Subgroups — C₃×#10592540 #13162881

Order 1728.

872 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3744	C₂	1	2	yes	maximal
2	C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
3	C₃×#64078	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63643	C₂	1	2	yes	maximal, complemented
6	#10592540	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#4841	—	3	4	no	maximal
8	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₃⋊D₃	—	2	1	no	—
10	C₂×C₃×#634	C₂²	1	2	yes	—
11	C₂×C₃³×C₈	C₂²	1	2	yes	—
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃×#3752	—	2	1	no	—
14	C₃×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
15	C₃×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
16	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
17	C₃×#3744	—	2	4	no	—
18	C₂×#3744	C₂×C₃	1	2	yes	—
19	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	1	yes	characteristic
20	#64078	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₃×#204	—	3	8	no	—
22	C₃×C₄×D₁₂	—	3	4	no	—
23	C₃×#869	—	3	4	no	—
24	C₃²×C₄⋊C₈	—	2	4	no	—
25	C₃²×C₄⋊C₈	D₃	1	4	yes	complemented
26	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	#63643	C₂×C₃	1	2	yes	complemented
28	C₃×#816	—	3	8	no	—
29	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
30	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
34	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
35	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
36	C₃×C₄⋊(C₃⋊D₃)	D₄	1	2	yes	—
37	C₃×C₄×C₃⋊D₃	—	4	2	no	—
38	C₃×#634	—	2	2	no	—
39	C₃×#634	—	2	2	no	—
40	C₃³×C₈	—	2	2	no	—
41	C₃×#461	—	9	1	no	—
42	#4841	—	3	4	no	—
43	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₄×C₃⋊D₃	—	2	1	no	—
45	C₂×#634	C₂²×C₃	1	2	yes	—
46	C₂×C₃×D₁₂	—	3	4	no	—
47	C₂×C₃×C₄×D₃	—	6	4	no	—
48	C₂×C₃×Dic₆	—	3	8	no	—
49	C₂×C₃²×C₈	—	2	8	no	—
50	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
51	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
52	C₃²×C₄²	—	2	4	no	—
53	C₃²×C₄²	C₂×D₃	1	4	yes	—
54	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
55	#3752	—	2	1	no	—
56	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
57	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
58	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
59	#3744	—	2	4	no	—
60	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
61	C₃×C₄⋊Dic₃	—	3	4	no	—
62	C₃×C₄⋊Dic₃	—	6	4	no	—
63	C₃×C₄⋊Dic₃	—	6	4	no	—
64	C₃×#204	—	6	16	no	—
65	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₃⋊D₃	—	2	2	no	—
67	C₂×C₃×C₃⋊D₃	—	4	1	no	—
68	C₂×C₃×C₃⋊D₃	—	4	1	no	—
69	C₃³×C₄	—	2	1	no	—
70	C₃³×C₄	SD₁₆	1	2	yes	complemented
71	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
72	C₃³×C₄	#41	1	1	yes	characteristic
73	C₃×C₃⋊Dic₃	—	4	2	no	—
74	C₃×C₃⋊Dic₃	—	4	1	no	—
75	C₂×C₃×SD₁₆	—	9	2	no	—
76	C₃×C₄⋊Q₈	—	9	1	no	—
77	C₃×C₄×D₄	—	9	1	no	—
78	C₂×#204	—	3	8	no	—
79	C₄×D₁₂	—	3	4	no	—
80	#869	—	3	4	no	—
81	C₃×C₄⋊C₈	—	2	4	no	—
82	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
83	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
84	C₃×#103	—	9	2	no	—
85	#816	—	3	8	no	—
86	C₂²×C₃⋊D₃	—	2	1	no	—
87	C₂²×C₃×D₃	—	6	4	no	—
88	C₂×C₃²×C₄	—	2	4	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	2	4	no	—
91	C₂×C₃²×C₄	D₁₂	1	4	yes	—
92	C₂×C₃²×C₄	D₁₂	1	4	yes	—
93	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
96	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₃⋊Dic₃	—	2	2	no	—
98	C₂×C₃⋊Dic₃	—	2	1	no	—
99	C₄⋊(C₃⋊D₃)	—	2	1	no	—
100	C₄⋊(C₃⋊D₃)	C₃×D₄	1	2	yes	—
101	C₄×C₃⋊D₃	—	4	2	no	—
102	#634	—	2	2	no	—
103	#634	—	2	2	no	—
104	C₂×C₃×Dic₃	—	6	4	no	—
105	C₂×C₃×Dic₃	—	6	8	no	—
106	C₃×D₁₂	—	3	8	no	—
107	C₃×D₁₂	—	6	4	no	—
108	C₃×C₄×D₃	—	12	8	no	—
109	C₃×Dic₆	—	6	8	no	—
110	C₃×Dic₆	—	6	8	no	—
111	C₃²×C₈	—	2	2	no	—
112	C₃²×C₈	—	2	8	no	—
113	C₃²×C₈	—	4	8	no	—
114	#461	—	9	1	no	Sylow
115	C₂×C₃³	#121	1	1	yes	characteristic
116	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₃³	#137	1	1	yes	characteristic
118	C₃×C₃⋊D₃	—	4	2	no	—
119	C₂×C₃×Q₈	—	9	2	no	—
120	C₂×C₃×D₄	—	9	1	no	—
121	C₂²×C₃×C₄	—	18	1	no	—
122	C₂×D₁₂	—	3	4	no	—
123	C₂×C₄×D₃	—	6	4	no	—
124	C₂×Dic₆	—	3	8	no	—
125	C₃×SD₁₆	—	18	4	no	—
126	C₂×C₃×C₈	—	2	8	no	—
127	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
128	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
129	C₃×C₄⋊C₄	—	9	1	no	—
130	C₃×C₄⋊C₄	—	18	1	no	—
131	C₃×C₄⋊C₄	—	18	1	no	—
132	C₃×C₂²⋊C₄	—	18	1	no	—
133	C₃×C₄²	—	2	4	no	—
134	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
135	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
136	(C₂×D₃)⋊C₄	—	6	4	no	—
137	C₄⋊Dic₃	—	3	4	no	—
138	C₄⋊Dic₃	—	6	4	no	—
139	C₄⋊Dic₃	—	6	4	no	—
140	#204	—	6	16	no	—
141	C₂²×C₃²	—	2	4	no	—
142	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
143	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
144	C₂×C₃⋊D₃	—	2	2	no	—
145	C₂×C₃⋊D₃	—	4	1	no	—
146	C₂×C₃⋊D₃	—	4	1	no	—
147	C₂×C₃×D₃	—	6	8	no	—
148	C₂×C₃×D₃	—	12	4	no	—
149	C₂×C₃×D₃	—	12	4	no	—
150	C₃²×C₄	—	2	1	no	—
151	C₃²×C₄	—	2	8	no	—
152	C₃²×C₄	—	2	4	no	—
153	C₃²×C₄	—	2	4	no	—
154	C₃²×C₄	—	2	4	no	—
155	C₃²×C₄	—	4	4	no	—
156	C₃²×C₄	#204	1	8	yes	complemented
157	C₃²×C₄	C₃×SD₁₆	1	2	yes	complemented
158	C₃²×C₄	D₃×D₄	1	4	yes	—
159	C₃²×C₄	#239	1	4	yes	—
160	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
161	C₃²×C₄	C₃×#41	1	1	yes	characteristic
162	C₃⋊Dic₃	—	4	2	no	—
163	C₃⋊Dic₃	—	4	1	no	—
164	C₃×Dic₃	—	12	8	no	—
165	C₃×Dic₃	—	12	4	no	—
166	C₂×SD₁₆	—	9	2	no	—
167	C₄⋊Q₈	—	9	1	no	—
168	C₄×D₄	—	9	1	no	—
169	C₄⋊C₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
170	#103	—	9	2	no	—
171	C₃³	#461	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₃	—	18	1	no	—
173	C₂²×D₃	—	6	4	no	—
174	C₃×Q₈	—	18	2	no	—
175	C₃×Q₈	—	18	2	no	—
176	C₃×D₄	—	9	2	no	—
177	C₃×D₄	—	18	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	1	no	—
182	C₂×C₃×C₄	—	18	2	no	—
183	C₂×C₃×C₄	—	36	2	no	—
184	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
185	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
186	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
187	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
189	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
190	C₂×Dic₃	—	6	4	no	—
191	C₂×Dic₃	—	6	8	no	—
192	D₁₂	—	3	8	no	—
193	D₁₂	—	6	4	no	—
194	C₄×D₃	—	12	8	no	—
195	Dic₆	—	6	8	no	—
196	Dic₆	—	6	8	no	—
197	C₃×C₈	—	2	8	no	—
198	C₃×C₈	—	2	2	no	—
199	C₃×C₈	—	4	8	no	—
200	C₂×C₃²	—	2	4	no	—
201	C₂×C₃²	—	2	4	no	—
202	C₂×C₃²	—	2	4	no	—
203	C₂×C₃²	#895	1	4	yes	—
204	C₂×C₃²	C₂×#204	1	4	yes	—
205	C₂×C₃²	#909	1	4	yes	—
206	C₂×C₃²	C₃×#121	1	1	yes	characteristic
207	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
208	C₂×C₃²	C₃×#137	1	1	yes	characteristic
209	C₃⋊D₃	—	4	2	no	—
210	C₃×D₃	—	12	8	no	—
211	C₂×Q₈	—	9	2	no	—
212	C₂×D₄	—	9	1	no	—
213	C₂²×C₄	—	18	1	no	—
214	SD₁₆	—	18	4	no	—
215	C₂×C₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
216	C₄⋊C₄	—	9	1	no	—
217	C₄⋊C₄	—	18	1	no	—
218	C₄⋊C₄	—	18	1	no	—
219	C₂²⋊C₄	—	18	1	no	—
220	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
221	C₂²×C₃	—	2	4	no	—
222	C₂²×C₃	—	18	2	no	—
223	C₂²×C₃	—	36	1	no	—
224	C₂²×C₃	—	36	1	no	—
225	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
226	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
227	C₂×D₃	—	6	8	no	—
228	C₂×D₃	—	12	4	no	—
229	C₂×D₃	—	12	4	no	—
230	C₃×C₄	—	2	4	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	8	no	—
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	2	4	no	—
235	C₃×C₄	—	4	4	no	—
236	C₃×C₄	—	36	2	no	—
237	C₃×C₄	—	36	1	no	—
238	C₃×C₄	C₃×#204	1	8	yes	complemented
239	C₃×C₄	#3744	1	2	yes	complemented
240	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
241	C₃×C₄	C₃×#239	1	4	yes	—
242	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
243	C₃×C₄	#3832	1	1	yes	characteristic
244	Dic₃	—	12	8	no	—
245	Dic₃	—	12	4	no	—
246	C₃²	—	2	4	no	—
247	C₃²	#4841	1	4	yes	complemented
248	C₃²	C₃×#461	1	1	yes	characteristic, complemented
249	C₂³	—	18	1	no	—
250	Q₈	—	18	2	no	—
251	Q₈	—	18	2	no	—
252	D₄	—	9	2	no	—
253	D₄	—	18	1	no	—
254	C₂×C₄	—	18	1	no	—
255	C₂×C₄	—	18	2	no	—
256	C₂×C₄	—	36	2	no	—
257	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
258	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₂×C₃	—	2	4	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	2	4	no	—
264	C₂×C₃	—	36	2	no	—
265	C₂×C₃	C₃×#895	1	4	yes	—
266	C₂×C₃	C₂×C₃×#204	1	4	yes	—
267	C₂×C₃	C₃×#909	1	4	yes	—
268	C₂×C₃	#64104	1	1	yes	characteristic
269	C₂×C₃	C₂×#3744	1	1	yes	characteristic
270	C₂×C₃	#64118	1	1	yes	characteristic
271	D₃	—	12	8	no	—
272	C₂²	—	18	2	no	—
273	C₂²	—	36	1	no	—
274	C₂²	—	36	1	no	—
275	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	36	2	no	—
278	C₄	—	36	1	no	—
279	C₄	C₃×#3744	1	2	yes	complemented
280	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
281	C₄	C₃×#3832	1	1	yes	characteristic
282	C₃	—	2	4	no	—
283	C₃	C₃×#4841	1	4	yes	complemented
284	C₃	#10592540	1	1	yes	characteristic, direct product factor
285	C₂	—	36	2	no	—
286	C₂	C₃×#64104	1	1	yes	characteristic
287	C₂	C₂×C₃×#3744	1	1	yes	characteristic
288	C₂	C₃×#64118	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²
hypercenter	C₃×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#461
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#64078	C₂	semidirect
4. C₃³×C₄⋊C₈	C₂	semidirect
5. C₃×#63643	C₂	semidirect
6. #10592540	C₃	direct (N × H)
20. #64078	C₂×C₃	semidirect
25. C₃²×C₄⋊C₈	D₃	semidirect
26. C₃²×C₄⋊C₈	C₂×C₃	semidirect
27. #63643	C₂×C₃	semidirect
70. C₃³×C₄	SD₁₆	semidirect
82. C₃×C₄⋊C₈	C₃×D₃	semidirect
83. C₃×C₄⋊C₈	C₃⋊D₃	semidirect
156. C₃²×C₄	#204	semidirect
157. C₃²×C₄	C₃×SD₁₆	semidirect
169. C₄⋊C₈	C₃×C₃⋊D₃	semidirect
171. C₃³	#461	semidirect
238. C₃×C₄	C₃×#204	semidirect
239. C₃×C₄	#3744	semidirect
247. C₃²	#4841	semidirect
248. C₃²	C₃×#461	semidirect
279. C₄	C₃×#3744	semidirect
283. C₃	C₃×#4841	semidirect
284. C₃	#10592540	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3744	2
C₃×C₄×C₄⋊(C₃⋊D₃)	2
C₃×#64078	2
C₃³×C₄⋊C₈	2
C₃×#63643	2
#10592540	3
C₃×#4841	3

Subgroups — C3×#10592540 #13162881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592540 #13162881