Subgroups — C₃×#70512 #11790775

Order 1152.

783 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5477	C₂	1	1	yes	maximal, characteristic
2	C₃²×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#825	C₂	1	4	yes	maximal, complemented
4	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1962	—	3	1	no	maximal
6	#70512	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×#235	C₂²	1	2	yes	—
8	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₃×#930	—	2	2	no	—
11	C₂×C₃×C₄×Dic₃	—	2	2	no	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
13	C₃×#875	C₄	1	2	yes	complemented
14	C₃×#874	C₄	1	1	yes	characteristic, complemented
15	C₃×C₄×D₁₂	—	2	2	no	—
16	C₃×#869	C₄	1	1	yes	characteristic, complemented
17	C₃²×C₄⋊C₈	—	2	2	no	—
18	C₃×#825	—	2	8	no	—
19	C₃×#532	—	3	1	no	—
20	#5477	C₂×C₃	1	1	yes	characteristic
21	C₃×#423	—	2	1	no	—
22	C₃×#423	D₃	1	1	yes	characteristic, complemented
23	C₃×#423	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×#104	—	3	4	no	—
25	C₃×C₄³	—	3	1	no	—
26	C₂×#825	C₂×C₃	1	4	yes	complemented
27	C₄²×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×C₃²×C₄	D₄	1	2	yes	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃×#67	—	2	2	no	—
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₃×#235	—	2	4	no	—
33	C₃×#235	—	2	4	no	—
34	C₂×C₃×D₁₂	—	2	2	no	—
35	C₂×C₃×D₁₂	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×D₃	—	2	2	no	—
37	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
38	C₃²×M₁₆	—	2	4	no	—
39	C₂×C₃²×C₈	—	2	2	no	—
40	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₃²×C₄²	D₄	1	2	yes	—
43	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄×Dic₃	—	2	8	no	—
46	C₃×C₄×Dic₃	—	2	2	no	—
47	C₃×C₄×Dic₃	D₄	1	4	yes	complemented
48	#1962	—	3	1	no	Sylow
49	C₂×C₃×#41	—	3	2	no	—
50	C₂×#235	C₂²×C₃	1	2	yes	—
51	C₂×C₃×M₁₆	—	2	2	no	—
52	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
53	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
54	C₃×C₄⋊Q₈	—	3	1	no	—
55	C₃×#127	—	3	1	no	—
56	C₃×#124	—	3	2	no	—
57	C₃×#121	—	6	2	no	—
58	C₃×C₄×D₄	—	6	2	no	—
59	C₂×C₃×C₄²	—	2	1	no	—
60	C₂×C₃×C₄²	—	3	2	no	—
61	C₂×C₃×C₄²	—	6	2	no	—
62	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
64	#930	—	2	2	no	—
65	C₂×C₄×Dic₃	—	2	2	no	—
66	C₂×C₄×Dic₃	C₂²×C₃	1	2	yes	—
67	#875	C₃×C₄	1	2	yes	complemented
68	#874	C₃×C₄	1	1	yes	characteristic, complemented
69	C₄×D₁₂	—	2	2	no	—
70	#869	C₃×C₄	1	1	yes	characteristic, complemented
71	C₃×C₄⋊C₈	—	2	4	no	—
72	C₃×C₄⋊C₈	—	2	2	no	—
73	C₃×C₄⋊C₈	—	2	2	no	—
74	C₃×#104	—	6	8	no	—
75	#825	—	2	8	no	—
76	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×D₃	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	4	no	—
79	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
80	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
81	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
83	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	#41	1	2	yes	—
85	C₃×#67	—	4	4	no	—
86	C₂×C₃×Dic₃	—	2	8	no	—
87	C₂×C₃×Dic₃	—	2	4	no	—
88	C₂×C₃×Dic₃	—	2	2	no	—
89	C₃×D₁₂	—	2	4	no	—
90	C₃×D₁₂	—	4	4	no	—
91	C₃×C₄×D₃	—	4	2	no	—
92	C₃×C₄×D₃	—	4	2	no	—
93	C₃×Dic₆	—	2	4	no	—
94	C₃²×C₈	—	4	2	no	—
95	#532	—	3	1	no	—
96	#423	C₃×D₃	1	1	yes	characteristic, complemented
97	C₂×#104	—	3	4	no	—
98	C₄³	—	3	1	no	—
99	C₃×#41	—	6	4	no	—
100	C₃×#41	—	6	4	no	—
101	C₂×C₃×Q₈	—	3	2	no	—
102	C₂×C₃×D₄	—	3	2	no	—
103	C₂×C₃×D₄	—	6	2	no	—
104	C₂×C₃×D₄	—	6	2	no	—
105	C₂²×C₃×C₄	—	2	2	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	—	6	2	no	—
108	C₂²×C₃×C₄	—	6	2	no	—
109	C₂²×C₃×C₄	D₁₂	1	2	yes	—
110	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
111	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂×#67	—	2	2	no	—
114	C₂²×Dic₃	—	2	2	no	—
115	#235	—	2	4	no	—
116	#235	—	2	4	no	—
117	C₂×D₁₂	—	2	2	no	—
118	C₂×D₁₂	C₂×C₃×C₄	1	2	yes	—
119	C₂×C₄×D₃	—	2	2	no	—
120	C₂×Dic₆	C₂×C₃×C₄	1	2	yes	—
121	C₃×M₁₆	—	2	4	no	—
122	C₃×M₁₆	—	2	4	no	—
123	C₃×M₁₆	—	4	4	no	—
124	C₂×C₃×C₈	—	2	4	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₃×C₄⋊C₄	—	6	2	no	—
128	C₃×C₂²⋊C₄	—	6	4	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	—	2	2	no	—
132	C₃×C₄²	—	3	4	no	—
133	C₃×C₄²	—	6	8	no	—
134	C₃×C₄²	—	6	2	no	—
135	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
136	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
137	C₃×C₄²	#67	1	2	yes	—
138	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
139	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
140	C₃×C₄²	C₃×D₄	1	2	yes	—
141	(C₂×D₃)⋊C₄	—	2	4	no	—
142	C₄⋊Dic₃	—	2	2	no	—
143	C₄×Dic₃	—	2	8	no	—
144	C₄×Dic₃	—	2	2	no	—
145	C₄×Dic₃	C₃×D₄	1	4	yes	complemented
146	C₂²×C₃²	—	2	2	no	—
147	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₃²	#121	1	2	yes	—
149	C₂×C₃×D₃	—	4	2	no	—
150	C₂×C₃×D₃	—	4	4	no	—
151	C₃²×C₄	—	2	2	no	—
152	C₃²×C₄	#104	1	4	yes	complemented
153	C₃²×C₄	C₄×D₄	1	2	yes	—
154	C₃²×C₄	#124	1	1	yes	characteristic
155	C₃²×C₄	#127	1	1	yes	characteristic
156	C₃×Dic₃	—	2	8	no	—
157	C₃×Dic₃	—	4	2	no	—
158	C₂×#41	—	3	2	no	—
159	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
160	C₄⋊Q₈	—	3	1	no	—
161	#127	—	3	1	no	—
162	#124	—	3	2	no	—
163	#121	—	6	2	no	—
164	C₄×D₄	—	6	2	no	—
165	C₂×C₄²	—	3	2	no	—
166	C₂×C₄²	—	6	2	no	—
167	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
168	C₄⋊C₈	—	2	2	no	—
169	#104	—	6	8	no	—
170	C₂³×C₃	—	2	1	no	—
171	C₂³×C₃	—	6	2	no	—
172	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
173	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂²×D₃	—	2	2	no	—
175	C₃×Q₈	—	6	4	no	—
176	C₃×D₄	—	6	4	no	—
177	C₃×D₄	—	12	4	no	—
178	C₃×D₄	—	12	4	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	4	4	no	—
188	C₂×C₃×C₄	—	6	8	no	—
189	C₂×C₃×C₄	—	6	4	no	—
190	C₂×C₃×C₄	—	6	2	no	—
191	C₂×C₃×C₄	—	12	2	no	—
192	C₂×C₃×C₄	—	12	2	no	—
193	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
194	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
195	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
196	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
197	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
199	C₂×C₃×C₄	#239	1	2	yes	—
200	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
203	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₃×#41	1	2	yes	—
205	#67	—	4	4	no	—
206	C₂×Dic₃	—	2	8	no	—
207	C₂×Dic₃	—	2	4	no	—
208	C₂×Dic₃	—	2	2	no	—
209	D₁₂	—	2	4	no	—
210	D₁₂	—	4	4	no	—
211	C₄×D₃	—	4	2	no	—
212	C₄×D₃	—	4	2	no	—
213	Dic₆	—	2	4	no	—
214	C₃×C₈	—	4	4	no	—
215	C₃×C₈	—	4	2	no	—
216	C₃×C₈	—	4	2	no	—
217	C₂×C₃²	—	2	2	no	—
218	C₂×C₃²	#390	1	1	yes	characteristic
219	C₂×C₃²	C₂×#104	1	2	yes	—
220	C₃×D₃	—	8	2	no	—
221	#41	—	6	4	no	—
222	#41	—	6	4	no	—
223	C₂×Q₈	—	3	2	no	—
224	C₂×D₄	—	3	2	no	—
225	C₂×D₄	—	6	2	no	—
226	C₂×D₄	—	6	2	no	—
227	C₂²×C₄	—	6	2	no	—
228	C₂²×C₄	—	6	2	no	—
229	C₂²×C₄	C₃×D₁₂	1	2	yes	—
230	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
231	M₁₆	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₄⋊C₄	—	6	2	no	—
234	C₂²⋊C₄	—	6	4	no	—
235	C₄²	—	3	4	no	—
236	C₄²	—	6	8	no	—
237	C₄²	—	6	2	no	—
238	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
239	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
240	C₄²	C₃×#67	1	2	yes	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	4	2	no	—
246	C₂²×C₃	—	12	4	no	—
247	C₂²×C₃	—	12	2	no	—
248	C₂²×C₃	#895	1	2	yes	—
249	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
250	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
251	C₂²×C₃	C₃×#121	1	2	yes	—
252	C₂×D₃	—	4	2	no	—
253	C₂×D₃	—	4	4	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	4	2	no	—
261	C₃×C₄	—	6	8	no	—
262	C₃×C₄	—	12	2	no	—
263	C₃×C₄	#825	1	4	yes	complemented
264	C₃×C₄	C₃×#104	1	4	yes	complemented
265	C₃×C₄	#893	1	2	yes	—
266	C₃×C₄	#940	1	1	yes	characteristic
267	C₃×C₄	#947	1	1	yes	characteristic
268	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
269	C₃×C₄	C₃×#124	1	1	yes	characteristic
270	C₃×C₄	C₃×#127	1	1	yes	characteristic
271	Dic₃	—	2	8	no	—
272	Dic₃	—	4	2	no	—
273	C₃²	#1962	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	6	2	no	—
275	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
276	Q₈	—	6	4	no	—
277	D₄	—	6	4	no	—
278	D₄	—	12	4	no	—
279	D₄	—	12	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	6	8	no	—
282	C₂×C₄	—	6	4	no	—
283	C₂×C₄	—	6	2	no	—
284	C₂×C₄	—	12	2	no	—
285	C₂×C₄	—	12	2	no	—
286	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
287	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
288	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
289	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
290	C₂×C₄	C₃×#239	1	2	yes	—
291	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
292	C₈	—	4	2	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	2	2	no	—
297	C₂×C₃	—	4	2	no	—
298	C₂×C₃	—	24	2	no	—
299	C₂×C₃	#4988	1	1	yes	characteristic
300	C₂×C₃	C₂×#825	1	2	yes	—
301	C₂×C₃	C₃×#390	1	1	yes	characteristic
302	C₂×C₃	C₂×C₃×#104	1	2	yes	—
303	D₃	—	8	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	12	2	no	—
306	C₂²	—	12	4	no	—
307	C₂²	C₃×#895	1	2	yes	—
308	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	6	8	no	—
311	C₄	—	12	2	no	—
312	C₄	C₃×#825	1	4	yes	complemented
313	C₄	C₃×#893	1	2	yes	—
314	C₄	C₃×#940	1	1	yes	characteristic
315	C₄	C₃×#947	1	1	yes	characteristic
316	C₃	—	2	1	no	—
317	C₃	#70512	1	1	yes	characteristic, direct product factor
318	C₃	C₃×#1962	1	1	yes	characteristic, complemented
319	C₂	—	2	2	no	—
320	C₂	—	24	2	no	—
321	C₂	C₃×#4988	1	1	yes	characteristic
322	C₂	C₂×C₃×#825	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#423	C₂
socle	C₂²×C₃²	C₂×C₂²⋊C₄
prime radical	C₃×#5477	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#423	D₃
hypocenter	C₃	C₃×#1962
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#423

Complements

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

N	H ≅ G/N	type
2. C₃²×#423	C₂	semidirect
3. C₂×C₃×#825	C₂	semidirect
4. C₃×C₄²×Dic₃	C₂	semidirect
6. #70512	C₃	direct (N × H)
13. C₃×#875	C₄	semidirect
14. C₃×#874	C₄	semidirect
16. C₃×#869	C₄	semidirect
22. C₃×#423	D₃	semidirect
23. C₃×#423	C₂×C₃	semidirect
26. C₂×#825	C₂×C₃	semidirect
27. C₄²×Dic₃	C₂×C₃	semidirect
47. C₃×C₄×Dic₃	D₄	semidirect
67. #875	C₃×C₄	semidirect
68. #874	C₃×C₄	semidirect
70. #869	C₃×C₄	semidirect
96. #423	C₃×D₃	semidirect
145. C₄×Dic₃	C₃×D₄	semidirect
152. C₃²×C₄	#104	semidirect
263. C₃×C₄	#825	semidirect
264. C₃×C₄	C₃×#104	semidirect
273. C₃²	#1962	semidirect
312. C₄	C₃×#825	semidirect
317. C₃	#70512	direct (N × H)
318. C₃	C₃×#1962	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5477	2
C₃²×#423	2
C₂×C₃×#825	2
C₃×C₄²×Dic₃	2
C₃×#1962	3
#70512	3

Subgroups — C3×#70512 #11790775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70512 #11790775