Subgroups — C₃×#69745 #11797323

Order 1152.

385 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#4549	C₂	1	2	yes	maximal
3	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
4	C₃×#4461	C₂	1	2	yes	maximal, complemented
5	C₃×#2163	—	3	1	no	maximal
6	#69745	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
10	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₃⋊C₁₆	C₂²	1	4	yes	—
15	C₂×C₃×C₄×C₈	—	2	1	no	—
16	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
17	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
18	C₃×C₄⋊C₁₆	—	3	2	no	—
19	C₃×C₂²⋊C₁₆	—	3	2	no	—
20	C₃×#346	—	3	2	no	—
21	#4549	C₂×C₃	1	2	yes	—
22	C₄⋊(C₃⋊C₁₆)	C₂×C₃	1	2	yes	complemented
23	#4461	C₂×C₃	1	2	yes	complemented
24	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	—	2	2	no	—
28	C₂×C₃²×C₈	—	2	2	no	—
29	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	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	C₈	1	1	yes	characteristic
34	C₃²×C₄²	C₈	1	1	yes	characteristic
35	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₃×C₃⋊C₁₆	—	2	4	no	—
38	#2163	—	3	1	no	Sylow
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	—	2	1	no	—
41	C₂²×C₃×C₈	Dic₃	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₃×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃×C₄²	—	2	1	no	—
46	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
47	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
48	C₂×C₃×C₁₆	—	3	4	no	—
49	C₃×C₄×C₈	—	2	1	no	—
50	C₃×C₄×C₈	—	2	1	no	—
51	C₃×C₄×C₈	—	2	1	no	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×C₄×C₈	Dic₃	1	1	yes	characteristic
54	C₃×C₄×C₈	Dic₃	1	1	yes	characteristic
55	C₃×C₄×C₈	C₃×C₄	1	1	yes	characteristic
56	C₃×C₄×C₈	C₃×C₄	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃⋊C₁₆	C₂²×C₃	1	4	yes	—
62	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
63	C₂×C₃²×C₄	—	2	2	no	—
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	—	2	1	no	—
66	C₂×C₃²×C₄	—	2	1	no	—
67	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₃²×C₈	—	2	1	no	—
74	C₃²×C₈	—	2	1	no	—
75	C₃²×C₈	M₁₆	1	2	yes	—
76	C₃²×C₈	#41	1	2	yes	—
77	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
78	C₄⋊C₁₆	—	3	2	no	—
79	C₂²⋊C₁₆	—	3	2	no	—
80	#346	—	3	2	no	—
81	C₂²×C₃×C₄	—	2	1	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	C₃⋊C₈	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₃⋊C₈	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₃×C₈	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₃×C₈	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₃×C₈	—	2	4	no	—
91	C₂×C₃×C₈	—	2	4	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	1	no	—
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	—	2	1	no	—
100	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
108	C₃×C₄²	—	2	1	no	—
109	C₃×C₄²	—	2	1	no	—
110	C₃×C₄²	—	2	1	no	—
111	C₃×C₄²	—	2	1	no	—
112	C₃×C₄²	C₃⋊C₈	1	1	yes	characteristic
113	C₃×C₄²	C₃⋊C₈	1	1	yes	characteristic
114	C₃×C₄²	C₃×C₈	1	1	yes	characteristic
115	C₃×C₄²	C₃×C₈	1	1	yes	characteristic
116	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
117	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
118	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
119	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
120	C₃×C₁₆	—	6	4	no	—
121	C₃⋊C₁₆	—	2	4	no	—
122	C₂²×C₃²	—	2	1	no	—
123	C₂²×C₃²	—	2	1	no	—
124	C₂²×C₃²	M₃₂	1	2	yes	—
125	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
126	C₃²×C₄	—	2	1	no	—
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	—	2	1	no	—
129	C₃²×C₄	—	2	1	no	—
130	C₃²×C₄	M₃₂	1	2	yes	—
131	C₃²×C₄	#117	1	1	yes	characteristic
132	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
133	C₂²×C₈	C₃×Dic₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
135	C₂×C₄²	C₃×Dic₃	1	1	yes	characteristic
136	C₂×C₁₆	—	3	4	no	—
137	C₄×C₈	C₃×Dic₃	1	1	yes	characteristic
138	C₄×C₈	C₃×Dic₃	1	1	yes	characteristic
139	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
140	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
141	C₂³×C₃	—	2	1	no	—
142	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
143	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₃×C₈	—	2	2	no	—
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	—	2	1	no	—
178	C₃×C₈	—	2	1	no	—
179	C₃×C₈	—	2	1	no	—
180	C₃×C₈	—	2	2	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃×C₈	#208	1	2	yes	—
183	C₃×C₈	C₃×M₁₆	1	2	yes	—
184	C₃×C₈	#235	1	2	yes	—
185	C₃×C₈	C₃×#41	1	2	yes	—
186	C₂×C₃²	—	2	2	no	—
187	C₂×C₃²	#431	1	1	yes	characteristic
188	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
189	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
190	C₂²×C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
191	C₂²×C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
198	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
199	C₄²	C₃×C₃⋊C₈	1	1	yes	characteristic
200	C₄²	C₃×C₃⋊C₈	1	1	yes	characteristic
201	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
202	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
203	C₁₆	—	6	4	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	#812	1	2	yes	—
213	C₂²×C₃	C₃×M₃₂	1	2	yes	—
214	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
215	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	#812	1	2	yes	—
232	C₃×C₄	C₃×M₃₂	1	2	yes	—
233	C₃×C₄	C₂×#208	1	1	yes	characteristic
234	C₃×C₄	#926	1	1	yes	characteristic
235	C₃×C₄	C₃×#117	1	1	yes	characteristic
236	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
237	C₃²	#2163	1	1	yes	characteristic, complemented, Sylow
238	C₂³	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
244	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
245	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
246	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
247	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	C₃×#208	1	2	yes	—
252	C₈	C₃×#235	1	2	yes	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	—	2	2	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	#4925	1	1	yes	characteristic
260	C₂×C₃	C₂×#812	1	1	yes	characteristic
261	C₂×C₃	C₂×#812	1	1	yes	characteristic
262	C₂×C₃	C₃×#431	1	1	yes	characteristic
263	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
264	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	C₃×#812	1	2	yes	—
268	C₂²	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	C₃×#812	1	2	yes	—
274	C₄	C₂×C₃×#208	1	1	yes	characteristic
275	C₄	C₃×#926	1	1	yes	characteristic
276	C₃	—	2	1	no	—
277	C₃	#69745	1	1	yes	characteristic, direct product factor
278	C₃	C₃×#2163	1	1	yes	characteristic, complemented
279	C₂	—	2	2	no	—
280	C₂	C₃×#4925	1	1	yes	characteristic
281	C₂	C₂×C₃×#812	1	1	yes	characteristic
282	C₂	C₂×C₃×#812	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	C₂×C₃²×C₄×C₈	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#2163
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
4. C₃×#4461	C₂	semidirect
6. #69745	C₃	direct (N × H)
22. C₄⋊(C₃⋊C₁₆)	C₂×C₃	semidirect
23. #4461	C₂×C₃	semidirect
237. C₃²	#2163	semidirect
277. C₃	#69745	direct (N × H)
278. 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
C₂×C₃²×C₄×C₈	2
C₃×#4549	2
C₃×C₄⋊(C₃⋊C₁₆)	2
C₃×#4461	2
C₃×#2163	3
#69745	3

Subgroups — C3×#69745 #11797323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69745 #11797323