Subgroups — C₃×#10265 #10628872

Order 768.

574 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#462	C₂	1	2	yes	maximal
3	C₃×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#332	C₂	1	2	yes	maximal
5	C₃×#1277	C₂	1	1	yes	maximal, characteristic
6	#10265	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₁₆	—	2	2	no	—
8	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₃×#500	—	2	4	no	—
11	C₃×#462	—	2	4	no	—
12	C₃×#462	—	2	4	no	—
13	C₂×C₃×#107	—	2	2	no	—
14	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃×#103	—	2	2	no	—
17	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₃×#336	—	2	2	no	—
19	C₃×#332	—	2	4	no	—
20	C₂×#500	C₂×C₃	1	1	yes	characteristic
21	C₂×#462	C₂×C₃	1	2	yes	—
22	#1984	C₂×C₃	1	1	yes	characteristic
23	C₂×#332	C₂×C₃	1	2	yes	—
24	#1277	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₃×Q₈	—	2	2	no	—
26	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
27	C₂×C₃×Q₁₆	—	2	8	no	—
28	C₂×C₃×Q₁₆	—	4	4	no	—
29	C₂×C₃×Q₁₆	—	4	4	no	—
30	C₂²×C₃×C₈	—	2	2	no	—
31	C₂²×C₃×C₈	—	2	1	no	—
32	C₂²×C₃×C₈	D₄	1	2	yes	—
33	C₃×C₄⋊Q₈	—	2	2	no	—
34	C₃×C₄⋊Q₈	—	2	2	no	—
35	C₃×C₄⋊Q₈	—	2	2	no	—
36	C₃×C₄⋊Q₈	D₄	1	4	yes	—
37	C₃×C₄×Q₈	—	2	2	no	—
38	C₃×C₄×Q₈	—	2	2	no	—
39	C₂×C₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄²	—	2	1	no	—
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₃×#107	—	2	8	no	—
46	C₃×C₄⋊C₈	—	2	2	no	—
47	C₃×C₄⋊C₈	—	2	2	no	—
48	C₃×C₄⋊C₈	D₄	1	4	yes	—
49	C₃×#103	—	2	8	no	—
50	C₃×C₄×C₈	—	2	2	no	—
51	C₂²×Q₁₆	—	2	2	no	—
52	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
53	C₂×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
54	#500	—	2	4	no	—
55	#462	—	2	4	no	—
56	#462	—	2	4	no	—
57	C₂×#107	—	2	2	no	—
58	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
60	C₂×#103	—	2	2	no	—
61	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
62	#336	—	2	2	no	—
63	#332	—	2	4	no	—
64	C₂×C₃×Q₈	—	2	1	no	—
65	C₂×C₃×Q₈	—	2	2	no	—
66	C₂×C₃×Q₈	—	2	4	no	—
67	C₂×C₃×Q₈	—	2	2	no	—
68	C₂×C₃×Q₈	—	4	2	no	—
69	C₂×C₃×Q₈	—	4	2	no	—
70	C₂×C₃×Q₈	—	4	2	no	—
71	C₂×C₃×Q₈	—	4	2	no	—
72	C₂×C₃×Q₈	—	4	2	no	—
73	C₂×C₃×Q₈	D₈	1	2	yes	—
74	C₂²×C₃×C₄	—	2	1	no	—
75	C₂²×C₃×C₄	—	2	1	no	—
76	C₂²×C₃×C₄	—	4	2	no	—
77	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₃×C₄	#41	1	1	yes	characteristic
80	C₃×Q₁₆	—	4	16	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	2	no	—
83	C₂×C₃×C₈	—	2	4	no	—
84	C₂×C₃×C₈	—	4	2	no	—
85	C₂×C₃×C₈	—	4	2	no	—
86	C₂×C₃×C₈	—	4	1	no	—
87	C₂×C₃×C₈	—	4	2	no	—
88	C₂×C₃×C₈	Q₁₆	1	4	yes	complemented
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	—	2	4	no	—
91	C₃×C₄⋊C₄	—	2	4	no	—
92	C₃×C₄⋊C₄	—	4	2	no	—
93	C₃×C₄⋊C₄	—	4	2	no	—
94	C₃×C₄⋊C₄	—	4	2	no	—
95	C₃×C₄²	—	4	1	no	—
96	C₃×C₄²	—	4	1	no	—
97	C₃×C₄²	C₂×D₄	1	2	yes	—
98	C₃×C₄²	C₂×D₄	1	2	yes	—
99	C₂²×Q₈	—	2	2	no	—
100	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
101	C₂×Q₁₆	—	2	8	no	—
102	C₂×Q₁₆	—	4	4	no	—
103	C₂×Q₁₆	—	4	4	no	—
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	C₃×D₄	1	2	yes	—
107	C₄⋊Q₈	—	2	2	no	—
108	C₄⋊Q₈	—	2	2	no	—
109	C₄⋊Q₈	—	2	2	no	—
110	C₄⋊Q₈	C₃×D₄	1	4	yes	—
111	C₄×Q₈	—	2	2	no	—
112	C₄×Q₈	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
117	C₂×C₄²	—	2	1	no	—
118	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
119	#107	—	2	8	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	C₃×D₄	1	4	yes	—
123	#103	—	2	8	no	—
124	C₄×C₈	—	2	2	no	—
125	C₂³×C₃	#121	1	1	yes	characteristic
126	C₃×Q₈	—	2	4	no	—
127	C₃×Q₈	—	4	2	no	—
128	C₃×Q₈	—	4	4	no	—
129	C₃×Q₈	—	4	4	no	—
130	C₃×Q₈	—	8	2	no	—
131	C₃×Q₈	—	8	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	4	1	no	—
135	C₂×C₃×C₄	—	4	1	no	—
136	C₂×C₃×C₄	—	4	1	no	—
137	C₂×C₃×C₄	—	4	4	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	8	2	no	—
140	C₂×C₃×C₄	—	8	2	no	—
141	C₂×C₃×C₄	#120	1	2	yes	—
142	C₂×C₃×C₄	#120	1	1	yes	characteristic
143	C₂×C₃×C₄	#121	1	2	yes	—
144	C₂×C₃×C₄	#127	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
147	C₂×C₃×C₄	#135	1	1	yes	characteristic
148	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₃×C₄	#137	1	2	yes	—
150	C₃×C₈	—	2	4	no	—
151	C₃×C₈	—	4	4	no	—
152	C₃×C₈	—	8	1	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	C₃×D₈	1	2	yes	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂²×C₄	C₃×#41	1	1	yes	characteristic
169	Q₁₆	—	4	16	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	C₃×Q₁₆	1	4	yes	complemented
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄²	—	4	1	no	—
185	C₄²	—	4	1	no	—
186	C₄²	C₂×C₃×D₄	1	2	yes	—
187	C₄²	C₂×C₃×D₄	1	2	yes	—
188	C₂²×C₃	#392	1	1	yes	characteristic
189	C₂²×C₃	#462	1	2	yes	—
190	C₂²×C₃	#466	1	1	yes	characteristic
191	C₂²×C₃	#469	1	1	yes	characteristic
192	C₂²×C₃	#473	1	2	yes	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	4	2	no	—
195	C₃×C₄	—	8	4	no	—
196	C₃×C₄	—	8	1	no	—
197	C₃×C₄	#447	1	2	yes	—
198	C₃×C₄	#449	1	2	yes	—
199	C₃×C₄	#451	1	2	yes	—
200	C₃×C₄	#494	1	1	yes	characteristic
201	C₃×C₄	#497	1	1	yes	characteristic
202	C₂³	C₃×#121	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	4	4	no	—
206	Q₈	—	4	4	no	—
207	Q₈	—	8	2	no	—
208	Q₈	—	8	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₃×#120	1	2	yes	—
219	C₂×C₄	C₃×#120	1	1	yes	characteristic
220	C₂×C₄	C₃×#121	1	2	yes	—
221	C₂×C₄	C₃×#127	1	1	yes	characteristic
222	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
223	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
224	C₂×C₄	C₃×#135	1	1	yes	characteristic
225	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
226	C₂×C₄	C₃×#137	1	2	yes	—
227	C₈	—	2	4	no	—
228	C₈	—	4	4	no	—
229	C₈	—	8	1	no	—
230	C₂×C₃	#1674	1	2	yes	—
231	C₂×C₃	#1694	1	2	yes	—
232	C₂×C₃	#1999	1	1	yes	characteristic
233	C₂×C₃	#2016	1	1	yes	characteristic
234	C₂×C₃	#2020	1	1	yes	characteristic
235	C₂²	C₃×#392	1	1	yes	characteristic
236	C₂²	C₃×#462	1	2	yes	—
237	C₂²	C₃×#466	1	1	yes	characteristic
238	C₂²	C₃×#469	1	1	yes	characteristic
239	C₂²	C₃×#473	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	8	4	no	—
243	C₄	—	8	1	no	—
244	C₄	C₃×#447	1	2	yes	—
245	C₄	C₃×#449	1	2	yes	—
246	C₄	C₃×#451	1	2	yes	—
247	C₄	C₃×#494	1	1	yes	characteristic
248	C₄	C₃×#497	1	1	yes	characteristic
249	C₃	#10265	1	1	yes	characteristic, direct product factor, Sylow
250	C₂	C₃×#1674	1	2	yes	—
251	C₂	C₃×#1694	1	2	yes	—
252	C₂	C₃×#1999	1	1	yes	characteristic
253	C₂	C₃×#2016	1	1	yes	characteristic
254	C₂	C₃×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
6. #10265	C₃	direct (N × H)
88. C₂×C₃×C₈	Q₁₆	semidirect
177. C₂×C₈	C₃×Q₁₆	semidirect
249. C₃	#10265	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×#500	2
C₂×C₃×#462	2
C₃×#1984	2
C₂×C₃×#332	2
C₃×#1277	2
#10265	3

Subgroups — C3×#10265 #10628872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10265 #10628872