Subgroups — #11776873

Order 1152.

395 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4837	C₂	1	1	yes	maximal, characteristic
2	#10590336	C₂	1	1	yes	maximal, characteristic
3	C₄×D₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10589346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4480	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589098	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589097	C₂	1	1	yes	maximal, characteristic, complemented
8	#70490	—	3	1	no	maximal
9	#68529	—	3	1	no	maximal
10	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
11	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
12	C₂×#3711	—	2	1	no	—
13	C₂×D₃×C₃⋊C₈	C₄	1	1	yes	characteristic
14	C₂×D₃×C₃⋊C₈	C₄	1	1	yes	characteristic
15	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
17	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃×Dic₃⋊C₈	—	2	1	no	—
19	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	Dic₃⋊(C₃⋊C₈)	—	2	1	no	—
23	Dic₃×C₃⋊C₈	C₄	1	1	yes	characteristic
24	Dic₃×C₃⋊C₈	C₄	1	1	yes	characteristic
25	C₃×#423	—	3	1	no	—
26	#4924	—	3	1	no	—
27	C₂×C₄×C₃⋊C₈	—	3	1	no	—
28	#4837	—	3	1	no	—
29	#4837	D₃	1	1	yes	characteristic
30	C₄×D₃×C₈	—	3	1	no	—
31	C₃×#334	—	3	1	no	—
32	#4480	—	3	1	no	—
33	#4480	—	3	1	no	—
34	#4480	—	3	1	no	—
35	#4480	—	3	1	no	—
36	#4480	—	3	1	no	—
37	#4480	—	6	1	no	—
38	#4480	D₃	1	1	yes	characteristic, complemented
39	#4457	—	3	1	no	—
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	Q₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	—	2	1	no	—
44	C₃²×C₄²	C₂³	1	1	yes	characteristic
45	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
46	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×Dic₃	D₄	1	1	yes	characteristic
50	C₃×C₄×Dic₃	Q₈	1	1	yes	characteristic
51	C₂×C₃×C₃⋊C₈	—	2	1	no	—
52	C₂×C₃×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₃×ΓL(1,𝔽₂₅)	—	4	2	no	—
56	#3711	—	4	2	no	—
57	D₃×C₃⋊C₈	—	2	2	no	—
58	D₃×C₃⋊C₈	—	2	2	no	—
59	#1566	—	9	1	no	Sylow
60	C₂×C₃×M₁₆	—	6	1	no	—
61	C₂×C₃×C₄²	—	3	1	no	—
62	C₂×#208	—	6	1	no	—
63	C₂²×C₃⋊C₈	—	3	1	no	—
64	C₂²×C₃⋊C₈	—	3	1	no	—
65	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
66	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
67	C₂×D₃×C₈	—	3	1	no	—
68	C₂×D₃×C₈	—	3	1	no	—
69	C₄²×D₃	C₂×D₃	1	1	yes	characteristic
70	C₃×C₄⋊C₈	—	2	1	no	—
71	C₃×C₄⋊C₈	—	3	1	no	—
72	C₃×C₄⋊C₈	—	3	1	no	—
73	C₃×C₄⋊C₈	—	6	1	no	—
74	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
75	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
76	C₃×C₄×C₈	—	3	1	no	—
77	Dic₃⋊C₈	—	2	1	no	—
78	Dic₃⋊C₈	—	6	1	no	—
79	C₈×Dic₃	—	3	1	no	—
80	C₈×Dic₃	—	3	1	no	—
81	C₄⋊(C₃⋊C₈)	—	3	1	no	—
82	C₄⋊(C₃⋊C₈)	—	3	1	no	—
83	C₄⋊(C₃⋊C₈)	—	6	1	no	—
84	C₄⋊(C₃⋊C₈)	—	6	1	no	—
85	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
86	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
87	C₄×C₃⋊C₈	—	3	1	no	—
88	C₄×C₃⋊C₈	—	3	1	no	—
89	C₄×C₃⋊C₈	—	3	1	no	—
90	C₄×C₃⋊C₈	—	3	1	no	—
91	C₄×C₃⋊C₈	—	6	1	no	—
92	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
93	C₂²×C₃×D₃	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
97	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
100	C₃×C₄×D₃	—	2	1	no	—
101	C₃×C₄×D₃	—	2	1	no	—
102	C₃×C₄×D₃	—	2	2	no	—
103	C₃×C₄×D₃	SD₁₆	1	2	yes	—
104	C₃×C₄×D₃	SD₁₆	1	2	yes	—
105	C₃²×C₈	—	4	1	no	—
106	C₃⋊(C₃⋊C₈)	—	2	1	no	—
107	C₃⋊(C₃⋊C₈)	—	4	1	no	—
108	C₃⋊(C₃⋊C₈)	M₁₆	1	2	yes	complemented
109	C₃×C₃⋊C₈	—	2	1	no	—
110	C₃×C₃⋊C₈	—	4	1	no	—
111	C₃×C₃⋊C₈	—	4	1	no	—
112	C₃×C₃⋊C₈	M₁₆	1	2	yes	complemented
113	#423	—	3	1	no	—
114	#423	—	9	1	no	—
115	C₂×C₄×C₈	—	9	1	no	—
116	#334	—	3	1	no	—
117	#334	—	9	1	no	—
118	#334	—	9	1	no	—
119	#334	—	9	1	no	—
120	C₂²×C₃×C₄	—	3	1	no	—
121	C₂²×C₃×C₄	—	3	1	no	—
122	C₂²×C₃×C₄	—	3	1	no	—
123	C₂×C₄×D₃	Dic₆	1	1	yes	characteristic
124	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
125	C₂×C₄×D₃	#67	1	1	yes	characteristic
126	C₃×M₁₆	—	12	2	no	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	—	3	1	no	—
130	C₂×C₃×C₈	—	3	1	no	—
131	C₂×C₃×C₈	—	4	1	no	—
132	C₂×C₃×C₈	—	6	1	no	—
133	C₂×C₃×C₈	—	6	1	no	—
134	C₃×C₄²	—	2	1	no	—
135	C₃×C₄²	—	3	1	no	—
136	C₃×C₄²	—	3	1	no	—
137	C₃×C₄²	—	3	1	no	—
138	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
139	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
140	C₄×Dic₃	Dic₆	1	1	yes	characteristic
141	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
142	C₄×Dic₃	#67	1	1	yes	characteristic
143	#208	—	12	2	no	—
144	C₂×C₃⋊C₈	—	2	1	no	—
145	C₂×C₃⋊C₈	—	2	1	no	—
146	C₂×C₃⋊C₈	—	3	1	no	—
147	C₂×C₃⋊C₈	—	3	1	no	—
148	C₂×C₃⋊C₈	—	3	1	no	—
149	C₂×C₃⋊C₈	—	3	1	no	—
150	C₂×C₃⋊C₈	—	6	1	no	—
151	C₂×C₃⋊C₈	—	6	1	no	—
152	C₂×C₃⋊C₈	—	6	2	no	—
153	C₂×C₃⋊C₈	—	6	2	no	—
154	C₂×C₃⋊C₈	—	6	1	no	—
155	C₂×C₃⋊C₈	—	6	1	no	—
156	C₂×C₃⋊C₈	—	12	1	no	—
157	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
158	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
159	ΓL(1,𝔽₂₅)	—	4	2	no	—
160	ΓL(1,𝔽₂₅)	—	12	2	no	—
161	D₃×C₈	—	6	2	no	—
162	D₃×C₈	—	6	2	no	—
163	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₃×D₃	—	2	1	no	—
165	C₂×C₃×D₃	—	2	1	no	—
166	C₂×C₃×D₃	#106	1	2	yes	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
169	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
170	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
171	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
172	C₃×Dic₃	—	2	1	no	—
173	C₃×Dic₃	#106	1	2	yes	—
174	C₃×Dic₃	#108	1	2	yes	—
175	C₂×M₁₆	—	6	1	no	—
176	C₂×M₁₆	—	18	1	no	—
177	C₂²×C₈	—	9	1	no	—
178	C₂²×C₈	—	9	1	no	—
179	C₂×C₄²	—	3	1	no	—
180	C₄⋊C₈	—	3	1	no	—
181	C₄⋊C₈	—	3	1	no	—
182	C₄⋊C₈	—	6	1	no	—
183	C₄⋊C₈	—	9	1	no	—
184	C₄⋊C₈	—	18	1	no	—
185	C₄⋊C₈	D₃²	1	1	yes	characteristic
186	C₄×C₈	—	3	1	no	—
187	C₄×C₈	—	9	1	no	—
188	C₄×C₈	—	9	1	no	—
189	C₄×C₈	—	9	1	no	—
190	C₂³×C₃	—	3	1	no	—
191	C₂²×D₃	Dic₃⋊C₄	1	1	yes	characteristic
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	—	3	1	no	—
196	C₂×C₃×C₄	—	3	1	no	—
197	C₂×C₃×C₄	—	3	2	no	—
198	C₂×C₃×C₄	—	3	2	no	—
199	C₂×C₃×C₄	—	3	1	no	—
200	C₂×C₃×C₄	—	6	2	no	—
201	C₂×C₃×C₄	—	6	1	no	—
202	C₂×C₃×C₄	—	6	1	no	—
203	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
206	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
209	C₂×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
210	C₂×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
211	C₂×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
212	C₄×D₃	—	2	1	no	—
213	C₄×D₃	—	2	1	no	—
214	C₄×D₃	—	2	2	no	—
215	C₄×D₃	#214	1	2	yes	—
216	C₄×D₃	#215	1	2	yes	—
217	C₃×C₈	—	3	2	no	—
218	C₃×C₈	—	4	1	no	—
219	C₃×C₈	—	4	1	no	—
220	C₃×C₈	—	6	1	no	—
221	C₃×C₈	—	8	1	no	—
222	C₃×C₈	—	12	1	no	—
223	C₃×C₈	—	12	1	no	—
224	C₃⋊C₈	—	2	1	no	—
225	C₃⋊C₈	—	3	2	no	—
226	C₃⋊C₈	—	3	2	no	—
227	C₃⋊C₈	—	4	1	no	—
228	C₃⋊C₈	—	4	1	no	—
229	C₃⋊C₈	—	6	2	no	—
230	C₃⋊C₈	—	6	1	no	—
231	C₃⋊C₈	—	6	1	no	—
232	C₃⋊C₈	—	12	1	no	—
233	C₃⋊C₈	—	12	1	no	—
234	C₃⋊C₈	—	12	1	no	—
235	C₃⋊C₈	—	24	1	no	—
236	C₃⋊C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
237	C₂×C₃²	#423	1	1	yes	characteristic
238	C₂×C₃²	C₂×#106	1	1	yes	characteristic
239	C₂×C₃²	C₂×#108	1	1	yes	characteristic
240	C₃×D₃	—	2	2	no	—
241	C₂²×C₄	—	3	1	no	—
242	C₂²×C₄	—	3	1	no	—
243	C₂²×C₄	—	3	1	no	—
244	M₁₆	—	12	2	no	—
245	M₁₆	—	36	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	3	1	no	—
248	C₂×C₈	—	3	1	no	—
249	C₂×C₈	—	6	1	no	—
250	C₂×C₈	—	6	1	no	—
251	C₂×C₈	—	9	1	no	—
252	C₂×C₈	—	9	1	no	—
253	C₂×C₈	—	18	2	no	—
254	C₂×C₈	—	18	2	no	—
255	C₂×C₈	—	18	1	no	—
256	C₄²	—	3	1	no	—
257	C₄²	—	3	1	no	—
258	C₄²	—	3	1	no	—
259	C₄²	C₂×D₃²	1	1	yes	characteristic
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	3	2	no	—
262	C₂²×C₃	—	6	1	no	—
263	C₂²×C₃	—	6	1	no	—
264	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
265	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
266	C₂×D₃	—	2	1	no	—
267	C₂×D₃	—	2	1	no	—
268	C₂×D₃	#808	1	2	yes	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	3	2	no	—
276	C₃×C₄	—	3	2	no	—
277	C₃×C₄	—	4	1	no	—
278	C₃×C₄	—	6	1	no	—
279	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
280	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
281	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
282	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
283	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
284	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
285	C₃×C₄	C₂×#214	1	1	yes	characteristic
286	C₃×C₄	C₂×#215	1	1	yes	characteristic
287	Dic₃	—	2	1	no	—
288	Dic₃	#808	1	2	yes	—
289	Dic₃	#822	1	2	yes	—
290	C₃²	#1566	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	3	1	no	—
292	C₂×C₄	—	3	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	2	no	—
295	C₂×C₄	—	3	2	no	—
296	C₂×C₄	—	3	1	no	—
297	C₂×C₄	—	6	2	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	—	6	1	no	—
300	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
301	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
302	C₂×C₄	D₃×#67	1	1	yes	characteristic
303	C₈	—	3	2	no	—
304	C₈	—	4	1	no	—
305	C₈	—	6	1	no	—
306	C₈	—	9	2	no	—
307	C₈	—	12	1	no	—
308	C₈	—	12	1	no	—
309	C₈	—	18	1	no	—
310	C₈	—	36	1	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	2	1	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	6	2	no	—
315	C₂×C₃	#4837	1	1	yes	characteristic
316	C₂×C₃	D₃×#106	1	1	yes	characteristic
317	C₂×C₃	D₃×#108	1	1	yes	characteristic
318	C₂×C₃	C₂×#808	1	1	yes	characteristic
319	C₂×C₃	#5118	1	1	yes	characteristic
320	C₂×C₃	C₂×#822	1	1	yes	characteristic
321	D₃	—	2	2	no	—
322	C₂²	—	3	2	no	—
323	C₂²	—	6	1	no	—
324	C₂²	—	6	1	no	—
325	C₂²	D₃×Dic₃⋊C₄	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	3	2	no	—
328	C₄	—	3	2	no	—
329	C₄	—	6	1	no	—
330	C₄	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
331	C₄	#63791	1	1	yes	characteristic
332	C₄	D₃×#214	1	1	yes	characteristic
333	C₄	D₃×#215	1	1	yes	characteristic
334	C₃	—	2	1	no	—
335	C₃	#68529	1	1	yes	characteristic, complemented
336	C₃	#70490	1	1	yes	characteristic, complemented
337	C₂	—	6	2	no	—
338	C₂	D₃×#808	1	1	yes	characteristic
339	C₂	#10590663	1	1	yes	characteristic
340	C₂	D₃×#822	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×#67
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄⋊C₈	C₂²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₃	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄⋊C₈	D₃²
hypocenter	C₃²	#1566
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #10589346	C₂	semidirect
5. C₃×#4480	C₂	semidirect
6. #10589098	C₂	semidirect
7. #10589097	C₂	semidirect
38. #4480	D₃	semidirect
108. C₃⋊(C₃⋊C₈)	M₁₆	semidirect
112. C₃×C₃⋊C₈	M₁₆	semidirect
236. C₃⋊C₈	ΓL(1,𝔽₂₅)	semidirect
290. C₃²	#1566	semidirect
335. C₃	#68529	semidirect
336. C₃	#70490	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4837	2
#10590336	2
C₄×D₃×C₃⋊C₈	2
#10589346	2
C₃×#4480	2
#10589098	2
#10589097	2
#70490	3
#68529	3