃⋊(C₃⋊C₈)	2
#10589155	2
#69836	3
#69485	3