Subgroups — C₃×#20503 #10639110

Order 768.

1206 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#442	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#2275	C₂	1	2	yes	maximal, complemented
4	C₃×#1776	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1762	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×#102	C₂	1	4	yes	maximal, complemented
8	C₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	#20503	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×D₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄×D₄	C₂²	1	2	yes	—
13	C₃×#442	C₄	1	16	yes	complemented
14	C₃×C₄×D₈	C₄	1	8	yes	complemented
15	C₂×C₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃×#106	C₂²	1	2	yes	—
17	C₂×C₃×#102	C₂²	1	2	yes	—
18	C₂×C₃×#102	C₂²	1	4	yes	—
19	C₂×C₃×#97	C₂²	1	1	yes	characteristic
20	C₂×C₃×#97	C₂²	1	2	yes	—
21	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
22	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
24	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄×C₂²⋊C₄	—	2	2	no	—
26	C₃×C₄×C₂²⋊C₄	—	2	4	no	—
27	C₃×#376	C₂²	1	1	yes	characteristic
28	C₂×C₃×#95	—	2	2	no	—
29	C₃×#340	C₂²	1	2	yes	—
30	C₃×#340	C₂²	1	4	yes	—
31	C₂×#442	C₂×C₃	1	2	yes	—
32	C₂×C₄×D₈	C₂×C₃	1	1	yes	characteristic
33	#2275	C₂×C₃	1	2	yes	complemented
34	#1776	C₂×C₃	1	2	yes	complemented
35	C₄×#106	C₂×C₃	1	1	yes	characteristic, complemented
36	#1762	C₂×C₃	1	2	yes	complemented
37	C₄×#102	C₂×C₃	1	4	yes	complemented
38	#1726	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂²×C₃×D₄	C₂³	1	2	yes	—
40	C₂³×C₃×C₄	—	2	2	no	—
41	C₂³×C₃×C₄	—	2	4	no	—
42	C₂×C₃×D₈	C₂×C₄	1	4	yes	—
43	C₂×C₃×D₈	C₂×C₄	1	8	yes	—
44	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₂³	1	2	yes	—
46	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
47	C₃×C₄×D₄	—	2	8	no	—
48	C₃×C₄×D₄	—	2	4	no	—
49	C₃×C₄×D₄	C₂×C₄	1	8	yes	complemented
50	C₃×C₄×D₄	C₂×C₄	1	16	yes	complemented
51	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
52	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
54	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
55	C₂×C₃×C₄²	—	2	4	no	—
56	C₂×C₃×C₄²	—	2	2	no	—
57	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄²	C₂³	1	2	yes	—
59	C₃×#107	C₂×C₄	1	4	yes	complemented
60	C₃×#106	C₂×C₄	1	8	yes	complemented
61	C₃×#102	C₂×C₄	1	8	yes	complemented
62	C₃×#102	C₂×C₄	1	16	yes	complemented
63	C₃×#97	—	2	2	no	—
64	C₃×#97	C₂×C₄	1	8	yes	complemented
65	C₃×C₄×C₈	—	2	4	no	—
66	C₃×C₄×C₈	C₂×C₄	1	4	yes	complemented
67	C₃×#95	—	2	2	no	—
68	C₃×#95	—	2	4	no	—
69	C₃×#95	D₄	1	2	yes	—
70	C₃×#95	D₄	1	2	yes	—
71	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄×D₄	C₂²×C₃	1	2	yes	—
73	C₂×C₄×D₄	C₂²×C₃	1	4	yes	—
74	#442	C₃×C₄	1	16	yes	complemented
75	C₄×D₈	C₃×C₄	1	8	yes	complemented
76	C₂×#107	C₂²×C₃	1	1	yes	characteristic
77	C₂×#106	C₂²×C₃	1	2	yes	—
78	C₂×#102	C₂²×C₃	1	2	yes	—
79	C₂×#102	C₂²×C₃	1	4	yes	—
80	C₂×#97	C₂²×C₃	1	1	yes	characteristic
81	C₂×#97	C₂²×C₃	1	2	yes	—
82	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
83	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
84	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
85	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
86	C₄×C₂²⋊C₄	—	2	2	no	—
87	C₄×C₂²⋊C₄	—	2	4	no	—
88	#376	C₂²×C₃	1	1	yes	characteristic
89	C₂×#95	—	2	2	no	—
90	#340	C₂²×C₃	1	2	yes	—
91	#340	C₂²×C₃	1	4	yes	—
92	C₂⁴×C₃	—	2	2	no	—
93	C₂×C₃×D₄	—	2	4	no	—
94	C₂×C₃×D₄	—	2	2	no	—
95	C₂×C₃×D₄	C₂²×C₄	1	4	yes	—
96	C₂×C₃×D₄	C₂²×C₄	1	8	yes	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	—	2	16	no	—
100	C₂²×C₃×C₄	—	2	8	no	—
101	C₂²×C₃×C₄	—	8	4	no	—
102	C₂²×C₃×C₄	—	8	4	no	—
103	C₂²×C₃×C₄	—	8	4	no	—
104	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
106	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
108	C₃×D₈	C₄²	1	16	yes	complemented
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
113	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
114	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
116	C₃×C₄⋊C₄	—	2	4	no	—
117	C₃×C₄⋊C₄	—	2	2	no	—
118	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
119	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
120	C₃×C₂²⋊C₄	—	2	8	no	—
121	C₃×C₂²⋊C₄	—	2	16	no	—
122	C₃×C₄²	—	8	4	no	—
123	C₃×C₄²	—	8	2	no	—
124	C₃×C₄²	C₂²×C₄	1	2	yes	—
125	C₃×C₄²	C₂²×C₄	1	4	yes	—
126	C₃×C₄²	#41	1	4	yes	—
127	C₃×C₄²	#41	1	2	yes	—
128	C₂²×D₄	C₂³×C₃	1	2	yes	—
129	C₂³×C₄	—	2	2	no	—
130	C₂³×C₄	—	2	4	no	—
131	C₂×D₈	C₂×C₃×C₄	1	4	yes	—
132	C₂×D₈	C₂×C₃×C₄	1	8	yes	—
133	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₃	1	2	yes	—
135	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
136	C₄×D₄	—	2	8	no	—
137	C₄×D₄	—	2	4	no	—
138	C₄×D₄	C₂×C₃×C₄	1	8	yes	complemented
139	C₄×D₄	C₂×C₃×C₄	1	16	yes	complemented
140	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
141	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
142	C₂×C₂²⋊C₄	—	2	4	no	—
143	C₂×C₂²⋊C₄	—	2	2	no	—
144	C₂×C₄²	—	2	4	no	—
145	C₂×C₄²	—	2	2	no	—
146	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₃	1	2	yes	—
148	#107	C₂×C₃×C₄	1	4	yes	complemented
149	#106	C₂×C₃×C₄	1	8	yes	complemented
150	#102	C₂×C₃×C₄	1	8	yes	complemented
151	#102	C₂×C₃×C₄	1	16	yes	complemented
152	#97	—	2	2	no	—
153	#97	C₂×C₃×C₄	1	8	yes	complemented
154	C₄×C₈	—	2	4	no	—
155	C₄×C₈	C₂×C₃×C₄	1	4	yes	complemented
156	#95	—	2	2	no	—
157	#95	—	2	4	no	—
158	#95	C₃×D₄	1	2	yes	—
159	#95	C₃×D₄	1	2	yes	—
160	C₂³×C₃	—	2	8	no	—
161	C₂³×C₃	—	2	4	no	—
162	C₂³×C₃	—	4	4	no	—
163	C₂³×C₃	—	4	2	no	—
164	C₂³×C₃	—	4	2	no	—
165	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	8	no	—
168	C₃×D₄	C₂×C₄²	1	8	yes	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	8	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	4	4	no	—
176	C₂×C₃×C₄	—	4	4	no	—
177	C₂×C₃×C₄	—	4	4	no	—
178	C₂×C₃×C₄	—	8	16	no	—
179	C₂×C₃×C₄	—	8	8	no	—
180	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
181	C₂×C₃×C₄	C₄×D₄	1	8	yes	—
182	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
183	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
184	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×#41	1	2	yes	—
186	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	—	2	4	no	—
189	C₃×C₈	C₂×C₄²	1	4	yes	—
190	C₂⁴	—	2	2	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂²×C₃×C₄	1	4	yes	—
194	C₂×D₄	C₂²×C₃×C₄	1	8	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	16	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	8	4	no	—
200	C₂²×C₄	—	8	4	no	—
201	C₂²×C₄	—	8	4	no	—
202	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
204	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
206	D₈	C₃×C₄²	1	16	yes	complemented
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
211	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
212	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
213	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
217	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
218	C₂²⋊C₄	—	2	8	no	—
219	C₂²⋊C₄	—	2	16	no	—
220	C₄²	—	8	4	no	—
221	C₄²	—	8	2	no	—
222	C₄²	C₂²×C₃×C₄	1	2	yes	—
223	C₄²	C₂²×C₃×C₄	1	4	yes	—
224	C₄²	C₃×#41	1	4	yes	—
225	C₄²	C₃×#41	1	2	yes	—
226	C₂²×C₃	—	2	8	no	—
227	C₂²×C₃	—	4	8	no	—
228	C₂²×C₃	—	4	8	no	—
229	C₂²×C₃	—	4	4	no	—
230	C₂²×C₃	—	4	4	no	—
231	C₂²×C₃	C₂×C₄×D₄	1	2	yes	—
232	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
233	C₂²×C₃	#576	1	2	yes	—
234	C₂²×C₃	#577	1	2	yes	—
235	C₃×C₄	—	2	8	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	4	8	no	—
239	C₃×C₄	—	4	4	no	—
240	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
241	C₃×C₄	C₄×#41	1	1	yes	characteristic
242	C₃×C₄	C₄×#41	1	2	yes	—
243	C₂³	—	2	8	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
249	D₄	—	2	4	no	—
250	D₄	—	2	8	no	—
251	D₄	C₂×C₃×C₄²	1	8	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	8	16	no	—
262	C₂×C₄	—	8	8	no	—
263	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
264	C₂×C₄	C₃×C₄×D₄	1	8	yes	—
265	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
266	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
267	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₃×#41	1	2	yes	—
269	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	4	no	—
272	C₈	C₂×C₃×C₄²	1	4	yes	—
273	C₂×C₃	—	4	8	no	—
274	C₂×C₃	C₄²×D₄	1	1	yes	characteristic
275	C₂×C₃	#2946	1	4	yes	—
276	C₂×C₃	#2948	1	1	yes	characteristic
277	C₂×C₃	#2952	1	1	yes	characteristic
278	C₂²	—	2	8	no	—
279	C₂²	—	4	8	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	C₂×C₃×C₄×D₄	1	2	yes	—
284	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
285	C₂²	C₃×#576	1	2	yes	—
286	C₂²	C₃×#577	1	2	yes	—
287	C₄	—	2	8	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	8	no	—
291	C₄	—	4	4	no	—
292	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
293	C₄	C₃×C₄×#41	1	1	yes	characteristic
294	C₄	C₃×C₄×#41	1	2	yes	—
295	C₃	#20503	1	1	yes	characteristic, direct product factor, Sylow
296	C₂	—	4	8	no	—
297	C₂	C₃×C₄²×D₄	1	1	yes	characteristic
298	C₂	C₃×#2946	1	4	yes	—
299	C₂	C₃×#2948	1	1	yes	characteristic
300	C₂	C₃×#2952	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂²×C₃×D₈	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#2275	C₂	semidirect
4. C₃×#1776	C₂	semidirect
5. C₃×C₄×#106	C₂	semidirect
6. C₃×#1762	C₂	semidirect
7. C₃×C₄×#102	C₂	semidirect
8. C₃×#1726	C₂	semidirect
9. #20503	C₃	direct (N × H)
13. C₃×#442	C₄	semidirect
14. C₃×C₄×D₈	C₄	semidirect
33. #2275	C₂×C₃	semidirect
34. #1776	C₂×C₃	semidirect
35. C₄×#106	C₂×C₃	semidirect
36. #1762	C₂×C₃	semidirect
37. C₄×#102	C₂×C₃	semidirect
38. #1726	C₂×C₃	semidirect
49. C₃×C₄×D₄	C₂×C₄	semidirect
50. C₃×C₄×D₄	C₂×C₄	semidirect
59. C₃×#107	C₂×C₄	semidirect
60. C₃×#106	C₂×C₄	semidirect
61. C₃×#102	C₂×C₄	semidirect
62. C₃×#102	C₂×C₄	semidirect
64. C₃×#97	C₂×C₄	semidirect
66. C₃×C₄×C₈	C₂×C₄	semidirect
74. #442	C₃×C₄	semidirect
75. C₄×D₈	C₃×C₄	semidirect
108. C₃×D₈	C₄²	semidirect
138. C₄×D₄	C₂×C₃×C₄	semidirect
139. C₄×D₄	C₂×C₃×C₄	semidirect
148. #107	C₂×C₃×C₄	semidirect
149. #106	C₂×C₃×C₄	semidirect
150. #102	C₂×C₃×C₄	semidirect
151. #102	C₂×C₃×C₄	semidirect
153. #97	C₂×C₃×C₄	semidirect
155. C₄×C₈	C₂×C₃×C₄	semidirect
206. D₈	C₃×C₄²	semidirect
295. C₃	#20503	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#442	2
C₂×C₃×C₄×D₈	2
C₃×#2275	2
C₃×#1776	2
C₃×C₄×#106	2
C₃×#1762	2
C₃×C₄×#102	2
C₃×#1726	2
#20503	3

Subgroups — C3×#20503 #10639110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20503 #10639110