Subgroups — C₃×#81891 #11874966

Order 1152.

761 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5728	C₂	1	2	yes	maximal
2	C₃²×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5078	C₂	1	2	yes	maximal, complemented
4	C₃×#4860	C₂	1	4	yes	maximal
5	C₃×#4858	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
7	C₃×#4680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3160	—	3	1	no	maximal
9	#81891	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×#946	—	2	2	no	—
12	C₃×Q₈×Dic₃	—	2	2	no	—
13	C₂×C₃×D₃×C₈	C₂²	1	2	yes	—
14	C₃×#897	—	2	2	no	—
15	C₃×#892	C₂²	1	4	yes	—
16	C₃×#890	C₂²	1	4	yes	—
17	C₃×#887	—	2	2	no	—
18	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
19	C₃×#869	C₂²	1	2	yes	—
20	C₃²×#106	C₂²	1	4	yes	—
21	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₃×#817	C₂²	1	4	yes	—
23	C₃×C₈×Dic₃	C₂²	1	2	yes	—
24	C₃×#807	C₂²	1	8	yes	—
25	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₃×#533	—	3	2	no	—
27	#5728	C₂×C₃	1	2	yes	—
28	C₃×#500	—	3	2	no	—
29	C₃×#499	—	3	4	no	—
30	C₃×#498	—	2	1	no	—
31	C₃×#498	—	3	1	no	—
32	C₃×#498	D₃	1	1	yes	characteristic, complemented
33	C₃×#498	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×#427	—	3	4	no	—
35	C₂×C₃×C₄×C₈	—	3	1	no	—
36	#5078	C₂×C₃	1	2	yes	complemented
37	#4860	C₂×C₃	1	4	yes	—
38	#4858	C₂×C₃	1	4	yes	complemented
39	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
40	#4680	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₃²×Q₈	—	2	2	no	—
42	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
43	C₂×C₃×C₄×D₃	C₂³	1	1	yes	characteristic
44	C₂×C₃×Dic₆	—	2	2	no	—
45	C₂×C₃²×C₈	C₂³	1	2	yes	—
46	C₃²×C₄⋊C₄	—	2	2	no	—
47	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
48	C₃²×C₄²	C₂³	1	1	yes	characteristic
49	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
50	C₃×C₄⋊Dic₃	—	2	2	no	—
51	C₃×C₄⋊Dic₃	—	2	4	no	—
52	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
53	C₃×Dic₃⋊C₄	—	2	4	no	—
54	C₃×C₄×Dic₃	—	2	4	no	—
55	C₃×C₄×Dic₃	D₄	1	2	yes	—
56	C₃×C₄×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
58	C₃×D₃×C₈	Q₈	1	8	yes	complemented
59	#3160	—	3	1	no	Sylow
60	C₂²×C₃×C₈	—	3	2	no	—
61	C₃×C₄⋊Q₈	—	2	2	no	—
62	C₃×C₄⋊Q₈	—	3	2	no	—
63	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
64	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
65	C₃×#125	—	3	4	no	—
66	C₃×C₂²⋊Q₈	—	6	2	no	—
67	C₃×C₂²⋊Q₈	—	6	2	no	—
68	C₃×C₄×Q₈	—	6	2	no	—
69	C₃×C₄×Q₈	—	6	2	no	—
70	C₃×#117	—	3	4	no	—
71	C₂×C₃×C₄²	—	3	1	no	—
72	#946	—	2	2	no	—
73	Q₈×Dic₃	—	2	2	no	—
74	C₂×D₃×C₈	C₂²×C₃	1	2	yes	—
75	#897	—	2	2	no	—
76	#892	C₂²×C₃	1	4	yes	—
77	#890	C₂²×C₃	1	4	yes	—
78	#887	—	2	2	no	—
79	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
80	#869	C₂²×C₃	1	2	yes	—
81	C₃×#107	—	3	8	no	—
82	C₃×#106	—	2	4	no	—
83	C₃×#106	—	3	4	no	—
84	C₃×#106	C₂×D₃	1	4	yes	—
85	C₃×#106	C₂²×C₃	1	4	yes	—
86	C₃×C₄×C₈	—	2	1	no	—
87	C₃×C₄×C₈	—	3	2	no	—
88	C₃×C₄×C₈	—	3	1	no	—
89	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
90	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
91	#817	C₂²×C₃	1	4	yes	—
92	C₈×Dic₃	C₂²×C₃	1	2	yes	—
93	#807	C₂²×C₃	1	8	yes	—
94	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
95	C₂²×C₃×D₃	C₂×D₄	1	1	yes	characteristic
96	C₃²×Q₈	—	4	4	no	—
97	C₂×C₃²×C₄	—	2	4	no	—
98	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
99	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×Dic₃	—	2	4	no	—
101	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
102	C₂×C₃×Dic₃	C₂×D₄	1	1	yes	characteristic
103	C₃×C₄×D₃	—	2	4	no	—
104	C₃×C₄×D₃	C₂×Q₈	1	4	yes	—
105	C₃×Dic₆	—	4	4	no	—
106	C₃²×C₈	C₂×Q₈	1	4	yes	—
107	C₃×C₃⋊C₈	C₂×Q₈	1	4	yes	—
108	#533	—	3	2	no	—
109	#500	—	3	2	no	—
110	#499	—	3	4	no	—
111	#498	—	3	1	no	—
112	#498	C₃×D₃	1	1	yes	characteristic, complemented
113	#427	—	3	4	no	—
114	C₂×C₄×C₈	—	3	1	no	—
115	C₂×C₃×Q₈	—	2	2	no	—
116	C₂×C₃×Q₈	—	2	2	no	—
117	C₂×C₃×Q₈	—	4	2	no	—
118	C₂×C₃×Q₈	—	6	2	no	—
119	C₂²×C₃×C₄	—	3	1	no	—
120	C₂²×C₃×C₄	—	3	2	no	—
121	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
122	C₂×C₄×D₃	C₂³×C₃	1	1	yes	characteristic
123	C₂×Dic₆	—	2	2	no	—
124	C₂×C₃×C₈	—	2	2	no	—
125	C₂×C₃×C₈	—	3	2	no	—
126	C₂×C₃×C₈	—	3	8	no	—
127	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
128	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
129	C₃×C₄⋊C₄	—	2	2	no	—
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄⋊C₄	—	2	4	no	—
132	C₃×C₄⋊C₄	—	3	4	no	—
133	C₃×C₄⋊C₄	—	4	2	no	—
134	C₃×C₄⋊C₄	—	6	4	no	—
135	C₃×C₄⋊C₄	—	6	4	no	—
136	C₃×C₄⋊C₄	—	6	2	no	—
137	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
138	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
139	C₃×C₂²⋊C₄	—	6	4	no	—
140	C₃×C₄²	—	2	1	no	—
141	C₃×C₄²	—	3	2	no	—
142	C₃×C₄²	—	3	1	no	—
143	C₃×C₄²	—	6	4	no	—
144	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
145	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
146	(C₂×D₃)⋊C₄	—	2	4	no	—
147	C₄⋊Dic₃	—	2	2	no	—
148	C₄⋊Dic₃	—	2	4	no	—
149	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
150	Dic₃⋊C₄	—	2	4	no	—
151	C₄×Dic₃	—	2	4	no	—
152	C₄×Dic₃	C₃×D₄	1	2	yes	—
153	C₄×Dic₃	C₂³×C₃	1	1	yes	characteristic
154	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
155	D₃×C₈	C₃×Q₈	1	8	yes	complemented
156	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₃×D₃	—	2	2	no	—
158	C₂×C₃×D₃	C₄⋊Q₈	1	2	yes	—
159	C₃²×C₄	—	4	4	no	—
160	C₃²×C₄	#135	1	4	yes	—
161	C₃²×C₄	C₂²×Q₈	1	2	yes	—
162	C₃×Dic₃	—	2	2	no	—
163	C₃×Dic₃	—	4	4	no	—
164	C₃×Dic₃	C₄⋊Q₈	1	2	yes	—
165	C₂²×C₈	—	3	2	no	—
166	C₄⋊Q₈	—	3	2	no	—
167	C₄⋊Q₈	C₂×C₃×D₃	1	2	yes	—
168	#125	—	3	4	no	—
169	C₂²⋊Q₈	—	6	2	no	—
170	C₂²⋊Q₈	—	6	2	no	—
171	C₄×Q₈	—	6	2	no	—
172	C₄×Q₈	—	6	2	no	—
173	#117	—	3	4	no	—
174	C₂×C₄²	—	3	1	no	—
175	#107	—	3	8	no	—
176	#106	—	3	4	no	—
177	#106	C₂×C₃×D₃	1	4	yes	—
178	C₄×C₈	—	3	2	no	—
179	C₄×C₈	—	3	1	no	—
180	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
181	C₂³×C₃	—	3	1	no	—
182	C₂²×D₃	C₂×C₃×D₄	1	1	yes	characteristic
183	C₃×Q₈	—	4	4	no	—
184	C₃×Q₈	—	4	4	no	—
185	C₃×Q₈	—	8	4	no	—
186	C₃×Q₈	—	12	4	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	3	1	no	—
192	C₂×C₃×C₄	—	3	4	no	—
193	C₂×C₃×C₄	—	3	2	no	—
194	C₂×C₃×C₄	—	4	4	no	—
195	C₂×C₃×C₄	—	6	4	no	—
196	C₂×C₃×C₄	—	6	4	no	—
197	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
198	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
199	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
201	C₂×Dic₃	—	2	4	no	—
202	C₂×Dic₃	C₂×C₃×D₄	1	2	yes	—
203	C₂×Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
204	C₄×D₃	—	2	4	no	—
205	C₄×D₃	C₂×C₃×Q₈	1	4	yes	—
206	Dic₆	—	4	4	no	—
207	C₃×C₈	—	2	4	no	—
208	C₃×C₈	—	3	4	no	—
209	C₃×C₈	D₃×Q₈	1	4	yes	—
210	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
211	C₃⋊C₈	C₂×C₃×Q₈	1	4	yes	—
212	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
213	C₂×C₃²	C₂×#135	1	2	yes	—
214	C₃×D₃	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	6	2	no	—
217	C₂²×C₄	—	3	1	no	—
218	C₂²×C₄	—	3	2	no	—
219	C₂×C₈	—	3	2	no	—
220	C₂×C₈	—	3	8	no	—
221	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	3	4	no	—
224	C₄⋊C₄	—	6	4	no	—
225	C₄⋊C₄	—	6	4	no	—
226	C₄⋊C₄	—	6	2	no	—
227	C₄⋊C₄	C₂²×C₃×D₃	1	4	yes	—
228	C₂²⋊C₄	—	6	4	no	—
229	C₄²	—	3	1	no	—
230	C₄²	—	3	2	no	—
231	C₄²	—	6	4	no	—
232	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	3	2	no	—
235	C₂²×C₃	—	6	2	no	—
236	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
237	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
238	C₂×D₃	—	2	2	no	—
239	C₂×D₃	C₃×C₄⋊Q₈	1	2	yes	—
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	3	2	no	—
243	C₃×C₄	—	4	4	no	—
244	C₃×C₄	—	4	4	no	—
245	C₃×C₄	—	6	2	no	—
246	C₃×C₄	—	8	4	no	—
247	C₃×C₄	—	12	4	no	—
248	C₃×C₄	#916	1	4	yes	—
249	C₃×C₄	C₃×#135	1	4	yes	—
250	C₃×C₄	C₂×D₃×Q₈	1	2	yes	—
251	C₃×C₄	C₂²×C₃×Q₈	1	2	yes	—
252	Dic₃	—	2	2	no	—
253	Dic₃	—	4	4	no	—
254	Dic₃	C₃×C₄⋊Q₈	1	2	yes	—
255	C₃²	#3160	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	3	1	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	12	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	4	no	—
262	C₂×C₄	—	3	2	no	—
263	C₂×C₄	—	6	4	no	—
264	C₂×C₄	—	6	4	no	—
265	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
266	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
267	C₈	—	3	4	no	—
268	C₈	C₃×D₃×Q₈	1	4	yes	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	6	2	no	—
272	C₂×C₃	D₃×C₄⋊Q₈	1	1	yes	characteristic
273	C₂×C₃	C₂×#916	1	2	yes	—
274	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
275	C₂×C₃	C₂×C₃×#135	1	2	yes	—
276	D₃	—	2	2	no	—
277	C₂²	—	3	2	no	—
278	C₂²	—	6	2	no	—
279	C₂²	C₂×C₃×D₃×D₄	1	1	yes	characteristic
280	C₄	—	3	2	no	—
281	C₄	—	4	4	no	—
282	C₄	—	6	2	no	—
283	C₄	—	12	4	no	—
284	C₄	C₃×#916	1	4	yes	—
285	C₄	C₂×C₃×D₃×Q₈	1	2	yes	—
286	C₃	—	2	1	no	—
287	C₃	#81891	1	1	yes	characteristic, direct product factor
288	C₃	C₃×#3160	1	1	yes	characteristic, complemented
289	C₂	—	6	2	no	—
290	C₂	C₃×D₃×C₄⋊Q₈	1	1	yes	characteristic
291	C₂	C₂×C₃×#916	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#498	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂²×C₃×D₃	C₂×D₄
hypercenter	C₃×#498	D₃
hypocenter	C₃	C₃×#3160
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#498

Complements

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

N	H ≅ G/N	type
2. C₃²×#498	C₂	semidirect
3. C₃×#5078	C₂	semidirect
5. C₃×#4858	C₂	semidirect
7. C₃×#4680	C₂	semidirect
9. #81891	C₃	direct (N × H)
32. C₃×#498	D₃	semidirect
33. C₃×#498	C₂×C₃	semidirect
36. #5078	C₂×C₃	semidirect
38. #4858	C₂×C₃	semidirect
40. #4680	C₂×C₃	semidirect
58. C₃×D₃×C₈	Q₈	semidirect
112. #498	C₃×D₃	semidirect
155. D₃×C₈	C₃×Q₈	semidirect
255. C₃²	#3160	semidirect
287. C₃	#81891	direct (N × H)
288. C₃	C₃×#3160	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5728	2
C₃²×#498	2
C₃×#5078	2
C₃×#4860	2
C₃×#4858	2
C₃×C₄×D₃×C₈	2
C₃×#4680	2
C₃×#3160	3
#81891	3

Subgroups — C3×#81891 #11874966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#81891 #11874966