Subgroups — #10948294

Order 768.

600 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#405	C₂	1	1	yes	maximal, characteristic
2	#71963	C₂	1	1	yes	maximal, characteristic
3	C₂×#4549	C₂	1	1	yes	maximal, characteristic
4	#69883	C₂	1	4	yes	maximal, complemented
5	#69882	C₂	1	2	yes	maximal, complemented
6	#69881	C₂	1	2	yes	maximal, complemented
7	#69745	C₂	1	2	yes	maximal, complemented
8	#69744	C₂	1	2	yes	maximal, complemented
9	#33422	—	3	1	no	maximal, Sylow
10	C₂²×C₃×M₁₆	C₄	1	1	yes	characteristic
11	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×#117	C₄	1	1	yes	characteristic
13	C₃×#405	C₄	1	1	yes	characteristic
14	C₃×#405	C₄	1	2	yes	—
15	C₃×#405	C₄	1	1	yes	characteristic
16	C₃×#405	C₂²	1	1	yes	characteristic
17	C₃×#405	C₂²	1	1	yes	characteristic
18	C₃×#405	C₂²	1	2	yes	—
19	C₂×C₃×#97	C₄	1	2	yes	—
20	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
21	C₂×#812	C₂²	1	2	yes	—
22	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
23	#4549	C₂²	1	1	yes	characteristic
24	#4549	C₂²	1	1	yes	characteristic
25	#4549	C₂²	1	2	yes	—
26	#4549	C₂²	1	2	yes	—
27	#4549	C₂²	1	2	yes	—
28	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	complemented
29	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	complemented
30	#4461	C₂²	1	4	yes	complemented
31	C₄×C₃⋊C₁₆	C₂²	1	4	yes	complemented
32	C₂×#405	D₃	1	1	yes	characteristic
33	#2164	—	3	4	no	—
34	#2163	—	3	2	no	—
35	#2162	—	3	2	no	—
36	#2151	—	3	2	no	—
37	#2112	—	3	1	no	—
38	C₂×C₂²⋊C₁₆	—	3	1	no	—
39	#2108	—	3	2	no	—
40	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×M₁₆	—	2	4	no	—
42	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
44	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₃×C₈	—	2	4	no	—
46	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
53	C₃×#117	C₈	1	1	yes	characteristic
54	C₃×#117	C₈	1	1	yes	characteristic
55	C₃×#117	C₈	1	2	yes	—
56	C₃×#117	C₂×C₄	1	2	yes	—
57	C₃×#117	C₂×C₄	1	1	yes	characteristic
58	C₃×#117	C₂×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
60	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
61	C₂×C₃×C₂²⋊C₄	C₈	1	2	yes	—
62	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
63	C₃×#97	C₂×C₄	1	2	yes	—
64	C₃×#97	C₂×C₄	1	2	yes	—
65	C₃×#97	C₂×C₄	1	2	yes	—
66	C₃×#97	C₂×C₄	1	2	yes	—
67	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
68	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
69	C₃×C₄×C₈	C₂³	1	2	yes	—
70	C₃×C₄×C₈	C₂³	1	2	yes	—
71	#812	—	2	4	no	—
72	C₂×C₃⋊C₁₆	—	2	4	no	—
73	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
74	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
75	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
76	C₂²×M₁₆	Dic₃	1	1	yes	characteristic
77	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
78	C₂×#117	Dic₃	1	1	yes	characteristic
79	C₂×M₃₂	—	3	2	no	—
80	C₂²×C₁₆	—	3	2	no	—
81	#405	Dic₃	1	1	yes	characteristic
82	#405	Dic₃	1	2	yes	—
83	#405	Dic₃	1	1	yes	characteristic
84	#405	C₂×D₃	1	1	yes	characteristic
85	#405	C₂×D₃	1	1	yes	characteristic
86	#405	C₂×D₃	1	2	yes	—
87	C₂×#97	Dic₃	1	2	yes	—
88	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
89	C₄⋊C₁₆	—	3	4	no	—
90	C₄⋊C₁₆	—	3	4	no	—
91	C₂²⋊C₁₆	—	3	1	no	—
92	C₂²⋊C₁₆	—	3	1	no	—
93	C₂²⋊C₁₆	—	3	2	no	—
94	C₂²⋊C₁₆	—	3	2	no	—
95	C₂²⋊C₁₆	—	3	2	no	—
96	#346	—	3	4	no	—
97	C₄×C₁₆	—	3	4	no	—
98	C₂⁴×C₃	C₂×C₈	1	1	yes	characteristic
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	—	2	2	no	—
101	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
102	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
106	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
109	C₃×M₁₆	—	2	2	no	—
110	C₃×M₁₆	—	2	4	no	—
111	C₃×M₁₆	M₁₆	1	4	yes	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
120	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₃×C₈	#41	1	4	yes	—
124	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
125	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
126	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
127	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
128	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
129	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
130	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
131	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
132	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
133	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
134	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
135	C₃×C₄²	C₂×C₈	1	2	yes	—
136	C₃×C₄²	C₂×C₈	1	2	yes	—
137	C₃×C₄²	C₂²×C₄	1	2	yes	—
138	C₃×C₄²	C₂²×C₄	1	2	yes	—
139	C₃⋊C₁₆	—	2	2	no	—
140	C₃⋊C₁₆	—	2	2	no	—
141	C₃⋊C₁₆	—	2	4	no	—
142	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
145	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
146	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
149	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₃	1	2	yes	—
152	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
155	#117	C₃⋊C₈	1	1	yes	characteristic
156	#117	C₃⋊C₈	1	1	yes	characteristic
157	#117	C₃⋊C₈	1	2	yes	—
158	#117	C₂×Dic₃	1	2	yes	—
159	#117	C₂×Dic₃	1	1	yes	characteristic
160	#117	C₂×Dic₃	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	C₃⋊C₈	1	2	yes	—
164	C₂×C₄²	C₂×Dic₃	1	2	yes	—
165	M₃₂	—	6	4	no	—
166	C₂×C₁₆	—	3	2	no	—
167	C₂×C₁₆	—	3	4	no	—
168	C₂×C₁₆	—	3	2	no	—
169	C₂×C₁₆	—	6	4	no	—
170	#97	C₂×Dic₃	1	2	yes	—
171	#97	C₂×Dic₃	1	2	yes	—
172	#97	C₂×Dic₃	1	2	yes	—
173	#97	C₂×Dic₃	1	2	yes	—
174	C₄×C₈	C₂×Dic₃	1	2	yes	—
175	C₄×C₈	C₂×Dic₃	1	2	yes	—
176	C₄×C₈	C₂²×D₃	1	2	yes	—
177	C₄×C₈	C₂²×D₃	1	2	yes	—
178	C₂³×C₃	—	2	2	no	—
179	C₂³×C₃	—	2	2	no	—
180	C₂³×C₃	—	4	1	no	—
181	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
182	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
183	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	#117	1	2	yes	—
194	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
196	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
197	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
198	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
199	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
202	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
203	C₃×C₈	—	2	1	no	—
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	—	2	2	no	—
206	C₃×C₈	#117	1	2	yes	—
207	C₃×C₈	C₂×M₁₆	1	4	yes	—
208	C₃×C₈	C₂×#41	1	2	yes	—
209	C₂⁴	C₂×C₃⋊C₈	1	1	yes	characteristic
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
213	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	#208	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	#235	1	4	yes	—
230	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
231	C₂×C₈	C₂²×Dic₃	1	2	yes	—
232	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₃	1	2	yes	—
234	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
235	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
236	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
237	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
238	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
239	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
240	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
241	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
243	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
244	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
245	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
246	C₄²	C₂×C₃⋊C₈	1	2	yes	—
247	C₄²	C₂×C₃⋊C₈	1	2	yes	—
248	C₄²	C₂²×Dic₃	1	2	yes	—
249	C₄²	C₂²×Dic₃	1	2	yes	—
250	C₁₆	—	6	2	no	—
251	C₁₆	—	6	2	no	—
252	C₁₆	—	6	4	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	4	4	no	—
261	C₂²×C₃	#431	1	2	yes	—
262	C₂²×C₃	#504	1	2	yes	—
263	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	#431	1	2	yes	—
271	C₃×C₄	C₂×#117	1	1	yes	characteristic
272	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
277	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
278	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
290	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
291	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
292	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
293	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
295	C₂×C₄	C₂×#208	1	2	yes	—
296	C₂×C₄	#926	1	2	yes	—
297	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	C₂×#208	1	4	yes	—
302	C₈	#926	1	2	yes	—
303	C₈	C₂×#235	1	2	yes	—
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	2	2	no	—
306	C₂×C₃	—	4	1	no	—
307	C₂×C₃	C₂×#431	1	1	yes	characteristic
308	C₂×C₃	C₂×#504	1	1	yes	characteristic
309	C₂×C₃	#3407	1	1	yes	characteristic
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	#4925	1	2	yes	—
319	C₂²	#5140	1	2	yes	—
320	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	#4925	1	2	yes	—
328	C₄	C₂²×#208	1	1	yes	characteristic
329	C₄	C₂×#926	1	1	yes	characteristic
330	C₃	#33422	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	2	2	no	—
333	C₂	—	4	1	no	—
334	C₂	C₂×#4925	1	1	yes	characteristic
335	C₂	C₂×#5140	1	1	yes	characteristic
336	C₂	#83780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₂×C₃×#405	C₂
prime radical	C₂⁴×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×#405	D₃
hypocenter	C₃	#33422
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #69883	C₂	semidirect
5. #69882	C₂	semidirect
6. #69881	C₂	semidirect
7. #69745	C₂	semidirect
8. #69744	C₂	semidirect
28. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
29. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
30. #4461	C₂²	semidirect
31. C₄×C₃⋊C₁₆	C₂²	semidirect
330. C₃	#33422	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#405	2
#71963	2
C₂×#4549	2
#69883	2
#69882	2
#69881	2
#69745	2
#69744	2
#33422	3