Subgroups — #10948761

Order 768.

552 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	#70949	C₂	1	2	yes	maximal, complemented
3	#70857	C₂	1	4	yes	maximal, complemented
4	D₄×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	#69745	C₂	1	1	yes	maximal, characteristic, complemented
6	#69745	C₂	1	2	yes	maximal, complemented
7	C₄×#812	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4461	C₂	1	2	yes	maximal, complemented
9	#33484	—	3	1	no	maximal, Sylow
10	C₃×C₄×#41	C₄	1	1	yes	characteristic
11	C₃×C₈×Q₈	C₄	1	1	yes	characteristic
12	C₃×C₈×Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₈×D₄	C₄	1	2	yes	—
14	C₃×C₈×D₄	C₄	1	1	yes	characteristic
15	C₃×C₈×D₄	C₂²	1	1	yes	characteristic
16	C₃×C₈×D₄	C₂²	1	2	yes	—
17	C₃×#431	C₄	1	1	yes	characteristic
18	C₃×#431	C₄	1	2	yes	—
19	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
20	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×#812	C₂²	1	2	yes	—
22	C₂²×C₃⋊C₁₆	C₂²	1	4	yes	—
23	#4549	C₂²	1	4	yes	—
24	#4549	C₂²	1	2	yes	—
25	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
26	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	—
27	#4461	C₂²	1	4	yes	—
28	#4461	C₂²	1	1	yes	characteristic
29	#4461	C₂²	1	2	yes	—
30	#4461	C₂²	1	1	yes	characteristic
31	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
32	C₈×#41	D₃	1	1	yes	characteristic
33	#2183	—	3	2	no	—
34	#2168	—	3	4	no	—
35	D₄×C₁₆	—	3	2	no	—
36	#2163	—	3	1	no	—
37	#2163	—	3	2	no	—
38	C₄×M₃₂	—	3	1	no	—
39	C₂×#346	—	3	2	no	—
40	C₂×C₃×#41	C₈	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂³	1	2	yes	—
44	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₄×Q₈	C₈	1	1	yes	characteristic
46	C₃×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×D₄	C₈	1	2	yes	—
48	C₃×C₄×D₄	C₈	1	1	yes	characteristic
49	C₃×C₄×D₄	C₂×C₄	1	2	yes	—
50	C₃×C₄×D₄	C₂×C₄	1	1	yes	characteristic
51	C₃×#117	C₈	1	1	yes	characteristic
52	C₃×#117	C₈	1	2	yes	—
53	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
54	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
59	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
60	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
61	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
63	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
65	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₃×C₄×C₈	C₂³	1	2	yes	—
69	#812	—	2	4	no	—
70	C₂×C₃⋊C₁₆	—	2	8	no	—
71	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
72	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
73	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
74	C₄×#41	Dic₃	1	1	yes	characteristic
75	C₂×M₃₂	—	3	2	no	—
76	C₂²×C₁₆	—	3	4	no	—
77	C₈×Q₈	Dic₃	1	1	yes	characteristic
78	C₈×Q₈	C₂×D₃	1	1	yes	characteristic
79	C₈×D₄	Dic₃	1	2	yes	—
80	C₈×D₄	Dic₃	1	1	yes	characteristic
81	C₈×D₄	C₂×D₃	1	1	yes	characteristic
82	C₈×D₄	C₂×D₃	1	2	yes	—
83	#431	Dic₃	1	1	yes	characteristic
84	#431	Dic₃	1	2	yes	—
85	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
86	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
87	C₄⋊C₁₆	—	3	2	no	—
88	C₄⋊C₁₆	—	3	4	no	—
89	C₂²⋊C₁₆	—	3	4	no	—
90	C₂²⋊C₁₆	—	3	2	no	—
91	#346	—	3	1	no	—
92	#346	—	3	2	no	—
93	#346	—	3	4	no	—
94	#346	—	3	1	no	—
95	C₄×C₁₆	—	3	2	no	—
96	C₃×#41	—	2	4	no	—
97	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
98	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
99	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
101	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
104	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
106	C₂×C₃×C₈	—	2	4	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	4	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
116	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
117	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
118	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
119	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
120	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
121	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
122	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
123	C₃×C₄²	C₂×C₈	1	2	yes	—
124	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
125	C₃×C₄²	C₂×C₈	1	2	yes	—
126	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
127	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
128	C₃×C₄²	C₂²×C₄	1	2	yes	—
129	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
130	C₃⋊C₁₆	—	2	4	no	—
131	C₃⋊C₁₆	—	2	2	no	—
132	C₃⋊C₁₆	#41	1	4	yes	complemented
133	C₂×#41	C₃⋊C₈	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₃	1	2	yes	—
135	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₃	1	2	yes	—
137	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
138	C₄×Q₈	C₃⋊C₈	1	1	yes	characteristic
139	C₄×Q₈	C₂×Dic₃	1	1	yes	characteristic
140	C₄×D₄	C₃⋊C₈	1	2	yes	—
141	C₄×D₄	C₃⋊C₈	1	1	yes	characteristic
142	C₄×D₄	C₂×Dic₃	1	2	yes	—
143	C₄×D₄	C₂×Dic₃	1	1	yes	characteristic
144	#117	C₃⋊C₈	1	1	yes	characteristic
145	#117	C₃⋊C₈	1	2	yes	—
146	C₂×C₄²	C₂×Dic₃	1	2	yes	—
147	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
148	M₃₂	—	6	4	no	—
149	C₂×C₁₆	—	3	2	no	—
150	C₂×C₁₆	—	3	2	no	—
151	C₂×C₁₆	—	3	4	no	—
152	C₂×C₁₆	—	6	8	no	—
153	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
154	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
155	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
156	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
157	C₂²⋊C₈	C₂×Dic₃	1	2	yes	—
158	C₂²⋊C₈	C₂×Dic₃	1	2	yes	—
159	C₂²⋊C₈	C₂×Dic₃	1	2	yes	—
160	C₄×C₈	C₂×Dic₃	1	2	yes	—
161	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
162	C₄×C₈	C₂×Dic₃	1	2	yes	—
163	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
164	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
166	C₄×C₈	C₂²×D₃	1	2	yes	—
167	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
168	C₂³×C₃	C₂²×C₈	1	2	yes	—
169	C₃×Q₈	—	2	2	no	—
170	C₃×D₄	—	2	4	no	—
171	C₃×D₄	M₃₂	1	4	yes	complemented
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
183	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
185	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
187	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	—	2	1	no	—
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	#131	1	2	yes	—
194	C₃×C₈	C₂×#41	1	2	yes	—
195	#41	—	2	4	no	—
196	C₂×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
197	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
198	C₂×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
200	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂²×Dic₃	1	2	yes	—
210	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₃	1	2	yes	—
213	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
214	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
216	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
217	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
218	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
219	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
220	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
221	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
222	C₄²	C₂×C₃⋊C₈	1	2	yes	—
223	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
224	C₄²	C₂×C₃⋊C₈	1	2	yes	—
225	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
226	C₄²	C₂²×Dic₃	1	1	yes	characteristic
227	C₄²	C₂²×Dic₃	1	2	yes	—
228	C₄²	C₂²×Dic₃	1	1	yes	characteristic
229	C₁₆	—	3	4	no	—
230	C₁₆	—	6	4	no	—
231	C₁₆	—	6	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	C₂×M₃₂	1	4	yes	—
238	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	C₂×M₃₂	1	2	yes	—
247	C₃×C₄	C₄×#41	1	1	yes	characteristic
248	C₃×C₄	C₂×#131	1	1	yes	characteristic
249	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
250	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
251	Q₈	—	2	2	no	—
252	D₄	—	2	4	no	—
253	D₄	#812	1	4	yes	complemented
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
265	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
267	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
269	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₈	—	2	1	no	—
275	C₈	#948	1	2	yes	—
276	C₈	D₃×#41	1	2	yes	—
277	C₂×C₃	—	2	4	no	—
278	C₂×C₃	—	4	1	no	—
279	C₂×C₃	C₈×#41	1	1	yes	characteristic
280	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
281	C₂×C₃	#3407	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	C₂×#812	1	4	yes	—
288	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	1	no	—
296	C₄	C₂×#812	1	2	yes	—
297	C₄	C₂×#948	1	1	yes	characteristic
298	C₄	Dic₃×#41	1	1	yes	characteristic
299	C₃	#33484	1	1	yes	characteristic, complemented, Sylow
300	C₂	—	2	4	no	—
301	C₂	—	4	1	no	—
302	C₂	#41×C₃⋊C₈	1	1	yes	characteristic
303	C₂	C₂²×#812	1	1	yes	characteristic
304	C₂	#83780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×C₈×#41	C₂
prime radical	C₂×C₃×#41	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₈×#41	D₃
hypocenter	C₃	#33484
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #70949	C₂	semidirect
3. #70857	C₂	semidirect
4. D₄×C₃⋊C₁₆	C₂	semidirect
5. #69745	C₂	semidirect
6. #69745	C₂	semidirect
7. C₄×#812	C₂	semidirect
8. C₂×#4461	C₂	semidirect
132. C₃⋊C₁₆	#41	semidirect
171. C₃×D₄	M₃₂	semidirect
253. D₄	#812	semidirect
299. C₃	#33484	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈×#41	2
#70949	2
#70857	2
D₄×C₃⋊C₁₆	2
#69745	2
C₄×#812	2
C₂×#4461	2
#33484	3