Subgroups — C₃×#10589422 #13145981

Order 1728.

812 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#64135	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3748	C₂	1	1	yes	maximal, characteristic
4	#10589422	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×#4556	—	3	4	no	maximal
6	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₃³×M₁₆	—	2	1	no	—
8	C₃³×M₁₆	—	2	1	no	—
9	C₂×C₃³×C₈	C₄	1	2	yes	complemented
10	C₃×#3757	C₄	1	2	yes	—
11	C₃×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
12	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
13	C₃×#3748	—	2	1	no	—
14	C₃×#3748	C₄	1	2	yes	complemented
15	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
16	C₂×C₃²×M₁₆	—	2	4	no	—
17	C₂×C₃²×M₁₆	D₃	1	4	yes	—
18	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
19	#64135	C₂×C₃	1	1	yes	characteristic
20	C₂×#3748	C₂×C₃	1	1	yes	characteristic
21	C₃×#926	—	3	4	no	—
22	C₂×C₃×#208	—	3	4	no	—
23	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	Q₈	1	1	yes	characteristic
30	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
31	C₃³×C₈	—	2	2	no	—
32	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
33	C₃×#344	—	9	1	no	—
34	#4556	—	3	4	no	—
35	C₂²×C₃²×C₄	—	2	4	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
37	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
38	C₃²×M₁₆	—	2	8	no	—
39	C₃²×M₁₆	—	2	8	no	—
40	C₃²×M₁₆	—	2	4	no	—
41	C₃²×M₁₆	—	2	4	no	—
42	C₃²×M₁₆	—	2	1	no	—
43	C₃²×M₁₆	—	2	1	no	—
44	C₂×C₃²×C₈	—	2	8	no	—
45	C₂×C₃²×C₈	Dic₃	1	8	yes	complemented
46	C₂×C₃²×C₈	C₃×C₄	1	2	yes	complemented
47	#3757	C₃×C₄	1	2	yes	—
48	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
49	C₄×C₃⋊Dic₃	—	2	1	no	—
50	#3748	—	2	1	no	—
51	#3748	C₃×C₄	1	2	yes	complemented
52	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
53	C₃×#217	—	3	8	no	—
54	C₃×C₄⋊Dic₃	—	6	4	no	—
55	C₃×C₄×Dic₃	—	6	4	no	—
56	C₃×#208	—	3	8	no	—
57	C₃×#208	—	6	4	no	—
58	C₂×C₃×C₃⋊C₈	—	6	4	no	—
59	C₂²×C₃³	—	4	1	no	—
60	C₂²×C₃³	C₄²	1	1	yes	characteristic
61	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
63	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₃×C₃⋊Dic₃	—	4	2	no	—
68	C₂×C₃×M₁₆	—	2	4	no	—
69	C₂×C₃×M₁₆	—	9	1	no	—
70	C₂×C₃×M₁₆	C₃×D₃	1	4	yes	—
71	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
72	C₃×#117	—	9	1	no	—
73	#926	—	3	4	no	—
74	C₂×#208	—	3	4	no	—
75	C₂³×C₃²	—	2	4	no	—
76	C₂³×C₃²	C₄×D₃	1	4	yes	—
77	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	—	2	4	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	Dic₆	1	4	yes	—
85	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
86	C₂×C₃²×C₄	D₁₂	1	4	yes	—
87	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
88	C₂×C₃²×C₄	#67	1	4	yes	—
89	C₂×C₃²×C₄	#67	1	4	yes	—
90	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₃×Q₈	1	1	yes	characteristic
96	C₂×C₃⋊Dic₃	—	2	2	no	—
97	C₂×C₃×Dic₃	—	6	8	no	—
98	C₃²×C₈	—	2	16	no	—
99	C₃²×C₈	—	2	8	no	—
100	C₃²×C₈	—	2	2	no	—
101	C₃⋊(C₃⋊C₈)	—	2	2	no	—
102	C₃×C₃⋊C₈	—	6	8	no	—
103	#344	—	9	1	no	Sylow
104	C₂×C₃³	—	2	1	no	—
105	C₂×C₃³	—	2	1	no	—
106	C₂×C₃³	—	2	1	no	—
107	C₂×C₃³	#95	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	2	4	no	—
109	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
110	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
111	C₃×M₁₆	—	2	8	no	—
112	C₃×M₁₆	—	2	8	no	—
113	C₃×M₁₆	—	2	4	no	—
114	C₃×M₁₆	—	2	4	no	—
115	C₃×M₁₆	—	2	1	no	—
116	C₃×M₁₆	—	2	1	no	—
117	C₃×M₁₆	—	9	2	no	—
118	C₃×M₁₆	—	18	1	no	—
119	C₂×C₃×C₈	—	2	8	no	—
120	C₂×C₃×C₈	—	18	1	no	—
121	C₂×C₃×C₈	C₃×Dic₃	1	8	yes	complemented
122	C₂×C₃×C₈	C₃⋊Dic₃	1	2	yes	complemented
123	C₃×C₄⋊C₄	—	18	1	no	—
124	C₃×C₂²⋊C₄	—	9	2	no	—
125	C₃×C₄²	—	18	1	no	—
126	#217	—	3	8	no	—
127	C₄⋊Dic₃	—	6	4	no	—
128	C₄×Dic₃	—	6	4	no	—
129	#208	—	3	8	no	—
130	#208	—	6	4	no	—
131	C₂×C₃⋊C₈	—	6	4	no	—
132	C₂²×C₃²	—	2	4	no	—
133	C₂²×C₃²	—	2	4	no	—
134	C₂²×C₃²	—	2	4	no	—
135	C₂²×C₃²	—	4	4	no	—
136	C₂²×C₃²	—	4	1	no	—
137	C₂²×C₃²	—	4	8	no	—
138	C₂²×C₃²	C₄×Dic₃	1	4	yes	—
139	C₂²×C₃²	Dic₃⋊C₄	1	4	yes	—
140	C₂²×C₃²	(C₂×D₃)⋊C₄	1	4	yes	—
141	C₂²×C₃²	C₃×C₄²	1	1	yes	characteristic
142	C₂²×C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
144	C₃²×C₄	—	2	4	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	—	2	4	no	—
148	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
149	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
150	C₃²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
151	C₃²×C₄	#217	1	4	yes	—
152	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
153	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
154	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
155	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
156	C₃⋊Dic₃	—	4	2	no	—
157	C₃×Dic₃	—	12	8	no	—
158	C₂×M₁₆	—	9	1	no	—
159	C₂×M₁₆	C₃×C₃⋊D₃	1	1	yes	characteristic
160	#117	—	9	1	no	—
161	C₃³	#344	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	C₃×C₄×D₃	1	4	yes	—
164	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	18	2	no	—
172	C₂×C₃×C₄	C₃×Dic₆	1	4	yes	—
173	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
174	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
175	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
176	C₂×C₃×C₄	C₃×#67	1	4	yes	—
177	C₂×C₃×C₄	C₃×#67	1	4	yes	—
178	C₂×C₃×C₄	#634	1	1	yes	characteristic
179	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
180	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
182	C₂×C₃×C₄	#638	1	1	yes	characteristic
183	C₂×C₃×C₄	#638	1	1	yes	characteristic
184	C₂×Dic₃	—	6	8	no	—
185	C₃×C₈	—	2	16	no	—
186	C₃×C₈	—	2	8	no	—
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	—	18	2	no	—
189	C₃⋊C₈	—	6	8	no	—
190	C₂×C₃²	—	2	8	no	—
191	C₂×C₃²	—	2	4	no	—
192	C₂×C₃²	—	2	1	no	—
193	C₂×C₃²	—	2	1	no	—
194	C₂×C₃²	—	2	1	no	—
195	C₂×C₃²	—	2	4	no	—
196	C₂×C₃²	—	2	4	no	—
197	C₂×C₃²	—	2	4	no	—
198	C₂×C₃²	—	4	4	no	—
199	C₂×C₃²	—	4	4	no	—
200	C₂×C₃²	C₃⋊#95	1	4	yes	—
201	C₂×C₃²	C₃×#95	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	1	no	—
205	M₁₆	—	9	2	no	—
206	M₁₆	—	18	1	no	—
207	C₂×C₈	—	18	1	no	—
208	C₂×C₈	C₃×C₃⋊Dic₃	1	2	yes	complemented
209	C₄⋊C₄	—	18	1	no	—
210	C₂²⋊C₄	—	9	2	no	—
211	C₄²	—	18	1	no	—
212	C₂²×C₃	—	2	4	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	4	4	no	—
216	C₂²×C₃	—	4	1	no	—
217	C₂²×C₃	—	4	8	no	—
218	C₂²×C₃	C₃×C₄×Dic₃	1	4	yes	—
219	C₂²×C₃	C₃×Dic₃⋊C₄	1	4	yes	—
220	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
221	C₂²×C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic
222	C₂²×C₃	#3750	1	1	yes	characteristic
223	C₂²×C₃	#3752	1	1	yes	characteristic
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	2	4	no	—
227	C₃×C₄	—	2	4	no	—
228	C₃×C₄	—	36	2	no	—
229	C₃×C₄	C₃×Dic₃⋊C₄	1	4	yes	—
230	C₃×C₄	C₃×C₄⋊Dic₃	1	4	yes	—
231	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
232	C₃×C₄	C₃×#217	1	4	yes	—
233	C₃×C₄	#3750	1	1	yes	characteristic
234	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
235	C₃×C₄	#3752	1	1	yes	characteristic
236	C₃×C₄	#3757	1	1	yes	characteristic
237	Dic₃	—	12	8	no	—
238	C₃²	—	2	4	no	—
239	C₃²	#4556	1	4	yes	complemented
240	C₃²	C₃×#344	1	1	yes	characteristic, complemented
241	C₂³	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
242	C₂×C₄	—	18	2	no	—
243	C₂×C₄	C₃×#634	1	1	yes	characteristic
244	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
245	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
246	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
247	C₂×C₄	C₃×#638	1	1	yes	characteristic
248	C₂×C₄	C₃×#638	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	18	2	no	—
251	C₂×C₃	—	2	8	no	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	—	2	4	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	4	no	—
259	C₂×C₃	—	4	4	no	—
260	C₂×C₃	—	4	4	no	—
261	C₂×C₃	C₃×C₃⋊#95	1	4	yes	—
262	C₂×C₃	#63658	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	C₃×C₄×C₃⋊Dic₃	1	1	yes	characteristic
265	C₂²	C₃×#3750	1	1	yes	characteristic
266	C₂²	C₃×#3752	1	1	yes	characteristic
267	C₄	—	36	2	no	—
268	C₄	C₃×#3750	1	1	yes	characteristic
269	C₄	C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
270	C₄	C₃×#3752	1	1	yes	characteristic
271	C₄	C₃×#3757	1	1	yes	characteristic
272	C₃	—	2	4	no	—
273	C₃	C₃×#4556	1	4	yes	complemented
274	C₃	#10589422	1	1	yes	characteristic, direct product factor
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	C₃×#63658	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#3757
commutator	C₂²×C₃²	C₃×C₄²
Frattini	C₂²×C₄	C₂×C₃×C₃⋊D₃
Fitting	C₂×C₃³×M₁₆	C₂
socle	C₂×C₃³	#95
prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂²×C₃²×C₄	C₂²×C₃
hypercenter	C₂×C₃×M₁₆	C₃⋊D₃
hypocenter	C₃²	C₃×#344
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10589422	C₃	direct (N × H)
9. C₂×C₃³×C₈	C₄	semidirect
14. C₃×#3748	C₄	semidirect
45. C₂×C₃²×C₈	Dic₃	semidirect
46. C₂×C₃²×C₈	C₃×C₄	semidirect
51. #3748	C₃×C₄	semidirect
121. C₂×C₃×C₈	C₃×Dic₃	semidirect
122. C₂×C₃×C₈	C₃⋊Dic₃	semidirect
161. C₃³	#344	semidirect
208. C₂×C₈	C₃×C₃⋊Dic₃	semidirect
239. C₃²	#4556	semidirect
240. C₃²	C₃×#344	semidirect
273. C₃	C₃×#4556	semidirect
274. C₃	#10589422	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×C₃³×M₁₆	2
C₃×#64135	2
C₂×C₃×#3748	2
#10589422	3
C₃×#4556	3

Subgroups — C3×#10589422 #13145981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589422 #13145981