Subgroups — C₃×#9724 #10628331

Order 768.

444 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#1525	C₂	1	2	yes	maximal, complemented
4	C₃×#1484	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#9724	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#532	C₂²	1	2	yes	—
8	C₃×#500	C₄	1	1	yes	characteristic
9	C₃×#493	C₄	1	1	yes	characteristic
10	C₃×#486	C₄	1	2	yes	—
11	C₃×#466	—	2	1	no	—
12	C₃×#466	—	2	1	no	—
13	C₃×C₄×D₈	—	2	1	no	—
14	C₃×C₄×D₈	—	2	1	no	—
15	C₃×#431	C₂²	1	1	yes	characteristic
16	C₂×C₃×#108	—	2	1	no	—
17	C₃×C₄×M₁₆	C₂²	1	2	yes	—
18	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₃×#341	—	2	1	no	—
21	C₃×#329	—	2	4	no	—
22	C₃×D₄⋊C₈	—	2	4	no	—
23	C₃×#322	—	2	1	no	—
24	C₃×C₈²	—	2	1	no	—
25	#3169	C₂×C₃	1	1	yes	characteristic
26	C₄×#108	C₂×C₃	1	1	yes	characteristic, complemented
27	#1525	C₂×C₃	1	2	yes	complemented
28	#1484	C₂×C₃	1	2	yes	complemented
29	C₈×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×#41	—	2	2	no	—
31	C₂×C₃×D₈	—	2	1	no	—
32	C₂×C₃×D₈	—	2	1	no	—
33	C₂×C₃×M₁₆	—	2	1	no	—
34	C₂×C₃×M₁₆	—	2	2	no	—
35	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₃×#127	C₂×C₄	1	2	yes	—
39	C₃×#124	—	2	2	no	—
40	C₃×#121	—	2	2	no	—
41	C₃×#121	—	2	2	no	—
42	C₃×C₄×D₄	—	2	2	no	—
43	C₃×C₄×D₄	—	2	2	no	—
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₃×#108	—	2	4	no	—
46	C₃×#107	—	2	1	no	—
47	C₃×#107	—	2	1	no	—
48	C₃×C₄⋊C₈	D₄	1	2	yes	—
49	C₃×#102	—	2	2	no	—
50	C₃×#102	—	2	2	no	—
51	C₃×C₂²⋊C₈	—	2	1	no	—
52	C₃×#97	—	2	2	no	—
53	C₃×#97	D₄	1	2	yes	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	—	2	1	no	—
56	C₃×C₄×C₈	—	2	2	no	—
57	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₃×C₄×C₈	D₄	1	1	yes	characteristic
60	C₃×C₄×C₈	D₄	1	1	yes	characteristic
61	#532	C₂²×C₃	1	2	yes	—
62	#500	C₃×C₄	1	1	yes	characteristic
63	#493	C₃×C₄	1	1	yes	characteristic
64	#486	C₃×C₄	1	2	yes	—
65	#466	—	2	1	no	—
66	#466	—	2	1	no	—
67	C₄×D₈	—	2	1	no	—
68	C₄×D₈	—	2	1	no	—
69	#431	C₂²×C₃	1	1	yes	characteristic
70	C₂×#108	—	2	1	no	—
71	C₄×M₁₆	C₂²×C₃	1	2	yes	—
72	C₄×M₁₆	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
74	#341	—	2	1	no	—
75	#329	—	2	4	no	—
76	D₄⋊C₈	—	2	4	no	—
77	#322	—	2	1	no	—
78	C₈²	—	2	1	no	—
79	C₃×#41	—	8	4	no	—
80	C₂×C₃×Q₈	—	2	2	no	—
81	C₂×C₃×D₄	—	2	2	no	—
82	C₂×C₃×D₄	—	2	2	no	—
83	C₂×C₃×D₄	—	2	2	no	—
84	C₂×C₃×D₄	—	4	2	no	—
85	C₂²×C₃×C₄	—	4	2	no	—
86	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	#41	1	1	yes	characteristic
88	C₂²×C₃×C₄	#41	1	1	yes	characteristic
89	C₃×D₈	—	4	2	no	—
90	C₃×D₈	—	4	2	no	—
91	C₃×M₁₆	—	2	4	no	—
92	C₃×M₁₆	—	2	4	no	—
93	C₃×M₁₆	—	4	2	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	4	1	no	—
98	C₂×C₃×C₈	—	4	2	no	—
99	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₃×C₄⋊C₄	—	2	2	no	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₂²⋊C₄	—	4	2	no	—
106	C₃×C₂²⋊C₄	—	4	2	no	—
107	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
108	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
109	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
110	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
111	C₂×#41	—	2	2	no	—
112	C₂×D₈	—	2	1	no	—
113	C₂×D₈	—	2	1	no	—
114	C₂×M₁₆	—	2	1	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
117	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
118	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
119	#127	C₂×C₃×C₄	1	2	yes	—
120	#124	—	2	2	no	—
121	#121	—	2	2	no	—
122	#121	—	2	2	no	—
123	C₄×D₄	—	2	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
126	#108	—	2	4	no	—
127	#107	—	2	1	no	—
128	#107	—	2	1	no	—
129	C₄⋊C₈	C₃×D₄	1	2	yes	—
130	#102	—	2	2	no	—
131	#102	—	2	2	no	—
132	C₂²⋊C₈	—	2	1	no	—
133	#97	—	2	2	no	—
134	#97	C₃×D₄	1	2	yes	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
139	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
140	C₄×C₈	C₃×D₄	1	1	yes	characteristic
141	C₄×C₈	C₃×D₄	1	1	yes	characteristic
142	C₂³×C₃	—	4	2	no	—
143	C₂³×C₃	#124	1	1	yes	characteristic
144	C₃×Q₈	—	8	2	no	—
145	C₃×D₄	—	4	2	no	—
146	C₃×D₄	—	4	2	no	—
147	C₃×D₄	—	4	4	no	—
148	C₃×D₄	—	8	2	no	—
149	C₃×D₄	—	8	4	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	—	8	4	no	—
156	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	#121	1	1	yes	characteristic
160	C₂×C₃×C₄	#121	1	1	yes	characteristic
161	C₂×C₃×C₄	#127	1	1	yes	characteristic
162	C₃×C₈	—	2	4	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	2	1	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	4	4	no	—
167	C₃×C₈	—	4	2	no	—
168	C₃×C₈	#104	1	2	yes	—
169	#41	—	8	4	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂²×C₄	C₃×#41	1	1	yes	characteristic
178	C₂²×C₄	C₃×#41	1	1	yes	characteristic
179	D₈	—	4	2	no	—
180	D₈	—	4	2	no	—
181	M₁₆	—	2	4	no	—
182	M₁₆	—	2	4	no	—
183	M₁₆	—	4	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
198	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
199	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
200	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	8	2	no	—
205	C₂²×C₃	—	8	2	no	—
206	C₂²×C₃	—	8	2	no	—
207	C₂²×C₃	#390	1	1	yes	characteristic
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	8	2	no	—
212	C₃×C₄	C₂×#104	1	1	yes	characteristic
213	C₃×C₄	#421	1	1	yes	characteristic
214	C₃×C₄	#443	1	2	yes	—
215	C₃×C₄	#472	1	2	yes	—
216	C₂³	—	4	2	no	—
217	C₂³	C₃×#124	1	1	yes	characteristic
218	Q₈	—	8	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	4	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	4	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
232	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
233	C₂×C₄	C₃×#121	1	1	yes	characteristic
234	C₂×C₄	C₃×#121	1	1	yes	characteristic
235	C₂×C₄	C₃×#127	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	4	no	—
241	C₈	—	4	2	no	—
242	C₈	C₃×#104	1	2	yes	—
243	C₂×C₃	—	4	1	no	—
244	C₂×C₃	—	16	2	no	—
245	C₂×C₃	#1974	1	1	yes	characteristic
246	C₂×C₃	#1978	1	1	yes	characteristic
247	C₂×C₃	#1980	1	1	yes	characteristic
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	C₃×#390	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	8	2	no	—
259	C₄	C₂×C₃×#104	1	1	yes	characteristic
260	C₄	C₃×#421	1	1	yes	characteristic
261	C₄	C₃×#443	1	2	yes	—
262	C₄	C₃×#472	1	2	yes	—
263	C₃	#9724	1	1	yes	characteristic, direct product factor, Sylow
264	C₂	—	4	1	no	—
265	C₂	—	16	2	no	—
266	C₂	C₃×#1974	1	1	yes	characteristic
267	C₂	C₃×#1978	1	1	yes	characteristic
268	C₂	C₃×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	#121
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#390
prime radical	C₃×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×#108	C₂	semidirect
3. C₃×#1525	C₂	semidirect
4. C₃×#1484	C₂	semidirect
5. C₃×C₈×M₁₆	C₂	semidirect
6. #9724	C₃	direct (N × H)
26. C₄×#108	C₂×C₃	semidirect
27. #1525	C₂×C₃	semidirect
28. #1484	C₂×C₃	semidirect
29. C₈×M₁₆	C₂×C₃	semidirect
263. C₃	#9724	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×#3169	2
C₃×C₄×#108	2
C₃×#1525	2
C₃×#1484	2
C₃×C₈×M₁₆	2
#9724	3

Subgroups — C3×#9724 #10628331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9724 #10628331