Subgroups — C₃×#10592719 #13163058

Order 1728.

856 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#64078	C₂	1	1	yes	maximal, characteristic
3	C₃×#63662	C₂	1	2	yes	maximal
4	C₃×#63635	C₂	1	2	yes	maximal
5	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
6	#10592719	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5020	—	3	4	no	maximal
8	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×#634	—	2	1	no	—
10	C₃³×C₄⋊C₄	—	2	1	no	—
11	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃³×C₄²	—	2	1	no	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
15	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
16	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₃²×C₄×Q₈	—	2	4	no	—
18	C₃²×C₄×Q₈	D₃	1	4	yes	—
19	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
20	#64078	C₂×C₃	1	1	yes	characteristic
21	C₃×#869	—	3	4	no	—
22	#63662	C₂×C₃	1	2	yes	—
23	#63635	C₂×C₃	1	2	yes	—
24	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	1	yes	characteristic
25	C₃×#835	—	3	8	no	—
26	C₃×#808	—	3	8	no	—
27	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
28	C₃³×Q₈	—	2	1	no	—
29	C₃³×Q₈	Q₈	1	2	yes	complemented
30	C₂×C₃³×C₄	—	2	1	no	—
31	C₂×C₃³×C₄	—	2	1	no	—
32	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
33	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
34	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
36	C₃×#634	—	4	2	no	—
37	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
38	C₃×#475	—	9	1	no	—
39	#5020	—	3	4	no	—
40	C₂×C₃²×Q₈	—	2	4	no	—
41	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
42	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
43	C₂×#634	—	2	1	no	—
44	C₂×C₃×Dic₆	—	6	4	no	—
45	C₃²×C₄⋊C₄	—	2	8	no	—
46	C₃²×C₄⋊C₄	—	2	4	no	—
47	C₃²×C₄⋊C₄	—	2	4	no	—
48	C₃²×C₄⋊C₄	—	2	1	no	—
49	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
50	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
51	C₃²×C₄²	—	2	8	no	—
52	C₃²×C₄²	—	2	4	no	—
53	C₃²×C₄²	—	2	4	no	—
54	C₃²×C₄²	—	2	1	no	—
55	C₃²×C₄²	C₂×D₃	1	4	yes	—
56	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
57	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
58	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
59	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
60	C₃×C₄⋊Dic₃	—	3	8	no	—
61	C₃×C₄⋊Dic₃	—	6	4	no	—
62	C₂×C₃×C₃⋊C₈	—	3	8	no	—
63	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
64	C₃³×C₄	—	2	2	no	—
65	C₃³×C₄	—	2	1	no	—
66	C₃³×C₄	—	4	1	no	—
67	C₃³×C₄	SD₁₆	1	2	yes	—
68	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
69	C₃³×C₄	#41	1	1	yes	characteristic
70	C₃×C₃⋊Dic₃	—	4	2	no	—
71	C₃×C₄⋊Q₈	—	9	1	no	—
72	C₃×C₄×Q₈	—	2	4	no	—
73	C₃×C₄×Q₈	C₃×D₃	1	4	yes	—
74	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
75	#869	—	3	4	no	—
76	C₃×#106	—	9	2	no	—
77	C₃×C₄⋊C₈	—	9	1	no	—
78	C₃×#103	—	9	2	no	—
79	#835	—	3	8	no	—
80	#808	—	3	8	no	—
81	C₄⋊(C₃⋊C₈)	—	3	4	no	—
82	C₃²×Q₈	—	2	8	no	—
83	C₃²×Q₈	—	2	4	no	—
84	C₃²×Q₈	—	2	1	no	—
85	C₃²×Q₈	—	2	8	no	—
86	C₃²×Q₈	Dic₆	1	8	yes	complemented
87	C₃²×Q₈	C₃×Q₈	1	2	yes	complemented
88	C₂×C₃²×C₄	—	2	8	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	2	1	no	—
91	C₂×C₃²×C₄	—	2	8	no	—
92	C₂×C₃²×C₄	—	2	4	no	—
93	C₂×C₃²×C₄	—	2	1	no	—
94	C₂×C₃²×C₄	—	2	4	no	—
95	C₂×C₃²×C₄	—	2	4	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	#67	1	4	yes	—
98	C₂×C₃²×C₄	#67	1	4	yes	—
99	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
102	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₂×C₃⋊Dic₃	—	2	2	no	—
104	#634	—	4	2	no	—
105	C₂×C₃×Dic₃	—	6	8	no	—
106	C₃×Dic₆	—	12	8	no	—
107	C₃⋊(C₃⋊C₈)	—	2	2	no	—
108	C₃×C₃⋊C₈	—	6	8	no	—
109	#475	—	9	1	no	Sylow
110	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₃³	#137	1	1	yes	characteristic
113	C₂×C₃×Q₈	—	2	4	no	—
114	C₂×C₃×Q₈	—	18	1	no	—
115	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
116	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
117	C₂×Dic₆	—	6	4	no	—
118	C₂×C₃×C₈	—	9	2	no	—
119	C₃×C₄⋊C₄	—	2	8	no	—
120	C₃×C₄⋊C₄	—	2	4	no	—
121	C₃×C₄⋊C₄	—	2	1	no	—
122	C₃×C₄⋊C₄	—	2	4	no	—
123	C₃×C₄⋊C₄	—	9	2	no	—
124	C₃×C₄⋊C₄	—	18	1	no	—
125	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
126	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
127	C₃×C₄²	—	2	8	no	—
128	C₃×C₄²	—	2	4	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	—	2	4	no	—
131	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
132	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
133	C₄⋊Dic₃	—	3	8	no	—
134	C₄⋊Dic₃	—	6	4	no	—
135	C₂×C₃⋊C₈	—	3	8	no	—
136	C₂²×C₃²	—	2	4	no	—
137	C₂²×C₃²	C₂×#67	1	4	yes	—
138	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
139	C₃²×C₄	—	2	16	no	—
140	C₃²×C₄	—	2	8	no	—
141	C₃²×C₄	—	2	2	no	—
142	C₃²×C₄	—	2	1	no	—
143	C₃²×C₄	—	2	8	no	—
144	C₃²×C₄	—	2	4	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	—	4	4	no	—
148	C₃²×C₄	—	4	8	no	—
149	C₃²×C₄	—	4	4	no	—
150	C₃²×C₄	—	4	1	no	—
151	C₃²×C₄	#215	1	8	yes	—
152	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
153	C₃²×C₄	C₂×Dic₆	1	4	yes	—
154	C₃²×C₄	#235	1	4	yes	—
155	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
156	C₃²×C₄	C₃×#41	1	1	yes	characteristic
157	C₃⋊Dic₃	—	4	2	no	—
158	C₃×Dic₃	—	12	8	no	—
159	C₄⋊Q₈	—	9	1	no	—
160	C₄×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
161	#106	—	9	2	no	—
162	C₄⋊C₈	—	9	1	no	—
163	#103	—	9	2	no	—
164	C₃³	#475	1	1	yes	characteristic, complemented, Sylow
165	C₃×Q₈	—	2	8	no	—
166	C₃×Q₈	—	2	4	no	—
167	C₃×Q₈	—	2	1	no	—
168	C₃×Q₈	—	2	8	no	—
169	C₃×Q₈	—	36	2	no	—
170	C₃×Q₈	C₃×Dic₆	1	8	yes	complemented
171	C₃×Q₈	#634	1	2	yes	complemented
172	C₂×C₃×C₄	—	2	8	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	8	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	18	2	no	—
182	C₂×C₃×C₄	C₃×#67	1	4	yes	—
183	C₂×C₃×C₄	C₃×#67	1	4	yes	—
184	C₂×C₃×C₄	#638	1	1	yes	characteristic
185	C₂×C₃×C₄	#638	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
187	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
188	C₂×Dic₃	—	6	8	no	—
189	Dic₆	—	12	8	no	—
190	C₃×C₈	—	18	2	no	—
191	C₃⋊C₈	—	6	8	no	—
192	C₂×C₃²	—	2	4	no	—
193	C₂×C₃²	—	2	4	no	—
194	C₂×C₃²	—	2	4	no	—
195	C₂×C₃²	#924	1	4	yes	—
196	C₂×C₃²	C₂×#215	1	4	yes	—
197	C₂×C₃²	#951	1	4	yes	—
198	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
199	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
200	C₂×C₃²	C₃×#137	1	1	yes	characteristic
201	C₂×Q₈	—	18	1	no	—
202	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
203	C₂×C₈	—	9	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	9	2	no	—
206	C₄⋊C₄	—	18	1	no	—
207	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
208	C₄²	—	2	1	no	—
209	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
212	C₂²×C₃	C₂×#638	1	1	yes	characteristic
213	C₃×C₄	—	2	16	no	—
214	C₃×C₄	—	2	8	no	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	8	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	—	2	4	no	—
221	C₃×C₄	—	4	4	no	—
222	C₃×C₄	—	4	8	no	—
223	C₃×C₄	—	4	4	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	—	36	2	no	—
226	C₃×C₄	C₃×#215	1	8	yes	—
227	C₃×C₄	#3755	1	2	yes	—
228	C₃×C₄	C₂×C₃×Dic₆	1	4	yes	—
229	C₃×C₄	C₃×#235	1	4	yes	—
230	C₃×C₄	C₂×#634	1	1	yes	characteristic
231	C₃×C₄	#3828	1	1	yes	characteristic
232	Dic₃	—	12	8	no	—
233	C₃²	—	2	4	no	—
234	C₃²	#5020	1	4	yes	complemented
235	C₃²	C₃×#475	1	1	yes	characteristic, complemented
236	Q₈	—	2	1	no	—
237	Q₈	—	36	2	no	—
238	Q₈	C₃×#634	1	2	yes	complemented
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	18	2	no	—
242	C₂×C₄	C₃×#638	1	1	yes	characteristic
243	C₂×C₄	C₃×#638	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
245	C₈	—	18	2	no	—
246	C₂×C₃	—	2	4	no	—
247	C₂×C₃	—	2	4	no	—
248	C₂×C₃	—	2	4	no	—
249	C₂×C₃	C₃×#924	1	4	yes	—
250	C₂×C₃	C₂×C₃×#215	1	4	yes	—
251	C₂×C₃	C₃×#951	1	4	yes	—
252	C₂×C₃	#64133	1	1	yes	characteristic
253	C₂×C₃	C₂×#3755	1	1	yes	characteristic
254	C₂×C₃	#64160	1	1	yes	characteristic
255	C₂²	C₂×C₃×#638	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	36	2	no	—
260	C₄	C₃×#3755	1	2	yes	—
261	C₄	C₂×C₃×#634	1	1	yes	characteristic
262	C₄	C₃×#3828	1	1	yes	characteristic
263	C₃	—	2	4	no	—
264	C₃	C₃×#5020	1	4	yes	complemented
265	C₃	#10592719	1	1	yes	characteristic, direct product factor
266	C₂	C₃×#64133	1	1	yes	characteristic
267	C₂	C₂×C₃×#3755	1	1	yes	characteristic
268	C₂	C₃×#64160	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄×Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₃×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×#475
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
6. #10592719	C₃	direct (N × H)
29. C₃³×Q₈	Q₈	semidirect
86. C₃²×Q₈	Dic₆	semidirect
87. C₃²×Q₈	C₃×Q₈	semidirect
164. C₃³	#475	semidirect
170. C₃×Q₈	C₃×Dic₆	semidirect
171. C₃×Q₈	#634	semidirect
234. C₃²	#5020	semidirect
235. C₃²	C₃×#475	semidirect
238. Q₈	C₃×#634	semidirect
264. C₃	C₃×#5020	semidirect
265. C₃	#10592719	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄×Q₈	2
C₃×#64078	2
C₃×#63662	2
C₃×#63635	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10592719	3
C₃×#5020	3

Subgroups — C3×#10592719 #13163058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592719 #13163058