Subgroups — #11777354

Order 1152.

451 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#3714	C₂	1	1	yes	maximal, characteristic
4	C₃×#4454	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589097	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589082	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589081	C₂	1	1	yes	maximal, characteristic, complemented
8	#70840	—	3	1	no	maximal
9	#68077	—	3	1	no	maximal
10	C₂×C₈×C₃⋊D₃	—	2	1	no	—
11	C₄×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
12	C₂×C₃×D₃×C₈	—	2	1	no	—
13	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×#3714	C₄	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	#63647	—	2	1	no	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
18	C₃×(C₂×D₃)⋊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	#63575	C₄	1	1	yes	characteristic
23	#63564	C₄	1	1	yes	characteristic
24	#63563	C₄	1	1	yes	characteristic
25	C₃×C₈×D₄	—	3	1	no	—
26	C₄×#213	—	3	1	no	—
27	C₄×D₂₄	—	3	1	no	—
28	C₈×D₁₂	—	3	1	no	—
29	C₈×D₁₂	—	3	1	no	—
30	C₈×D₁₂	—	6	1	no	—
31	C₈×D₁₂	D₃	1	1	yes	characteristic, complemented
32	C₃×#322	—	3	1	no	—
33	D₄⋊(C₃⋊C₈)	—	3	1	no	—
34	#4483	—	3	1	no	—
35	#4480	—	3	1	no	—
36	#4459	—	3	1	no	—
37	#4459	—	3	1	no	—
38	#4457	—	3	1	no	—
39	#4454	D₃	1	1	yes	characteristic, complemented
40	C₂×C₄⋊(C₃⋊D₃)	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃⋊D₃	—	2	1	no	—
42	C₂×C₃×D₁₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×D₃	—	2	1	no	—
44	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
46	C₃²×C₄²	C₂³	1	1	yes	characteristic
47	#3752	—	2	1	no	—
48	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
50	C₈×C₃⋊D₃	—	4	2	no	—
51	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
52	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₃×D₃×C₈	—	4	2	no	—
58	#3714	C₈	1	4	yes	complemented
59	#1580	—	9	1	no	Sylow
60	C₂²×C₃×C₈	—	6	1	no	—
61	C₃×C₄×D₄	—	3	1	no	—
62	C₂×#213	—	3	1	no	—
63	C₂×D₂₄	—	3	1	no	—
64	C₂×D₃×C₈	—	2	1	no	—
65	C₂×D₃×C₈	—	6	2	no	—
66	C₂×D₃×C₈	—	6	1	no	—
67	C₂×D₃×C₈	—	6	1	no	—
68	C₄×D₁₂	—	3	1	no	—
69	C₄×D₁₂	—	3	1	no	—
70	C₄×D₁₂	—	6	1	no	—
71	C₄×D₁₂	C₂×D₃	1	1	yes	characteristic
72	C₃×C₄⋊C₈	—	3	1	no	—
73	C₃×C₂²⋊C₈	—	6	1	no	—
74	C₃×C₄×C₈	—	2	1	no	—
75	C₃×C₄×C₈	—	3	1	no	—
76	C₃×C₄×C₈	—	3	1	no	—
77	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
78	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
79	#832	—	3	1	no	—
80	#821	—	3	1	no	—
81	#821	—	3	1	no	—
82	(C₂×D₃)⋊C₈	—	2	1	no	—
83	(C₂×D₃)⋊C₈	—	6	2	no	—
84	(C₂×D₃)⋊C₈	—	6	1	no	—
85	(C₂×D₃)⋊C₈	—	6	1	no	—
86	#818	—	3	1	no	—
87	#809	—	3	1	no	—
88	#807	—	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	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
94	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
95	C₂²×C₃⋊D₃	—	2	1	no	—
96	C₂²×C₃×D₃	—	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	C₄⋊(C₃⋊D₃)	—	2	1	no	—
102	C₄⋊(C₃⋊D₃)	C₂×C₈	1	2	yes	complemented
103	C₄×C₃⋊D₃	—	4	1	no	—
104	C₄×C₃⋊D₃	—	4	1	no	—
105	C₂×C₃×Dic₃	—	2	1	no	—
106	C₃×D₁₂	—	2	1	no	—
107	C₃×D₁₂	C₂×C₈	1	2	yes	complemented
108	C₃×C₄×D₃	—	4	1	no	—
109	C₃×C₄×D₃	—	4	1	no	—
110	C₃²×C₈	—	2	1	no	—
111	C₃²×C₈	—	2	1	no	—
112	C₃⋊(C₃⋊C₈)	—	4	1	no	—
113	C₃×C₃⋊C₈	—	2	1	no	—
114	C₃×C₃⋊C₈	—	4	1	no	—
115	C₃×C₃⋊C₈	—	4	1	no	—
116	C₃×C₃⋊C₈	C₂×C₈	1	2	yes	complemented
117	C₄×D₈	—	9	1	no	—
118	C₈×D₄	—	3	1	no	—
119	C₈×D₄	—	9	1	no	—
120	#334	—	9	1	no	—
121	D₄⋊C₈	—	9	1	no	—
122	D₄⋊C₈	—	9	1	no	—
123	#322	—	3	1	no	—
124	C₂×C₃×D₄	—	3	1	no	—
125	C₂²×C₃×C₄	—	6	1	no	—
126	C₂×D₁₂	—	3	1	no	—
127	C₂×D₁₂	—	3	1	no	—
128	C₂×D₁₂	—	6	1	no	—
129	C₂×D₁₂	C₄×D₃	1	1	yes	characteristic
130	C₂×C₄×D₃	—	2	1	no	—
131	C₂×C₄×D₃	—	6	2	no	—
132	C₂×C₄×D₃	—	6	1	no	—
133	C₂×C₄×D₃	—	6	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	3	1	no	—
137	C₂×C₃×C₈	—	3	1	no	—
138	C₂×C₃×C₈	—	6	1	no	—
139	C₂×C₃×C₈	—	6	1	no	—
140	C₂×C₃×C₈	—	12	2	no	—
141	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
142	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
143	C₂×C₃×C₈	#67	1	1	yes	characteristic
144	C₂×C₃×C₈	#67	1	1	yes	characteristic
145	C₃×C₄⋊C₄	—	3	1	no	—
146	C₃×C₂²⋊C₄	—	6	1	no	—
147	C₃×C₄²	—	2	1	no	—
148	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
149	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
150	#213	—	3	4	no	—
151	(C₂×D₃)⋊C₄	—	2	1	no	—
152	(C₂×D₃)⋊C₄	—	6	2	no	—
153	(C₂×D₃)⋊C₄	—	6	1	no	—
154	(C₂×D₃)⋊C₄	—	6	1	no	—
155	C₄⋊Dic₃	—	3	1	no	—
156	C₄⋊Dic₃	—	3	1	no	—
157	C₄⋊Dic₃	—	6	1	no	—
158	C₄⋊Dic₃	C₄×D₃	1	1	yes	characteristic
159	C₂×C₃⋊C₈	—	2	1	no	—
160	C₂×C₃⋊C₈	—	2	1	no	—
161	C₂×C₃⋊C₈	—	6	2	no	—
162	C₂×C₃⋊C₈	—	6	1	no	—
163	C₂×C₃⋊C₈	—	6	1	no	—
164	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
165	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
166	D₂₄	—	3	4	no	—
167	D₃×C₈	—	4	2	no	—
168	D₃×C₈	—	12	4	no	—
169	D₃×C₈	—	12	2	no	—
170	D₃×C₈	—	12	2	no	—
171	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
172	C₂×C₃⋊D₃	—	2	2	no	—
173	C₂×C₃⋊D₃	—	4	1	no	—
174	C₂×C₃⋊D₃	—	4	1	no	—
175	C₂×C₃×D₃	—	2	2	no	—
176	C₂×C₃×D₃	—	4	1	no	—
177	C₂×C₃×D₃	—	4	1	no	—
178	C₃²×C₄	—	2	1	no	—
179	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
180	C₃²×C₄	#131	1	1	yes	characteristic
181	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₃⋊Dic₃	—	4	1	no	—
184	C₃×Dic₃	—	4	1	no	—
185	C₂×D₈	—	9	1	no	—
186	C₂²×C₈	—	6	1	no	—
187	C₂²×C₈	—	18	1	no	—
188	C₄×D₄	—	3	1	no	—
189	C₄×D₄	—	9	1	no	—
190	#107	—	9	1	no	—
191	C₄⋊C₈	—	3	1	no	—
192	C₄⋊C₈	—	9	1	no	—
193	#102	—	9	1	no	—
194	#102	—	9	1	no	—
195	C₂²⋊C₈	—	6	1	no	—
196	C₂²⋊C₈	—	18	1	no	—
197	C₄×C₈	—	3	1	no	—
198	C₄×C₈	—	3	1	no	—
199	C₄×C₈	D₃²	1	1	yes	characteristic
200	C₂³×C₃	—	6	1	no	—
201	C₂²×D₃	—	2	1	no	—
202	C₂²×D₃	—	6	2	no	—
203	C₂²×D₃	—	6	1	no	—
204	C₂²×D₃	—	6	1	no	—
205	C₃×D₄	—	3	2	no	—
206	C₃×D₄	—	6	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	6	1	no	—
211	C₂×C₃×C₄	—	12	1	no	—
212	C₂×C₃×C₄	—	12	1	no	—
213	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
216	C₂×C₃×C₄	#235	1	1	yes	characteristic
217	C₂×C₃×C₄	#235	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
219	C₂×Dic₃	—	2	1	no	—
220	C₂×Dic₃	—	6	2	no	—
221	C₂×Dic₃	—	6	1	no	—
222	C₂×Dic₃	—	6	1	no	—
223	D₁₂	—	2	1	no	—
224	D₁₂	—	3	2	no	—
225	D₁₂	—	3	2	no	—
226	D₁₂	—	6	2	no	—
227	D₁₂	—	6	2	no	—
228	D₁₂	—	6	1	no	—
229	D₁₂	—	6	1	no	—
230	D₁₂	D₃×C₈	1	2	yes	complemented
231	C₄×D₃	—	4	1	no	—
232	C₄×D₃	—	4	1	no	—
233	C₄×D₃	—	12	2	no	—
234	C₄×D₃	—	12	2	no	—
235	C₄×D₃	—	12	1	no	—
236	C₄×D₃	—	12	1	no	—
237	C₄×D₃	—	12	1	no	—
238	C₄×D₃	—	12	1	no	—
239	C₃×C₈	—	2	2	no	—
240	C₃×C₈	—	2	1	no	—
241	C₃×C₈	—	2	1	no	—
242	C₃×C₈	—	2	1	no	—
243	C₃×C₈	—	2	1	no	—
244	C₃×C₈	—	3	2	no	—
245	C₃×C₈	—	4	1	no	—
246	C₃×C₈	—	6	1	no	—
247	C₃×C₈	—	12	1	no	—
248	C₃×C₈	—	12	1	no	—
249	C₃⋊C₈	—	2	1	no	—
250	C₃⋊C₈	—	4	1	no	—
251	C₃⋊C₈	—	4	1	no	—
252	C₃⋊C₈	—	12	2	no	—
253	C₃⋊C₈	—	12	1	no	—
254	C₃⋊C₈	—	12	1	no	—
255	C₃⋊C₈	D₃×C₈	1	2	yes	complemented
256	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
257	C₂×C₃²	#442	1	1	yes	characteristic
258	C₂×C₃²	#444	1	1	yes	characteristic
259	C₃⋊D₃	—	4	2	no	—
260	C₃×D₃	—	4	2	no	—
261	C₂×D₄	—	3	1	no	—
262	C₂×D₄	—	9	1	no	—
263	C₂²×C₄	—	6	1	no	—
264	C₂²×C₄	—	18	1	no	—
265	D₈	—	9	4	no	—
266	C₂×C₈	—	3	1	no	—
267	C₂×C₈	—	3	1	no	—
268	C₂×C₈	—	6	1	no	—
269	C₂×C₈	—	6	1	no	—
270	C₂×C₈	—	12	2	no	—
271	C₂×C₈	—	18	1	no	—
272	C₂×C₈	—	36	2	no	—
273	C₂×C₈	#626	1	1	yes	characteristic
274	C₂×C₈	#626	1	1	yes	characteristic
275	C₄⋊C₄	—	3	1	no	—
276	C₄⋊C₄	—	9	1	no	—
277	C₂²⋊C₄	—	6	1	no	—
278	C₂²⋊C₄	—	18	1	no	—
279	C₄²	C₂×D₃²	1	1	yes	characteristic
280	C₂²×C₃	—	2	1	no	—
281	C₂²×C₃	—	6	2	no	—
282	C₂²×C₃	—	12	1	no	—
283	C₂²×C₃	—	12	1	no	—
284	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
285	C₂²×C₃	C₄×#67	1	1	yes	characteristic
286	C₂×D₃	—	2	2	no	—
287	C₂×D₃	—	4	1	no	—
288	C₂×D₃	—	4	1	no	—
289	C₂×D₃	—	6	4	no	—
290	C₂×D₃	—	6	2	no	—
291	C₂×D₃	—	6	2	no	—
292	C₂×D₃	—	12	2	no	—
293	C₂×D₃	—	12	2	no	—
294	C₂×D₃	—	12	1	no	—
295	C₂×D₃	—	12	1	no	—
296	C₂×D₃	—	12	1	no	—
297	C₂×D₃	—	12	1	no	—
298	C₃×C₄	—	2	1	no	—
299	C₃×C₄	—	2	1	no	—
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	4	1	no	—
305	C₃×C₄	—	12	1	no	—
306	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
307	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
308	C₃×C₄	#901	1	1	yes	characteristic
309	C₃×C₄	#901	1	1	yes	characteristic
310	C₃×C₄	#908	1	1	yes	characteristic
311	C₃×C₄	#908	1	1	yes	characteristic
312	C₃×C₄	#932	1	1	yes	characteristic
313	C₃×C₄	#949	1	1	yes	characteristic
314	Dic₃	—	4	1	no	—
315	Dic₃	—	12	2	no	—
316	Dic₃	—	12	1	no	—
317	Dic₃	—	12	1	no	—
318	C₃²	#1580	1	1	yes	characteristic, complemented, Sylow
319	C₂³	—	6	1	no	—
320	C₂³	—	18	1	no	—
321	D₄	—	3	2	no	—
322	D₄	—	6	1	no	—
323	D₄	—	9	2	no	—
324	D₄	—	18	1	no	—
325	C₂×C₄	—	6	1	no	—
326	C₂×C₄	—	12	1	no	—
327	C₂×C₄	—	12	1	no	—
328	C₂×C₄	—	18	1	no	—
329	C₂×C₄	—	36	1	no	—
330	C₂×C₄	—	36	1	no	—
331	C₂×C₄	#3798	1	1	yes	characteristic
332	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
333	C₂×C₄	C₂×#626	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	3	2	no	—
337	C₈	—	6	1	no	—
338	C₈	—	12	1	no	—
339	C₈	—	12	1	no	—
340	C₈	—	36	1	no	—
341	C₂×C₃	—	2	1	no	—
342	C₂×C₃	—	2	1	no	—
343	C₂×C₃	—	2	1	no	—
344	C₂×C₃	—	12	2	no	—
345	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
346	C₂×C₃	#4711	1	1	yes	characteristic
347	C₂×C₃	#4717	1	1	yes	characteristic
348	C₂×C₃	#5014	1	1	yes	characteristic
349	C₂×C₃	C₈×#67	1	1	yes	characteristic
350	C₂×C₃	#5145	1	1	yes	characteristic
351	D₃	—	4	2	no	—
352	D₃	—	12	4	no	—
353	D₃	—	12	2	no	—
354	D₃	—	12	2	no	—
355	C₂²	—	6	2	no	—
356	C₂²	—	12	1	no	—
357	C₂²	—	12	1	no	—
358	C₂²	—	18	2	no	—
359	C₂²	—	36	1	no	—
360	C₂²	—	36	1	no	—
361	C₂²	C₄×#626	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	12	1	no	—
364	C₄	—	36	1	no	—
365	C₄	D₃²×C₈	1	1	yes	characteristic
366	C₄	#63803	1	1	yes	characteristic
367	C₄	#63825	1	1	yes	characteristic
368	C₄	#63830	1	1	yes	characteristic
369	C₃	—	2	1	no	—
370	C₃	#68077	1	1	yes	characteristic, complemented
371	C₃	#70840	1	1	yes	characteristic, complemented
372	C₂	—	12	2	no	—
373	C₂	—	36	2	no	—
374	C₂	#10590346	1	1	yes	characteristic
375	C₂	C₈×#626	1	1	yes	characteristic
376	C₂	#10590924	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#626
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×#3714	C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#1580
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₈×C₄⋊(C₃⋊D₃)	C₂	semidirect
2. C₃×C₈×D₁₂	C₂	semidirect
4. C₃×#4454	C₂	semidirect
5. #10589097	C₂	semidirect
6. #10589082	C₂	semidirect
7. #10589081	C₂	semidirect
31. C₈×D₁₂	D₃	semidirect
39. #4454	D₃	semidirect
58. #3714	C₈	semidirect
102. C₄⋊(C₃⋊D₃)	C₂×C₈	semidirect
107. C₃×D₁₂	C₂×C₈	semidirect
116. C₃×C₃⋊C₈	C₂×C₈	semidirect
230. D₁₂	D₃×C₈	semidirect
255. C₃⋊C₈	D₃×C₈	semidirect
318. C₃²	#1580	semidirect
370. C₃	#68077	semidirect
371. C₃	#70840	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊(C₃⋊D₃)	2
C₃×C₈×D₁₂	2
C₄×#3714	2
C₃×#4454	2
#10589097	2
#10589082	2
#10589081	2
#70840	3
#68077	3