Subgroups — #10948293

Order 768.

600 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	#71431	C₂	1	1	yes	maximal, characteristic
4	#71419	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#71399	C₂	1	1	yes	maximal, characteristic, complemented
7	#69882	C₂	1	1	yes	maximal, characteristic, complemented
8	#69276	C₂	1	2	yes	maximal, complemented
9	#69276	C₂	1	2	yes	maximal, complemented
10	#69275	C₂	1	2	yes	maximal, complemented
11	#69274	C₂	1	2	yes	maximal, complemented
12	#33422	—	3	1	no	maximal, Sylow
13	C₂×D₃×M₁₆	C₄	1	1	yes	characteristic
14	C₂²×D₃×C₈	C₂²	1	1	yes	characteristic
15	D₃×#117	C₄	1	1	yes	characteristic
16	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₃×#405	C₂²	1	1	yes	characteristic
19	#5122	C₄	1	1	yes	characteristic
20	#5101	C₂²	1	1	yes	characteristic
21	C₂×#812	C₂²	1	1	yes	characteristic
22	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
23	#4997	C₄	1	1	yes	characteristic
24	#4708	C₄	1	2	yes	—
25	D₃×#97	C₄	1	2	yes	—
26	#4689	C₂²	1	2	yes	—
27	C₄×D₃×C₈	C₂²	1	2	yes	—
28	C₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	C₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
30	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
31	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
32	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
33	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
34	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
35	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
36	#4502	C₂²	1	2	yes	complemented
37	#4502	C₂²	1	2	yes	complemented
38	Dic₃×C₁₆	C₂²	1	2	yes	complemented
39	Dic₃×C₁₆	C₂²	1	2	yes	complemented
40	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	complemented
41	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	complemented
42	C₂×#405	—	3	1	no	—
43	#2164	—	3	2	no	—
44	#2164	—	3	2	no	—
45	#2163	—	3	2	no	—
46	#2162	—	3	2	no	—
47	#2151	—	3	1	no	—
48	#2151	D₃	1	1	yes	characteristic, complemented
49	#2112	—	3	1	no	—
50	C₂×C₂²⋊C₁₆	—	3	1	no	—
51	#2108	—	3	1	no	—
52	#2108	—	3	1	no	—
53	C₂²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₃×#117	C₂×C₄	1	1	yes	characteristic
57	#926	C₂×C₄	1	1	yes	characteristic
58	C₂×#208	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
60	D₃×M₁₆	—	2	4	no	—
61	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
62	C₂×D₃×C₈	—	2	2	no	—
63	C₂×D₃×C₈	—	2	2	no	—
64	C₂×D₃×C₈	C₂×C₄	1	2	yes	—
65	C₂×D₃×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×D₃×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×D₃×C₈	C₂³	1	1	yes	characteristic
68	C₂×D₃×C₈	C₂³	1	1	yes	characteristic
69	#892	C₈	1	1	yes	characteristic
70	#892	C₈	1	1	yes	characteristic
71	D₃×C₄⋊C₄	C₈	1	1	yes	characteristic
72	D₃×C₄⋊C₄	C₈	1	1	yes	characteristic
73	D₃×C₂²⋊C₄	C₈	1	2	yes	—
74	#877	C₈	1	2	yes	—
75	#872	C₂×C₄	1	2	yes	—
76	C₄²×D₃	C₂×C₄	1	2	yes	—
77	C₃×M₃₂	—	2	2	no	—
78	C₂×C₃×C₁₆	—	2	2	no	—
79	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₃×C₁₆	C₂³	1	2	yes	—
82	C₃×#97	C₂×C₄	1	2	yes	—
83	C₃×C₄×C₈	C₂³	1	2	yes	—
84	#815	C₂×C₄	1	2	yes	—
85	#815	C₂×C₄	1	2	yes	—
86	C₈×Dic₃	C₂×C₄	1	2	yes	—
87	C₈×Dic₃	C₂³	1	2	yes	—
88	#812	—	2	2	no	—
89	C₂×C₃⋊C₁₆	—	2	2	no	—
90	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
91	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
92	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
93	#803	C₂×C₄	1	2	yes	—
94	C₄×C₃⋊C₈	C₂×C₄	1	2	yes	—
95	C₂²×M₁₆	—	3	1	no	—
96	C₂³×C₈	—	3	1	no	—
97	C₂×#117	—	3	1	no	—
98	C₂×M₃₂	—	3	1	no	—
99	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
100	C₂²×C₁₆	—	3	1	no	—
101	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
102	#405	—	3	1	no	—
103	#405	—	3	1	no	—
104	#405	—	3	1	no	—
105	#405	—	3	2	no	—
106	#405	—	3	2	no	—
107	#405	C₂×D₃	1	1	yes	characteristic
108	C₂×#97	—	3	2	no	—
109	C₂×C₄×C₈	—	3	2	no	—
110	C₄⋊C₁₆	—	3	2	no	—
111	C₄⋊C₁₆	—	3	2	no	—
112	C₄⋊C₁₆	C₂×D₃	1	2	yes	complemented
113	C₄⋊C₁₆	C₂×D₃	1	2	yes	complemented
114	C₂²⋊C₁₆	—	3	1	no	—
115	C₂²⋊C₁₆	—	3	1	no	—
116	C₂²⋊C₁₆	—	3	1	no	—
117	C₂²⋊C₁₆	—	3	1	no	—
118	C₂²⋊C₁₆	—	3	2	no	—
119	C₂²⋊C₁₆	—	3	2	no	—
120	#346	—	3	2	no	—
121	#346	—	3	2	no	—
122	C₄×C₁₆	—	3	2	no	—
123	C₄×C₁₆	—	3	2	no	—
124	C₂³×D₃	C₂×C₈	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
126	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
127	C₂×C₄×D₃	—	2	2	no	—
128	C₂×C₄×D₃	—	2	2	no	—
129	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
130	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
131	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
132	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
133	C₂×C₄×D₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×D₃	C₂²×C₄	1	1	yes	characteristic
135	C₃×M₁₆	—	2	2	no	—
136	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
137	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₃×C₈	#41	1	2	yes	—
139	C₂×C₃×C₈	#41	1	2	yes	—
140	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
141	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
142	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
143	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
144	C₃×C₄²	C₂²×C₄	1	2	yes	—
145	#217	C₂×C₈	1	2	yes	—
146	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
147	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
148	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
149	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
150	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
151	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
152	C₄×Dic₃	C₂×C₈	1	2	yes	—
153	C₄×Dic₃	C₂×C₈	1	2	yes	—
154	C₄×Dic₃	C₂²×C₄	1	2	yes	—
155	#208	M₁₆	1	4	yes	—
156	C₂×C₃⋊C₈	—	2	2	no	—
157	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
160	ΓL(1,𝔽₂₅)	—	2	4	no	—
161	D₃×C₈	—	2	2	no	—
162	D₃×C₈	—	2	2	no	—
163	D₃×C₈	—	2	1	no	—
164	D₃×C₈	—	2	1	no	—
165	D₃×C₈	—	2	1	no	—
166	D₃×C₈	—	2	1	no	—
167	D₃×C₈	—	2	4	no	—
168	C₃×C₁₆	—	2	1	no	—
169	C₃×C₁₆	—	2	1	no	—
170	C₃×C₁₆	—	2	2	no	—
171	C₃⋊C₁₆	—	2	1	no	—
172	C₃⋊C₁₆	—	2	1	no	—
173	C₃⋊C₁₆	—	2	2	no	—
174	C₂³×C₄	—	3	1	no	—
175	C₂×M₁₆	—	3	1	no	—
176	C₂×M₁₆	—	3	2	no	—
177	C₂×M₁₆	—	6	4	no	—
178	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
179	C₂²×C₈	—	3	1	no	—
180	C₂²×C₈	—	3	1	no	—
181	C₂²×C₈	—	3	1	no	—
182	C₂²×C₈	—	3	2	no	—
183	C₂²×C₈	—	3	1	no	—
184	C₂²×C₈	—	3	1	no	—
185	C₂²×C₈	—	6	2	no	—
186	C₂²×C₈	—	6	2	no	—
187	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
188	#117	—	3	1	no	—
189	#117	—	3	1	no	—
190	#117	—	3	1	no	—
191	#117	—	3	2	no	—
192	#117	—	3	2	no	—
193	#117	C₄×D₃	1	1	yes	characteristic
194	C₂×C₄⋊C₄	—	3	1	no	—
195	C₂×C₄⋊C₄	—	3	1	no	—
196	C₂×C₂²⋊C₄	—	3	2	no	—
197	C₂×C₄²	—	3	2	no	—
198	M₃₂	—	2	2	no	—
199	M₃₂	—	6	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	3	1	no	—
202	C₂×C₁₆	—	3	2	no	—
203	C₂×C₁₆	—	3	1	no	—
204	C₂×C₁₆	—	6	2	no	—
205	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₃	1	2	yes	—
207	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
208	#97	—	3	2	no	—
209	#97	—	3	2	no	—
210	#97	—	3	2	no	—
211	#97	C₄×D₃	1	2	yes	—
212	C₄×C₈	—	3	2	no	—
213	C₄×C₈	—	3	2	no	—
214	C₄×C₈	—	3	2	no	—
215	C₄×C₈	C₂²×D₃	1	2	yes	—
216	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
217	C₂²×D₃	—	2	2	no	—
218	C₂²×D₃	—	2	2	no	—
219	C₂²×D₃	—	4	1	no	—
220	C₂²×D₃	C₂²×C₈	1	1	yes	characteristic
221	C₂²×D₃	C₂²×C₈	1	1	yes	characteristic
222	C₂×C₃×C₄	#117	1	2	yes	—
223	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
224	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
225	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
227	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
228	C₂×Dic₃	—	2	2	no	—
229	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
230	C₂×Dic₃	C₂²×C₈	1	2	yes	—
231	C₂×Dic₃	C₂²×C₈	1	2	yes	—
232	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
233	C₄×D₃	—	2	4	no	—
234	C₄×D₃	—	2	1	no	—
235	C₄×D₃	—	2	1	no	—
236	C₄×D₃	—	2	1	no	—
237	C₄×D₃	—	2	1	no	—
238	C₄×D₃	—	2	2	no	—
239	C₄×D₃	—	2	2	no	—
240	C₄×D₃	—	4	2	no	—
241	C₃×C₈	—	2	2	no	—
242	C₃×C₈	#117	1	2	yes	—
243	C₃×C₈	C₂×#41	1	1	yes	characteristic
244	C₃×C₈	C₂×#41	1	1	yes	characteristic
245	C₃⋊C₈	—	2	1	no	—
246	C₃⋊C₈	—	2	1	no	—
247	C₃⋊C₈	C₂×M₁₆	1	4	yes	—
248	C₂⁴	—	3	1	no	—
249	C₂²×C₄	—	3	1	no	—
250	C₂²×C₄	—	3	2	no	—
251	C₂²×C₄	—	3	1	no	—
252	C₂²×C₄	—	3	1	no	—
253	C₂²×C₄	—	3	1	no	—
254	C₂²×C₄	—	3	1	no	—
255	C₂²×C₄	—	3	2	no	—
256	C₂²×C₄	—	6	2	no	—
257	C₂²×C₄	—	6	2	no	—
258	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
259	M₁₆	—	2	2	no	—
260	M₁₆	—	3	4	no	—
261	M₁₆	—	6	4	no	—
262	C₂×C₈	—	3	2	no	—
263	C₂×C₈	—	3	1	no	—
264	C₂×C₈	—	3	1	no	—
265	C₂×C₈	—	6	4	no	—
266	C₂×C₈	—	6	2	no	—
267	C₂×C₈	—	6	2	no	—
268	C₂×C₈	—	6	2	no	—
269	C₂×C₈	—	6	1	no	—
270	C₂×C₈	—	6	1	no	—
271	C₂×C₈	—	6	1	no	—
272	C₂×C₈	—	6	1	no	—
273	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
274	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
275	C₂×C₈	#237	1	2	yes	—
276	C₂×C₈	#239	1	2	yes	—
277	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
278	C₄⋊C₄	—	3	2	no	—
279	C₄⋊C₄	—	3	2	no	—
280	C₄⋊C₄	—	3	1	no	—
281	C₄⋊C₄	—	3	1	no	—
282	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
283	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
284	C₂²⋊C₄	—	3	2	no	—
285	C₂²⋊C₄	—	3	2	no	—
286	C₂²⋊C₄	—	3	2	no	—
287	C₂²⋊C₄	D₃×C₈	1	2	yes	—
288	C₄²	—	3	2	no	—
289	C₄²	—	3	2	no	—
290	C₄²	—	3	2	no	—
291	C₄²	C₂×C₄×D₃	1	2	yes	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₁₆	—	2	2	no	—
295	C₁₆	—	6	1	no	—
296	C₁₆	—	6	1	no	—
297	C₁₆	—	6	2	no	—
298	C₂²×C₃	—	2	1	no	—
299	C₂²×C₃	—	2	1	no	—
300	C₂²×C₃	#431	1	2	yes	—
301	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
302	C₂×D₃	—	2	1	no	—
303	C₂×D₃	—	2	1	no	—
304	C₂×D₃	—	2	1	no	—
305	C₂×D₃	—	2	1	no	—
306	C₂×D₃	—	2	1	no	—
307	C₂×D₃	—	4	4	no	—
308	C₂×D₃	#504	1	2	yes	—
309	C₃×C₄	—	2	2	no	—
310	C₃×C₄	—	2	2	no	—
311	C₃×C₄	#431	1	2	yes	—
312	C₃×C₄	C₂×#117	1	1	yes	characteristic
313	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
314	Dic₃	—	2	2	no	—
315	Dic₃	—	2	2	no	—
316	Dic₃	—	2	1	no	—
317	Dic₃	—	2	1	no	—
318	C₂³	—	3	1	no	—
319	C₂³	—	3	1	no	—
320	C₂³	—	6	2	no	—
321	C₂³	—	6	2	no	—
322	C₂³	—	12	1	no	—
323	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
324	C₂×C₄	—	3	1	no	—
325	C₂×C₄	—	3	2	no	—
326	C₂×C₄	—	3	2	no	—
327	C₂×C₄	—	3	1	no	—
328	C₂×C₄	—	6	4	no	—
329	C₂×C₄	—	6	2	no	—
330	C₂×C₄	—	6	1	no	—
331	C₂×C₄	—	6	1	no	—
332	C₂×C₄	—	6	1	no	—
333	C₂×C₄	—	6	1	no	—
334	C₂×C₄	—	6	2	no	—
335	C₂×C₄	—	6	2	no	—
336	C₂×C₄	—	12	2	no	—
337	C₂×C₄	#892	1	2	yes	—
338	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
339	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
340	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
341	C₂×C₄	D₃×M₁₆	1	2	yes	—
342	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
343	C₈	—	2	2	no	—
344	C₈	—	3	4	no	—
345	C₈	—	6	1	no	—
346	C₈	—	6	1	no	—
347	C₈	#892	1	2	yes	—
348	C₈	C₂×#237	1	1	yes	characteristic
349	C₈	C₂×#239	1	1	yes	characteristic
350	C₂×C₃	—	2	2	no	—
351	C₂×C₃	C₂×#431	1	1	yes	characteristic
352	C₂×C₃	C₂×#504	1	1	yes	characteristic
353	C₂×C₃	#3407	1	1	yes	characteristic
354	D₃	—	2	2	no	—
355	D₃	—	4	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	3	2	no	—
359	C₂²	—	6	1	no	—
360	C₂²	—	6	1	no	—
361	C₂²	—	6	1	no	—
362	C₂²	—	6	1	no	—
363	C₂²	—	6	1	no	—
364	C₂²	—	12	4	no	—
365	C₂²	#4833	1	2	yes	—
366	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	2	no	—
369	C₄	—	6	2	no	—
370	C₄	—	6	2	no	—
371	C₄	—	6	1	no	—
372	C₄	—	6	1	no	—
373	C₄	#4833	1	2	yes	—
374	C₄	C₂×#892	1	1	yes	characteristic
375	C₄	C₂×D₃×M₁₆	1	1	yes	characteristic
376	C₃	#33422	1	1	yes	characteristic, complemented, Sylow
377	C₂	—	2	2	no	—
378	C₂	—	6	2	no	—
379	C₂	—	12	1	no	—
380	C₂	C₂×#4833	1	1	yes	characteristic
381	C₂	D₃×#504	1	1	yes	characteristic
382	C₂	#82012	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2151	C₂
prime radical	C₂³×D₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2151	D₃
hypocenter	C₃	#33422
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
2. C₃×#2151	C₂	semidirect
4. #71419	C₂	semidirect
6. #71399	C₂	semidirect
7. #69882	C₂	semidirect
8. #69276	C₂	semidirect
9. #69276	C₂	semidirect
10. #69275	C₂	semidirect
11. #69274	C₂	semidirect
28. C₃×C₄⋊C₁₆	C₂²	semidirect
29. C₃×C₄⋊C₁₆	C₂²	semidirect
36. #4502	C₂²	semidirect
37. #4502	C₂²	semidirect
38. Dic₃×C₁₆	C₂²	semidirect
39. Dic₃×C₁₆	C₂²	semidirect
40. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
41. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
48. #2151	D₃	semidirect
112. C₄⋊C₁₆	C₂×D₃	semidirect
113. C₄⋊C₁₆	C₂×D₃	semidirect
376. C₃	#33422	semidirect

Maximal subgroups

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

subgroup	index
D₃×#405	2
C₃×#2151	2
#71431	2
#71419	2
C₂×(C₂×D₃)⋊C₁₆	2
#71399	2
#69882	2
#69276	2
#69275	2
#69274	2
#33422	3