Subgroups — C₃×#70555 #11786243

Order 1152.

561 conjugacy classes of subgroups.

The subgroup lattice has 287 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×#4977	C₂	1	2	yes	maximal
5	C₂×C₃×#807	C₂	1	2	yes	maximal
6	C₃×#1846	—	3	1	no	maximal
7	#70555	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃²×#106	—	2	2	no	—
14	C₃×C₃⋊#95	—	2	4	no	—
15	C₃×#817	—	2	2	no	—
16	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
17	C₃×#807	—	2	4	no	—
18	C₂×C₃×#107	—	3	2	no	—
19	C₂×C₃×#106	—	2	1	no	—
20	C₂×C₃×#106	—	3	1	no	—
21	C₂×C₃×#106	D₃	1	1	yes	characteristic
22	C₂×C₃×#106	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×C₄×C₈	—	3	1	no	—
24	C₃×#382	—	3	2	no	—
25	C₂×#817	C₂×C₃	1	1	yes	characteristic
26	C₂×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
27	#4977	C₂×C₃	1	2	yes	—
28	C₂×#807	C₂×C₃	1	2	yes	—
29	C₂²×C₃²×C₄	—	2	2	no	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂²×C₃×Dic₃	D₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₂×C₃²×C₈	Q₈	1	2	yes	—
36	C₃²×C₄⋊C₄	—	2	2	no	—
37	C₃²×C₄⋊C₄	—	2	2	no	—
38	C₃×C₄⋊Dic₃	—	2	2	no	—
39	C₃×C₄⋊Dic₃	—	2	2	no	—
40	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
41	C₃×C₄×Dic₃	Q₈	1	2	yes	—
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
45	#1846	—	3	1	no	Sylow
46	C₂²×C₃×C₈	—	2	1	no	—
47	C₂²×C₃×C₈	—	3	1	no	—
48	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
53	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₃×C₄²	—	3	1	no	—
55	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
56	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
58	C₃×#107	—	6	4	no	—
59	C₃×#106	—	2	4	no	—
60	C₃×#106	—	2	2	no	—
61	C₃×#106	—	2	2	no	—
62	C₃×#106	—	6	2	no	—
63	C₃×C₄×C₈	—	3	4	no	—
64	C₃×#95	—	6	4	no	—
65	C₃⋊#95	—	2	4	no	—
66	#817	—	2	2	no	—
67	C₈×Dic₃	C₃×C₄	1	4	yes	complemented
68	#807	—	2	4	no	—
69	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	—	4	2	no	—
71	C₂×C₃²×C₄	—	4	4	no	—
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃×Dic₃	—	2	4	no	—
79	C₂×C₃×Dic₃	—	4	2	no	—
80	C₂×C₃×Dic₃	—	4	4	no	—
81	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
82	C₃²×C₈	C₄⋊C₄	1	4	yes	—
83	C₃×C₃⋊C₈	C₄⋊C₄	1	4	yes	—
84	C₂×#107	—	3	2	no	—
85	C₂×#106	—	3	1	no	—
86	C₂×#106	C₃×D₃	1	1	yes	characteristic
87	C₂×C₄×C₈	—	3	1	no	—
88	#382	—	3	2	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	—	3	2	no	—
93	C₂²×C₃×C₄	—	4	2	no	—
94	C₂²×C₃×C₄	—	6	2	no	—
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
97	C₂²×Dic₃	—	2	2	no	—
98	C₂²×Dic₃	C₃×D₄	1	2	yes	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	3	2	no	—
103	C₂×C₃×C₈	—	3	2	no	—
104	C₂×C₃×C₈	—	3	2	no	—
105	C₂×C₃×C₈	Dic₆	1	2	yes	—
106	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
107	C₂×C₃×C₈	#67	1	2	yes	—
108	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
110	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
111	C₃×C₄⋊C₄	—	2	4	no	—
112	C₃×C₄⋊C₄	—	2	2	no	—
113	C₃×C₄⋊C₄	—	2	2	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	—	2	2	no	—
116	C₃×C₄⋊C₄	—	4	2	no	—
117	C₃×C₄⋊C₄	—	6	2	no	—
118	C₃×C₄⋊C₄	—	6	2	no	—
119	C₃×C₄²	—	3	2	no	—
120	C₃×C₄²	—	3	2	no	—
121	C₄⋊Dic₃	—	2	2	no	—
122	C₄⋊Dic₃	—	2	2	no	—
123	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
124	C₄×Dic₃	C₃×Q₈	1	2	yes	—
125	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
126	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
127	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
128	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₃²	#127	1	1	yes	characteristic
130	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
131	C₂²×C₃²	#135	1	4	yes	—
132	C₃²×C₄	—	8	2	no	—
133	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
136	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
137	C₃×Dic₃	—	2	4	no	—
138	C₃×Dic₃	—	8	2	no	—
139	C₂²×C₈	—	3	1	no	—
140	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	3	2	no	—
142	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
143	C₂×C₄²	—	3	1	no	—
144	#107	—	6	4	no	—
145	#106	—	2	2	no	—
146	#106	—	6	2	no	—
147	C₄×C₈	—	3	4	no	—
148	#95	—	6	4	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
151	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	3	4	no	—
159	C₂×C₃×C₄	—	4	4	no	—
160	C₂×C₃×C₄	—	4	4	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	4	no	—
164	C₂×C₃×C₄	—	6	4	no	—
165	C₂×C₃×C₄	—	8	4	no	—
166	C₂×C₃×C₄	—	12	4	no	—
167	C₂×C₃×C₄	—	12	2	no	—
168	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
172	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
180	C₂×Dic₃	—	2	4	no	—
181	C₂×Dic₃	—	4	2	no	—
182	C₂×Dic₃	—	4	4	no	—
183	C₂×Dic₃	C₃×C₄⋊C₄	1	4	yes	—
184	C₃×C₈	—	2	4	no	—
185	C₃×C₈	—	3	4	no	—
186	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
187	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
188	C₃⋊C₈	C₃×C₄⋊C₄	1	4	yes	—
189	C₂×C₃²	#384	1	1	yes	characteristic
190	C₂×C₃²	#427	1	2	yes	—
191	C₂×C₃²	#495	1	2	yes	—
192	C₂×C₃²	#499	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	3	2	no	—
195	C₂²×C₄	—	6	2	no	—
196	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
197	C₂×C₈	—	3	2	no	—
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	3	2	no	—
200	C₂×C₈	C₃×Dic₆	1	2	yes	—
201	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
202	C₂×C₈	C₃×#67	1	2	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	6	2	no	—
206	C₄⋊C₄	—	6	2	no	—
207	C₄²	—	3	2	no	—
208	C₄²	—	3	2	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	4	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	#888	1	1	yes	characteristic
214	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
215	C₂²×C₃	#916	1	4	yes	—
216	C₂²×C₃	#940	1	1	yes	characteristic
217	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
218	C₂²×C₃	C₃×#127	1	1	yes	characteristic
219	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
220	C₂²×C₃	C₃×#135	1	4	yes	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	6	4	no	—
226	C₃×C₄	—	8	4	no	—
227	C₃×C₄	—	8	2	no	—
228	C₃×C₄	—	8	2	no	—
229	C₃×C₄	—	24	2	no	—
230	C₃×C₄	#888	1	1	yes	characteristic
231	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
232	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
233	C₃×C₄	#944	1	1	yes	characteristic
234	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
235	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
236	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
237	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
238	Dic₃	—	2	4	no	—
239	Dic₃	—	8	2	no	—
240	C₃²	#1846	1	1	yes	characteristic, complemented, Sylow
241	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
242	C₂×C₄	—	3	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	6	4	no	—
246	C₂×C₄	—	12	4	no	—
247	C₂×C₄	—	12	2	no	—
248	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
249	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
250	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
251	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
252	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
254	C₈	—	3	4	no	—
255	C₈	C₃×Dic₃⋊C₄	1	4	yes	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	2	no	—
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	#4858	1	2	yes	—
261	C₂×C₃	#4860	1	2	yes	—
262	C₂×C₃	#4975	1	1	yes	characteristic
263	C₂×C₃	#5168	1	2	yes	—
264	C₂×C₃	C₃×#384	1	1	yes	characteristic
265	C₂×C₃	C₃×#427	1	2	yes	—
266	C₂×C₃	C₃×#495	1	2	yes	—
267	C₂×C₃	C₃×#499	1	2	yes	—
268	C₂²	C₃×#888	1	1	yes	characteristic
269	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
270	C₂²	C₃×#916	1	4	yes	—
271	C₂²	C₃×#940	1	1	yes	characteristic
272	C₄	—	6	4	no	—
273	C₄	—	8	2	no	—
274	C₄	—	24	2	no	—
275	C₄	C₃×#888	1	1	yes	characteristic
276	C₄	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
277	C₄	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
278	C₄	C₃×#944	1	1	yes	characteristic
279	C₃	—	2	1	no	—
280	C₃	#70555	1	1	yes	characteristic, direct product factor
281	C₃	C₃×#1846	1	1	yes	characteristic, complemented
282	C₂	C₃×#4858	1	2	yes	—
283	C₂	C₃×#4860	1	2	yes	—
284	C₂	C₃×#4975	1	1	yes	characteristic
285	C₂	C₃×#5168	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#106	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#106	D₃
hypocenter	C₃	C₃×#1846
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#106

Complements

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

N	H ≅ G/N	type
7. #70555	C₃	direct (N × H)
16. C₃×C₈×Dic₃	C₄	semidirect
67. C₈×Dic₃	C₃×C₄	semidirect
240. C₃²	#1846	semidirect
280. C₃	#70555	direct (N × H)
281. C₃	C₃×#1846	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#106	2
C₂×C₃×#817	2
C₂×C₃×C₈×Dic₃	2
C₃×#4977	2
C₂×C₃×#807	2
C₃×#1846	3
#70555	3

Subgroups — C3×#70555 #11786243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70555 #11786243