Subgroups — #11777650

Order 1152.

391 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₂₄	C₂	1	1	yes	maximal, characteristic
2	#10590337	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10589323	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4454	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589081	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589081	C₂	1	1	yes	maximal, characteristic, complemented
8	#70880	—	3	1	no	maximal
9	#68081	—	3	1	no	maximal
10	C₂×C₃×D₂₄	C₄	1	1	yes	characteristic
11	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
12	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×#3711	—	2	1	no	—
14	C₂×D₃×C₃⋊C₈	—	2	1	no	—
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×#821	C₄	1	1	yes	characteristic
19	C₃×#821	C₄	1	1	yes	characteristic
20	C₃×#818	C₄	1	1	yes	characteristic
21	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	(C₂×D₃)⋊(C₃⋊C₈)	—	2	1	no	—
24	(C₂×D₃)⋊(C₃⋊C₈)	—	2	1	no	—
25	C₃×C₄×D₈	—	3	1	no	—
26	#5012	—	3	1	no	—
27	D₄×C₃⋊C₈	—	3	1	no	—
28	C₄×D₂₄	D₃	1	1	yes	characteristic
29	#4688	—	3	1	no	—
30	C₈×D₁₂	—	3	1	no	—
31	C₃×#322	—	3	1	no	—
32	D₄⋊(C₃⋊C₈)	—	3	1	no	—
33	D₄⋊(C₃⋊C₈)	—	3	1	no	—
34	#4459	—	3	1	no	—
35	#4459	—	3	1	no	—
36	#4457	—	3	1	no	—
37	#4457	—	3	1	no	—
38	#4457	—	6	1	no	—
39	#4454	D₃	1	1	yes	characteristic, complemented
40	C₂×C₃×D₁₂	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	C₂×C₃×C₄×D₃	—	2	1	no	—
44	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
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×(C₂×D₃)⋊C₄	—	2	1	no	—
50	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
51	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
55	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
56	C₃×D₂₄	—	2	2	no	—
57	#3711	—	4	2	no	—
58	D₃×C₃⋊C₈	—	4	2	no	—
59	#1586	—	9	1	no	Sylow
60	C₂×C₃×D₈	—	3	1	no	—
61	C₃×C₄×D₄	—	3	1	no	—
62	C₃×C₄×D₄	—	3	1	no	—
63	C₂×#208	—	6	1	no	—
64	C₂²×C₃⋊C₈	—	6	1	no	—
65	C₂×D₂₄	Dic₃	1	1	yes	characteristic
66	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
67	C₂×D₃×C₈	—	6	1	no	—
68	C₄×D₁₂	C₂×D₃	1	1	yes	characteristic
69	C₄×D₁₂	C₂×D₃	1	1	yes	characteristic
70	C₃×#107	—	3	1	no	—
71	C₃×#102	—	3	1	no	—
72	C₃×#102	—	3	1	no	—
73	C₃×C₄×C₈	—	2	1	no	—
74	C₃×C₄×C₈	—	3	1	no	—
75	C₃×C₄×C₈	—	3	1	no	—
76	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
77	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
78	#830	—	6	1	no	—
79	#830	—	6	1	no	—
80	#821	Dic₃	1	1	yes	characteristic
81	#821	Dic₃	1	1	yes	characteristic
82	(C₂×D₃)⋊C₈	—	6	1	no	—
83	(C₂×D₃)⋊C₈	—	6	1	no	—
84	#818	Dic₃	1	1	yes	characteristic
85	C₄⋊(C₃⋊C₈)	—	3	1	no	—
86	C₄⋊(C₃⋊C₈)	—	3	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)	—	6	1	no	—
91	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
92	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
93	C₂²×C₃×D₃	—	2	1	no	—
94	C₂²×C₃×D₃	—	2	1	no	—
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	#41	1	1	yes	characteristic
98	C₂×C₃×Dic₃	—	2	1	no	—
99	C₂×C₃×Dic₃	—	2	1	no	—
100	C₃×D₁₂	—	2	1	no	—
101	C₃×D₁₂	—	2	1	no	—
102	C₃×D₁₂	—	2	1	no	—
103	C₃×D₁₂	M₁₆	1	2	yes	complemented
104	C₃×C₄×D₃	—	4	1	no	—
105	C₃×C₄×D₃	—	4	1	no	—
106	C₃×C₄×D₃	—	4	1	no	—
107	C₃×C₄×D₃	—	4	1	no	—
108	C₃²×C₈	—	2	1	no	—
109	C₃²×C₈	—	2	1	no	—
110	C₃⋊(C₃⋊C₈)	—	4	1	no	—
111	C₃⋊(C₃⋊C₈)	—	4	1	no	—
112	C₃×C₃⋊C₈	—	2	2	no	—
113	C₃×C₃⋊C₈	—	2	1	no	—
114	C₃×C₃⋊C₈	—	2	1	no	—
115	C₄×D₈	—	3	1	no	—
116	#436	—	9	1	no	—
117	C₈×D₄	—	9	1	no	—
118	#334	—	9	1	no	—
119	D₄⋊C₈	—	9	1	no	—
120	D₄⋊C₈	—	9	1	no	—
121	#322	—	3	1	no	—
122	C₂×C₃×D₄	—	3	1	no	—
123	C₂×C₃×D₄	—	3	1	no	—
124	C₂²×C₃×C₄	—	6	1	no	—
125	C₂²×C₃×C₄	—	6	1	no	—
126	C₂×D₁₂	C₂×Dic₃	1	1	yes	characteristic
127	C₂×D₁₂	C₂×Dic₃	1	1	yes	characteristic
128	C₂×C₄×D₃	—	2	1	no	—
129	C₂×C₄×D₃	—	2	1	no	—
130	C₃×D₈	—	6	2	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	3	1	no	—
134	C₂×C₃×C₈	—	3	1	no	—
135	C₂×C₃×C₈	—	6	1	no	—
136	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
137	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
140	C₃×C₄⋊C₄	—	3	1	no	—
141	C₃×C₄⋊C₄	—	3	1	no	—
142	C₃×C₂²⋊C₄	—	6	1	no	—
143	C₃×C₂²⋊C₄	—	6	1	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
146	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
147	(C₂×D₃)⋊C₄	—	2	1	no	—
148	(C₂×D₃)⋊C₄	—	2	1	no	—
149	C₄⋊Dic₃	C₂×Dic₃	1	1	yes	characteristic
150	C₄⋊Dic₃	C₂×Dic₃	1	1	yes	characteristic
151	#208	—	12	2	no	—
152	C₂×C₃⋊C₈	—	2	1	no	—
153	C₂×C₃⋊C₈	—	6	1	no	—
154	C₂×C₃⋊C₈	—	6	1	no	—
155	C₂×C₃⋊C₈	—	6	1	no	—
156	C₂×C₃⋊C₈	—	6	1	no	—
157	C₂×C₃⋊C₈	—	12	1	no	—
158	C₂×C₃⋊C₈	—	12	1	no	—
159	C₂×C₃⋊C₈	—	12	2	no	—
160	C₂×C₃⋊C₈	D₁₂	1	1	yes	characteristic
161	C₂×C₃⋊C₈	D₁₂	1	1	yes	characteristic
162	D₂₄	—	2	2	no	—
163	ΓL(1,𝔽₂₅)	—	12	2	no	—
164	D₃×C₈	—	12	2	no	—
165	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
166	C₂×C₃×D₃	—	2	2	no	—
167	C₂×C₃×D₃	—	4	1	no	—
168	C₂×C₃×D₃	—	4	1	no	—
169	C₂×C₃×D₃	—	4	1	no	—
170	C₂×C₃×D₃	—	4	1	no	—
171	C₂×C₃×D₃	—	4	1	no	—
172	C₃²×C₄	—	2	1	no	—
173	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
174	C₃²×C₄	#131	1	1	yes	characteristic
175	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
176	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₃×Dic₃	—	4	1	no	—
178	C₃×Dic₃	—	4	1	no	—
179	C₂×D₈	—	3	1	no	—
180	C₂×M₁₆	—	18	1	no	—
181	C₂²×C₈	—	18	1	no	—
182	C₄×D₄	—	3	1	no	—
183	C₄×D₄	—	3	1	no	—
184	#107	—	3	1	no	—
185	C₄⋊C₈	—	9	1	no	—
186	C₄⋊C₈	—	9	1	no	—
187	#102	—	3	1	no	—
188	#102	—	3	1	no	—
189	C₂²⋊C₈	—	18	1	no	—
190	C₂²⋊C₈	—	18	1	no	—
191	C₄×C₈	—	3	1	no	—
192	C₄×C₈	—	3	1	no	—
193	C₄×C₈	D₃²	1	1	yes	characteristic
194	C₂³×C₃	—	6	1	no	—
195	C₂³×C₃	—	6	1	no	—
196	C₂²×D₃	—	2	1	no	—
197	C₂²×D₃	—	2	1	no	—
198	C₃×D₄	—	3	2	no	—
199	C₃×D₄	—	6	1	no	—
200	C₃×D₄	—	6	1	no	—
201	C₃×D₄	—	6	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	—	6	1	no	—
206	C₂×C₃×C₄	—	6	1	no	—
207	C₂×C₃×C₄	—	12	1	no	—
208	C₂×C₃×C₄	—	12	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	C₂×D₁₂	1	1	yes	characteristic
213	C₂×C₃×C₄	#235	1	1	yes	characteristic
214	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	#237	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
217	C₂×Dic₃	—	2	1	no	—
218	C₂×Dic₃	—	2	1	no	—
219	D₁₂	—	2	1	no	—
220	D₁₂	—	2	1	no	—
221	D₁₂	—	2	1	no	—
222	D₁₂	#208	1	2	yes	complemented
223	C₄×D₃	—	4	1	no	—
224	C₄×D₃	—	4	1	no	—
225	C₄×D₃	—	4	1	no	—
226	C₄×D₃	—	4	1	no	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃×C₈	—	4	1	no	—
233	C₃×C₈	—	6	2	no	—
234	C₃×C₈	—	6	1	no	—
235	C₃×C₈	—	6	1	no	—
236	C₃⋊C₈	—	2	2	no	—
237	C₃⋊C₈	—	2	1	no	—
238	C₃⋊C₈	—	2	1	no	—
239	C₃⋊C₈	—	12	1	no	—
240	C₃⋊C₈	—	12	1	no	—
241	C₃⋊C₈	—	12	1	no	—
242	C₃⋊C₈	—	12	1	no	—
243	C₃⋊C₈	—	24	1	no	—
244	C₃⋊C₈	—	24	1	no	—
245	C₂×C₃²	#435	1	1	yes	characteristic
246	C₂×C₃²	#442	1	1	yes	characteristic
247	C₂×C₃²	#443	1	1	yes	characteristic
248	C₃×D₃	—	4	2	no	—
249	C₃×D₃	—	8	1	no	—
250	C₂×D₄	—	3	1	no	—
251	C₂×D₄	—	3	1	no	—
252	C₂²×C₄	—	6	1	no	—
253	C₂²×C₄	—	6	1	no	—
254	D₈	—	6	2	no	—
255	M₁₆	—	36	2	no	—
256	C₂×C₈	—	3	1	no	—
257	C₂×C₈	—	3	1	no	—
258	C₂×C₈	—	6	1	no	—
259	C₂×C₈	—	18	1	no	—
260	C₂×C₈	—	18	1	no	—
261	C₂×C₈	—	36	2	no	—
262	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
263	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
264	C₄⋊C₄	—	3	1	no	—
265	C₄⋊C₄	—	3	1	no	—
266	C₂²⋊C₄	—	6	1	no	—
267	C₂²⋊C₄	—	6	1	no	—
268	C₄²	C₂×D₃²	1	1	yes	characteristic
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	6	2	no	—
271	C₂²×C₃	—	12	1	no	—
272	C₂²×C₃	—	12	1	no	—
273	C₂²×C₃	—	12	1	no	—
274	C₂²×C₃	—	12	1	no	—
275	C₂²×C₃	—	12	1	no	—
276	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
277	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
278	C₂×D₃	—	2	2	no	—
279	C₂×D₃	—	4	1	no	—
280	C₂×D₃	—	4	1	no	—
281	C₂×D₃	—	4	1	no	—
282	C₂×D₃	—	4	1	no	—
283	C₂×D₃	—	4	1	no	—
284	C₃×C₄	—	2	1	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	—	2	1	no	—
290	C₃×C₄	—	4	1	no	—
291	C₃×C₄	—	12	1	no	—
292	C₃×C₄	—	12	1	no	—
293	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
294	C₃×C₄	#907	1	1	yes	characteristic
295	C₃×C₄	#908	1	1	yes	characteristic
296	C₃×C₄	#908	1	1	yes	characteristic
297	C₃×C₄	#911	1	1	yes	characteristic
298	C₃×C₄	#911	1	1	yes	characteristic
299	C₃×C₄	C₂×#208	1	1	yes	characteristic
300	C₃×C₄	#948	1	1	yes	characteristic
301	Dic₃	—	4	1	no	—
302	Dic₃	—	4	1	no	—
303	C₃²	#1586	1	1	yes	characteristic, complemented, Sylow
304	C₂³	—	6	1	no	—
305	C₂³	—	6	1	no	—
306	D₄	—	3	2	no	—
307	D₄	—	6	1	no	—
308	D₄	—	6	1	no	—
309	D₄	—	6	1	no	—
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	—	6	1	no	—
312	C₂×C₄	—	12	1	no	—
313	C₂×C₄	—	12	1	no	—
314	C₂×C₄	—	12	1	no	—
315	C₂×C₄	—	12	1	no	—
316	C₂×C₄	#3795	1	1	yes	characteristic
317	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
318	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	6	2	no	—
322	C₈	—	6	1	no	—
323	C₈	—	6	1	no	—
324	C₈	—	36	1	no	—
325	C₈	—	36	1	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	12	2	no	—
330	C₂×C₃	—	24	1	no	—
331	C₂×C₃	#4700	1	1	yes	characteristic
332	C₂×C₃	#4706	1	1	yes	characteristic
333	C₂×C₃	#4711	1	1	yes	characteristic
334	C₂×C₃	#5011	1	1	yes	characteristic
335	C₂×C₃	#5152	1	1	yes	characteristic
336	C₂×C₃	#5196	1	1	yes	characteristic
337	D₃	—	4	2	no	—
338	D₃	—	8	1	no	—
339	C₂²	—	6	2	no	—
340	C₂²	—	12	1	no	—
341	C₂²	—	12	1	no	—
342	C₂²	—	12	1	no	—
343	C₂²	—	12	1	no	—
344	C₂²	—	12	1	no	—
345	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	12	1	no	—
348	C₄	—	12	1	no	—
349	C₄	#63795	1	1	yes	characteristic
350	C₄	#63795	1	1	yes	characteristic
351	C₄	D₃×#208	1	1	yes	characteristic
352	C₄	#63815	1	1	yes	characteristic
353	C₃	—	2	1	no	—
354	C₃	#68081	1	1	yes	characteristic, complemented
355	C₃	#70880	1	1	yes	characteristic, complemented
356	C₂	—	12	2	no	—
357	C₂	—	24	1	no	—
358	C₂	#10590326	1	1	yes	characteristic
359	C₂	#10590771	1	1	yes	characteristic
360	C₂	#10590915	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₁₂
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×D₂₄	C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#1586
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
2. #10590337	C₂	semidirect
3. C₃⋊C₈×D₁₂	C₂	semidirect
4. #10589323	C₂	semidirect
5. C₃×#4454	C₂	semidirect
6. #10589081	C₂	semidirect
7. #10589081	C₂	semidirect
39. #4454	D₃	semidirect
103. C₃×D₁₂	M₁₆	semidirect
222. D₁₂	#208	semidirect
303. C₃²	#1586	semidirect
354. C₃	#68081	semidirect
355. C₃	#70880	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₂₄	2
#10590337	2
C₃⋊C₈×D₁₂	2
#10589323	2
C₃×#4454	2
#10589081	2
#70880	3
#68081	3