Subgroups — C₃×#21144 #10639751

Order 768.

982 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#424	C₂	1	2	yes	maximal
2	C₂×C₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₃×#1940	C₂	1	8	yes	maximal, complemented
6	C₃×#1729	C₂	1	2	yes	maximal
7	#21144	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×#117	C₂²	1	2	yes	—
12	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₃×#424	—	2	8	no	—
14	C₃×#423	—	2	4	no	—
15	C₂×C₃×C₄⋊C₈	C₂²	1	4	yes	—
16	C₂×C₃×C₄⋊C₈	C₂²	1	8	yes	—
17	C₂×C₃×#97	—	2	4	no	—
18	C₂×C₃×#97	C₂²	1	4	yes	—
19	C₃×#384	C₄	1	2	yes	—
20	C₃×#382	C₄	1	2	yes	—
21	C₃×#380	C₄	1	4	yes	—
22	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
23	C₃×#336	C₂²	1	8	yes	—
24	C₂×#424	C₂×C₃	1	2	yes	—
25	C₂×#423	C₂×C₃	1	1	yes	characteristic
26	C₂²×#97	C₂×C₃	1	1	yes	characteristic
27	#2285	C₂×C₃	1	1	yes	characteristic
28	#1940	C₂×C₃	1	8	yes	complemented
29	#1729	C₂×C₃	1	2	yes	—
30	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
31	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃×M₁₆	—	2	4	no	—
33	C₂×C₃×M₁₆	—	2	4	no	—
34	C₂×C₃×M₁₆	—	2	4	no	—
35	C₂²×C₃×C₈	—	2	2	no	—
36	C₂²×C₃×C₈	—	2	2	no	—
37	C₂²×C₃×C₈	—	2	2	no	—
38	C₂²×C₃×C₈	—	2	2	no	—
39	C₂²×C₃×C₈	D₄	1	4	yes	—
40	C₂²×C₃×C₈	Q₈	1	4	yes	—
41	C₂²×C₃×C₈	C₂³	1	4	yes	—
42	C₂²×C₃×C₈	C₂³	1	2	yes	—
43	C₂²×C₃×C₈	C₂³	1	2	yes	—
44	C₃×#117	—	2	4	no	—
45	C₃×#117	—	2	4	no	—
46	C₂×C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
47	C₂×C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
48	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄²	D₄	1	4	yes	—
52	C₂×C₃×C₄²	Q₈	1	4	yes	—
53	C₂×C₃×C₄²	C₂³	1	4	yes	—
54	C₂×C₃×C₄²	C₂³	1	2	yes	—
55	C₃×C₄⋊C₈	—	2	16	no	—
56	C₃×C₄⋊C₈	—	2	8	no	—
57	C₃×#97	—	2	8	no	—
58	C₃×#95	C₂×C₄	1	8	yes	—
59	C₂²×M₁₆	C₂²×C₃	1	2	yes	—
60	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×#117	C₂²×C₃	1	2	yes	—
63	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
64	#424	—	2	8	no	—
65	#423	—	2	4	no	—
66	C₂×C₄⋊C₈	C₂²×C₃	1	4	yes	—
67	C₂×C₄⋊C₈	C₂²×C₃	1	8	yes	—
68	C₂×#97	—	2	4	no	—
69	C₂×#97	C₂²×C₃	1	4	yes	—
70	#384	C₃×C₄	1	2	yes	—
71	#382	C₃×C₄	1	2	yes	—
72	#380	C₃×C₄	1	4	yes	—
73	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
74	#336	C₂²×C₃	1	8	yes	—
75	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	—	4	4	no	—
77	C₂²×C₃×C₄	C₄⋊C₄	1	8	yes	—
78	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
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	1	yes	characteristic
82	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
84	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
85	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
86	C₂²×C₃×C₄	#41	1	2	yes	—
87	C₂²×C₃×C₄	#41	1	2	yes	—
88	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
89	C₃×M₁₆	—	4	8	no	—
90	C₂×C₃×C₈	—	2	8	no	—
91	C₂×C₃×C₈	—	2	8	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	4	no	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	—	2	4	no	—
99	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
100	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
101	C₃×C₄⋊C₄	—	2	8	no	—
102	C₃×C₄⋊C₄	—	4	8	no	—
103	C₃×C₂²⋊C₄	—	2	8	no	—
104	C₃×C₄²	—	2	4	no	—
105	C₃×C₄²	—	2	4	no	—
106	C₃×C₄²	C₄⋊C₄	1	4	yes	—
107	C₃×C₄²	C₄⋊C₄	1	4	yes	—
108	C₃×C₄²	C₂×D₄	1	4	yes	—
109	C₃×C₄²	C₂×Q₈	1	4	yes	—
110	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
111	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂×M₁₆	—	2	4	no	—
113	C₂×M₁₆	—	2	4	no	—
114	C₂×M₁₆	—	2	4	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	C₃×D₄	1	4	yes	—
120	C₂²×C₈	C₃×Q₈	1	4	yes	—
121	C₂²×C₈	C₂³×C₃	1	4	yes	—
122	C₂²×C₈	C₂³×C₃	1	2	yes	—
123	C₂²×C₈	C₂³×C₃	1	2	yes	—
124	#117	—	2	4	no	—
125	#117	—	2	4	no	—
126	C₂×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
127	C₂×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
128	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
129	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
130	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
131	C₂×C₄²	C₃×D₄	1	4	yes	—
132	C₂×C₄²	C₃×Q₈	1	4	yes	—
133	C₂×C₄²	C₂³×C₃	1	4	yes	—
134	C₂×C₄²	C₂³×C₃	1	2	yes	—
135	C₄⋊C₈	—	2	16	no	—
136	C₄⋊C₈	—	2	8	no	—
137	#97	—	2	8	no	—
138	#95	C₂×C₃×C₄	1	8	yes	—
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	C₂×C₄⋊C₄	1	2	yes	—
141	C₂³×C₃	C₄×D₄	1	2	yes	—
142	C₂³×C₃	C₄×Q₈	1	2	yes	—
143	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	8	no	—
147	C₂×C₃×C₄	—	2	16	no	—
148	C₂×C₃×C₄	—	4	8	no	—
149	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
150	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
151	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
153	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
154	C₂×C₃×C₄	#121	1	4	yes	—
155	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
156	C₂×C₃×C₄	#125	1	4	yes	—
157	C₂×C₃×C₄	C₄⋊Q₈	1	4	yes	—
158	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
164	C₃×C₈	—	2	4	no	—
165	C₃×C₈	—	2	4	no	—
166	C₃×C₈	—	4	4	no	—
167	C₂⁴	C₂²×C₃×C₄	1	1	yes	characteristic
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	C₃×C₄⋊C₄	1	8	yes	—
170	C₂²×C₄	C₂²×C₃×C₄	1	8	yes	—
171	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
172	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
173	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
175	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
176	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
177	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
178	C₂²×C₄	C₃×#41	1	2	yes	—
179	C₂²×C₄	C₃×#41	1	2	yes	—
180	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
181	M₁₆	—	4	8	no	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
192	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₂²⋊C₄	—	2	8	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	C₃×C₄⋊C₄	1	4	yes	—
199	C₄²	C₃×C₄⋊C₄	1	4	yes	—
200	C₄²	C₂×C₃×D₄	1	4	yes	—
201	C₄²	C₂×C₃×Q₈	1	4	yes	—
202	C₂²×C₃	—	2	4	no	—
203	C₂²×C₃	—	2	4	no	—
204	C₂²×C₃	—	2	4	no	—
205	C₂²×C₃	#389	1	4	yes	—
206	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
207	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂²×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
209	C₂²×C₃	#568	1	4	yes	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	4	8	no	—
213	C₃×C₄	#389	1	4	yes	—
214	C₃×C₄	C₂×#121	1	1	yes	characteristic
215	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
216	C₃×C₄	C₂×#125	1	1	yes	characteristic
217	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
218	C₂³	—	2	4	no	—
219	C₂³	C₂×C₃×C₄⋊C₄	1	2	yes	—
220	C₂³	C₃×C₄×D₄	1	2	yes	—
221	C₂³	C₃×C₄×Q₈	1	2	yes	—
222	C₂³	C₂³×C₃×C₄	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
229	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
230	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
231	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
232	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
233	C₂×C₄	C₃×#121	1	4	yes	—
234	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
235	C₂×C₄	C₃×#125	1	4	yes	—
236	C₂×C₄	C₃×C₄⋊Q₈	1	4	yes	—
237	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
241	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
242	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	—	2	4	no	—
245	C₈	—	4	4	no	—
246	C₂×C₃	—	2	4	no	—
247	C₂×C₃	C₂×#389	1	1	yes	characteristic
248	C₂×C₃	#2905	1	2	yes	—
249	C₂×C₃	#2932	1	2	yes	—
250	C₂×C₃	#2961	1	2	yes	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	C₃×#389	1	4	yes	—
255	C₂²	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
256	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
257	C₂²	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
258	C₂²	C₃×#568	1	4	yes	—
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	C₃×#389	1	4	yes	—
263	C₄	C₂×C₃×#121	1	1	yes	characteristic
264	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
265	C₄	C₂×C₃×#125	1	1	yes	characteristic
266	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
267	C₃	#21144	1	1	yes	characteristic, direct product factor, Sylow
268	C₂	—	2	4	no	—
269	C₂	C₂×C₃×#389	1	1	yes	characteristic
270	C₂	C₃×#2905	1	2	yes	—
271	C₂	C₃×#2932	1	2	yes	—
272	C₂	C₃×#2961	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂⁴
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 ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃×#1940	C₂	semidirect
7. #21144	C₃	direct (N × H)
28. #1940	C₂×C₃	semidirect
267. C₃	#21144	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×#424	2
C₂×C₃×#423	2
C₂²×C₃×#97	2
C₃×#2285	2
C₃×#1940	2
C₃×#1729	2
#21144	3

Subgroups — C3×#21144 #10639751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21144 #10639751