Subgroups — C₃×#20782 #10639389

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#117	C₂	1	1	yes	maximal, characteristic
2	C₃×#1840	C₂	1	2	yes	maximal, complemented
3	C₃×#1832	C₂	1	2	yes	maximal, complemented
4	C₃×#1818	C₂	1	2	yes	maximal
5	C₃×#1798	C₂	1	2	yes	maximal
6	C₃×C₄×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×C₂²⋊C₈	C₂	1	2	yes	maximal
8	C₃×#1744	C₂	1	2	yes	maximal, complemented
9	#20782	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#117	C₂²	1	1	yes	characteristic
11	C₂×C₃×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₃×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₃×C₄×C₈	C₂²	1	4	yes	—
16	C₃×C₄×C₄⋊C₄	C₄	1	4	yes	—
17	C₃×C₄×C₂²⋊C₄	C₄	1	4	yes	—
18	C₃×#376	C₂²	1	1	yes	characteristic
19	C₃×#376	C₂²	1	1	yes	characteristic
20	C₃×C₄³	C₂²	1	1	yes	characteristic
21	C₃×C₄³	C₂²	1	1	yes	characteristic
22	C₃×#336	C₂²	1	4	yes	—
23	C₃×#336	C₂²	1	4	yes	—
24	C₃×#336	C₂²	1	4	yes	—
25	C₃×#336	C₂²	1	4	yes	—
26	C₄×#117	C₂×C₃	1	1	yes	characteristic
27	#1840	C₂×C₃	1	2	yes	complemented
28	#1832	C₂×C₃	1	2	yes	complemented
29	#1818	C₂×C₃	1	2	yes	—
30	#1798	C₂×C₃	1	2	yes	—
31	C₄×C₄⋊C₈	C₂×C₃	1	2	yes	complemented
32	C₄×C₂²⋊C₈	C₂×C₃	1	2	yes	—
33	#1744	C₂×C₃	1	2	yes	complemented
34	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
35	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂³	1	8	yes	—
37	C₃×#117	—	2	4	no	—
38	C₃×#117	—	2	4	no	—
39	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₄²	—	2	4	no	—
44	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
45	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
46	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
47	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
53	C₃×C₄⋊C₈	—	2	8	no	—
54	C₃×C₂²⋊C₈	—	2	8	no	—
55	C₃×C₄×C₈	—	2	8	no	—
56	C₃×#95	C₂×C₄	1	4	yes	—
57	C₃×#95	C₂×C₄	1	4	yes	—
58	C₂×#117	C₂²×C₃	1	1	yes	characteristic
59	C₂×#117	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄⋊C₈	C₂²×C₃	1	4	yes	—
62	C₂×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
63	C₂×C₄×C₈	C₂²×C₃	1	4	yes	—
64	C₄×C₄⋊C₄	C₃×C₄	1	4	yes	—
65	C₄×C₂²⋊C₄	C₃×C₄	1	4	yes	—
66	#376	C₂²×C₃	1	1	yes	characteristic
67	#376	C₂²×C₃	1	1	yes	characteristic
68	C₄³	C₂²×C₃	1	1	yes	characteristic
69	C₄³	C₂²×C₃	1	1	yes	characteristic
70	#336	C₂²×C₃	1	4	yes	—
71	#336	C₂²×C₃	1	4	yes	—
72	#336	C₂²×C₃	1	4	yes	—
73	#336	C₂²×C₃	1	4	yes	—
74	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	—	2	2	no	—
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	—	2	4	no	—
78	C₂²×C₃×C₄	—	2	8	no	—
79	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
80	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
81	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
82	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
84	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
86	C₂²×C₃×C₄	#41	1	4	yes	—
87	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
88	C₂×C₃×C₈	—	2	8	no	—
89	C₂×C₃×C₈	—	2	8	no	—
90	C₂×C₃×C₈	—	2	8	no	—
91	C₃×C₄⋊C₄	—	2	4	no	—
92	C₃×C₄⋊C₄	—	2	4	no	—
93	C₃×C₂²⋊C₄	—	2	4	no	—
94	C₃×C₂²⋊C₄	—	2	4	no	—
95	C₃×C₄²	—	2	4	no	—
96	C₃×C₄²	—	2	8	no	—
97	C₃×C₄²	—	2	8	no	—
98	C₃×C₄²	—	2	2	no	—
99	C₃×C₄²	—	2	2	no	—
100	C₃×C₄²	—	2	2	no	—
101	C₃×C₄²	—	2	2	no	—
102	C₃×C₄²	#41	1	4	yes	—
103	C₃×C₄²	#41	1	4	yes	—
104	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
105	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₃	1	8	yes	—
107	#117	—	2	4	no	—
108	#117	—	2	4	no	—
109	C₂×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
110	C₂×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₄²	—	2	4	no	—
114	C₂×C₄²	C₂×C₃×C₄	1	4	yes	—
115	C₂×C₄²	C₂×C₃×C₄	1	4	yes	—
116	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
117	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
118	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
119	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	8	no	—
124	C₂²⋊C₈	—	2	8	no	—
125	C₄×C₈	—	2	8	no	—
126	#95	C₂×C₃×C₄	1	4	yes	—
127	#95	C₂×C₃×C₄	1	4	yes	—
128	C₂³×C₃	—	2	1	no	—
129	C₂³×C₃	—	2	1	no	—
130	C₂³×C₃	—	2	4	no	—
131	C₂³×C₃	#117	1	2	yes	—
132	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	2	8	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	4	8	no	—
146	C₂×C₃×C₄	#117	1	4	yes	—
147	C₂×C₃×C₄	#117	1	2	yes	—
148	C₂×C₃×C₄	#117	1	4	yes	—
149	C₂×C₃×C₄	#131	1	8	yes	—
150	C₂×C₃×C₄	C₂×#41	1	2	yes	—
151	C₂×C₃×C₄	C₂×#41	1	2	yes	—
152	C₂×C₃×C₄	C₂×#41	1	2	yes	—
153	C₃×C₈	—	4	8	no	—
154	C₂⁴	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
160	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
161	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
162	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
163	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
164	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
166	C₂²×C₄	C₃×#41	1	4	yes	—
167	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	2	8	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	8	no	—
177	C₄²	—	2	8	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	C₃×#41	1	4	yes	—
183	C₄²	C₃×#41	1	4	yes	—
184	C₂²×C₃	—	2	4	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	4	2	no	—
187	C₂²×C₃	—	4	2	no	—
188	C₂²×C₃	C₂×#117	1	1	yes	characteristic
189	C₂²×C₃	C₂×#117	1	1	yes	characteristic
190	C₂²×C₃	C₂×#117	1	1	yes	characteristic
191	C₂²×C₃	C₂×#131	1	4	yes	—
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	4	2	no	—
195	C₃×C₄	—	4	2	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	#433	1	8	yes	—
199	C₃×C₄	#529	1	4	yes	—
200	C₂³	—	2	4	no	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	C₃×#117	1	2	yes	—
204	C₂³	C₂³×C₃×C₄	1	1	yes	characteristic
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	C₃×#117	1	4	yes	—
219	C₂×C₄	C₃×#117	1	2	yes	—
220	C₂×C₄	C₃×#117	1	4	yes	—
221	C₂×C₄	C₃×#131	1	8	yes	—
222	C₂×C₄	C₂×C₃×#41	1	2	yes	—
223	C₂×C₄	C₂×C₃×#41	1	2	yes	—
224	C₂×C₄	C₂×C₃×#41	1	2	yes	—
225	C₈	—	4	8	no	—
226	C₂×C₃	—	4	2	no	—
227	C₂×C₃	#2283	1	1	yes	characteristic
228	C₂×C₃	C₂×#433	1	2	yes	—
229	C₂×C₃	#2922	1	2	yes	—
230	C₂×C₃	#2922	1	2	yes	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	C₂×C₃×#117	1	1	yes	characteristic
236	C₂²	C₂×C₃×#117	1	1	yes	characteristic
237	C₂²	C₂×C₃×#117	1	1	yes	characteristic
238	C₂²	C₂×C₃×#131	1	4	yes	—
239	C₄	—	2	4	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	C₃×#433	1	8	yes	—
246	C₄	C₃×#529	1	4	yes	—
247	C₃	#20782	1	1	yes	characteristic, direct product factor, Sylow
248	C₂	—	4	2	no	—
249	C₂	C₃×#2283	1	1	yes	characteristic
250	C₂	C₂×C₃×#433	1	2	yes	—
251	C₂	C₃×#2922	1	2	yes	—
252	C₂	C₃×#2922	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃×#1840	C₂	semidirect
3. C₃×#1832	C₂	semidirect
6. C₃×C₄×C₄⋊C₈	C₂	semidirect
8. C₃×#1744	C₂	semidirect
9. #20782	C₃	direct (N × H)
27. #1840	C₂×C₃	semidirect
28. #1832	C₂×C₃	semidirect
31. C₄×C₄⋊C₈	C₂×C₃	semidirect
33. #1744	C₂×C₃	semidirect
247. C₃	#20782	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×#117	2
C₃×#1840	2
C₃×#1832	2
C₃×#1818	2
C₃×#1798	2
C₃×C₄×C₄⋊C₈	2
C₃×C₄×C₂²⋊C₈	2
C₃×#1744	2
#20782	3

Subgroups — C3×#20782 #10639389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20782 #10639389