Subgroups — C₃×#69894 #11790489

Order 1152.

457 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
4	C₃×#4995	C₂	1	2	yes	maximal
5	C₃×#4554	C₂	1	2	yes	maximal
6	C₃×#1952	—	3	1	no	maximal
7	#69894	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₃²×#117	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	2	yes	—
13	C₃²×#97	—	2	1	no	—
14	C₃²×C₄×C₈	—	2	1	no	—
15	C₃×#819	C₄	1	4	yes	complemented
16	C₃×#818	—	2	1	no	—
17	C₃×#818	—	2	1	no	—
18	C₃×#808	—	2	2	no	—
19	C₃×#807	—	2	2	no	—
20	C₂×C₃×#108	—	3	1	no	—
21	C₃×#428	—	3	2	no	—
22	C₂×C₃×#107	—	3	1	no	—
23	C₃×#405	—	2	1	no	—
24	C₃×#405	D₃	1	1	yes	characteristic
25	C₃×#405	C₂×C₃	1	1	yes	characteristic
26	C₂×#819	C₂×C₃	1	1	yes	characteristic
27	C₂×#818	C₂×C₃	1	1	yes	characteristic
28	#4995	C₂×C₃	1	2	yes	—
29	C₃×#343	—	3	2	no	—
30	#4554	C₂×C₃	1	2	yes	—
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	—	2	2	no	—
33	C₃²×M₁₆	—	2	1	no	—
34	C₃²×M₁₆	Q₈	1	2	yes	—
35	C₂×C₃²×C₈	—	2	1	no	—
36	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	Q₈	1	2	yes	—
40	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₃²×C₂²⋊C₄	—	2	1	no	—
43	C₃²×C₄²	—	2	1	no	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄⋊Dic₃	—	2	2	no	—
46	C₃×#208	—	2	2	no	—
47	C₃×#208	C₂×C₄	1	4	yes	—
48	C₂×C₃×C₃⋊C₈	—	2	2	no	—
49	#1952	—	3	1	no	Sylow
50	C₂×C₃×M₁₆	—	2	1	no	—
51	C₂×C₃×M₁₆	—	3	2	no	—
52	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₃×C₈	—	2	1	no	—
55	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₃×#117	—	2	1	no	—
58	C₃×#117	C₂×D₃	1	1	yes	characteristic
59	C₃×#117	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₃×C₄⋊C₄	—	3	2	no	—
61	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
62	C₂×#208	C₂²×C₃	1	2	yes	—
63	C₃×#108	—	3	4	no	—
64	C₃×#107	—	6	1	no	—
65	C₃×#107	—	6	1	no	—
66	C₃×#107	—	6	2	no	—
67	C₃×#106	—	6	2	no	—
68	C₃×#97	—	2	2	no	—
69	C₃×#97	—	2	1	no	—
70	C₃×#97	—	2	1	no	—
71	C₃×C₄×C₈	—	2	2	no	—
72	C₃×C₄×C₈	—	2	1	no	—
73	C₃×C₄×C₈	—	2	1	no	—
74	#819	C₃×C₄	1	4	yes	complemented
75	#818	—	2	1	no	—
76	#818	—	2	1	no	—
77	#808	—	2	2	no	—
78	#807	—	2	2	no	—
79	C₂³×C₃²	#41	1	1	yes	characteristic
80	C₂×C₃²×C₄	—	2	1	no	—
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₃²×C₄	#41	1	1	yes	characteristic
88	C₂×C₃×Dic₃	—	4	2	no	—
89	C₂×C₃×Dic₃	—	8	2	no	—
90	C₃²×C₈	—	2	2	no	—
91	C₃²×C₈	C₄⋊C₄	1	2	yes	—
92	C₃²×C₈	C₄⋊C₄	1	2	yes	—
93	C₃×C₃⋊C₈	—	2	4	no	—
94	C₂×#108	—	3	1	no	—
95	#428	—	3	2	no	—
96	C₂×#107	—	3	1	no	—
97	#405	C₃×D₃	1	1	yes	characteristic
98	#343	—	3	2	no	—
99	C₂²×C₃×C₄	—	2	1	no	—
100	C₂²×C₃×C₄	—	6	2	no	—
101	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₂²×Dic₃	—	2	2	no	—
104	C₃×M₁₆	—	2	2	no	—
105	C₃×M₁₆	—	2	1	no	—
106	C₃×M₁₆	—	2	1	no	—
107	C₃×M₁₆	—	2	2	no	—
108	C₃×M₁₆	—	3	4	no	—
109	C₃×M₁₆	—	6	2	no	—
110	C₃×M₁₆	Dic₆	1	2	yes	—
111	C₃×M₁₆	C₃×Q₈	1	2	yes	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	2	1	no	—
119	C₂×C₃×C₈	—	6	2	no	—
120	C₂×C₃×C₈	Dic₆	1	2	yes	—
121	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
122	C₂×C₃×C₈	#67	1	1	yes	characteristic
123	C₂×C₃×C₈	#67	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
125	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
126	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
127	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
128	C₃×C₄⋊C₄	—	2	1	no	—
129	C₃×C₄⋊C₄	—	2	1	no	—
130	C₃×C₄⋊C₄	—	6	2	no	—
131	C₃×C₄⋊C₄	—	6	2	no	—
132	C₃×C₄⋊C₄	D₁₂	1	1	yes	characteristic
133	C₃×C₄⋊C₄	D₁₂	1	1	yes	characteristic
134	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
135	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
136	C₃×C₂²⋊C₄	—	2	2	no	—
137	C₃×C₂²⋊C₄	—	2	1	no	—
138	C₃×C₂²⋊C₄	—	2	1	no	—
139	C₃×C₄²	—	2	2	no	—
140	C₃×C₄²	—	2	1	no	—
141	C₃×C₄²	—	2	1	no	—
142	C₄⋊Dic₃	—	2	2	no	—
143	C₄⋊Dic₃	—	2	2	no	—
144	#208	—	2	2	no	—
145	#208	C₂×C₃×C₄	1	4	yes	—
146	C₂×C₃⋊C₈	—	2	2	no	—
147	C₂²×C₃²	—	2	1	no	—
148	C₂²×C₃²	—	2	1	no	—
149	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
150	C₂²×C₃²	#121	1	1	yes	characteristic
151	C₂²×C₃²	#125	1	1	yes	characteristic
152	C₃²×C₄	—	4	1	no	—
153	C₃²×C₄	—	4	1	no	—
154	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
156	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
157	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
158	C₃×Dic₃	—	8	2	no	—
159	C₂×M₁₆	—	3	2	no	—
160	C₂×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
162	#117	C₂×C₃×D₃	1	1	yes	characteristic
163	C₂×C₄⋊C₄	—	3	2	no	—
164	#108	—	3	4	no	—
165	#107	—	6	1	no	—
166	#107	—	6	1	no	—
167	#107	—	6	2	no	—
168	#106	—	6	2	no	—
169	#97	—	2	1	no	—
170	C₄×C₈	—	2	1	no	—
171	C₂³×C₃	—	2	1	no	—
172	C₂³×C₃	#235	1	1	yes	characteristic
173	C₂³×C₃	C₃×#41	1	1	yes	characteristic
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	12	2	no	—
187	C₂×C₃×C₄	—	24	2	no	—
188	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
192	C₂×C₃×C₄	#235	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
199	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
200	C₂×Dic₃	—	4	2	no	—
201	C₂×Dic₃	—	8	2	no	—
202	C₃×C₈	—	2	4	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	2	2	no	—
206	C₃×C₈	—	2	2	no	—
207	C₃×C₈	—	6	4	no	—
208	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
209	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
210	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
211	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
212	C₃⋊C₈	—	2	4	no	—
213	C₂×C₃²	—	2	1	no	—
214	C₂×C₃²	—	2	1	no	—
215	C₂×C₃²	#389	1	1	yes	characteristic
216	C₂×C₃²	#472	1	1	yes	characteristic
217	C₂×C₃²	#473	1	1	yes	characteristic
218	C₂²×C₄	—	6	2	no	—
219	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
220	M₁₆	—	2	1	no	—
221	M₁₆	—	3	4	no	—
222	M₁₆	—	6	2	no	—
223	M₁₆	C₃×Dic₆	1	2	yes	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	6	2	no	—
226	C₂×C₈	C₃×Dic₆	1	2	yes	—
227	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
228	C₂×C₈	C₃×#67	1	1	yes	characteristic
229	C₂×C₈	C₃×#67	1	1	yes	characteristic
230	C₄⋊C₄	—	6	2	no	—
231	C₄⋊C₄	—	6	2	no	—
232	C₄⋊C₄	C₃×D₁₂	1	1	yes	characteristic
233	C₄⋊C₄	C₃×D₁₂	1	1	yes	characteristic
234	C₂²⋊C₄	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	4	1	no	—
244	C₂²×C₃	—	4	1	no	—
245	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
246	C₂²×C₃	#870	1	1	yes	characteristic
247	C₂²×C₃	#930	1	1	yes	characteristic
248	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
249	C₂²×C₃	C₃×#121	1	1	yes	characteristic
250	C₂²×C₃	C₃×#125	1	1	yes	characteristic
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	4	2	no	—
256	C₃×C₄	—	4	2	no	—
257	C₃×C₄	—	4	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	—	24	2	no	—
262	C₃×C₄	#869	1	1	yes	characteristic
263	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
264	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
265	C₃×C₄	#924	1	1	yes	characteristic
266	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
267	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
268	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
269	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
270	Dic₃	—	8	2	no	—
271	C₃²	#1952	1	1	yes	characteristic, complemented, Sylow
272	C₂³	C₃×#235	1	1	yes	characteristic
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	12	2	no	—
276	C₂×C₄	—	24	2	no	—
277	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
278	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
279	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
280	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
281	C₂×C₄	C₃×#235	1	1	yes	characteristic
282	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	6	4	no	—
285	C₈	C₃×Dic₃⋊C₄	1	2	yes	—
286	C₈	C₃×Dic₃⋊C₄	1	2	yes	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	4	1	no	—
295	C₂×C₃	—	4	1	no	—
296	C₂×C₃	#4928	1	1	yes	characteristic
297	C₂×C₃	#5142	1	1	yes	characteristic
298	C₂×C₃	#5143	1	1	yes	characteristic
299	C₂×C₃	C₃×#389	1	1	yes	characteristic
300	C₂×C₃	C₃×#472	1	1	yes	characteristic
301	C₂×C₃	C₃×#473	1	1	yes	characteristic
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	C₃×C₄×Dic₆	1	1	yes	characteristic
305	C₂²	C₃×#870	1	1	yes	characteristic
306	C₂²	C₃×#930	1	1	yes	characteristic
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	24	2	no	—
310	C₄	C₃×#869	1	1	yes	characteristic
311	C₄	C₃×C₄×D₁₂	1	1	yes	characteristic
312	C₄	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
313	C₄	C₃×#924	1	1	yes	characteristic
314	C₃	—	2	1	no	—
315	C₃	#69894	1	1	yes	characteristic, direct product factor
316	C₃	C₃×#1952	1	1	yes	characteristic, complemented
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	C₃×#4928	1	1	yes	characteristic
320	C₂	C₃×#5142	1	1	yes	characteristic
321	C₂	C₃×#5143	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#930
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#405	C₂
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#405	D₃
hypocenter	C₃	C₃×#1952
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#405

Complements

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

N	H ≅ G/N	type
7. #69894	C₃	direct (N × H)
15. C₃×#819	C₄	semidirect
74. #819	C₃×C₄	semidirect
271. C₃²	#1952	semidirect
315. C₃	#69894	direct (N × H)
316. C₃	C₃×#1952	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#405	2
C₂×C₃×#819	2
C₂×C₃×#818	2
C₃×#4995	2
C₃×#4554	2
C₃×#1952	3
#69894	3

Subgroups — C3×#69894 #11790489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69894 #11790489