Subgroups — C₃×#10588609 #13142379

Order 1728.

666 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉⋊#95	C₂	1	4	yes	maximal
5	C₃×#4929	—	3	1	no	maximal
6	#10588609	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
11	C₂×C₃×Dic₁₈	—	2	2	no	—
12	C₂×C₃×Dic₁₈	—	2	2	no	—
13	C₂×C₃×Dic₁₈	C₄	1	4	yes	complemented
14	C₃×C₄²×C₉	—	2	2	no	—
15	C₃×C₄⋊Dic₉	—	2	2	no	—
16	C₂²×C₃×Dic₆	—	3	1	no	—
17	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
18	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
19	C₂²×Dic₁₈	C₂×C₃	1	1	yes	characteristic
20	C₃×C₃⋊#95	—	3	4	no	—
21	C₂×C₄²×C₉	—	2	1	no	—
22	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
23	C₂×C₄⋊Dic₉	C₂×C₃	1	1	yes	characteristic
24	C₉⋊#95	C₂×C₃	1	4	yes	—
25	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	Q₈	1	4	yes	complemented
32	C₂×C₃×Dic₉	—	2	2	no	—
33	C₂×C₃×Dic₉	—	2	2	no	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	C₂×C₃×Dic₉	—	2	4	no	—
36	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
37	C₃×Dic₁₈	—	2	8	no	—
38	C₃×#391	—	9	1	no	—
39	#4929	—	3	1	no	—
40	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
41	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃×Dic₃	—	3	2	no	—
43	C₂²×C₃×Dic₃	—	3	2	no	—
44	C₂×C₃×Dic₆	—	3	4	no	—
45	C₂×C₃×Dic₆	—	6	2	no	—
46	C₂×C₃×Dic₆	—	6	2	no	—
47	C₃²×C₄²	—	2	2	no	—
48	C₃×C₄⋊Dic₃	—	6	2	no	—
49	C₂²×C₄×C₉	—	2	2	no	—
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
52	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
53	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
54	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
55	C₂×Dic₁₈	—	2	2	no	—
56	C₂×Dic₁₈	—	2	2	no	—
57	C₂×Dic₁₈	C₃×C₄	1	4	yes	complemented
58	C₄²×C₉	—	2	4	no	—
59	C₄²×C₉	—	2	2	no	—
60	C₄⋊Dic₉	—	2	2	no	—
61	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
65	C₂²×C₃×C₉	#41	1	2	yes	—
66	C₃×C₄×C₉	—	2	4	no	—
67	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	—
68	C₃×Dic₉	—	2	4	no	—
69	C₃×Dic₉	—	4	2	no	—
70	C₂²×C₃×Q₈	—	9	1	no	—
71	C₂²×Dic₆	—	3	1	no	—
72	C₂×C₃×C₄⋊C₄	—	9	1	no	—
73	C₂×C₃×C₄²	—	2	1	no	—
74	C₂×C₃×C₄²	D₉	1	1	yes	characteristic
75	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
76	C₂×C₄⋊Dic₃	—	3	1	no	—
77	C₃×#95	—	9	4	no	—
78	C₃⋊#95	—	3	4	no	—
79	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	Dic₆	1	4	yes	—
83	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
84	C₂×C₃²×C₄	D₁₂	1	2	yes	—
85	C₂×C₃²×C₄	#67	1	2	yes	—
86	C₂×C₃×Dic₃	—	3	4	no	—
87	C₂×C₃×Dic₃	—	6	2	no	—
88	C₂×C₃×Dic₃	—	6	2	no	—
89	C₂×C₃×Dic₃	—	6	2	no	—
90	C₂×C₃×Dic₃	—	6	4	no	—
91	C₃×Dic₆	—	6	8	no	—
92	C₂³×C₉	—	2	1	no	—
93	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₄×C₉	—	2	4	no	—
95	C₂×C₄×C₉	—	2	4	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	4	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
103	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
104	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
105	C₂×C₄×C₉	C₃×Q₈	1	4	yes	complemented
106	C₂×Dic₉	—	2	2	no	—
107	C₂×Dic₉	—	2	2	no	—
108	C₂×Dic₉	—	2	2	no	—
109	C₂×Dic₉	—	2	4	no	—
110	C₂×Dic₉	C₂×C₃×C₄	1	4	yes	—
111	Dic₁₈	—	2	8	no	—
112	#391	—	9	1	no	Sylow
113	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₃×C₉	C₄×Q₈	1	2	yes	—
115	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₃×C₉	#124	1	1	yes	characteristic
117	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₃×Q₈	—	9	4	no	—
119	C₂×C₃×Q₈	—	18	2	no	—
120	C₂×C₃×Q₈	—	18	2	no	—
121	C₂²×C₃×C₄	—	2	2	no	—
122	C₂²×C₃×C₄	—	2	1	no	—
123	C₂²×C₃×C₄	—	9	2	no	—
124	C₂²×C₃×C₄	—	9	2	no	—
125	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
126	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
128	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
129	C₂²×Dic₃	—	3	2	no	—
130	C₂²×Dic₃	—	3	2	no	—
131	C₂×Dic₆	—	3	4	no	—
132	C₂×Dic₆	—	6	2	no	—
133	C₂×Dic₆	—	6	2	no	—
134	C₃×C₄⋊C₄	—	18	2	no	—
135	C₃×C₄²	—	2	4	no	—
136	C₃×C₄²	—	2	2	no	—
137	C₃×C₄²	—	2	2	no	—
138	C₄⋊Dic₃	—	6	2	no	—
139	C₂²×C₃²	C₂×Dic₆	1	2	yes	—
140	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
142	C₂²×C₃²	#235	1	2	yes	—
143	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
144	C₃²×C₄	—	2	4	no	—
145	C₃²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
146	C₃×Dic₃	—	6	4	no	—
147	C₃×Dic₃	—	12	2	no	—
148	C₂²×C₉	—	2	2	no	—
149	C₂²×C₉	—	2	1	no	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	—	2	2	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
155	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂²×C₉	C₂×C₃×Q₈	1	2	yes	—
157	C₂²×C₉	C₃×#41	1	2	yes	—
158	C₄×C₉	—	2	8	no	—
159	C₄×C₉	—	2	4	no	—
160	C₄×C₉	—	2	4	no	—
161	C₄×C₉	C₃×C₂²⋊C₄	1	4	yes	—
162	Dic₉	—	2	4	no	—
163	Dic₉	—	4	2	no	—
164	C₂²×Q₈	—	9	1	no	—
165	C₂×C₄⋊C₄	—	9	1	no	—
166	C₂×C₄²	C₃×D₉	1	1	yes	characteristic
167	#95	—	9	4	no	—
168	C₃×C₉	#391	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₃	—	2	1	no	—
170	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
171	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
172	C₃×Q₈	—	18	8	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	9	4	no	—
184	C₂×C₃×C₄	—	18	2	no	—
185	C₂×C₃×C₄	—	18	2	no	—
186	C₂×C₃×C₄	—	18	2	no	—
187	C₂×C₃×C₄	—	18	4	no	—
188	C₂×C₃×C₄	Dic₁₈	1	4	yes	complemented
189	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
190	C₂×C₃×C₄	D₃₆	1	2	yes	—
191	C₂×C₃×C₄	#611	1	2	yes	—
192	C₂×C₃×C₄	C₃×Dic₆	1	4	yes	—
193	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
194	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
195	C₂×C₃×C₄	C₃×#67	1	2	yes	—
196	C₂×Dic₃	—	3	4	no	—
197	C₂×Dic₃	—	6	2	no	—
198	C₂×Dic₃	—	6	2	no	—
199	C₂×Dic₃	—	6	2	no	—
200	C₂×Dic₃	—	6	4	no	—
201	Dic₆	—	6	8	no	—
202	C₂×C₃²	C₄×Dic₆	1	2	yes	—
203	C₂×C₃²	#869	1	1	yes	characteristic
204	C₂×C₃²	#875	1	1	yes	characteristic
205	C₂×C₃²	#924	1	2	yes	—
206	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₉	—	2	2	no	—
208	C₂×C₉	—	2	1	no	—
209	C₂×C₉	—	2	2	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₉	C₃×C₄×Q₈	1	2	yes	—
214	C₂×C₉	C₃×C₂²⋊Q₈	1	2	yes	—
215	C₂×C₉	C₃×#124	1	1	yes	characteristic
216	C₂×C₉	C₃×C₄⋊Q₈	1	1	yes	characteristic
217	C₂×Q₈	—	9	4	no	—
218	C₂×Q₈	—	18	2	no	—
219	C₂×Q₈	—	18	2	no	—
220	C₂²×C₄	—	9	2	no	—
221	C₂²×C₄	—	9	2	no	—
222	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
223	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
224	C₄⋊C₄	—	18	2	no	—
225	C₄²	—	2	2	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	C₂×Dic₁₈	1	2	yes	—
232	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
233	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
234	C₂²×C₃	#3697	1	2	yes	—
235	C₂²×C₃	C₂×#611	1	1	yes	characteristic
236	C₂²×C₃	C₂×C₃×Dic₆	1	2	yes	—
237	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
239	C₂²×C₃	C₃×#235	1	2	yes	—
240	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
241	C₃×C₄	—	2	8	no	—
242	C₃×C₄	—	2	4	no	—
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	2	4	no	—
245	C₃×C₄	—	18	4	no	—
246	C₃×C₄	—	36	2	no	—
247	C₃×C₄	(C₂×D₉)⋊C₄	1	4	yes	—
248	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
249	Dic₃	—	6	4	no	—
250	Dic₃	—	12	2	no	—
251	C₃²	#4929	1	1	yes	characteristic
252	C₉	—	2	1	no	—
253	C₉	C₃×#391	1	1	yes	characteristic, complemented
254	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
255	Q₈	—	18	8	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	9	4	no	—
259	C₂×C₄	—	18	2	no	—
260	C₂×C₄	—	18	2	no	—
261	C₂×C₄	—	18	2	no	—
262	C₂×C₄	—	18	4	no	—
263	C₂×C₄	C₃×Dic₁₈	1	4	yes	complemented
264	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
265	C₂×C₄	C₃×D₃₆	1	2	yes	—
266	C₂×C₄	C₃×#611	1	2	yes	—
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	C₄×Dic₁₈	1	2	yes	—
273	C₂×C₃	#63431	1	1	yes	characteristic
274	C₂×C₃	#63437	1	1	yes	characteristic
275	C₂×C₃	#63486	1	2	yes	—
276	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
277	C₂×C₃	C₃×C₄×Dic₆	1	2	yes	—
278	C₂×C₃	C₃×#869	1	1	yes	characteristic
279	C₂×C₃	C₃×#875	1	1	yes	characteristic
280	C₂×C₃	C₃×#924	1	2	yes	—
281	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
282	C₂²	C₂×C₃×Dic₁₈	1	2	yes	—
283	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
284	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
285	C₂²	C₃×#3697	1	2	yes	—
286	C₂²	C₂×C₃×#611	1	1	yes	characteristic
287	C₄	—	2	4	no	—
288	C₄	—	18	4	no	—
289	C₄	—	36	2	no	—
290	C₄	C₃×(C₂×D₉)⋊C₄	1	4	yes	—
291	C₃	—	2	1	no	—
292	C₃	#10588609	1	1	yes	characteristic, direct product factor
293	C₃	C₃×#4929	1	1	yes	characteristic
294	C₂	C₃×C₄×Dic₁₈	1	2	yes	—
295	C₂	C₃×#63431	1	1	yes	characteristic
296	C₂	C₃×#63437	1	1	yes	characteristic
297	C₂	C₃×#63486	1	2	yes	—
298	C₂	C₂×C₃×(C₂×D₉)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₉
hypocenter	C₉	C₃×#391
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
6. #10588609	C₃	direct (N × H)
13. C₂×C₃×Dic₁₈	C₄	semidirect
31. C₂×C₃×C₄×C₉	Q₈	semidirect
57. C₂×Dic₁₈	C₃×C₄	semidirect
105. C₂×C₄×C₉	C₃×Q₈	semidirect
168. C₃×C₉	#391	semidirect
188. C₂×C₃×C₄	Dic₁₈	semidirect
253. C₉	C₃×#391	semidirect
263. C₂×C₄	C₃×Dic₁₈	semidirect
292. C₃	#10588609	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×Dic₁₈	2
C₂×C₃×C₄²×C₉	2
C₂×C₃×C₄⋊Dic₉	2
C₃×C₉⋊#95	2
C₃×#4929	3
#10588609	3

Subgroups — C3×#10588609 #13142379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588609 #13142379