Subgroups — C₃×#69395 #11790778

Order 1152.

849 conjugacy classes of subgroups.

The subgroup lattice has 328 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×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#805	C₂	1	4	yes	maximal, complemented
4	C₃×#4924	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1962	—	3	1	no	maximal
6	#69395	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×#235	C₂²	1	2	yes	—
8	C₂×C₃²×C₄²	—	2	2	no	—
9	C₂×C₃²×C₄²	C₂²	1	2	yes	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₃×#930	—	2	2	no	—
12	C₂×C₃×#208	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	C₃×#805	—	2	8	no	—
18	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
19	C₃×#532	—	3	1	no	—
20	#5477	C₂×C₃	1	1	yes	characteristic
21	C₃×#423	—	3	1	no	—
22	C₂×C₃×#104	—	3	4	no	—
23	C₃×C₄³	—	2	1	no	—
24	C₃×C₄³	D₃	1	1	yes	characteristic, complemented
25	C₃×C₄³	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×#805	C₂×C₃	1	4	yes	complemented
27	#4924	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×C₃²×C₄	—	2	2	no	—
29	C₂²×C₃²×C₄	D₄	1	2	yes	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃×#67	—	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×C₄²	—	2	8	no	—
39	C₃²×C₄²	—	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	C₃²×C₄²	D₄	1	4	yes	complemented
44	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
45	C₃×C₄⋊Dic₃	—	2	2	no	—
46	C₃×#208	—	2	4	no	—
47	C₂×C₃×C₃⋊C₈	—	2	2	no	—
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	—	3	2	no	—
52	C₃×C₄⋊Q₈	—	3	1	no	—
53	C₃×#127	—	3	1	no	—
54	C₃×#124	—	3	2	no	—
55	C₃×#121	—	6	2	no	—
56	C₃×C₄×D₄	—	6	2	no	—
57	C₂×C₃×C₄²	—	2	4	no	—
58	C₂×C₃×C₄²	—	2	2	no	—
59	C₂×C₃×C₄²	—	2	2	no	—
60	C₂×C₃×C₄²	—	2	2	no	—
61	C₂×C₃×C₄²	—	2	1	no	—
62	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
63	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
64	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
65	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
66	#930	—	2	2	no	—
67	C₂×#208	C₂²×C₃	1	2	yes	—
68	#875	C₃×C₄	1	2	yes	complemented
69	#874	C₃×C₄	1	1	yes	characteristic, complemented
70	C₄×D₁₂	—	2	2	no	—
71	#869	C₃×C₄	1	1	yes	characteristic, complemented
72	C₃×C₄⋊C₈	—	6	2	no	—
73	C₃×#104	—	6	8	no	—
74	#805	—	2	8	no	—
75	C₄⋊(C₃⋊C₈)	—	2	2	no	—
76	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×D₃	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	8	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
82	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
83	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
85	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	#41	1	2	yes	—
87	C₃×#67	—	4	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	C₃×C₃⋊C₈	—	4	2	no	—
95	#532	—	3	1	no	—
96	#423	—	3	1	no	—
97	C₂×#104	—	3	4	no	—
98	C₄³	C₃×D₃	1	1	yes	characteristic, complemented
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	4	no	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	—	2	2	no	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	6	2	no	—
111	C₂²×C₃×C₄	#67	1	2	yes	—
112	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
113	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₂×#67	—	2	2	no	—
116	#235	—	2	4	no	—
117	#235	—	2	4	no	—
118	C₂×D₁₂	—	2	2	no	—
119	C₂×D₁₂	C₂×C₃×C₄	1	2	yes	—
120	C₂×C₄×D₃	—	2	2	no	—
121	C₂×Dic₆	C₂×C₃×C₄	1	2	yes	—
122	C₃×M₁₆	—	6	4	no	—
123	C₂×C₃×C₈	—	6	2	no	—
124	C₃×C₄⋊C₄	—	6	2	no	—
125	C₃×C₂²⋊C₄	—	6	4	no	—
126	C₃×C₄²	—	2	16	no	—
127	C₃×C₄²	—	2	8	no	—
128	C₃×C₄²	—	2	8	no	—
129	C₃×C₄²	—	2	4	no	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	—	2	2	no	—
132	C₃×C₄²	—	2	1	no	—
133	C₃×C₄²	—	2	1	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₃×C₄²	—	2	4	no	—
136	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
137	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
138	C₃×C₄²	D₁₂	1	2	yes	—
139	C₃×C₄²	D₁₂	1	4	yes	complemented
140	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
141	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
142	C₃×C₄²	C₃×D₄	1	2	yes	—
143	C₃×C₄²	C₃×D₄	1	4	yes	complemented
144	(C₂×D₃)⋊C₄	—	2	4	no	—
145	C₄⋊Dic₃	—	2	2	no	—
146	#208	—	2	4	no	—
147	C₂×C₃⋊C₈	—	2	2	no	—
148	C₂²×C₃²	—	2	2	no	—
149	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₃²	#121	1	2	yes	—
151	C₂×C₃×D₃	—	4	2	no	—
152	C₂×C₃×D₃	—	4	4	no	—
153	C₃²×C₄	—	2	8	no	—
154	C₃²×C₄	—	2	2	no	—
155	C₃²×C₄	#104	1	4	yes	complemented
156	C₃²×C₄	C₄×D₄	1	2	yes	—
157	C₃²×C₄	#124	1	1	yes	characteristic
158	C₃²×C₄	#127	1	1	yes	characteristic
159	C₃×Dic₃	—	4	2	no	—
160	C₂×#41	—	3	2	no	—
161	C₂×M₁₆	—	3	2	no	—
162	C₄⋊Q₈	—	3	1	no	—
163	#127	—	3	1	no	—
164	#124	—	3	2	no	—
165	#121	—	6	2	no	—
166	C₄×D₄	—	6	2	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂×C₃×D₃	1	2	yes	—
169	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
170	C₄⋊C₈	—	6	2	no	—
171	#104	—	6	8	no	—
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	—	6	2	no	—
174	C₂³×C₃	C₂×#67	1	1	yes	characteristic
175	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂²×D₃	—	2	2	no	—
177	C₃×Q₈	—	6	4	no	—
178	C₃×D₄	—	6	4	no	—
179	C₃×D₄	—	12	4	no	—
180	C₃×D₄	—	12	4	no	—
181	C₂×C₃×C₄	—	2	16	no	—
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	8	no	—
184	C₂×C₃×C₄	—	2	8	no	—
185	C₂×C₃×C₄	—	2	8	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	6	2	no	—
197	C₂×C₃×C₄	—	12	2	no	—
198	C₂×C₃×C₄	—	12	2	no	—
199	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
200	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
201	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
202	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
203	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
205	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
206	C₂×C₃×C₄	#235	1	2	yes	—
207	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
209	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₃×#41	1	2	yes	—
211	#67	—	4	4	no	—
212	C₂×Dic₃	—	2	2	no	—
213	D₁₂	—	2	4	no	—
214	D₁₂	—	4	4	no	—
215	C₄×D₃	—	4	2	no	—
216	C₄×D₃	—	4	2	no	—
217	Dic₆	—	2	4	no	—
218	C₃×C₈	—	12	2	no	—
219	C₃⋊C₈	—	4	2	no	—
220	C₂×C₃²	—	2	2	no	—
221	C₂×C₃²	#390	1	1	yes	characteristic
222	C₂×C₃²	C₂×#104	1	2	yes	—
223	C₃×D₃	—	8	2	no	—
224	#41	—	6	4	no	—
225	#41	—	6	4	no	—
226	C₂×Q₈	—	3	2	no	—
227	C₂×D₄	—	3	2	no	—
228	C₂×D₄	—	6	2	no	—
229	C₂×D₄	—	6	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	6	2	no	—
232	C₂²×C₄	C₃×#67	1	2	yes	—
233	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
234	M₁₆	—	6	4	no	—
235	C₂×C₈	—	6	2	no	—
236	C₄⋊C₄	—	6	2	no	—
237	C₂²⋊C₄	—	6	4	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	2	2	no	—
240	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
241	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
242	C₄²	C₃×D₁₂	1	2	yes	—
243	C₄²	C₃×D₁₂	1	4	yes	complemented
244	C₂²×C₃	—	2	4	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	12	4	no	—
250	C₂²×C₃	—	12	2	no	—
251	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
252	C₂²×C₃	#930	1	2	yes	—
253	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₂²×C₃	C₃×#121	1	2	yes	—
255	C₂×D₃	—	4	2	no	—
256	C₂×D₃	—	4	4	no	—
257	C₃×C₄	—	2	16	no	—
258	C₃×C₄	—	2	8	no	—
259	C₃×C₄	—	2	8	no	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	4	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	12	2	no	—
268	C₃×C₄	#805	1	4	yes	complemented
269	C₃×C₄	C₃×#104	1	4	yes	complemented
270	C₃×C₄	C₄×D₁₂	1	2	yes	—
271	C₃×C₄	#874	1	1	yes	characteristic
272	C₃×C₄	#875	1	1	yes	characteristic
273	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
274	C₃×C₄	C₃×#124	1	1	yes	characteristic
275	C₃×C₄	C₃×#127	1	1	yes	characteristic
276	Dic₃	—	4	2	no	—
277	C₃²	#1962	1	1	yes	characteristic, complemented, Sylow
278	C₂³	—	6	2	no	—
279	C₂³	C₂×C₃×#67	1	1	yes	characteristic
280	Q₈	—	6	4	no	—
281	D₄	—	6	4	no	—
282	D₄	—	12	4	no	—
283	D₄	—	12	4	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	6	2	no	—
288	C₂×C₄	—	12	2	no	—
289	C₂×C₄	—	12	2	no	—
290	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
291	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
292	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
294	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
295	C₂×C₄	C₃×#235	1	2	yes	—
296	C₈	—	12	2	no	—
297	C₂×C₃	—	2	4	no	—
298	C₂×C₃	—	2	2	no	—
299	C₂×C₃	—	2	2	no	—
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	24	2	no	—
303	C₂×C₃	C₂×#805	1	2	yes	—
304	C₂×C₃	#4939	1	1	yes	characteristic
305	C₂×C₃	C₃×#390	1	1	yes	characteristic
306	C₂×C₃	C₂×C₃×#104	1	2	yes	—
307	D₃	—	8	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	12	4	no	—
310	C₂²	—	12	2	no	—
311	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
312	C₂²	C₃×#930	1	2	yes	—
313	C₄	—	2	8	no	—
314	C₄	—	2	2	no	—
315	C₄	—	12	2	no	—
316	C₄	C₃×#805	1	4	yes	complemented
317	C₄	C₃×C₄×D₁₂	1	2	yes	—
318	C₄	C₃×#874	1	1	yes	characteristic
319	C₄	C₃×#875	1	1	yes	characteristic
320	C₃	—	2	1	no	—
321	C₃	#69395	1	1	yes	characteristic, direct product factor
322	C₃	C₃×#1962	1	1	yes	characteristic, complemented
323	C₂	—	2	2	no	—
324	C₂	—	24	2	no	—
325	C₂	C₂×C₃×#805	1	2	yes	—
326	C₂	C₃×#4939	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×C₄³	C₂
socle	C₂²×C₃²	C₂×C₂²⋊C₄
prime radical	C₃×#5477	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×C₄³	D₃
hypocenter	C₃	C₃×#1962
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×C₄³

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄³	C₂	semidirect
3. C₂×C₃×#805	C₂	semidirect
4. C₃×#4924	C₂	semidirect
6. #69395	C₃	direct (N × H)
13. C₃×#875	C₄	semidirect
14. C₃×#874	C₄	semidirect
16. C₃×#869	C₄	semidirect
24. C₃×C₄³	D₃	semidirect
25. C₃×C₄³	C₂×C₃	semidirect
26. C₂×#805	C₂×C₃	semidirect
27. #4924	C₂×C₃	semidirect
43. C₃²×C₄²	D₄	semidirect
68. #875	C₃×C₄	semidirect
69. #874	C₃×C₄	semidirect
71. #869	C₃×C₄	semidirect
98. C₄³	C₃×D₃	semidirect
139. C₃×C₄²	D₁₂	semidirect
143. C₃×C₄²	C₃×D₄	semidirect
155. C₃²×C₄	#104	semidirect
243. C₄²	C₃×D₁₂	semidirect
268. C₃×C₄	#805	semidirect
269. C₃×C₄	C₃×#104	semidirect
277. C₃²	#1962	semidirect
316. C₄	C₃×#805	semidirect
321. C₃	#69395	direct (N × H)
322. 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×C₄³	2
C₂×C₃×#805	2
C₃×#4924	2
C₃×#1962	3
#69395	3

Subgroups — C3×#69395 #11790778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69395 #11790778