Subgroups — C₃×#69312 #11797118

Order 1152.

309 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃×D₃×#108	C₂	1	1	yes	maximal, characteristic
3	C₃×#4860	C₂	1	1	yes	maximal, characteristic
4	C₃²×#365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4513	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2155	—	3	1	no	maximal
9	#69312	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃×D₃×M₁₆	—	2	1	no	—
11	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×#892	—	2	1	no	—
13	C₃²×M₃₂	C₂²	1	1	yes	characteristic
14	C₃²×#108	C₂²	1	1	yes	characteristic
15	C₃²×#106	C₂²	1	1	yes	characteristic
16	C₃×#822	—	2	1	no	—
17	C₃×#819	C₂²	1	1	yes	characteristic
18	C₃×#817	C₂²	1	1	yes	characteristic
19	C₃×#812	C₂²	1	1	yes	characteristic
20	C₃×#807	—	2	1	no	—
21	C₃×#798	C₄	1	2	yes	complemented
22	C₃×D₃×C₁₆	C₄	1	2	yes	complemented
23	C₂×C₃×M₃₂	—	3	1	no	—
24	C₂×C₃×#108	—	3	1	no	—
25	C₃×#427	—	3	1	no	—
26	D₃×M₃₂	C₂×C₃	1	1	yes	characteristic
27	D₃×#108	C₂×C₃	1	1	yes	characteristic
28	#4860	C₂×C₃	1	1	yes	characteristic
29	C₃×#365	—	2	1	no	—
30	C₃×#365	—	3	1	no	—
31	C₃×#365	—	3	1	no	—
32	C₃×#365	—	3	1	no	—
33	C₃×#365	D₃	1	1	yes	characteristic, complemented
34	C₃×#365	C₂×C₃	1	1	yes	characteristic, complemented
35	#4513	C₂×C₃	1	1	yes	characteristic, complemented
36	#4494	C₂×C₃	1	1	yes	characteristic, complemented
37	#4487	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
39	C₃²×M₁₆	—	2	1	no	—
40	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
41	C₃²×C₄⋊C₄	—	2	1	no	—
42	C₃×(C₂×D₃)⋊C₄	—	4	1	no	—
43	C₃×C₄⋊Dic₃	—	2	1	no	—
44	C₃×C₄×Dic₃	—	4	1	no	—
45	C₃×#208	—	2	1	no	—
46	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₃×ΓL(1,𝔽₂₅)	—	4	1	no	—
48	C₃×D₃×C₈	—	4	1	no	—
49	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
50	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
51	C₃×D₃×C₈	Q₈	1	1	yes	characteristic
52	C₃×D₃×C₈	Q₈	1	1	yes	characteristic
53	C₃²×C₁₆	C₂×C₄	1	2	yes	—
54	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
55	#2155	—	3	1	no	Sylow
56	C₂×C₃×M₁₆	—	6	1	no	—
57	C₂²×C₃×C₈	—	3	1	no	—
58	C₃×#117	—	6	1	no	—
59	D₃×M₁₆	—	2	1	no	—
60	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
61	#892	—	2	1	no	—
62	C₃×M₃₂	—	2	1	no	—
63	C₃×M₃₂	—	3	1	no	—
64	C₃×M₃₂	—	3	2	no	—
65	C₃×M₃₂	C₂×D₃	1	1	yes	characteristic
66	C₃×M₃₂	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×C₁₆	—	3	2	no	—
68	C₃×#108	—	2	1	no	—
69	C₃×#108	—	3	1	no	—
70	C₃×#108	—	6	1	no	—
71	C₃×#108	C₂×D₃	1	1	yes	characteristic
72	C₃×#108	C₂²×C₃	1	1	yes	characteristic
73	C₃×#107	—	6	1	no	—
74	C₃×#106	—	2	1	no	—
75	C₃×#106	—	3	1	no	—
76	C₃×#106	C₂×D₃	1	1	yes	characteristic
77	C₃×#106	C₂²×C₃	1	1	yes	characteristic
78	#822	—	2	1	no	—
79	#819	C₂²×C₃	1	1	yes	characteristic
80	#817	C₂²×C₃	1	1	yes	characteristic
81	#812	C₂²×C₃	1	1	yes	characteristic
82	#807	—	2	1	no	—
83	#798	C₃×C₄	1	2	yes	complemented
84	D₃×C₁₆	C₃×C₄	1	2	yes	complemented
85	C₂²×C₃×D₃	SD₁₆	1	1	yes	characteristic
86	C₂×C₃²×C₄	—	4	1	no	—
87	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×Dic₃	—	4	1	no	—
89	C₂×C₃×Dic₃	SD₁₆	1	1	yes	characteristic
90	C₃×C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
91	C₃×C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
92	C₃×C₄×D₃	SD₁₆	1	1	yes	characteristic
93	C₃×C₄×D₃	SD₁₆	1	1	yes	characteristic
94	C₃²×C₈	—	4	1	no	—
95	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₃×C₃⋊C₈	—	4	1	no	—
98	C₃×C₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₃×C₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂×M₃₂	—	3	1	no	—
101	C₂×#108	—	3	1	no	—
102	#427	—	3	1	no	—
103	#365	—	3	1	no	—
104	#365	—	3	1	no	—
105	#365	—	3	1	no	—
106	#365	C₃×D₃	1	1	yes	characteristic, complemented
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂×C₄×D₃	C₃×D₄	1	1	yes	characteristic
109	C₃×M₁₆	—	2	1	no	—
110	C₃×M₁₆	—	2	1	no	—
111	C₃×M₁₆	—	4	1	no	—
112	C₃×M₁₆	—	6	1	no	—
113	C₃×M₁₆	—	12	1	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	—	3	1	no	—
116	C₂×C₃×C₈	—	3	1	no	—
117	C₂×C₃×C₈	—	3	1	no	—
118	C₂×C₃×C₈	—	3	1	no	—
119	C₂×C₃×C₈	—	3	1	no	—
120	C₂×C₃×C₈	—	12	1	no	—
121	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
123	C₃×C₄⋊C₄	—	2	1	no	—
124	C₃×C₄⋊C₄	—	2	1	no	—
125	C₃×C₄⋊C₄	—	4	1	no	—
126	C₃×C₄⋊C₄	—	6	1	no	—
127	C₃×C₂²⋊C₄	—	12	1	no	—
128	C₃×C₄²	—	12	1	no	—
129	(C₂×D₃)⋊C₄	—	4	1	no	—
130	C₄⋊Dic₃	—	2	1	no	—
131	C₄×Dic₃	—	4	1	no	—
132	#208	—	2	1	no	—
133	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
134	ΓL(1,𝔽₂₅)	—	4	1	no	—
135	D₃×C₈	—	4	1	no	—
136	D₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
137	D₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
138	D₃×C₈	C₃×Q₈	1	1	yes	characteristic
139	D₃×C₈	C₃×Q₈	1	1	yes	characteristic
140	C₃×C₁₆	—	2	2	no	—
141	C₃×C₁₆	—	3	2	no	—
142	C₃×C₁₆	C₄×D₃	1	2	yes	—
143	C₃×C₁₆	C₂×C₃×C₄	1	2	yes	—
144	C₃⋊C₁₆	C₂×C₃×C₄	1	2	yes	—
145	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₃×D₃	—	4	1	no	—
147	C₂×C₃×D₃	#106	1	1	yes	characteristic
148	C₂×C₃×D₃	#106	1	1	yes	characteristic
149	C₃²×C₄	—	8	1	no	—
150	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
152	C₃×Dic₃	—	8	1	no	—
153	C₃×Dic₃	#106	1	1	yes	characteristic
154	C₃×Dic₃	#106	1	1	yes	characteristic
155	C₂×M₁₆	—	6	1	no	—
156	C₂²×C₈	—	3	1	no	—
157	#117	—	6	1	no	—
158	M₃₂	—	3	1	no	—
159	M₃₂	—	3	2	no	—
160	M₃₂	C₂×C₃×D₃	1	1	yes	characteristic
161	C₂×C₁₆	—	3	2	no	—
162	#108	—	3	1	no	—
163	#108	—	6	1	no	—
164	#108	C₂×C₃×D₃	1	1	yes	characteristic
165	#107	—	6	1	no	—
166	#106	—	3	1	no	—
167	#106	C₂×C₃×D₃	1	1	yes	characteristic
168	C₂³×C₃	—	3	1	no	—
169	C₂²×D₃	C₃×SD₁₆	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	3	1	no	—
172	C₂×C₃×C₄	—	3	1	no	—
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	—	3	1	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	4	1	no	—
177	C₂×C₃×C₄	—	4	1	no	—
178	C₂×C₃×C₄	—	8	1	no	—
179	C₂×C₃×C₄	—	12	1	no	—
180	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×Dic₃	—	4	1	no	—
183	C₂×Dic₃	C₃×SD₁₆	1	1	yes	characteristic
184	C₄×D₃	C₃×C₄⋊C₄	1	1	yes	characteristic
185	C₄×D₃	C₃×C₄⋊C₄	1	1	yes	characteristic
186	C₄×D₃	C₃×SD₁₆	1	1	yes	characteristic
187	C₄×D₃	C₃×SD₁₆	1	1	yes	characteristic
188	C₃×C₈	—	2	1	no	—
189	C₃×C₈	—	2	1	no	—
190	C₃×C₈	—	3	1	no	—
191	C₃×C₈	—	3	1	no	—
192	C₃×C₈	—	4	1	no	—
193	C₃×C₈	—	4	1	no	—
194	C₃×C₈	—	8	1	no	—
195	C₃×C₈	—	12	1	no	—
196	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
197	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
198	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
199	C₃×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
200	C₃⋊C₈	—	4	1	no	—
201	C₃⋊C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
202	C₃⋊C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₃²	—	2	1	no	—
204	C₂×C₃²	C₂×#106	1	1	yes	characteristic
205	C₃×D₃	—	2	1	no	—
206	C₃×D₃	—	2	1	no	—
207	C₂²×C₄	—	3	1	no	—
208	M₁₆	—	2	1	no	—
209	M₁₆	—	6	1	no	—
210	M₁₆	—	12	1	no	—
211	C₂×C₈	—	3	1	no	—
212	C₂×C₈	—	3	1	no	—
213	C₂×C₈	—	3	1	no	—
214	C₂×C₈	—	3	1	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	—	12	1	no	—
217	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	6	1	no	—
220	C₂²⋊C₄	—	12	1	no	—
221	C₄²	—	12	1	no	—
222	C₁₆	—	3	2	no	—
223	C₁₆	C₃×C₄×D₃	1	2	yes	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	3	1	no	—
226	C₂²×C₃	—	3	1	no	—
227	C₂²×C₃	—	12	1	no	—
228	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
229	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
230	C₂×D₃	—	4	1	no	—
231	C₂×D₃	C₃×#106	1	1	yes	characteristic
232	C₂×D₃	C₃×#106	1	1	yes	characteristic
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	3	1	no	—
236	C₃×C₄	—	3	1	no	—
237	C₃×C₄	—	8	1	no	—
238	C₃×C₄	—	8	1	no	—
239	C₃×C₄	—	16	1	no	—
240	C₃×C₄	—	24	1	no	—
241	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
242	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
243	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
244	C₃×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
245	Dic₃	—	8	1	no	—
246	Dic₃	C₃×#106	1	1	yes	characteristic
247	Dic₃	C₃×#106	1	1	yes	characteristic
248	C₃²	#2155	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	3	1	no	—
250	C₂×C₄	—	3	1	no	—
251	C₂×C₄	—	3	1	no	—
252	C₂×C₄	—	3	1	no	—
253	C₂×C₄	—	3	1	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	12	1	no	—
257	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
258	C₈	—	3	1	no	—
259	C₈	—	3	1	no	—
260	C₈	—	4	1	no	—
261	C₈	—	12	1	no	—
262	C₈	C₂×C₃×C₄×D₃	1	1	yes	characteristic
263	C₈	C₃×D₃×Q₈	1	1	yes	characteristic
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	4	1	no	—
268	C₂×C₃	—	6	1	no	—
269	C₂×C₃	—	6	1	no	—
270	C₂×C₃	D₃×#106	1	1	yes	characteristic
271	C₂×C₃	C₂×C₃×#106	1	1	yes	characteristic
272	D₃	—	2	1	no	—
273	D₃	—	2	1	no	—
274	C₂²	—	3	1	no	—
275	C₂²	—	3	1	no	—
276	C₂²	—	12	1	no	—
277	C₂²	C₃×D₃×SD₁₆	1	1	yes	characteristic
278	C₄	—	3	1	no	—
279	C₄	—	3	1	no	—
280	C₄	—	8	1	no	—
281	C₄	—	24	1	no	—
282	C₄	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
283	C₄	C₃×D₃×SD₁₆	1	1	yes	characteristic
284	C₃	—	2	1	no	—
285	C₃	#69312	1	1	yes	characteristic, direct product factor
286	C₃	C₃×#2155	1	1	yes	characteristic, complemented
287	C₂	—	2	1	no	—
288	C₂	—	6	1	no	—
289	C₂	—	6	1	no	—
290	C₂	C₃×D₃×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	D₃×#106
commutator	C₃×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×#365	C₂
socle	C₂×C₃²	C₂×#106
prime radical	C₂²×C₃×D₃	SD₁₆
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×#365	D₃
hypocenter	C₃	C₃×#2155
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#365

Complements

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

N	H ≅ G/N	type
4. C₃²×#365	C₂	semidirect
5. C₃×#4513	C₂	semidirect
6. C₃×#4494	C₂	semidirect
7. C₃×#4487	C₂	semidirect
9. #69312	C₃	direct (N × H)
21. C₃×#798	C₄	semidirect
22. C₃×D₃×C₁₆	C₄	semidirect
33. C₃×#365	D₃	semidirect
34. C₃×#365	C₂×C₃	semidirect
35. #4513	C₂×C₃	semidirect
36. #4494	C₂×C₃	semidirect
37. #4487	C₂×C₃	semidirect
83. #798	C₃×C₄	semidirect
84. D₃×C₁₆	C₃×C₄	semidirect
106. #365	C₃×D₃	semidirect
248. C₃²	#2155	semidirect
285. C₃	#69312	direct (N × H)
286. C₃	C₃×#2155	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×M₃₂	2
C₃×D₃×#108	2
C₃×#4860	2
C₃²×#365	2
C₃×#4513	2
C₃×#4494	2
C₃×#4487	2
C₃×#2155	3
#69312	3

Subgroups — C3×#69312 #11797118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69312 #11797118