Subgroups — C₃×#68710 #11785484

Order 1152.

567 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	C₂×C₃×#907	C₂	1	1	yes	maximal, characteristic
2	C₃×#5006	C₂	1	2	yes	maximal
3	C₃×#4998	C₂	1	2	yes	maximal, complemented
4	C₃²×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4557	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1812	—	3	1	no	maximal
7	#68710	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#235	C₂²	1	1	yes	characteristic
9	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
10	C₃²×#117	C₂²	1	2	yes	—
11	C₃×#926	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×#907	—	2	2	no	—
14	C₃×#907	—	2	2	no	—
15	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	2	yes	complemented
16	C₂×C₃×D₃×C₈	C₄	1	2	yes	complemented
17	C₃×#808	—	2	2	no	—
18	C₃×#807	—	2	2	no	—
19	C₃×#806	—	2	4	no	—
20	C₂×C₃×#131	—	3	1	no	—
21	C₂×#907	C₂×C₃	1	1	yes	characteristic
22	C₃×#427	—	3	2	no	—
23	C₃×#410	—	3	2	no	—
24	#5006	C₂×C₃	1	2	yes	—
25	#4998	C₂×C₃	1	2	yes	complemented
26	C₃×H(3,ℤ/4)	—	2	1	no	—
27	C₃×H(3,ℤ/4)	—	3	1	no	—
28	C₃×H(3,ℤ/4)	D₃	1	1	yes	characteristic, complemented
29	C₃×H(3,ℤ/4)	C₂×C₃	1	1	yes	characteristic, complemented
30	#4557	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃×#67	D₄	1	2	yes	—
33	C₃×#235	—	2	2	no	—
34	C₃×#235	—	2	2	no	—
35	C₂×C₃×D₁₂	Q₈	1	1	yes	characteristic
36	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
37	C₂×C₃×Dic₆	Q₈	1	1	yes	characteristic
38	C₃²×M₁₆	—	2	2	no	—
39	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
40	C₃²×C₄⋊C₄	—	2	2	no	—
41	C₃²×C₂²⋊C₄	—	2	2	no	—
42	C₃²×C₄²	—	2	2	no	—
43	C₃×#217	—	2	2	no	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄×Dic₃	—	2	2	no	—
46	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
49	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
50	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
51	C₃×D₃×C₈	—	2	2	no	—
52	C₃×D₃×C₈	—	2	2	no	—
53	#1812	—	3	1	no	Sylow
54	C₂×C₃×#41	—	3	1	no	—
55	C₂×#235	C₂²×C₃	1	1	yes	characteristic
56	C₃×#131	—	6	2	no	—
57	C₃×#131	—	6	2	no	—
58	C₂×C₃×M₁₆	—	2	1	no	—
59	C₂×C₃×M₁₆	—	3	2	no	—
60	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₃×C₈	—	3	1	no	—
63	C₂²×C₃×C₈	—	3	2	no	—
64	C₃×#117	—	2	2	no	—
65	C₃×#117	—	3	2	no	—
66	C₃×#117	C₂×D₃	1	2	yes	—
67	C₃×#117	C₂²×C₃	1	2	yes	—
68	#926	C₂²×C₃	1	2	yes	—
69	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
70	#907	—	2	2	no	—
71	#907	—	2	2	no	—
72	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	2	yes	complemented
73	C₂×D₃×C₈	C₃×C₄	1	2	yes	complemented
74	C₃×#107	—	6	2	no	—
75	C₃×#106	—	6	2	no	—
76	C₃×#99	—	6	4	no	—
77	#808	—	2	2	no	—
78	#807	—	2	2	no	—
79	#806	—	2	4	no	—
80	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₃×D₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₃²×C₄	—	4	2	no	—
83	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
85	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₃²×C₄	#41	1	2	yes	—
87	C₃×#67	—	2	4	no	—
88	C₂×C₃×Dic₃	—	4	2	no	—
89	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
90	C₃×D₁₂	—	2	2	no	—
91	C₃×C₄×D₃	—	2	2	no	—
92	C₃×C₄×D₃	—	2	2	no	—
93	C₃×Dic₆	—	2	2	no	—
94	C₃²×C₈	—	2	2	no	—
95	C₃×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₃×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₃×C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
98	C₂×#131	—	3	1	no	—
99	#427	—	3	2	no	—
100	#410	—	3	2	no	—
101	H(3,ℤ/4)	—	3	1	no	—
102	H(3,ℤ/4)	C₃×D₃	1	1	yes	characteristic, complemented
103	C₃×#41	—	6	2	no	—
104	C₃×#41	—	6	2	no	—
105	C₂×C₃×Q₈	—	3	1	no	—
106	C₂×C₃×D₄	—	3	2	no	—
107	C₂×C₃×D₄	—	3	1	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	3	2	no	—
110	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂×#67	C₃×D₄	1	2	yes	—
113	#235	—	2	2	no	—
114	#235	—	2	2	no	—
115	C₂×D₁₂	C₃×Q₈	1	1	yes	characteristic
116	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
117	C₂×Dic₆	C₃×Q₈	1	1	yes	characteristic
118	C₃×M₁₆	—	2	2	no	—
119	C₃×M₁₆	—	2	2	no	—
120	C₃×M₁₆	—	4	2	no	—
121	C₃×M₁₆	—	6	4	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	3	1	no	—
124	C₂×C₃×C₈	—	3	1	no	—
125	C₂×C₃×C₈	—	3	2	no	—
126	C₂×C₃×C₈	—	3	2	no	—
127	C₂×C₃×C₈	—	6	2	no	—
128	C₂×C₃×C₈	—	6	2	no	—
129	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
130	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
131	C₃×C₄⋊C₄	—	2	2	no	—
132	C₃×C₄⋊C₄	—	2	2	no	—
133	C₃×C₄⋊C₄	—	4	2	no	—
134	C₃×C₄⋊C₄	—	6	2	no	—
135	C₃×C₂²⋊C₄	—	2	2	no	—
136	C₃×C₂²⋊C₄	—	2	2	no	—
137	C₃×C₂²⋊C₄	—	4	2	no	—
138	C₃×C₂²⋊C₄	—	6	2	no	—
139	C₃×C₄²	—	2	4	no	—
140	C₃×C₄²	—	2	2	no	—
141	C₃×C₄²	—	2	2	no	—
142	C₃×C₄²	—	6	2	no	—
143	#217	—	2	2	no	—
144	C₄⋊Dic₃	—	2	2	no	—
145	C₄×Dic₃	—	2	2	no	—
146	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
147	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
148	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
149	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
150	ΓL(1,𝔽₂₅)	—	2	4	no	—
151	D₃×C₈	—	2	2	no	—
152	D₃×C₈	—	2	2	no	—
153	C₂²×C₃²	—	4	1	no	—
154	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₃²	#121	1	2	yes	—
156	C₂×C₃×D₃	—	2	2	no	—
157	C₂×C₃×D₃	—	4	2	no	—
158	C₃²×C₄	—	8	2	no	—
159	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₃²×C₄	#117	1	1	yes	characteristic
161	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
162	C₃×Dic₃	—	2	2	no	—
163	C₃×Dic₃	—	8	2	no	—
164	C₂×#41	—	3	1	no	—
165	#131	—	6	2	no	—
166	#131	—	6	2	no	—
167	C₂×M₁₆	—	3	2	no	—
168	C₂×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
169	C₂²×C₈	—	3	1	no	—
170	C₂²×C₈	—	3	2	no	—
171	#117	—	3	2	no	—
172	#117	C₂×C₃×D₃	1	2	yes	—
173	#107	—	6	2	no	—
174	#106	—	6	2	no	—
175	#99	—	6	4	no	—
176	C₂³×C₃	—	2	1	no	—
177	C₂³×C₃	—	3	2	no	—
178	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
179	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂²×D₃	C₃×C₄⋊C₄	1	2	yes	—
181	C₃×Q₈	—	6	2	no	—
182	C₃×D₄	—	6	4	no	—
183	C₃×D₄	—	6	2	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	3	2	no	—
189	C₂×C₃×C₄	—	4	4	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	6	2	no	—
193	C₂×C₃×C₄	—	6	2	no	—
194	C₂×C₃×C₄	—	12	2	no	—
195	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
196	C₂×C₃×C₄	#235	1	2	yes	—
197	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
198	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
201	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
202	C₂×C₃×C₄	C₃×#41	1	2	yes	—
203	#67	—	2	4	no	—
204	C₂×Dic₃	—	4	2	no	—
205	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
206	D₁₂	—	2	2	no	—
207	C₄×D₃	—	2	2	no	—
208	C₄×D₃	—	2	2	no	—
209	Dic₆	—	2	2	no	—
210	C₃×C₈	—	2	2	no	—
211	C₃×C₈	—	2	2	no	—
212	C₃×C₈	—	3	1	no	—
213	C₃×C₈	—	3	1	no	—
214	C₃×C₈	—	3	2	no	—
215	C₃×C₈	—	4	2	no	—
216	C₃⋊C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
217	C₃⋊C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
218	C₃⋊C₈	C₃×C₂²⋊C₄	1	2	yes	—
219	C₂×C₃²	—	2	2	no	—
220	C₂×C₃²	—	2	1	no	—
221	C₂×C₃²	#380	1	1	yes	characteristic
222	C₃×D₃	—	4	2	no	—
223	#41	—	6	2	no	—
224	#41	—	6	2	no	—
225	C₂×Q₈	—	3	1	no	—
226	C₂×D₄	—	3	2	no	—
227	C₂×D₄	—	3	1	no	—
228	C₂²×C₄	—	3	2	no	—
229	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
230	M₁₆	—	2	2	no	—
231	M₁₆	—	6	4	no	—
232	C₂×C₈	—	3	1	no	—
233	C₂×C₈	—	3	1	no	—
234	C₂×C₈	—	3	2	no	—
235	C₂×C₈	—	3	2	no	—
236	C₂×C₈	—	6	2	no	—
237	C₂×C₈	—	6	2	no	—
238	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	6	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	6	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	6	2	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	4	1	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	6	2	no	—
250	C₂²×C₃	—	8	1	no	—
251	C₂²×C₃	—	12	2	no	—
252	C₂²×C₃	#884	1	2	yes	—
253	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
254	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
255	C₂²×C₃	C₃×#121	1	2	yes	—
256	C₂×D₃	—	2	2	no	—
257	C₂×D₃	—	4	2	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	6	2	no	—
262	C₃×C₄	—	8	2	no	—
263	C₃×C₄	—	8	2	no	—
264	C₃×C₄	—	8	4	no	—
265	C₃×C₄	—	24	2	no	—
266	C₃×C₄	#872	1	1	yes	characteristic
267	C₃×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
268	C₃×C₄	#896	1	2	yes	—
269	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
270	C₃×C₄	C₃×#117	1	1	yes	characteristic
271	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
272	Dic₃	—	2	2	no	—
273	Dic₃	—	8	2	no	—
274	C₃²	#1812	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	3	2	no	—
276	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
277	Q₈	—	6	2	no	—
278	D₄	—	6	4	no	—
279	D₄	—	6	2	no	—
280	C₂×C₄	—	3	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	6	2	no	—
283	C₂×C₄	—	6	2	no	—
284	C₂×C₄	—	12	2	no	—
285	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
286	C₂×C₄	C₃×#235	1	2	yes	—
287	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
288	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	3	1	no	—
291	C₈	—	3	1	no	—
292	C₈	—	3	2	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	4	2	no	—
299	C₂×C₃	—	4	1	no	—
300	C₂×C₃	—	12	2	no	—
301	C₂×C₃	#4667	1	1	yes	characteristic
302	C₂×C₃	C₃×#380	1	1	yes	characteristic
303	D₃	—	4	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	6	2	no	—
306	C₂²	—	12	2	no	—
307	C₂²	C₃×#884	1	2	yes	—
308	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
309	C₄	—	6	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	24	2	no	—
312	C₄	C₃×#872	1	1	yes	characteristic
313	C₄	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
314	C₄	C₃×#896	1	2	yes	—
315	C₃	—	2	1	no	—
316	C₃	#68710	1	1	yes	characteristic, direct product factor
317	C₃	C₃×#1812	1	1	yes	characteristic, complemented
318	C₂	—	2	2	no	—
319	C₂	—	2	1	no	—
320	C₂	—	12	2	no	—
321	C₂	C₃×#4667	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#872
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×H(3,ℤ/4)	C₂
socle	C₂×C₃²	#380
prime radical	C₂×C₃×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×H(3,ℤ/4)	D₃
hypocenter	C₃	C₃×#1812
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×M₁₆ ◁ C₃×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
3. C₃×#4998	C₂	semidirect
4. C₃²×H(3,ℤ/4)	C₂	semidirect
5. C₃×#4557	C₂	semidirect
7. #68710	C₃	direct (N × H)
15. C₂×C₃×ΓL(1,𝔽₂₅)	C₄	semidirect
16. C₂×C₃×D₃×C₈	C₄	semidirect
25. #4998	C₂×C₃	semidirect
28. C₃×H(3,ℤ/4)	D₃	semidirect
29. C₃×H(3,ℤ/4)	C₂×C₃	semidirect
30. #4557	C₂×C₃	semidirect
72. C₂×ΓL(1,𝔽₂₅)	C₃×C₄	semidirect
73. C₂×D₃×C₈	C₃×C₄	semidirect
102. H(3,ℤ/4)	C₃×D₃	semidirect
274. C₃²	#1812	semidirect
316. C₃	#68710	direct (N × H)
317. C₃	C₃×#1812	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#907	2
C₃×#5006	2
C₃×#4998	2
C₃²×H(3,ℤ/4)	2
C₃×#4557	2
C₃×#1812	3
#68710	3

Subgroups — C3×#68710 #11785484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68710 #11785484