Subgroups — C₃×#20505 #10639112

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#441	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#2276	C₂	1	2	yes	maximal
4	C₃×#1776	C₂	1	2	yes	maximal
5	C₃×C₄×#106	C₂	1	1	yes	maximal, characteristic
6	C₃×#1763	C₂	1	2	yes	maximal
7	C₃×C₄×#103	C₂	1	4	yes	maximal
8	C₃×#1726	C₂	1	1	yes	maximal, characteristic
9	#20505	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×Q₈	C₂²	1	4	yes	—
12	C₂×C₃×C₄×Q₈	C₂²	1	2	yes	—
13	C₃×#441	C₄	1	16	yes	complemented
14	C₃×C₄×Q₁₆	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×#103	C₂²	1	2	yes	—
18	C₂×C₃×#103	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	—	2	2	no	—
24	C₃×C₄×C₄⋊C₄	—	2	4	no	—
25	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
26	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
27	C₃×#376	—	2	2	no	—
28	C₃×#376	C₂²	1	1	yes	characteristic
29	C₃×#340	C₂²	1	2	yes	—
30	C₃×#340	C₂²	1	4	yes	—
31	C₂×#441	C₂×C₃	1	2	yes	—
32	C₂×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
33	#2276	C₂×C₃	1	2	yes	—
34	#1776	C₂×C₃	1	2	yes	—
35	C₄×#106	C₂×C₃	1	1	yes	characteristic
36	#1763	C₂×C₃	1	2	yes	—
37	C₄×#103	C₂×C₃	1	4	yes	—
38	#1726	C₂×C₃	1	1	yes	characteristic
39	C₂²×C₃×Q₈	C₂³	1	2	yes	—
40	C₂×C₃×Q₁₆	C₂×C₄	1	4	yes	—
41	C₂×C₃×Q₁₆	C₂×C₄	1	8	yes	—
42	C₂²×C₃×C₈	C₂³	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	—	2	8	no	—
46	C₃×C₄×Q₈	—	2	4	no	—
47	C₃×C₄×Q₈	C₂×C₄	1	8	yes	—
48	C₃×C₄×Q₈	C₂×C₄	1	16	yes	—
49	C₂×C₃×C₄⋊C₄	—	2	4	no	—
50	C₂×C₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
52	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₃×C₄²	—	2	2	no	—
54	C₂×C₃×C₄²	—	2	4	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	—
60	C₃×#106	C₂×C₄	1	8	yes	—
61	C₃×#103	C₂×C₄	1	8	yes	—
62	C₃×#103	C₂×C₄	1	16	yes	—
63	C₃×#97	—	2	2	no	—
64	C₃×#97	C₂×C₄	1	8	yes	—
65	C₃×C₄×C₈	—	2	4	no	—
66	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
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×Q₁₆	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄×Q₈	C₂²×C₃	1	2	yes	—
73	C₂×C₄×Q₈	C₂²×C₃	1	4	yes	—
74	#441	C₃×C₄	1	16	yes	complemented
75	C₄×Q₁₆	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×#103	C₂²×C₃	1	2	yes	—
79	C₂×#103	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	—	2	2	no	—
85	C₄×C₄⋊C₄	—	2	4	no	—
86	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
87	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
88	#376	—	2	2	no	—
89	#376	C₂²×C₃	1	1	yes	characteristic
90	#340	C₂²×C₃	1	2	yes	—
91	#340	C₂²×C₃	1	4	yes	—
92	C₂×C₃×Q₈	—	2	4	no	—
93	C₂×C₃×Q₈	—	2	2	no	—
94	C₂×C₃×Q₈	C₂²×C₄	1	4	yes	—
95	C₂×C₃×Q₈	C₂²×C₄	1	8	yes	—
96	C₂²×C₃×C₄	—	2	2	no	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
101	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
103	C₃×Q₁₆	C₄²	1	16	yes	complemented
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
109	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
111	C₃×C₄⋊C₄	—	2	16	no	—
112	C₃×C₄⋊C₄	—	2	8	no	—
113	C₃×C₄⋊C₄	—	2	4	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
116	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
117	C₃×C₄²	—	2	8	no	—
118	C₃×C₄²	—	2	16	no	—
119	C₃×C₄²	—	8	4	no	—
120	C₃×C₄²	—	8	2	no	—
121	C₃×C₄²	C₂²×C₄	1	2	yes	—
122	C₃×C₄²	C₂²×C₄	1	4	yes	—
123	C₃×C₄²	#41	1	4	yes	—
124	C₃×C₄²	#41	1	2	yes	—
125	C₂²×Q₈	C₂³×C₃	1	2	yes	—
126	C₂×Q₁₆	C₂×C₃×C₄	1	4	yes	—
127	C₂×Q₁₆	C₂×C₃×C₄	1	8	yes	—
128	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₃	1	2	yes	—
130	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
131	C₄×Q₈	—	2	8	no	—
132	C₄×Q₈	—	2	4	no	—
133	C₄×Q₈	C₂×C₃×C₄	1	8	yes	—
134	C₄×Q₈	C₂×C₃×C₄	1	16	yes	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
138	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	—	2	4	no	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
144	C₂×C₄²	C₂³×C₃	1	2	yes	—
145	#107	C₂×C₃×C₄	1	4	yes	—
146	#106	C₂×C₃×C₄	1	8	yes	—
147	#103	C₂×C₃×C₄	1	8	yes	—
148	#103	C₂×C₃×C₄	1	16	yes	—
149	#97	—	2	2	no	—
150	#97	C₂×C₃×C₄	1	8	yes	—
151	C₄×C₈	—	2	4	no	—
152	C₄×C₈	C₂×C₃×C₄	1	4	yes	—
153	#95	—	2	2	no	—
154	#95	—	2	4	no	—
155	#95	C₃×D₄	1	2	yes	—
156	#95	C₃×D₄	1	2	yes	—
157	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
158	C₃×Q₈	—	2	4	no	—
159	C₃×Q₈	—	2	8	no	—
160	C₃×Q₈	C₂×C₄²	1	8	yes	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	8	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	4	4	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	—	4	4	no	—
173	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
174	C₂×C₃×C₄	C₄×D₄	1	8	yes	—
175	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
176	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
177	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×#41	1	2	yes	—
179	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
180	C₃×C₈	—	2	2	no	—
181	C₃×C₈	—	2	4	no	—
182	C₃×C₈	C₂×C₄²	1	4	yes	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	C₂²×C₃×C₄	1	4	yes	—
186	C₂×Q₈	C₂²×C₃×C₄	1	8	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
192	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
193	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
194	Q₁₆	C₃×C₄²	1	16	yes	complemented
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
199	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
200	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
201	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
207	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
208	C₄²	—	2	8	no	—
209	C₄²	—	2	16	no	—
210	C₄²	—	8	4	no	—
211	C₄²	—	8	2	no	—
212	C₄²	C₂²×C₃×C₄	1	2	yes	—
213	C₄²	C₂²×C₃×C₄	1	4	yes	—
214	C₄²	C₃×#41	1	4	yes	—
215	C₄²	C₃×#41	1	2	yes	—
216	C₂²×C₃	C₂×C₄×D₄	1	2	yes	—
217	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂²×C₃	#577	1	2	yes	—
219	C₂²×C₃	#578	1	2	yes	—
220	C₃×C₄	—	2	8	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	8	no	—
224	C₃×C₄	—	4	8	no	—
225	C₃×C₄	—	4	4	no	—
226	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
227	C₃×C₄	C₄×#41	1	1	yes	characteristic
228	C₃×C₄	C₄×#41	1	2	yes	—
229	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	2	8	no	—
232	Q₈	C₂×C₃×C₄²	1	8	yes	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
246	C₂×C₄	C₃×C₄×D₄	1	8	yes	—
247	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
248	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
249	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×#41	1	2	yes	—
251	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	2	4	no	—
254	C₈	C₂×C₃×C₄²	1	4	yes	—
255	C₂×C₃	C₄²×D₄	1	1	yes	characteristic
256	C₂×C₃	#2947	1	4	yes	—
257	C₂×C₃	#2950	1	1	yes	characteristic
258	C₂×C₃	#2952	1	1	yes	characteristic
259	C₂²	C₂×C₃×C₄×D₄	1	2	yes	—
260	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
261	C₂²	C₃×#577	1	2	yes	—
262	C₂²	C₃×#578	1	2	yes	—
263	C₄	—	2	8	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	8	no	—
267	C₄	—	4	8	no	—
268	C₄	—	4	4	no	—
269	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
270	C₄	C₃×C₄×#41	1	1	yes	characteristic
271	C₄	C₃×C₄×#41	1	2	yes	—
272	C₃	#20505	1	1	yes	characteristic, direct product factor, Sylow
273	C₂	C₃×C₄²×D₄	1	1	yes	characteristic
274	C₂	C₃×#2947	1	4	yes	—
275	C₂	C₃×#2950	1	1	yes	characteristic
276	C₂	C₃×#2952	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
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
9. #20505	C₃	direct (N × H)
13. C₃×#441	C₄	semidirect
14. C₃×C₄×Q₁₆	C₄	semidirect
74. #441	C₃×C₄	semidirect
75. C₄×Q₁₆	C₃×C₄	semidirect
103. C₃×Q₁₆	C₄²	semidirect
194. Q₁₆	C₃×C₄²	semidirect
272. C₃	#20505	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×#441	2
C₂×C₃×C₄×Q₁₆	2
C₃×#2276	2
C₃×#1776	2
C₃×C₄×#106	2
C₃×#1763	2
C₃×C₄×#103	2
C₃×#1726	2
#20505	3

Subgroups — C3×#20505 #10639112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20505 #10639112