Subgroups — #11777488

Order 1152.

367 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#215	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#10590337	C₂	1	1	yes	maximal, characteristic
4	#10589401	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589097	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589080	C₂	1	1	yes	maximal, characteristic
7	#10589078	C₂	1	1	yes	maximal, characteristic, complemented
8	#70933	—	3	1	no	maximal
9	#70896	—	3	1	no	maximal
10	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×#215	C₄	1	1	yes	characteristic
12	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×#3711	—	2	1	no	—
14	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₃×#835	C₄	1	1	yes	characteristic
17	C₃×#809	C₄	1	1	yes	characteristic
18	C₃×#808	C₄	1	1	yes	characteristic
19	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₃⋊C₈	—	2	1	no	—
22	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	(C₂×D₃)⋊(C₃⋊C₈)	—	2	1	no	—
25	C₃×C₈×Q₈	—	3	1	no	—
26	C₃×C₄×SD₁₆	—	3	1	no	—
27	C₄×#215	D₃	1	1	yes	characteristic
28	Q₈×C₃⋊C₈	D₃	1	1	yes	characteristic, complemented
29	#5012	—	3	1	no	—
30	#4688	—	3	1	no	—
31	#4535	—	3	1	no	—
32	#4535	—	3	1	no	—
33	#4535	—	6	1	no	—
34	D₄⋊(C₃⋊C₈)	—	3	1	no	—
35	#4483	—	3	1	no	—
36	#4480	—	3	1	no	—
37	#4457	—	3	1	no	—
38	#4455	—	3	1	no	—
39	#4454	—	3	1	no	—
40	C₂×C₃²×Q₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×D₁₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	—	2	1	no	—
43	C₃²×C₄⋊C₄	—	2	1	no	—
44	C₃²×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₃²×C₄²	—	2	1	no	—
46	C₃²×C₄²	C₂³	1	1	yes	characteristic
47	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
49	C₃×#215	—	2	2	no	—
50	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
51	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
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	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
57	#3711	—	4	2	no	—
58	#1583	—	9	1	no	Sylow
59	C₂×C₃×SD₁₆	—	3	1	no	—
60	C₃×C₄×Q₈	—	2	1	no	—
61	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
62	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
63	C₃×C₄×D₄	—	3	1	no	—
64	C₂×#215	Dic₃	1	1	yes	characteristic
65	C₂×#208	—	6	1	no	—
66	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
67	C₄×D₁₂	C₂×D₃	1	1	yes	characteristic
68	C₃×#106	—	3	1	no	—
69	C₃×C₄⋊C₈	—	3	1	no	—
70	C₃×C₄⋊C₈	—	6	1	no	—
71	C₃×#103	—	3	1	no	—
72	C₃×#102	—	3	1	no	—
73	C₃×C₄×C₈	—	3	1	no	—
74	C₃×C₄×C₈	—	3	1	no	—
75	C₃×C₄×C₈	—	6	1	no	—
76	#835	Dic₃	1	1	yes	characteristic
77	#830	—	6	1	no	—
78	(C₂×D₃)⋊C₈	—	6	1	no	—
79	#809	Dic₃	1	1	yes	characteristic
80	#808	Dic₃	1	1	yes	characteristic
81	C₄⋊(C₃⋊C₈)	—	2	1	no	—
82	C₄⋊(C₃⋊C₈)	—	3	1	no	—
83	C₄⋊(C₃⋊C₈)	—	3	1	no	—
84	C₄⋊(C₃⋊C₈)	—	6	1	no	—
85	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
86	C₄×C₃⋊C₈	—	2	1	no	—
87	C₄×C₃⋊C₈	—	3	1	no	—
88	C₄×C₃⋊C₈	—	3	1	no	—
89	C₄×C₃⋊C₈	—	6	1	no	—
90	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
91	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
92	C₂²×C₃×D₃	—	2	1	no	—
93	C₃²×Q₈	—	2	1	no	—
94	C₃²×Q₈	M₁₆	1	2	yes	complemented
95	C₂×C₃²×C₄	—	2	1	no	—
96	C₂×C₃²×C₄	—	2	1	no	—
97	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃²×C₄	#41	1	1	yes	characteristic
100	C₂×C₃×Dic₃	—	2	1	no	—
101	C₃×D₁₂	—	2	1	no	—
102	C₃×D₁₂	—	2	1	no	—
103	C₃×C₄×D₃	—	4	1	no	—
104	C₃×C₄×D₃	—	4	1	no	—
105	C₃⋊(C₃⋊C₈)	—	4	1	no	—
106	C₃⋊(C₃⋊C₈)	—	4	1	no	—
107	C₃×C₃⋊C₈	—	2	1	no	—
108	C₃×C₃⋊C₈	—	2	1	no	—
109	C₃×C₃⋊C₈	—	2	1	no	—
110	C₃×C₃⋊C₈	—	4	1	no	—
111	C₃×C₃⋊C₈	SD₁₆	1	2	yes	complemented
112	C₈×Q₈	—	3	1	no	—
113	C₄×SD₁₆	—	3	1	no	—
114	#436	—	9	1	no	—
115	#334	—	9	1	no	—
116	Q₈⋊C₈	—	9	1	no	—
117	D₄⋊C₈	—	9	1	no	—
118	#322	—	9	1	no	—
119	C₂×C₃×Q₈	—	2	1	no	—
120	C₂×C₃×Q₈	C₄×D₃	1	1	yes	characteristic
121	C₂×C₃×Q₈	C₂×Dic₃	1	1	yes	characteristic
122	C₂×C₃×D₄	—	3	1	no	—
123	C₂²×C₃×C₄	—	6	1	no	—
124	C₂×D₁₂	C₂×Dic₃	1	1	yes	characteristic
125	C₂×C₄×D₃	—	2	1	no	—
126	C₃×SD₁₆	—	6	2	no	—
127	C₂×C₃×C₈	—	3	1	no	—
128	C₂×C₃×C₈	—	3	1	no	—
129	C₂×C₃×C₈	—	3	1	no	—
130	C₂×C₃×C₈	—	3	1	no	—
131	C₂×C₃×C₈	—	6	1	no	—
132	C₃×C₄⋊C₄	—	2	1	no	—
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₄⋊C₄	—	2	1	no	—
135	C₃×C₄⋊C₄	—	3	1	no	—
136	C₃×C₄⋊C₄	—	4	1	no	—
137	C₃×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
138	C₃×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
139	C₃×C₂²⋊C₄	—	6	1	no	—
140	C₃×C₄²	—	2	1	no	—
141	C₃×C₄²	—	2	1	no	—
142	C₃×C₄²	—	2	1	no	—
143	C₃×C₄²	—	4	1	no	—
144	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
145	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
146	#215	—	2	2	no	—
147	(C₂×D₃)⋊C₄	—	2	1	no	—
148	C₄⋊Dic₃	C₂×Dic₃	1	1	yes	characteristic
149	#208	—	12	2	no	—
150	C₂×C₃⋊C₈	—	2	1	no	—
151	C₂×C₃⋊C₈	—	6	1	no	—
152	C₂×C₃⋊C₈	—	6	1	no	—
153	C₂×C₃⋊C₈	—	6	1	no	—
154	C₂×C₃⋊C₈	—	6	1	no	—
155	C₂×C₃⋊C₈	—	12	1	no	—
156	C₂×C₃⋊C₈	—	12	1	no	—
157	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
158	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
159	C₂×C₃⋊C₈	#67	1	1	yes	characteristic
160	C₂×C₃⋊C₈	#67	1	1	yes	characteristic
161	ΓL(1,𝔽₂₅)	—	12	2	no	—
162	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
163	C₂×C₃×D₃	—	4	1	no	—
164	C₂×C₃×D₃	—	4	1	no	—
165	C₂×C₃×D₃	—	4	1	no	—
166	C₃²×C₄	—	2	2	no	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	—	4	1	no	—
169	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
170	C₃²×C₄	#131	1	1	yes	characteristic
171	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
172	C₃²×C₄	#135	1	1	yes	characteristic
173	C₃×Dic₃	—	4	1	no	—
174	C₂×SD₁₆	—	3	1	no	—
175	C₂×M₁₆	—	18	1	no	—
176	C₄×Q₈	D₃²	1	1	yes	characteristic
177	C₄×D₄	—	3	1	no	—
178	#106	—	3	1	no	—
179	C₄⋊C₈	—	3	1	no	—
180	C₄⋊C₈	—	6	1	no	—
181	C₄⋊C₈	—	9	1	no	—
182	#103	—	3	1	no	—
183	#102	—	3	1	no	—
184	C₂²⋊C₈	—	18	1	no	—
185	C₄×C₈	—	3	1	no	—
186	C₄×C₈	—	3	1	no	—
187	C₄×C₈	—	6	1	no	—
188	C₄×C₈	—	9	1	no	—
189	C₂³×C₃	—	6	1	no	—
190	C₂²×D₃	—	2	1	no	—
191	C₃×Q₈	—	2	1	no	—
192	C₃×Q₈	—	2	1	no	—
193	C₃×Q₈	—	2	2	no	—
194	C₃×Q₈	—	4	1	no	—
195	C₃×Q₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
196	C₃×Q₈	#208	1	2	yes	complemented
197	C₃×D₄	—	6	1	no	—
198	C₃×D₄	—	6	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	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	—	4	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	—	6	1	no	—
209	C₂×C₃×C₄	—	12	1	no	—
210	C₂×C₃×C₄	—	12	1	no	—
211	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
212	C₂×C₃×C₄	#235	1	1	yes	characteristic
213	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	#237	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
217	C₂×Dic₃	—	2	1	no	—
218	D₁₂	—	2	1	no	—
219	D₁₂	—	2	1	no	—
220	C₄×D₃	—	4	1	no	—
221	C₄×D₃	—	4	1	no	—
222	C₃×C₈	—	3	2	no	—
223	C₃×C₈	—	6	1	no	—
224	C₃×C₈	—	6	1	no	—
225	C₃×C₈	—	6	1	no	—
226	C₃×C₈	—	12	1	no	—
227	C₃⋊C₈	—	2	1	no	—
228	C₃⋊C₈	—	2	1	no	—
229	C₃⋊C₈	—	2	1	no	—
230	C₃⋊C₈	—	4	1	no	—
231	C₃⋊C₈	—	12	1	no	—
232	C₃⋊C₈	—	12	1	no	—
233	C₃⋊C₈	—	12	2	no	—
234	C₃⋊C₈	—	12	1	no	—
235	C₃⋊C₈	—	12	1	no	—
236	C₃⋊C₈	—	24	1	no	—
237	C₃⋊C₈	#215	1	2	yes	complemented
238	C₂×C₃²	#435	1	1	yes	characteristic
239	C₂×C₃²	C₄×SD₁₆	1	1	yes	characteristic
240	C₂×C₃²	#444	1	1	yes	characteristic
241	C₃×D₃	—	8	1	no	—
242	C₂×Q₈	D₃×Dic₃	1	1	yes	characteristic
243	C₂×D₄	—	3	1	no	—
244	C₂²×C₄	—	6	1	no	—
245	SD₁₆	—	6	2	no	—
246	M₁₆	—	36	2	no	—
247	C₂×C₈	—	3	1	no	—
248	C₂×C₈	—	3	1	no	—
249	C₂×C₈	—	3	1	no	—
250	C₂×C₈	—	3	1	no	—
251	C₂×C₈	—	6	1	no	—
252	C₂×C₈	—	18	1	no	—
253	C₂×C₈	—	18	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	3	1	no	—
256	C₄⋊C₄	D₃×Dic₃	1	1	yes	characteristic
257	C₂²⋊C₄	—	6	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	C₂×D₃²	1	1	yes	characteristic
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	12	1	no	—
262	C₂²×C₃	—	12	1	no	—
263	C₂²×C₃	—	12	1	no	—
264	C₂²×C₃	C₄×#67	1	1	yes	characteristic
265	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
266	C₂×D₃	—	4	1	no	—
267	C₂×D₃	—	4	1	no	—
268	C₂×D₃	—	4	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	4	1	no	—
278	C₃×C₄	—	4	2	no	—
279	C₃×C₄	—	4	1	no	—
280	C₃×C₄	—	4	1	no	—
281	C₃×C₄	—	8	1	no	—
282	C₃×C₄	—	12	1	no	—
283	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
284	C₃×C₄	#901	1	1	yes	characteristic
285	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
286	C₃×C₄	#916	1	1	yes	characteristic
287	C₃×C₄	C₂×#208	1	1	yes	characteristic
288	C₃×C₄	C₂×#215	1	1	yes	characteristic
289	C₃×C₄	#948	1	1	yes	characteristic
290	C₃×C₄	#950	1	1	yes	characteristic
291	Dic₃	—	4	1	no	—
292	C₃²	#1583	1	1	yes	characteristic, complemented, Sylow
293	C₂³	—	6	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	#3711	1	2	yes	complemented
296	D₄	—	6	1	no	—
297	D₄	—	6	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	6	1	no	—
301	C₂×C₄	—	12	1	no	—
302	C₂×C₄	—	12	1	no	—
303	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
304	C₂×C₄	#3804	1	1	yes	characteristic
305	C₂×C₄	D₃×#67	1	1	yes	characteristic
306	C₈	—	3	2	no	—
307	C₈	—	6	1	no	—
308	C₈	—	6	1	no	—
309	C₈	—	6	1	no	—
310	C₈	—	12	1	no	—
311	C₈	—	36	1	no	—
312	C₈	—	36	1	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	—	24	1	no	—
317	C₂×C₃	#5011	1	1	yes	characteristic
318	C₂×C₃	C₄×#215	1	1	yes	characteristic
319	C₂×C₃	#5111	1	1	yes	characteristic
320	C₂×C₃	#5145	1	1	yes	characteristic
321	C₂×C₃	Dic₃×SD₁₆	1	1	yes	characteristic
322	C₂×C₃	#5197	1	1	yes	characteristic
323	D₃	—	8	1	no	—
324	C₂²	—	12	1	no	—
325	C₂²	—	12	1	no	—
326	C₂²	—	12	1	no	—
327	C₂²	Dic₃×#67	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	12	1	no	—
332	C₄	#63814	1	1	yes	characteristic
333	C₄	C₂×#3711	1	1	yes	characteristic
334	C₄	D₃×#215	1	1	yes	characteristic
335	C₄	#63947	1	1	yes	characteristic
336	C₃	—	2	1	no	—
337	C₃	#70896	1	1	yes	characteristic, complemented
338	C₃	#70933	1	1	yes	characteristic, complemented
339	C₂	—	24	1	no	—
340	C₂	#10591155	1	1	yes	characteristic
341	C₂	Dic₃×#215	1	1	yes	characteristic
342	C₂	#10591674	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×Q₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×D₁₂	C₂×C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×Q₈	D₃²
hypocenter	C₃²	#1583
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×C₃⋊C₈	C₂	semidirect
4. #10589401	C₂	semidirect
5. #10589097	C₂	semidirect
7. #10589078	C₂	semidirect
28. Q₈×C₃⋊C₈	D₃	semidirect
94. C₃²×Q₈	M₁₆	semidirect
111. C₃×C₃⋊C₈	SD₁₆	semidirect
195. C₃×Q₈	ΓL(1,𝔽₂₅)	semidirect
196. C₃×Q₈	#208	semidirect
237. C₃⋊C₈	#215	semidirect
292. C₃²	#1583	semidirect
295. Q₈	#3711	semidirect
337. C₃	#70896	semidirect
338. C₃	#70933	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#215	2
C₃×Q₈×C₃⋊C₈	2
#10590337	2
#10589401	2
#10589097	2
#10589080	2
#10589078	2
#70933	3
#70896	3