Subgroups — C₃×#69275 #11797320

Order 1152.

329 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	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×(C₂×D₃)⋊C₁₆	C₂	1	2	yes	maximal
4	C₃×#4502	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2163	—	3	1	no	maximal
7	#69275	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₃×C₈	C₄	1	1	yes	characteristic
9	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄²×D₃	C₄	1	1	yes	characteristic
11	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₈×Dic₃	C₄	1	1	yes	characteristic
14	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₃×C₄×C₈	—	3	1	no	—
18	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃×C₄⋊C₁₆	—	2	1	no	—
20	C₃×C₄⋊C₁₆	—	3	1	no	—
21	C₃×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
22	C₃×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₂²⋊C₁₆	—	3	2	no	—
24	C₃×#346	—	3	2	no	—
25	(C₂×D₃)⋊C₁₆	C₂×C₃	1	2	yes	—
26	#4502	C₂×C₃	1	2	yes	complemented
27	C₄⋊(C₃⋊C₁₆)	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₃×C₄×D₃	C₈	1	1	yes	characteristic
29	C₂×C₃×C₄×D₃	C₈	1	1	yes	characteristic
30	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
33	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×Dic₃	C₈	1	1	yes	characteristic
35	C₃×C₄×Dic₃	C₈	1	1	yes	characteristic
36	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×D₃×C₈	—	2	1	no	—
40	C₃×D₃×C₈	—	2	1	no	—
41	C₃×D₃×C₈	—	2	2	no	—
42	C₃²×C₁₆	—	2	2	no	—
43	C₃×C₃⋊C₁₆	—	2	2	no	—
44	#2163	—	3	1	no	Sylow
45	C₂²×C₃×C₈	—	3	1	no	—
46	C₂²×C₃×C₈	—	3	1	no	—
47	C₂×C₃×C₄²	—	3	1	no	—
48	C₂×D₃×C₈	C₃×C₄	1	1	yes	characteristic
49	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₄²×D₃	C₃×C₄	1	1	yes	characteristic
51	C₂×C₃×C₁₆	—	2	2	no	—
52	C₂×C₃×C₁₆	—	3	2	no	—
53	C₂×C₃×C₁₆	C₂×D₃	1	2	yes	—
54	C₂×C₃×C₁₆	C₂²×C₃	1	2	yes	—
55	C₃×C₄×C₈	—	2	1	no	—
56	C₃×C₄×C₈	—	3	1	no	—
57	C₃×C₄×C₈	—	3	1	no	—
58	C₃×C₄×C₈	—	3	1	no	—
59	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₈×Dic₃	C₃×C₄	1	1	yes	characteristic
62	C₈×Dic₃	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	—
64	C₄×C₃⋊C₈	C₃×C₄	1	1	yes	characteristic
65	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
72	C₃×C₄×D₃	—	2	2	no	—
73	C₃×C₄×D₃	—	2	2	no	—
74	C₃×C₄×D₃	—	2	1	no	—
75	C₃×C₄×D₃	—	2	1	no	—
76	C₃²×C₈	—	2	1	no	—
77	C₃²×C₈	#41	1	1	yes	characteristic
78	C₃²×C₈	#41	1	1	yes	characteristic
79	C₃×C₃⋊C₈	—	2	1	no	—
80	C₃×C₃⋊C₈	M₁₆	1	2	yes	—
81	C₂×C₄×C₈	—	3	1	no	—
82	C₄⋊C₁₆	—	3	1	no	—
83	C₄⋊C₁₆	C₃×D₃	1	1	yes	characteristic, complemented
84	C₂²⋊C₁₆	—	3	2	no	—
85	#346	—	3	2	no	—
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂²×C₃×C₄	—	3	1	no	—
88	C₂²×C₃×C₄	—	3	1	no	—
89	C₂×C₄×D₃	C₃×C₈	1	1	yes	characteristic
90	C₂×C₄×D₃	C₃×C₈	1	1	yes	characteristic
91	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	3	1	no	—
95	C₂×C₃×C₈	—	3	1	no	—
96	C₂×C₃×C₈	—	6	2	no	—
97	C₂×C₃×C₈	—	6	1	no	—
98	C₂×C₃×C₈	—	6	1	no	—
99	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	—	3	1	no	—
105	C₃×C₄²	—	3	1	no	—
106	C₃×C₄²	—	3	1	no	—
107	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
108	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
109	C₄×Dic₃	C₃×C₈	1	1	yes	characteristic
110	C₄×Dic₃	C₃×C₈	1	1	yes	characteristic
111	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
114	D₃×C₈	—	2	1	no	—
115	D₃×C₈	—	2	1	no	—
116	D₃×C₈	—	2	2	no	—
117	C₃×C₁₆	—	2	2	no	—
118	C₃×C₁₆	—	2	2	no	—
119	C₃×C₁₆	—	4	2	no	—
120	C₃×C₁₆	—	6	2	no	—
121	C₃⋊C₁₆	—	2	2	no	—
122	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₃×D₃	—	2	1	no	—
124	C₂×C₃×D₃	—	2	1	no	—
125	C₂×C₃×D₃	M₃₂	1	2	yes	—
126	C₃²×C₄	—	2	1	no	—
127	C₃²×C₄	M₃₂	1	2	yes	—
128	C₃²×C₄	#117	1	1	yes	characteristic
129	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
130	C₃×Dic₃	—	2	1	no	—
131	C₃×Dic₃	—	2	1	no	—
132	C₃×Dic₃	—	2	1	no	—
133	C₂²×C₈	—	3	1	no	—
134	C₂²×C₈	—	3	1	no	—
135	C₂×C₄²	—	3	1	no	—
136	C₂×C₁₆	—	3	2	no	—
137	C₂×C₁₆	C₂×C₃×D₃	1	2	yes	—
138	C₄×C₈	—	3	1	no	—
139	C₄×C₈	—	3	1	no	—
140	C₄×C₈	—	3	1	no	—
141	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
142	C₂³×C₃	—	3	1	no	—
143	C₂²×D₃	C₂×C₃×C₈	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	3	1	no	—
148	C₂×C₃×C₄	—	3	1	no	—
149	C₂×C₃×C₄	—	3	1	no	—
150	C₂×C₃×C₄	—	6	2	no	—
151	C₂×C₃×C₄	—	6	2	no	—
152	C₂×C₃×C₄	—	6	1	no	—
153	C₂×C₃×C₄	—	6	1	no	—
154	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
155	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
160	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
161	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
162	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
163	C₄×D₃	—	2	2	no	—
164	C₄×D₃	—	2	2	no	—
165	C₄×D₃	—	2	1	no	—
166	C₄×D₃	—	2	1	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	1	no	—
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	—	2	1	no	—
171	C₃×C₈	—	3	2	no	—
172	C₃×C₈	—	4	1	no	—
173	C₃×C₈	—	6	1	no	—
174	C₃×C₈	#237	1	1	yes	characteristic
175	C₃×C₈	#239	1	1	yes	characteristic
176	C₃×C₈	C₃×#41	1	1	yes	characteristic
177	C₃×C₈	C₃×#41	1	1	yes	characteristic
178	C₃⋊C₈	—	2	1	no	—
179	C₃⋊C₈	C₃×M₁₆	1	2	yes	—
180	C₂×C₃²	#431	1	1	yes	characteristic
181	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
182	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
183	C₃×D₃	—	2	2	no	—
184	C₂²×C₄	—	3	1	no	—
185	C₂²×C₄	—	3	1	no	—
186	C₂²×C₄	—	3	1	no	—
187	C₂×C₈	—	3	1	no	—
188	C₂×C₈	—	3	1	no	—
189	C₂×C₈	—	6	2	no	—
190	C₂×C₈	—	6	1	no	—
191	C₂×C₈	—	6	1	no	—
192	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
193	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
194	C₄²	—	3	1	no	—
195	C₄²	—	3	1	no	—
196	C₄²	—	3	1	no	—
197	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
198	C₁₆	—	2	2	no	—
199	C₁₆	—	6	2	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	3	2	no	—
202	C₂²×C₃	—	6	1	no	—
203	C₂²×C₃	—	6	1	no	—
204	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
205	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
206	C₂×D₃	—	2	1	no	—
207	C₂×D₃	—	2	1	no	—
208	C₂×D₃	C₃×M₃₂	1	2	yes	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	—	6	1	no	—
216	C₃×C₄	—	6	1	no	—
217	C₃×C₄	—	6	1	no	—
218	C₃×C₄	#798	1	2	yes	—
219	C₃×C₄	C₃×M₃₂	1	2	yes	—
220	C₃×C₄	#892	1	1	yes	characteristic
221	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
222	C₃×C₄	C₃×#117	1	1	yes	characteristic
223	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
224	Dic₃	—	2	1	no	—
225	Dic₃	—	2	1	no	—
226	Dic₃	—	2	1	no	—
227	C₃²	#2163	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	3	1	no	—
229	C₂×C₄	—	3	1	no	—
230	C₂×C₄	—	3	1	no	—
231	C₂×C₄	—	3	1	no	—
232	C₂×C₄	—	6	2	no	—
233	C₂×C₄	—	6	1	no	—
234	C₂×C₄	—	6	1	no	—
235	C₂×C₄	—	6	2	no	—
236	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
237	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
238	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	3	2	no	—
241	C₈	—	6	1	no	—
242	C₈	C₃×#237	1	1	yes	characteristic
243	C₈	C₃×#239	1	1	yes	characteristic
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	6	2	no	—
248	C₂×C₃	#4833	1	1	yes	characteristic
249	C₂×C₃	C₂×#798	1	1	yes	characteristic
250	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
251	C₂×C₃	C₃×#431	1	1	yes	characteristic
252	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
253	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
254	D₃	—	2	2	no	—
255	C₂²	—	3	2	no	—
256	C₂²	—	6	1	no	—
257	C₂²	—	6	1	no	—
258	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	6	1	no	—
261	C₄	—	6	1	no	—
262	C₄	—	6	1	no	—
263	C₄	C₃×#798	1	2	yes	—
264	C₄	C₃×#892	1	1	yes	characteristic
265	C₄	C₃×D₃×M₁₆	1	1	yes	characteristic
266	C₃	—	2	1	no	—
267	C₃	#69275	1	1	yes	characteristic, direct product factor
268	C₃	C₃×#2163	1	1	yes	characteristic, complemented
269	C₂	—	6	2	no	—
270	C₂	C₃×#4833	1	1	yes	characteristic
271	C₂	C₂×C₃×#798	1	1	yes	characteristic
272	C₂	C₃×D₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₄⋊C₁₆	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂²×C₃×D₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₄⋊C₁₆	D₃
hypocenter	C₃	C₃×#2163
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄⋊C₁₆	C₂	semidirect
4. C₃×#4502	C₂	semidirect
5. C₃×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
7. #69275	C₃	direct (N × H)
21. C₃×C₄⋊C₁₆	D₃	semidirect
22. C₃×C₄⋊C₁₆	C₂×C₃	semidirect
26. #4502	C₂×C₃	semidirect
27. C₄⋊(C₃⋊C₁₆)	C₂×C₃	semidirect
83. C₄⋊C₁₆	C₃×D₃	semidirect
227. C₃²	#2163	semidirect
267. C₃	#69275	direct (N × H)
268. C₃	C₃×#2163	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₃×C₈	2
C₃²×C₄⋊C₁₆	2
C₃×(C₂×D₃)⋊C₁₆	2
C₃×#4502	2
C₃×C₄⋊(C₃⋊C₁₆)	2
C₃×#2163	3
#69275	3

Subgroups — C3×#69275 #11797320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69275 #11797320