Subgroups — C₃×#10588752 #13142974

Order 1728.

492 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3672	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63392	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5075	—	3	1	no	maximal
7	#10588752	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₄×C₉	—	2	1	no	—
9	C₂×C₃×D₄×C₉	—	2	1	no	—
10	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
11	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₃×#3672	—	2	4	no	—
14	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
15	C₃×Dic₉⋊C₄	—	2	1	no	—
16	C₃×C₄×Dic₉	—	2	1	no	—
17	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
18	C₃²×#127	D₃	1	1	yes	characteristic
19	C₂×C₃×#214	—	3	2	no	—
20	C₃×C₄×Dic₆	—	3	1	no	—
21	C₃×#832	—	3	2	no	—
22	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
23	C₉×#127	—	2	1	no	—
24	C₉×#127	C₂×C₃	1	1	yes	characteristic
25	C₂×#3672	C₂×C₃	1	2	yes	complemented
26	C₄×Dic₁₈	C₂×C₃	1	1	yes	characteristic, complemented
27	#63392	C₂×C₃	1	2	yes	complemented
28	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃×C₉	—	2	2	no	—
30	C₃×D₄×C₉	—	2	2	no	—
31	C₃×D₄×C₉	—	2	2	no	—
32	C₃×D₄×C₉	—	4	2	no	—
33	C₃×D₄×C₉	—	4	2	no	—
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂×C₃×Dic₉	—	2	1	no	—
38	C₂×C₃×Dic₉	—	2	1	no	—
39	C₃×Dic₁₈	—	2	1	no	—
40	C₃×Dic₁₈	D₄	1	2	yes	complemented
41	C₃×C₉⋊C₈	—	2	2	no	—
42	C₃×#460	—	9	1	no	—
43	#5075	—	3	1	no	—
44	C₂×C₃²×D₄	—	2	1	no	—
45	C₂×C₃²×D₄	—	2	1	no	—
46	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
47	C₂×C₃×Dic₆	—	3	1	no	—
48	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₃×#214	—	6	4	no	—
50	C₃×C₄⋊Dic₃	—	3	1	no	—
51	C₃×Dic₃⋊C₄	—	6	1	no	—
52	C₃×C₄×Dic₃	—	6	1	no	—
53	C₂×C₃×C₃⋊C₈	—	3	2	no	—
54	C₂×D₄×C₉	—	2	2	no	—
55	C₂×D₄×C₉	—	2	2	no	—
56	C₂×D₄×C₉	—	2	2	no	—
57	C₂×D₄×C₉	—	2	1	no	—
58	C₂×D₄×C₉	—	2	1	no	—
59	C₂×D₄×C₉	C₂²×C₃	1	2	yes	—
60	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
61	C₄²×C₉	—	2	1	no	—
62	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
63	#3672	—	2	4	no	—
64	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
65	Dic₉⋊C₄	—	2	1	no	—
66	C₄×Dic₉	—	2	1	no	—
67	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
68	C₂²×C₃×C₉	—	4	2	no	—
69	C₂²×C₃×C₉	—	4	2	no	—
70	C₂²×C₃×C₉	—	4	2	no	—
71	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄×C₉	—	2	1	no	—
73	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
74	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄×C₉	#41	1	1	yes	characteristic
76	C₃×Dic₉	—	2	2	no	—
77	C₃×Dic₉	—	4	1	no	—
78	C₂×C₃×SD₁₆	—	9	2	no	—
79	C₃×#127	—	2	1	no	—
80	C₃×#127	D₉	1	1	yes	characteristic
81	C₃×#127	C₃×D₃	1	1	yes	characteristic
82	C₃×C₄×Q₈	—	9	1	no	—
83	C₂×#214	—	3	2	no	—
84	C₄×Dic₆	—	3	1	no	—
85	C₃×C₄⋊C₈	—	9	1	no	—
86	C₃×#102	—	9	2	no	—
87	#832	—	3	2	no	—
88	C₄⋊(C₃⋊C₈)	—	3	1	no	—
89	C₂³×C₃²	—	2	2	no	—
90	C₃²×D₄	—	2	2	no	—
91	C₃²×D₄	—	2	2	no	—
92	C₃²×D₄	—	4	2	no	—
93	C₃²×D₄	—	4	2	no	—
94	C₂×C₃²×C₄	#67	1	1	yes	characteristic
95	C₂×C₃²×C₄	#67	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃×Dic₃	—	6	1	no	—
98	C₂×C₃×Dic₃	—	6	1	no	—
99	C₃×Dic₆	—	3	2	no	—
100	C₃×Dic₆	—	6	1	no	—
101	C₂³×C₉	—	2	4	no	—
102	C₂³×C₉	—	2	2	no	—
103	C₃×C₃⋊C₈	—	6	2	no	—
104	D₄×C₉	—	2	2	no	—
105	D₄×C₉	—	2	2	no	—
106	D₄×C₉	—	4	2	no	—
107	D₄×C₉	—	4	2	no	—
108	D₄×C₉	—	4	4	no	—
109	D₄×C₉	—	4	2	no	—
110	D₄×C₉	—	4	4	no	—
111	D₄×C₉	—	4	2	no	—
112	C₂×C₄×C₉	—	2	1	no	—
113	C₂×C₄×C₉	—	2	1	no	—
114	C₂×C₄×C₉	—	2	1	no	—
115	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
116	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
117	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
118	C₂×Dic₉	—	2	1	no	—
119	C₂×Dic₉	—	2	1	no	—
120	Dic₁₈	—	2	1	no	—
121	Dic₁₈	C₃×D₄	1	2	yes	complemented
122	C₉⋊C₈	—	2	2	no	—
123	#460	—	9	1	no	Sylow
124	C₂×C₃×C₉	—	8	2	no	—
125	C₂×C₃×C₉	#121	1	1	yes	characteristic
126	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₃×Q₈	—	9	1	no	—
129	C₂×C₃×D₄	—	2	2	no	—
130	C₂×C₃×D₄	—	2	2	no	—
131	C₂×C₃×D₄	—	2	1	no	—
132	C₂×C₃×D₄	—	2	1	no	—
133	C₂×C₃×D₄	—	2	1	no	—
134	C₂×C₃×D₄	—	2	1	no	—
135	C₂×C₃×D₄	—	2	2	no	—
136	C₂×C₃×D₄	C₂×D₉	1	2	yes	—
137	C₂×C₃×D₄	C₂×C₃×D₃	1	2	yes	—
138	C₂×Dic₆	—	3	1	no	—
139	C₃×SD₁₆	—	18	4	no	—
140	C₂×C₃×C₈	—	9	2	no	—
141	C₃×C₄⋊C₄	—	9	1	no	—
142	C₃×C₄⋊C₄	—	18	1	no	—
143	C₃×C₄²	—	2	1	no	—
144	C₃×C₄²	—	18	1	no	—
145	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
146	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
147	#214	—	6	4	no	—
148	C₄⋊Dic₃	—	3	1	no	—
149	Dic₃⋊C₄	—	6	1	no	—
150	C₄×Dic₃	—	6	1	no	—
151	C₂×C₃⋊C₈	—	3	2	no	—
152	C₂²×C₃²	—	4	2	no	—
153	C₂²×C₃²	—	4	2	no	—
154	C₂²×C₃²	—	4	2	no	—
155	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
156	C₃²×C₄	—	2	1	no	—
157	C₃²×C₄	#214	1	2	yes	—
158	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
159	C₃²×C₄	#237	1	1	yes	characteristic
160	C₃×Dic₃	—	6	2	no	—
161	C₃×Dic₃	—	12	1	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	—	4	4	no	—
164	C₂²×C₉	—	4	4	no	—
165	C₂²×C₉	—	4	4	no	—
166	C₂²×C₉	—	4	2	no	—
167	C₂²×C₉	—	4	2	no	—
168	C₂²×C₉	—	4	2	no	—
169	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
170	C₄×C₉	—	2	2	no	—
171	C₄×C₉	—	2	1	no	—
172	C₄×C₉	—	2	2	no	—
173	C₄×C₉	—	2	1	no	—
174	C₄×C₉	—	2	1	no	—
175	C₄×C₉	C₃×SD₁₆	1	2	yes	complemented
176	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
177	C₄×C₉	C₃×#41	1	1	yes	characteristic
178	Dic₉	—	2	2	no	—
179	Dic₉	—	4	1	no	—
180	C₂×SD₁₆	—	9	2	no	—
181	#127	C₃×D₉	1	1	yes	characteristic
182	C₄×Q₈	—	9	1	no	—
183	C₄⋊C₈	—	9	1	no	—
184	#102	—	9	2	no	—
185	C₃×C₉	#460	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₃	—	2	4	no	—
187	C₂³×C₃	—	2	2	no	—
188	C₂³×C₃	—	2	2	no	—
189	C₃×Q₈	—	9	2	no	—
190	C₃×Q₈	—	18	1	no	—
191	C₃×D₄	—	2	2	no	—
192	C₃×D₄	—	2	2	no	—
193	C₃×D₄	—	2	2	no	—
194	C₃×D₄	—	2	2	no	—
195	C₃×D₄	—	4	2	no	—
196	C₃×D₄	—	4	2	no	—
197	C₃×D₄	—	4	2	no	—
198	C₃×D₄	—	4	4	no	—
199	C₃×D₄	—	4	2	no	—
200	C₃×D₄	—	4	4	no	—
201	C₃×D₄	—	4	2	no	—
202	C₃×D₄	—	4	2	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	18	1	no	—
207	C₂×C₃×C₄	—	18	1	no	—
208	C₂×C₃×C₄	#611	1	1	yes	characteristic
209	C₂×C₃×C₄	#611	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
211	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
212	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
214	C₂×Dic₃	—	6	1	no	—
215	C₂×Dic₃	—	6	1	no	—
216	Dic₆	—	3	2	no	—
217	Dic₆	—	6	1	no	—
218	C₃×C₈	—	18	2	no	—
219	C₃⋊C₈	—	6	2	no	—
220	C₂×C₃²	—	8	2	no	—
221	C₂×C₃²	#932	1	1	yes	characteristic
222	C₂×C₃²	C₂×#214	1	1	yes	characteristic
223	C₂×C₃²	#938	1	1	yes	characteristic
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	—	2	1	no	—
227	C₂×C₉	—	8	4	no	—
228	C₂×C₉	—	8	2	no	—
229	C₂×C₉	C₃×#121	1	1	yes	characteristic
230	C₂×C₉	C₂×C₃×SD₁₆	1	1	yes	characteristic
231	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₂×Q₈	—	9	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	C₂×C₃×D₉	1	2	yes	—
236	SD₁₆	—	18	4	no	—
237	C₂×C₈	—	9	2	no	—
238	C₄⋊C₄	—	9	1	no	—
239	C₄⋊C₄	—	18	1	no	—
240	C₄²	—	18	1	no	—
241	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	4	4	no	—
244	C₂²×C₃	—	4	4	no	—
245	C₂²×C₃	—	4	4	no	—
246	C₂²×C₃	—	4	2	no	—
247	C₂²×C₃	—	4	2	no	—
248	C₂²×C₃	—	4	2	no	—
249	C₂²×C₃	—	4	2	no	—
250	C₂²×C₃	—	4	2	no	—
251	C₂²×C₃	—	4	2	no	—
252	C₂²×C₃	C₂×#611	1	1	yes	characteristic
253	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	18	2	no	—
261	C₃×C₄	—	36	1	no	—
262	C₃×C₄	#3672	1	2	yes	complemented
263	C₃×C₄	D₄×D₉	1	1	yes	characteristic
264	C₃×C₄	#3699	1	1	yes	characteristic
265	C₃×C₄	C₃×#214	1	2	yes	—
266	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
267	C₃×C₄	C₃×#237	1	1	yes	characteristic
268	Dic₃	—	6	2	no	—
269	Dic₃	—	12	1	no	—
270	C₃²	#5075	1	1	yes	characteristic
271	C₉	—	2	1	no	—
272	C₉	C₃×#460	1	1	yes	characteristic, complemented
273	C₂³	—	2	2	no	—
274	Q₈	—	9	2	no	—
275	Q₈	—	18	1	no	—
276	D₄	—	2	2	no	—
277	D₄	—	2	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	—	18	1	no	—
281	C₂×C₄	—	18	1	no	—
282	C₂×C₄	C₃×#611	1	1	yes	characteristic
283	C₂×C₄	C₃×#611	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
285	C₈	—	18	2	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	8	4	no	—
290	C₂×C₃	—	8	2	no	—
291	C₂×C₃	—	8	2	no	—
292	C₂×C₃	C₂×#3672	1	1	yes	characteristic
293	C₂×C₃	#63495	1	1	yes	characteristic
294	C₂×C₃	#63500	1	1	yes	characteristic
295	C₂×C₃	C₃×#932	1	1	yes	characteristic
296	C₂×C₃	C₂×C₃×#214	1	1	yes	characteristic
297	C₂×C₃	C₃×#938	1	1	yes	characteristic
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₂×C₃×#611	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	18	2	no	—
304	C₄	—	36	1	no	—
305	C₄	C₃×#3672	1	2	yes	complemented
306	C₄	C₃×D₄×D₉	1	1	yes	characteristic
307	C₄	C₃×#3699	1	1	yes	characteristic
308	C₃	—	2	1	no	—
309	C₃	#10588752	1	1	yes	characteristic, direct product factor
310	C₃	C₃×#5075	1	1	yes	characteristic
311	C₂	—	8	2	no	—
312	C₂	C₂×C₃×#3672	1	1	yes	characteristic
313	C₂	C₃×#63495	1	1	yes	characteristic
314	C₂	C₃×#63500	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#127	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₃²×#127	D₃
hypercenter	C₃×#127	D₉
hypocenter	C₉	C₃×#460
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3672	C₂	semidirect
3. C₃×C₄×Dic₁₈	C₂	semidirect
4. C₃×#63392	C₂	semidirect
5. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
7. #10588752	C₃	direct (N × H)
25. C₂×#3672	C₂×C₃	semidirect
26. C₄×Dic₁₈	C₂×C₃	semidirect
27. #63392	C₂×C₃	semidirect
28. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
40. C₃×Dic₁₈	D₄	semidirect
73. C₃×C₄×C₉	SD₁₆	semidirect
121. Dic₁₈	C₃×D₄	semidirect
175. C₄×C₉	C₃×SD₁₆	semidirect
185. C₃×C₉	#460	semidirect
262. C₃×C₄	#3672	semidirect
272. C₉	C₃×#460	semidirect
305. C₄	C₃×#3672	semidirect
309. C₃	#10588752	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×C₉×#127	2
C₂×C₃×#3672	2
C₃×C₄×Dic₁₈	2
C₃×#63392	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#5075	3
#10588752	3

Subgroups — C3×#10588752 #13142974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588752 #13142974