Subgroups — #11797448

Order 1152.

415 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#638	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63542	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4549	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589133	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589108	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×#67	—	3	1	no	maximal
9	#67⋊C₁₆	—	3	1	no	maximal
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₄×#638	C₄	1	1	yes	characteristic
12	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
13	#63657	C₄	1	1	yes	characteristic
14	#63647	C₄	1	1	yes	characteristic
15	#63641	C₄	1	1	yes	characteristic
16	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
22	#63542	—	2	2	no	—
23	C₂²×C₃×C₁₆	—	3	1	no	—
24	C₈×#67	—	3	1	no	—
25	C₈×#67	—	3	1	no	—
26	C₈×#67	—	6	1	no	—
27	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic, complemented
28	C₂×D₃×C₁₆	—	3	1	no	—
29	C₂×D₃×C₁₆	—	3	1	no	—
30	C₃×C₂²⋊C₁₆	—	3	1	no	—
31	#4549	D₃	1	1	yes	characteristic, complemented
32	(C₂×D₃)⋊C₁₆	—	3	1	no	—
33	(C₂×D₃)⋊C₁₆	—	3	1	no	—
34	Dic₃⋊C₁₆	—	3	1	no	—
35	Dic₃⋊C₁₆	—	3	1	no	—
36	Dic₃×C₁₆	—	3	1	no	—
37	Dic₃×C₁₆	—	3	1	no	—
38	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×#638	C₈	1	1	yes	characteristic
40	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	—	2	1	no	—
42	C₂×C₃²×C₈	—	2	1	no	—
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
45	#3757	C₈	1	1	yes	characteristic
46	#3752	C₈	1	1	yes	characteristic
47	#3750	C₈	1	1	yes	characteristic
48	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
51	C₈×C₃⋊D₃	—	2	1	no	—
52	C₈×C₃⋊D₃	—	2	1	no	—
53	C₃×C₃⋊C₁₆	—	2	1	no	—
54	C₃×C₃⋊C₁₆	—	2	1	no	—
55	C₃×C₃⋊C₁₆	—	2	1	no	—
56	C₃×C₃⋊C₁₆	D₄	1	2	yes	complemented
57	D₄×C₁₆	—	9	1	no	Sylow
58	C₂²×C₃×C₈	—	2	1	no	—
59	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₄×#67	—	3	1	no	—
62	C₄×#67	—	3	1	no	—
63	C₄×#67	—	6	1	no	—
64	C₂×D₃×C₈	—	3	1	no	—
65	C₂×D₃×C₈	—	3	1	no	—
66	C₂×D₃×C₈	—	6	1	no	—
67	C₂×C₃×C₁₆	—	3	1	no	—
68	C₂×C₃×C₁₆	—	3	1	no	—
69	C₂×C₃×C₁₆	—	3	1	no	—
70	C₂×C₃×C₁₆	—	3	1	no	—
71	C₂×C₃×C₁₆	—	6	2	no	—
72	#830	—	3	1	no	—
73	#830	—	3	1	no	—
74	#830	—	6	1	no	—
75	(C₂×D₃)⋊C₈	—	3	1	no	—
76	(C₂×D₃)⋊C₈	—	3	1	no	—
77	(C₂×D₃)⋊C₈	—	6	1	no	—
78	Dic₃⋊C₈	—	3	1	no	—
79	Dic₃⋊C₈	—	3	1	no	—
80	Dic₃⋊C₈	—	6	1	no	—
81	C₈×Dic₃	—	3	1	no	—
82	C₈×Dic₃	—	3	1	no	—
83	C₈×Dic₃	—	6	1	no	—
84	C₂×C₃⋊C₁₆	—	2	2	no	—
85	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
86	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
87	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
88	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
89	D₃×C₁₆	—	6	2	no	—
90	D₃×C₁₆	—	6	2	no	—
91	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
93	C₂×C₃²×C₄	—	2	1	no	—
94	C₂×C₃²×C₄	—	2	1	no	—
95	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
97	#638	C₁₆	1	4	yes	complemented
98	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
99	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
100	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
101	C₄×C₃⋊D₃	—	2	1	no	—
102	C₄×C₃⋊D₃	—	2	1	no	—
103	C₃²×C₈	—	2	1	no	—
104	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
105	C₃²×C₈	#41	1	1	yes	characteristic
106	C₃⋊(C₃⋊C₈)	—	2	1	no	—
107	C₃⋊(C₃⋊C₈)	—	2	1	no	—
108	C₂²×C₁₆	—	3	1	no	—
109	C₂²×C₁₆	—	9	1	no	—
110	C₈×D₄	—	9	1	no	—
111	C₄⋊C₁₆	—	9	1	no	—
112	C₂²⋊C₁₆	—	3	1	no	—
113	C₂²⋊C₁₆	—	9	1	no	—
114	C₄×C₁₆	—	9	1	no	—
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
118	C₂×#67	—	3	1	no	—
119	C₂×#67	—	3	1	no	—
120	C₂×#67	—	6	1	no	—
121	C₂×C₄×D₃	—	3	1	no	—
122	C₂×C₄×D₃	—	3	1	no	—
123	C₂×C₄×D₃	—	6	1	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	2	1	no	—
126	C₂×C₃×C₈	—	2	1	no	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	4	1	no	—
131	C₂×C₃×C₈	—	4	1	no	—
132	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
133	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
136	#217	—	3	1	no	—
137	#217	—	3	1	no	—
138	#217	—	6	1	no	—
139	(C₂×D₃)⋊C₄	—	3	1	no	—
140	(C₂×D₃)⋊C₄	—	3	1	no	—
141	(C₂×D₃)⋊C₄	—	6	1	no	—
142	Dic₃⋊C₄	—	3	1	no	—
143	Dic₃⋊C₄	—	3	1	no	—
144	Dic₃⋊C₄	—	6	1	no	—
145	C₄×Dic₃	—	3	1	no	—
146	C₄×Dic₃	—	3	1	no	—
147	C₄×Dic₃	—	6	1	no	—
148	C₂×C₃⋊C₈	—	3	1	no	—
149	C₂×C₃⋊C₈	—	3	1	no	—
150	C₂×C₃⋊C₈	—	3	1	no	—
151	C₂×C₃⋊C₈	—	3	1	no	—
152	C₂×C₃⋊C₈	—	6	1	no	—
153	C₂×C₃⋊C₈	—	6	1	no	—
154	D₃×C₈	—	6	1	no	—
155	D₃×C₈	—	6	1	no	—
156	D₃×C₈	—	6	1	no	—
157	D₃×C₈	—	6	1	no	—
158	D₃×C₈	—	12	1	no	—
159	D₃×C₈	—	12	1	no	—
160	C₃×C₁₆	—	3	2	no	—
161	C₃×C₁₆	—	6	1	no	—
162	C₃×C₁₆	—	6	1	no	—
163	C₃×C₁₆	—	6	1	no	—
164	C₃⋊C₁₆	—	2	1	no	—
165	C₃⋊C₁₆	—	2	1	no	—
166	C₃⋊C₁₆	—	2	1	no	—
167	C₃⋊C₁₆	#67	1	2	yes	complemented
168	C₂²×C₃²	—	2	1	no	—
169	C₂²×C₃²	—	2	1	no	—
170	C₂²×C₃²	C₂×C₁₆	1	2	yes	complemented
171	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
172	C₂×C₃⋊D₃	—	2	1	no	—
173	C₂×C₃⋊D₃	—	2	1	no	—
174	C₂×C₃⋊D₃	C₂×C₁₆	1	2	yes	complemented
175	C₃²×C₄	—	2	1	no	—
176	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
177	C₃²×C₄	#131	1	1	yes	characteristic
178	C₃⋊Dic₃	—	2	1	no	—
179	C₃⋊Dic₃	—	2	1	no	—
180	C₃⋊Dic₃	C₂×C₁₆	1	2	yes	complemented
181	C₂²×C₈	—	9	1	no	—
182	C₂²×C₈	D₃²	1	1	yes	characteristic
183	C₄×D₄	—	9	1	no	—
184	C₂×C₁₆	—	3	1	no	—
185	C₂×C₁₆	—	3	1	no	—
186	C₂×C₁₆	—	3	1	no	—
187	C₂×C₁₆	—	3	1	no	—
188	C₂×C₁₆	—	6	2	no	—
189	C₂×C₁₆	—	18	2	no	—
190	C₄⋊C₈	—	9	1	no	—
191	C₂²⋊C₈	—	9	1	no	—
192	C₂²⋊C₈	—	9	1	no	—
193	C₄×C₈	—	9	1	no	—
194	C₂³×C₃	—	2	1	no	—
195	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
196	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
197	C₂²×D₃	—	3	1	no	—
198	C₂²×D₃	—	3	1	no	—
199	C₂²×D₃	—	6	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	D₃×C₈	1	1	yes	characteristic
209	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
212	#67	—	3	4	no	—
213	#67	—	3	4	no	—
214	#67	—	6	4	no	—
215	C₂×Dic₃	—	3	1	no	—
216	C₂×Dic₃	—	3	1	no	—
217	C₂×Dic₃	—	3	1	no	—
218	C₂×Dic₃	—	3	1	no	—
219	C₂×Dic₃	—	3	1	no	—
220	C₂×Dic₃	—	3	1	no	—
221	C₂×Dic₃	—	6	1	no	—
222	C₂×Dic₃	—	6	1	no	—
223	C₂×Dic₃	—	6	1	no	—
224	C₄×D₃	—	6	1	no	—
225	C₄×D₃	—	6	1	no	—
226	C₄×D₃	—	6	1	no	—
227	C₄×D₃	—	6	1	no	—
228	C₄×D₃	—	12	1	no	—
229	C₄×D₃	—	12	1	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	2	1	no	—
234	C₃×C₈	—	4	1	no	—
235	C₃×C₈	#235	1	1	yes	characteristic
236	C₃×C₈	D₃×D₄	1	1	yes	characteristic
237	C₃×C₈	#237	1	1	yes	characteristic
238	C₃×C₈	C₂×#67	1	1	yes	characteristic
239	C₃⋊C₈	—	6	1	no	—
240	C₃⋊C₈	—	6	1	no	—
241	C₃⋊C₈	—	6	1	no	—
242	C₃⋊C₈	—	6	1	no	—
243	C₃⋊C₈	—	12	1	no	—
244	C₃⋊C₈	—	12	1	no	—
245	C₂×C₃²	—	2	2	no	—
246	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
247	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
248	C₂×C₃²	#504	1	1	yes	characteristic
249	C₃⋊D₃	—	2	2	no	—
250	C₂×D₄	—	9	1	no	—
251	C₂²×C₄	—	9	1	no	—
252	C₂²×C₄	#624	1	1	yes	characteristic
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	9	1	no	—
256	C₂×C₈	—	9	1	no	—
257	C₂×C₈	—	18	1	no	—
258	C₂×C₈	—	18	1	no	—
259	C₂×C₈	#624	1	1	yes	characteristic
260	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
261	C₄⋊C₄	—	9	1	no	—
262	C₂²⋊C₄	—	9	1	no	—
263	C₂²⋊C₄	—	9	1	no	—
264	C₄²	—	9	1	no	—
265	C₁₆	—	3	2	no	—
266	C₁₆	—	6	1	no	—
267	C₁₆	—	6	1	no	—
268	C₁₆	—	6	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	2	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	4	1	no	—
276	C₂²×C₃	—	4	1	no	—
277	C₂²×C₃	D₃×C₁₆	1	2	yes	complemented
278	C₂²×C₃	D₃×C₁₆	1	2	yes	complemented
279	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
280	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
281	C₂×D₃	—	3	2	no	—
282	C₂×D₃	—	3	2	no	—
283	C₂×D₃	—	6	2	no	—
284	C₂×D₃	—	6	1	no	—
285	C₂×D₃	—	6	1	no	—
286	C₂×D₃	—	6	1	no	—
287	C₂×D₃	—	6	1	no	—
288	C₂×D₃	—	12	1	no	—
289	C₂×D₃	—	12	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	4	1	no	—
295	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
296	C₃×C₄	#901	1	1	yes	characteristic
297	C₃×C₄	#907	1	1	yes	characteristic
298	C₃×C₄	C₄×#67	1	1	yes	characteristic
299	Dic₃	—	3	2	no	—
300	Dic₃	—	3	2	no	—
301	Dic₃	—	6	2	no	—
302	Dic₃	—	6	1	no	—
303	Dic₃	—	6	1	no	—
304	Dic₃	—	6	1	no	—
305	Dic₃	—	6	1	no	—
306	Dic₃	—	12	1	no	—
307	Dic₃	—	12	1	no	—
308	C₃²	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
309	C₂³	—	9	1	no	—
310	C₂³	#3710	1	1	yes	characteristic
311	D₄	—	9	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	9	1	no	—
315	C₂×C₄	—	9	1	no	—
316	C₂×C₄	—	9	1	no	—
317	C₂×C₄	—	18	1	no	—
318	C₂×C₄	—	18	1	no	—
319	C₂×C₄	#3710	1	1	yes	characteristic
320	C₂×C₄	C₂×#624	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	18	1	no	—
323	C₈	—	18	1	no	—
324	C₈	#3804	1	1	yes	characteristic
325	C₈	D₃×#67	1	1	yes	characteristic
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	—	2	2	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	4	2	no	—
332	C₂×C₃	#67⋊C₈	1	1	yes	characteristic
333	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
334	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
335	C₂×C₃	#4901	1	1	yes	characteristic
336	C₂×C₃	#4907	1	1	yes	characteristic
337	C₂×C₃	C₈×#67	1	1	yes	characteristic
338	D₃	—	6	2	no	—
339	D₃	—	6	2	no	—
340	D₃	—	12	2	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	9	2	no	—
344	C₂²	—	18	1	no	—
345	C₂²	—	18	1	no	—
346	C₂²	#63542	1	2	yes	complemented
347	C₂²	C₂×#3710	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	9	2	no	—
350	C₄	—	18	1	no	—
351	C₄	—	18	1	no	—
352	C₄	#63817	1	1	yes	characteristic
353	C₄	#63952	1	1	yes	characteristic
354	C₃	—	2	1	no	—
355	C₃	#67⋊C₁₆	1	1	yes	characteristic, complemented
356	C₃	C₁₆×#67	1	1	yes	characteristic, complemented
357	C₂	—	2	2	no	—
358	C₂	—	18	2	no	—
359	C₂	C₂×#63542	1	1	yes	characteristic
360	C₂	#10590624	1	1	yes	characteristic
361	C₂	#10591148	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₁₆
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×#638	C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	D₄×C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₃⋊C₁₆	C₂	semidirect
3. C₂×#63542	C₂	semidirect
4. C₃×#4549	C₂	semidirect
5. #10589133	C₂	semidirect
6. #10589108	C₂	semidirect
7. #10589102	C₂	semidirect
27. C₂²×C₃⋊C₁₆	D₃	semidirect
31. #4549	D₃	semidirect
56. C₃×C₃⋊C₁₆	D₄	semidirect
97. #638	C₁₆	semidirect
167. C₃⋊C₁₆	#67	semidirect
170. C₂²×C₃²	C₂×C₁₆	semidirect
174. C₂×C₃⋊D₃	C₂×C₁₆	semidirect
180. C₃⋊Dic₃	C₂×C₁₆	semidirect
277. C₂²×C₃	D₃×C₁₆	semidirect
278. C₂²×C₃	D₃×C₁₆	semidirect
308. C₃²	D₄×C₁₆	semidirect
346. C₂²	#63542	semidirect
355. C₃	#67⋊C₁₆	semidirect
356. C₃	C₁₆×#67	semidirect

Maximal subgroups

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

subgroup	index
C₈×#638	2
C₂²×C₃×C₃⋊C₁₆	2
C₂×#63542	2
C₃×#4549	2
#10589133	2
#10589108	2
#10589102	2
C₁₆×#67	3
#67⋊C₁₆	3