Subgroups — C₃×#70701 #11790492

Order 1152.

445 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
2	C₃×#5121	C₂	1	2	yes	maximal
3	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
4	C₃×#5101	C₂	1	1	yes	maximal, characteristic
5	C₃×#4554	C₂	1	2	yes	maximal
6	C₃×#1952	—	3	1	no	maximal
7	#70701	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₃×#926	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	C₃²×#108	C₄	1	4	yes	complemented
14	C₃×#818	—	2	1	no	—
15	C₃×#818	—	2	1	no	—
16	C₃×#818	—	2	2	no	—
17	C₃×#817	—	2	2	no	—
18	C₃×#815	—	2	1	no	—
19	C₃×C₈×Dic₃	—	2	1	no	—
20	C₂×C₃×#108	—	2	1	no	—
21	C₂×C₃×#108	D₃	1	1	yes	characteristic
22	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic
23	C₃×#428	—	3	2	no	—
24	C₂×C₃×#107	—	3	1	no	—
25	C₃×#405	—	3	1	no	—
26	#5121	C₂×C₃	1	2	yes	—
27	C₂×#818	C₂×C₃	1	1	yes	characteristic
28	#5101	C₂×C₃	1	1	yes	characteristic
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	2	no	—
34	C₃²×M₁₆	C₂×C₄	1	4	yes	—
35	C₂×C₃²×C₈	—	2	2	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	C₃×#217	—	2	1	no	—
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₃×C₄⋊Dic₃	—	2	2	no	—
43	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
44	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
45	C₃×C₄×Dic₃	—	2	1	no	—
46	C₃×#208	—	2	1	no	—
47	C₃×#208	Q₈	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	—	2	1	no	—
49	#1952	—	3	1	no	Sylow
50	C₂×C₃×M₁₆	—	2	2	no	—
51	C₂×C₃×M₁₆	—	3	1	no	—
52	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
53	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
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	—	3	1	no	—
58	C₂×C₃×C₄⋊C₄	—	3	2	no	—
59	#926	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
61	C₂×#208	C₂²×C₃	1	1	yes	characteristic
62	C₃×#108	—	2	4	no	—
63	C₃×#108	Dic₃	1	4	yes	complemented
64	C₃×#108	C₃×C₄	1	4	yes	complemented
65	C₃×#107	—	6	2	no	—
66	C₃×#107	—	6	1	no	—
67	C₃×#107	—	6	1	no	—
68	C₃×#106	—	6	2	no	—
69	C₃×#97	—	6	1	no	—
70	C₃×C₄×C₈	—	6	1	no	—
71	#818	—	2	1	no	—
72	#818	—	2	1	no	—
73	#818	—	2	2	no	—
74	#817	—	2	2	no	—
75	#815	—	2	1	no	—
76	C₈×Dic₃	—	2	1	no	—
77	C₂³×C₃²	#41	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	#41	1	1	yes	characteristic
84	C₂×C₃×Dic₃	—	2	1	no	—
85	C₂×C₃×Dic₃	—	2	1	no	—
86	C₂×C₃×Dic₃	—	4	2	no	—
87	C₂×C₃×Dic₃	—	8	2	no	—
88	C₃²×C₈	—	2	4	no	—
89	C₃²×C₈	C₄⋊C₄	1	2	yes	—
90	C₃²×C₈	C₄⋊C₄	1	2	yes	—
91	C₃×C₃⋊C₈	—	2	2	no	—
92	C₂×#108	C₃×D₃	1	1	yes	characteristic
93	#428	—	3	2	no	—
94	C₂×#107	—	3	1	no	—
95	#405	—	3	1	no	—
96	#343	—	3	2	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	6	2	no	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	2	2	no	—
102	C₃×M₁₆	—	2	2	no	—
103	C₃×M₁₆	—	2	2	no	—
104	C₃×M₁₆	—	2	4	no	—
105	C₃×M₁₆	—	3	2	no	—
106	C₃×M₁₆	—	4	2	no	—
107	C₃×M₁₆	—	6	1	no	—
108	C₃×M₁₆	C₂×Dic₃	1	4	yes	—
109	C₃×M₁₆	C₂×C₃×C₄	1	4	yes	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
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	—	4	2	no	—
117	C₂×C₃×C₈	—	6	1	no	—
118	C₂×C₃×C₈	Dic₆	1	2	yes	—
119	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
120	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
122	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
123	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
126	C₃×C₄⋊C₄	—	3	1	no	—
127	C₃×C₄⋊C₄	—	3	1	no	—
128	C₃×C₄⋊C₄	—	6	2	no	—
129	C₃×C₄⋊C₄	—	6	2	no	—
130	C₃×C₂²⋊C₄	—	6	1	no	—
131	C₃×C₄²	—	6	1	no	—
132	#217	—	2	1	no	—
133	C₄⋊Dic₃	—	2	2	no	—
134	C₄⋊Dic₃	—	2	2	no	—
135	C₄⋊Dic₃	C₃×D₄	1	1	yes	characteristic
136	C₄⋊Dic₃	C₃×D₄	1	1	yes	characteristic
137	C₄×Dic₃	—	2	1	no	—
138	#208	—	2	1	no	—
139	#208	C₃×Q₈	1	2	yes	—
140	C₂×C₃⋊C₈	—	2	1	no	—
141	C₂²×C₃²	—	2	1	no	—
142	C₂²×C₃²	—	2	1	no	—
143	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
144	C₂²×C₃²	#121	1	1	yes	characteristic
145	C₂²×C₃²	#125	1	1	yes	characteristic
146	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
148	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
149	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
150	C₃×Dic₃	—	4	1	no	—
151	C₃×Dic₃	—	4	1	no	—
152	C₃×Dic₃	—	8	2	no	—
153	C₂×M₁₆	—	3	1	no	—
154	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
155	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
156	#117	—	3	1	no	—
157	C₂×C₄⋊C₄	—	3	2	no	—
158	#108	C₃×Dic₃	1	4	yes	complemented
159	#107	—	6	2	no	—
160	#107	—	6	1	no	—
161	#107	—	6	1	no	—
162	#106	—	6	2	no	—
163	#97	—	6	1	no	—
164	C₄×C₈	—	6	1	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	#239	1	1	yes	characteristic
167	C₂³×C₃	C₃×#41	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	6	1	no	—
175	C₂×C₃×C₄	—	6	1	no	—
176	C₂×C₃×C₄	—	12	2	no	—
177	C₂×C₃×C₄	—	24	2	no	—
178	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
180	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	#237	1	1	yes	characteristic
182	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
189	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
190	C₂×Dic₃	—	2	1	no	—
191	C₂×Dic₃	—	2	1	no	—
192	C₂×Dic₃	—	4	2	no	—
193	C₂×Dic₃	—	8	2	no	—
194	C₃×C₈	—	2	4	no	—
195	C₃×C₈	—	2	4	no	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	4	4	no	—
199	C₃×C₈	—	6	2	no	—
200	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
201	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
202	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
203	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
204	C₃⋊C₈	—	2	2	no	—
205	C₂×C₃²	—	2	1	no	—
206	C₂×C₃²	—	2	1	no	—
207	C₂×C₃²	#389	1	1	yes	characteristic
208	C₂×C₃²	#472	1	1	yes	characteristic
209	C₂×C₃²	#473	1	1	yes	characteristic
210	C₂²×C₄	—	6	2	no	—
211	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
212	M₁₆	—	2	2	no	—
213	M₁₆	—	3	2	no	—
214	M₁₆	—	6	1	no	—
215	M₁₆	C₂×C₃×Dic₃	1	4	yes	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	6	1	no	—
218	C₂×C₈	C₃×Dic₆	1	2	yes	—
219	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
220	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
221	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	—	3	1	no	—
224	C₄⋊C₄	—	6	2	no	—
225	C₄⋊C₄	—	6	2	no	—
226	C₂²⋊C₄	—	6	1	no	—
227	C₄²	—	6	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	4	1	no	—
236	C₂²×C₃	—	4	1	no	—
237	C₂²×C₃	#890	1	1	yes	characteristic
238	C₂²×C₃	#895	1	1	yes	characteristic
239	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
240	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
241	C₂²×C₃	C₃×#121	1	1	yes	characteristic
242	C₂²×C₃	C₃×#125	1	1	yes	characteristic
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	12	1	no	—
248	C₃×C₄	—	12	1	no	—
249	C₃×C₄	—	24	2	no	—
250	C₃×C₄	#888	1	1	yes	characteristic
251	C₃×C₄	#897	1	1	yes	characteristic
252	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
253	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
254	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
255	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
256	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
257	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
258	Dic₃	—	4	1	no	—
259	Dic₃	—	4	1	no	—
260	Dic₃	—	8	2	no	—
261	C₃²	#1952	1	1	yes	characteristic, complemented, Sylow
262	C₂³	C₃×#239	1	1	yes	characteristic
263	C₂×C₄	—	6	1	no	—
264	C₂×C₄	—	6	1	no	—
265	C₂×C₄	—	12	2	no	—
266	C₂×C₄	—	24	2	no	—
267	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
268	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
269	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
270	C₂×C₄	C₃×#237	1	1	yes	characteristic
271	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	6	2	no	—
275	C₈	C₃×C₄⋊Dic₃	1	2	yes	—
276	C₈	C₃×C₄⋊Dic₃	1	2	yes	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	—	4	1	no	—
286	C₂×C₃	#4896	1	1	yes	characteristic
287	C₂×C₃	#4897	1	1	yes	characteristic
288	C₂×C₃	#4984	1	1	yes	characteristic
289	C₂×C₃	C₃×#389	1	1	yes	characteristic
290	C₂×C₃	C₃×#472	1	1	yes	characteristic
291	C₂×C₃	C₃×#473	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	C₃×#890	1	1	yes	characteristic
295	C₂²	C₃×#895	1	1	yes	characteristic
296	C₂²	C₃×Q₈×Dic₃	1	1	yes	characteristic
297	C₄	—	12	1	no	—
298	C₄	—	12	1	no	—
299	C₄	—	24	2	no	—
300	C₄	C₃×#888	1	1	yes	characteristic
301	C₄	C₃×#897	1	1	yes	characteristic
302	C₄	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
303	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
304	C₃	—	2	1	no	—
305	C₃	#70701	1	1	yes	characteristic, direct product factor
306	C₃	C₃×#1952	1	1	yes	characteristic, complemented
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	C₃×#4896	1	1	yes	characteristic
310	C₂	C₃×#4897	1	1	yes	characteristic
311	C₂	C₃×#4984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#895
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#108	D₃
hypocenter	C₃	C₃×#1952
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#108

Complements

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

N	H ≅ G/N	type
7. #70701	C₃	direct (N × H)
13. C₃²×#108	C₄	semidirect
63. C₃×#108	Dic₃	semidirect
64. C₃×#108	C₃×C₄	semidirect
158. #108	C₃×Dic₃	semidirect
261. C₃²	#1952	semidirect
305. C₃	#70701	direct (N × H)
306. 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
C₂×C₃²×#108	2
C₃×#5121	2
C₂×C₃×#818	2
C₃×#5101	2
C₃×#4554	2
C₃×#1952	3
#70701	3

Subgroups — C3×#70701 #11790492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70701 #11790492