Subgroups — C₃×#10588531 #13142025

Order 1728.

344 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₃₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#63432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63380	C₂	1	1	yes	maximal, characteristic
5	C₃×#63380	C₂	1	1	yes	maximal, characteristic
6	C₃×#63378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4847	—	3	1	no	maximal
9	#10588531	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×D₃₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×D₉	—	2	1	no	—
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₃×(C₂×D₉)⋊C₄	—	2	1	no	—
16	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
19	C₃×Dic₉⋊C₄	—	2	1	no	—
20	C₃×Dic₉⋊C₄	—	2	1	no	—
21	C₃×C₄×D₁₂	—	3	1	no	—
22	C₃×#870	—	3	1	no	—
23	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
24	C₃×#821	—	3	1	no	—
25	C₃×#821	—	3	1	no	—
26	C₃×#818	—	3	1	no	—
27	C₃×#817	—	3	1	no	—
28	C₄×D₃₆	C₂×C₃	1	1	yes	characteristic
29	#63432	C₂×C₃	1	1	yes	characteristic, complemented
30	C₉×C₄⋊C₈	—	2	1	no	—
31	C₉×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
32	#63380	C₂×C₃	1	1	yes	characteristic
33	#63380	C₂×C₃	1	1	yes	characteristic
34	#63378	C₂×C₃	1	1	yes	characteristic, complemented
35	#63377	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂²×C₃×D₉	—	2	1	no	—
37	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₃×Dic₉	—	2	1	no	—
41	C₂×C₃×Dic₉	—	2	1	no	—
42	C₂×C₃×Dic₉	—	2	1	no	—
43	C₃×D₃₆	—	2	1	no	—
44	C₃×D₃₆	Q₈	1	2	yes	—
45	C₃×C₄×D₉	—	4	2	no	—
46	C₃×C₈×C₉	—	2	1	no	—
47	C₃×C₈×C₉	—	2	1	no	—
48	C₃×#478	—	9	1	no	—
49	#4847	—	3	1	no	—
50	C₂×C₃×D₁₂	—	3	1	no	—
51	C₂×C₃×C₄×D₃	—	6	1	no	—
52	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
56	C₃×C₄⋊Dic₃	—	3	1	no	—
57	C₃×C₄⋊Dic₃	—	3	1	no	—
58	C₃×C₄⋊Dic₃	—	3	1	no	—
59	C₃×Dic₃⋊C₄	—	6	1	no	—
60	C₃×Dic₃⋊C₄	—	6	1	no	—
61	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₄×D₉	—	2	1	no	—
63	C₂×C₈×C₉	—	2	1	no	—
64	C₂×C₈×C₉	—	2	1	no	—
65	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
67	C₄²×C₉	—	2	1	no	—
68	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
69	(C₂×D₉)⋊C₄	—	2	1	no	—
70	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
72	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
73	Dic₉⋊C₄	—	2	1	no	—
74	Dic₉⋊C₄	—	2	1	no	—
75	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×D₉	—	2	2	no	—
77	C₂×C₃×D₉	—	4	1	no	—
78	C₂×C₃×D₉	—	4	1	no	—
79	C₃×C₄×C₉	—	2	1	no	—
80	C₃×C₄×C₉	—	2	1	no	—
81	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
82	C₃×C₄×C₉	#41	1	1	yes	characteristic
83	C₃×Dic₉	—	4	1	no	—
84	C₃×Dic₉	—	4	1	no	—
85	C₃×Dic₉	—	4	1	no	—
86	C₃×#125	—	9	1	no	—
87	C₃×C₄×D₄	—	9	1	no	—
88	C₄×D₁₂	—	3	1	no	—
89	#870	—	3	1	no	—
90	C₃×#107	—	9	1	no	—
91	C₃×#106	—	9	1	no	—
92	C₃×C₄⋊C₈	—	2	1	no	—
93	C₃×C₄⋊C₈	D₉	1	1	yes	characteristic, complemented
94	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
95	C₃×#102	—	9	1	no	—
96	C₃×#102	—	9	1	no	—
97	#821	—	3	1	no	—
98	#821	—	3	1	no	—
99	#818	—	3	1	no	—
100	#817	—	3	1	no	—
101	C₂²×C₃×D₃	—	6	1	no	—
102	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
103	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃×Dic₃	—	6	1	no	—
106	C₂×C₃×Dic₃	—	6	1	no	—
107	C₂×C₃×Dic₃	—	6	1	no	—
108	C₃×D₁₂	—	3	2	no	—
109	C₃×D₁₂	—	6	1	no	—
110	C₃×C₄×D₃	—	12	2	no	—
111	C₂²×D₉	—	2	1	no	—
112	C₃²×C₈	—	2	1	no	—
113	C₃²×C₈	—	2	1	no	—
114	C₂×C₄×C₉	—	2	1	no	—
115	C₂×C₄×C₉	—	2	1	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
118	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
119	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
120	C₂×Dic₉	—	2	1	no	—
121	C₂×Dic₉	—	2	1	no	—
122	C₂×Dic₉	—	2	1	no	—
123	D₃₆	—	2	1	no	—
124	D₃₆	C₃×Q₈	1	2	yes	—
125	C₄×D₉	—	4	2	no	—
126	C₈×C₉	—	2	1	no	—
127	C₈×C₉	—	2	1	no	—
128	C₈×C₉	—	4	1	no	—
129	C₈×C₉	—	4	1	no	—
130	#478	—	9	1	no	Sylow
131	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₃×C₉	#135	1	1	yes	characteristic
133	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₃×D₉	—	4	2	no	—
135	C₂×C₃×D₄	—	9	1	no	—
136	C₂²×C₃×C₄	—	18	1	no	—
137	C₂×D₁₂	—	3	1	no	—
138	C₂×C₄×D₃	—	6	1	no	—
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	—	2	1	no	—
141	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
145	C₃×C₄⋊C₄	—	9	1	no	—
146	C₃×C₄⋊C₄	—	9	1	no	—
147	C₃×C₄⋊C₄	—	9	1	no	—
148	C₃×C₄⋊C₄	—	18	1	no	—
149	C₃×C₄⋊C₄	—	18	1	no	—
150	C₃×C₂²⋊C₄	—	18	1	no	—
151	C₃×C₄²	—	2	1	no	—
152	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
153	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
154	(C₂×D₃)⋊C₄	—	6	1	no	—
155	C₄⋊Dic₃	—	3	1	no	—
156	C₄⋊Dic₃	—	3	1	no	—
157	C₄⋊Dic₃	—	3	1	no	—
158	Dic₃⋊C₄	—	6	1	no	—
159	Dic₃⋊C₄	—	6	1	no	—
160	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
161	C₂×C₃×D₃	—	6	2	no	—
162	C₂×C₃×D₃	—	12	1	no	—
163	C₂×C₃×D₃	—	12	1	no	—
164	C₃²×C₄	—	2	1	no	—
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	#237	1	1	yes	characteristic
167	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
168	C₃×Dic₃	—	12	1	no	—
169	C₃×Dic₃	—	12	1	no	—
170	C₃×Dic₃	—	12	1	no	—
171	C₂²×C₉	—	2	1	no	—
172	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×D₉	—	2	2	no	—
174	C₂×D₉	—	4	1	no	—
175	C₂×D₉	—	4	1	no	—
176	C₄×C₉	—	2	2	no	—
177	C₄×C₉	—	2	1	no	—
178	C₄×C₉	—	2	1	no	—
179	C₄×C₉	—	2	1	no	—
180	C₄×C₉	—	2	1	no	—
181	C₄×C₉	—	4	1	no	—
182	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
183	C₄×C₉	C₃×#41	1	1	yes	characteristic
184	Dic₉	—	4	1	no	—
185	Dic₉	—	4	1	no	—
186	Dic₉	—	4	1	no	—
187	#125	—	9	1	no	—
188	C₄×D₄	—	9	1	no	—
189	#107	—	9	1	no	—
190	#106	—	9	1	no	—
191	C₄⋊C₈	C₃×D₉	1	1	yes	characteristic, complemented
192	#102	—	9	1	no	—
193	#102	—	9	1	no	—
194	C₃×C₉	#478	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₃	—	18	1	no	—
196	C₂²×D₃	—	6	1	no	—
197	C₃×D₄	—	9	2	no	—
198	C₃×D₄	—	18	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	18	1	no	—
203	C₂×C₃×C₄	—	18	1	no	—
204	C₂×C₃×C₄	—	18	1	no	—
205	C₂×C₃×C₄	—	36	2	no	—
206	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
207	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
209	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
210	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
212	C₂×Dic₃	—	6	1	no	—
213	C₂×Dic₃	—	6	1	no	—
214	C₂×Dic₃	—	6	1	no	—
215	D₁₂	—	3	2	no	—
216	D₁₂	—	6	1	no	—
217	C₄×D₃	—	12	2	no	—
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	—	2	1	no	—
220	C₃×C₈	—	2	1	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	—	4	1	no	—
223	C₃×C₈	—	4	1	no	—
224	C₂×C₃²	#897	1	1	yes	characteristic
225	C₂×C₃²	#904	1	1	yes	characteristic
226	C₂×C₃²	#908	1	1	yes	characteristic
227	C₃×D₃	—	12	2	no	—
228	C₂×C₉	—	2	1	no	—
229	C₂×C₉	—	2	1	no	—
230	C₂×C₉	—	2	1	no	—
231	C₂×C₉	C₃×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₉	C₃×#135	1	1	yes	characteristic
233	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
234	D₉	—	4	2	no	—
235	C₂×D₄	—	9	1	no	—
236	C₂²×C₄	—	18	1	no	—
237	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
238	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
239	C₄⋊C₄	—	9	1	no	—
240	C₄⋊C₄	—	9	1	no	—
241	C₄⋊C₄	—	9	1	no	—
242	C₄⋊C₄	—	18	1	no	—
243	C₄⋊C₄	—	18	1	no	—
244	C₂²⋊C₄	—	18	1	no	—
245	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	18	2	no	—
248	C₂²×C₃	—	36	1	no	—
249	C₂²×C₃	—	36	1	no	—
250	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
251	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
252	C₂×D₃	—	6	2	no	—
253	C₂×D₃	—	12	1	no	—
254	C₂×D₃	—	12	1	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	36	1	no	—
264	C₃×C₄	—	36	1	no	—
265	C₃×C₄	—	36	1	no	—
266	C₃×C₄	#3699	1	1	yes	characteristic
267	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
268	C₃×C₄	C₃×#237	1	1	yes	characteristic
269	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
270	Dic₃	—	12	1	no	—
271	Dic₃	—	12	1	no	—
272	Dic₃	—	12	1	no	—
273	C₃²	#4847	1	1	yes	characteristic
274	C₉	—	2	1	no	—
275	C₉	C₃×#478	1	1	yes	characteristic, complemented
276	C₂³	—	18	1	no	—
277	D₄	—	9	2	no	—
278	D₄	—	18	1	no	—
279	C₂×C₄	—	18	1	no	—
280	C₂×C₄	—	18	1	no	—
281	C₂×C₄	—	18	1	no	—
282	C₂×C₄	—	36	2	no	—
283	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
284	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	36	2	no	—
292	C₂×C₃	#63459	1	1	yes	characteristic
293	C₂×C₃	#63467	1	1	yes	characteristic
294	C₂×C₃	#63470	1	1	yes	characteristic
295	C₂×C₃	C₃×#897	1	1	yes	characteristic
296	C₂×C₃	C₃×#904	1	1	yes	characteristic
297	C₂×C₃	C₃×#908	1	1	yes	characteristic
298	D₃	—	12	2	no	—
299	C₂²	—	18	2	no	—
300	C₂²	—	36	1	no	—
301	C₂²	—	36	1	no	—
302	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	36	1	no	—
306	C₄	—	36	1	no	—
307	C₄	—	36	1	no	—
308	C₄	C₃×#3699	1	1	yes	characteristic
309	C₄	C₃×Q₈×D₉	1	1	yes	characteristic
310	C₃	—	2	1	no	—
311	C₃	#10588531	1	1	yes	characteristic, direct product factor
312	C₃	C₃×#4847	1	1	yes	characteristic
313	C₂	—	36	2	no	—
314	C₂	C₃×#63459	1	1	yes	characteristic
315	C₂	C₃×#63467	1	1	yes	characteristic
316	C₂	C₃×#63470	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₃₆
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×D₁₂
prime radical	C₂×C₃×D₃₆	C₂²
hypercenter	C₃×C₄⋊C₈	D₉
hypocenter	C₉	C₃×#478
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
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
2. C₃×#63432	C₂	semidirect
3. C₃×C₉×C₄⋊C₈	C₂	semidirect
6. C₃×#63378	C₂	semidirect
7. C₃×#63377	C₂	semidirect
9. #10588531	C₃	direct (N × H)
29. #63432	C₂×C₃	semidirect
31. C₉×C₄⋊C₈	C₂×C₃	semidirect
34. #63378	C₂×C₃	semidirect
35. #63377	C₂×C₃	semidirect
93. C₃×C₄⋊C₈	D₉	semidirect
191. C₄⋊C₈	C₃×D₉	semidirect
194. C₃×C₉	#478	semidirect
275. C₉	C₃×#478	semidirect
311. C₃	#10588531	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×D₃₆	2
C₃×#63432	2
C₃×C₉×C₄⋊C₈	2
C₃×#63380	2
C₃×#63378	2
C₃×#63377	2
C₃×#4847	3
#10588531	3

Subgroups — C3×#10588531 #13142025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588531 #13142025