Subgroups — C₃×#78540 #11877029

Order 1152.

787 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5470	C₂	1	2	yes	maximal
2	C₃²×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#5137	C₂	1	4	yes	maximal
4	C₃×#4716	C₂	1	4	yes	maximal, complemented
5	C₃×#4699	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4681	C₂	1	2	yes	maximal, complemented
7	C₃×#4680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3185	—	3	1	no	maximal
9	#78540	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₃×#926	—	2	2	no	—
13	C₃×#924	—	2	2	no	—
14	C₃×#924	C₂²	1	4	yes	—
15	C₂×C₃×Dic₁₂	C₂²	1	4	yes	—
16	C₃×#870	—	2	2	no	—
17	C₃×#869	C₂²	1	2	yes	—
18	C₃×#869	C₂²	1	2	yes	—
19	C₃×C₄×Dic₆	—	2	2	no	—
20	C₃×C₄×Dic₆	C₂²	1	4	yes	—
21	C₃²×#97	C₂²	1	2	yes	—
22	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃×#817	C₂²	1	4	yes	—
25	C₃×#816	C₂²	1	8	yes	—
26	C₃×#533	—	3	2	no	—
27	#5470	C₂×C₃	1	2	yes	—
28	C₃×#498	—	3	1	no	—
29	C₃×#494	—	3	1	no	—
30	C₃×#487	—	3	2	no	—
31	C₃×#470	—	3	4	no	—
32	C₃×#441	—	3	4	no	—
33	C₃×C₄×M₁₆	—	2	1	no	—
34	C₃×C₄×M₁₆	D₃	1	1	yes	characteristic
35	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
36	#5137	C₂×C₃	1	4	yes	—
37	#4716	C₂×C₃	1	4	yes	complemented
38	#4699	C₂×C₃	1	1	yes	characteristic, complemented
39	#4681	C₂×C₃	1	2	yes	complemented
40	#4680	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂²×C₃²×C₄	D₄	1	2	yes	—
42	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃×Dic₆	—	2	2	no	—
44	C₂×C₃×Dic₆	C₂³	1	4	yes	—
45	C₃²×M₁₆	—	2	4	no	—
46	C₂×C₃²×C₈	C₂³	1	2	yes	—
47	C₂×C₃²×C₈	C₂³	1	2	yes	—
48	C₃²×C₄²	D₄	1	2	yes	—
49	C₃²×C₄²	C₂³	1	1	yes	characteristic
50	C₃²×C₄²	C₂³	1	1	yes	characteristic
51	C₃×#217	—	2	4	no	—
52	C₃×C₄⋊Dic₃	—	2	2	no	—
53	C₃×C₄⋊Dic₃	—	2	2	no	—
54	C₃×C₄⋊Dic₃	—	2	2	no	—
55	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
56	C₃×Dic₃⋊C₄	—	2	4	no	—
57	C₃×Dic₃⋊C₄	—	2	4	no	—
58	C₃×C₄×Dic₃	—	2	4	no	—
59	C₃×Dic₁₂	—	2	8	no	—
60	#3185	—	3	1	no	Sylow
61	C₂×C₃×Q₁₆	—	3	4	no	—
62	C₂×C₃×M₁₆	—	2	2	no	—
63	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
64	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
65	C₃×C₄⋊Q₈	—	3	2	no	—
66	C₃×C₄⋊Q₈	—	3	2	no	—
67	C₃×#125	—	6	2	no	—
68	C₃×C₂²⋊Q₈	—	3	4	no	—
69	C₃×C₂²⋊Q₈	—	6	2	no	—
70	C₃×C₄×Q₈	—	3	4	no	—
71	C₃×C₄×Q₈	—	6	2	no	—
72	C₃×#117	—	6	2	no	—
73	C₂×C₃×C₄²	—	2	1	no	—
74	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
75	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
76	#926	—	2	2	no	—
77	#924	—	2	2	no	—
78	#924	C₂²×C₃	1	4	yes	—
79	C₂×Dic₁₂	C₂²×C₃	1	4	yes	—
80	#870	—	2	2	no	—
81	#869	C₂²×C₃	1	2	yes	—
82	#869	C₂²×C₃	1	2	yes	—
83	C₄×Dic₆	—	2	2	no	—
84	C₄×Dic₆	C₂²×C₃	1	4	yes	—
85	C₃×#106	—	3	4	no	—
86	C₃×#103	—	3	8	no	—
87	C₃×#97	—	2	2	no	—
88	C₃×#97	C₂×D₃	1	2	yes	—
89	C₃×#97	C₂²×C₃	1	2	yes	—
90	C₃×C₄×C₈	—	2	1	no	—
91	C₃×C₄×C₈	—	2	1	no	—
92	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
94	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
95	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
96	#817	C₂²×C₃	1	4	yes	—
97	#816	C₂²×C₃	1	8	yes	—
98	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃²×C₄	—	2	2	no	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
102	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
104	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
105	C₂×C₃×Dic₃	—	2	4	no	—
106	C₂×C₃×Dic₃	—	2	4	no	—
107	C₃×Dic₆	—	2	8	no	—
108	C₃×Dic₆	—	4	4	no	—
109	C₃²×C₈	—	2	2	no	—
110	C₃²×C₈	#41	1	4	yes	—
111	#533	—	3	2	no	—
112	#498	—	3	1	no	—
113	#494	—	3	1	no	—
114	#487	—	3	2	no	—
115	#470	—	3	4	no	—
116	#441	—	3	4	no	—
117	C₄×M₁₆	C₃×D₃	1	1	yes	characteristic
118	C₂×C₃×Q₈	—	3	4	no	—
119	C₂×C₃×Q₈	—	6	2	no	—
120	C₂²×C₃×C₄	—	2	2	no	—
121	C₂²×C₃×C₄	—	2	1	no	—
122	C₂²×C₃×C₄	D₁₂	1	2	yes	—
123	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
124	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₆	—	2	2	no	—
127	C₂×Dic₆	C₂³×C₃	1	4	yes	—
128	C₃×Q₁₆	—	6	8	no	—
129	C₃×M₁₆	—	2	8	no	—
130	C₃×M₁₆	—	2	4	no	—
131	C₃×M₁₆	—	2	4	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
135	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
136	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
137	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
138	C₃×C₄⋊C₄	—	3	4	no	—
139	C₃×C₄⋊C₄	—	6	4	no	—
140	C₃×C₄⋊C₄	—	6	4	no	—
141	C₃×C₄⋊C₄	—	6	2	no	—
142	C₃×C₄⋊C₄	—	6	2	no	—
143	C₃×C₄⋊C₄	—	6	2	no	—
144	C₃×C₂²⋊C₄	—	6	4	no	—
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	2	1	no	—
148	C₃×C₄²	—	6	4	no	—
149	C₃×C₄²	D₁₂	1	2	yes	—
150	C₃×C₄²	C₃×D₄	1	2	yes	—
151	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
152	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
153	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
154	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
155	#217	—	2	4	no	—
156	C₄⋊Dic₃	—	2	2	no	—
157	C₄⋊Dic₃	—	2	2	no	—
158	C₄⋊Dic₃	—	2	2	no	—
159	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
160	Dic₃⋊C₄	—	2	4	no	—
161	Dic₃⋊C₄	—	2	4	no	—
162	C₄×Dic₃	—	2	4	no	—
163	Dic₁₂	—	2	8	no	—
164	C₂²×C₃²	—	2	1	no	—
165	C₂²×C₃²	—	2	2	no	—
166	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	—	2	2	no	—
169	C₃²×C₄	#137	1	4	yes	—
170	C₃²×C₄	C₂×#41	1	2	yes	—
171	C₃×Dic₃	—	4	4	no	—
172	C₃×Dic₃	—	4	4	no	—
173	C₂×Q₁₆	—	3	4	no	—
174	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
175	C₄⋊Q₈	—	3	2	no	—
176	C₄⋊Q₈	—	3	2	no	—
177	#125	—	6	2	no	—
178	C₂²⋊Q₈	—	3	4	no	—
179	C₂²⋊Q₈	—	6	2	no	—
180	C₄×Q₈	—	3	4	no	—
181	C₄×Q₈	—	6	2	no	—
182	#117	—	6	2	no	—
183	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
184	#106	—	3	4	no	—
185	#103	—	3	8	no	—
186	#97	C₂×C₃×D₃	1	2	yes	—
187	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
188	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
189	C₂³×C₃	—	2	1	no	—
190	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
191	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
192	C₃×Q₈	—	6	8	no	—
193	C₃×Q₈	—	12	4	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	4	4	no	—
204	C₂×C₃×C₄	—	6	4	no	—
205	C₂×C₃×C₄	—	6	4	no	—
206	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
207	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
209	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
210	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
212	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
214	C₂×Dic₃	—	2	4	no	—
215	C₂×Dic₃	—	2	4	no	—
216	Dic₆	—	2	8	no	—
217	Dic₆	—	4	4	no	—
218	C₃×C₈	—	2	4	no	—
219	C₃×C₈	—	2	2	no	—
220	C₃×C₈	—	2	2	no	—
221	C₃×C₈	—	2	4	no	—
222	C₃×C₈	#235	1	4	yes	—
223	C₃×C₈	C₃×#41	1	4	yes	—
224	C₂×C₃²	—	4	1	no	—
225	C₂×C₃²	#532	1	1	yes	characteristic
226	C₂×C₃²	C₂×#137	1	2	yes	—
227	C₂×Q₈	—	3	4	no	—
228	C₂×Q₈	—	6	2	no	—
229	C₂²×C₄	C₃×D₁₂	1	2	yes	—
230	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
231	Q₁₆	—	6	8	no	—
232	M₁₆	—	2	4	no	—
233	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
234	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
235	C₄⋊C₄	—	3	4	no	—
236	C₄⋊C₄	—	6	2	no	—
237	C₄⋊C₄	—	6	2	no	—
238	C₄⋊C₄	—	6	4	no	—
239	C₄⋊C₄	—	6	4	no	—
240	C₄⋊C₄	—	6	2	no	—
241	C₂²⋊C₄	—	6	4	no	—
242	C₄²	—	6	4	no	—
243	C₄²	C₃×D₁₂	1	2	yes	—
244	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
245	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	2	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	4	2	no	—
253	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
254	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	4	2	no	—
263	C₃×C₄	—	12	4	no	—
264	C₃×C₄	—	12	4	no	—
265	C₃×C₄	#909	1	4	yes	—
266	C₃×C₄	C₃×#137	1	4	yes	—
267	C₃×C₄	C₂×#235	1	2	yes	—
268	C₃×C₄	C₂×C₃×#41	1	2	yes	—
269	Dic₃	—	4	4	no	—
270	Dic₃	—	4	4	no	—
271	C₃²	#3185	1	1	yes	characteristic, complemented, Sylow
272	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
273	Q₈	—	6	8	no	—
274	Q₈	—	12	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	6	4	no	—
278	C₂×C₄	—	6	4	no	—
279	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
280	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
281	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
282	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	C₃×#235	1	4	yes	—
285	C₂×C₃	—	2	2	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	4	2	no	—
288	C₂×C₃	—	4	1	no	—
289	C₂×C₃	—	4	1	no	—
290	C₂×C₃	#5477	1	1	yes	characteristic
291	C₂×C₃	C₂×#909	1	2	yes	—
292	C₂×C₃	C₃×#532	1	1	yes	characteristic
293	C₂×C₃	C₂×C₃×#137	1	2	yes	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	12	4	no	—
300	C₄	—	12	4	no	—
301	C₄	C₃×#909	1	4	yes	—
302	C₄	C₂×C₃×#235	1	2	yes	—
303	C₃	—	2	1	no	—
304	C₃	#78540	1	1	yes	characteristic, direct product factor
305	C₃	C₃×#3185	1	1	yes	characteristic, complemented
306	C₂	—	4	1	no	—
307	C₂	C₃×#5477	1	1	yes	characteristic
308	C₂	C₂×C₃×#909	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×C₄×M₁₆	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂³×C₃²	C₂×D₄
hypercenter	C₃×C₄×M₁₆	D₃
hypocenter	C₃	C₃×#3185
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₃×#4716	C₂	semidirect
5. C₃×#4699	C₂	semidirect
6. C₃×#4681	C₂	semidirect
7. C₃×#4680	C₂	semidirect
9. #78540	C₃	direct (N × H)
37. #4716	C₂×C₃	semidirect
38. #4699	C₂×C₃	semidirect
39. #4681	C₂×C₃	semidirect
40. #4680	C₂×C₃	semidirect
271. C₃²	#3185	semidirect
304. C₃	#78540	direct (N × H)
305. C₃	C₃×#3185	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5470	2
C₃²×C₄×M₁₆	2
C₃×#5137	2
C₃×#4716	2
C₃×#4699	2
C₃×#4681	2
C₃×#4680	2
C₃×#3185	3
#78540	3

Subgroups — C3×#78540 #11877029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#78540 #11877029