Subgroups — C₃×#10588389 #13141502

Order 1728.

430 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3663	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×D₉	C₂	1	1	yes	maximal, characteristic
3	C₃×#63434	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63375	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4708	—	3	1	no	maximal
9	#10588389	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₃×(C₂×D₉)⋊C₄	C₄	1	2	yes	—
15	C₃×Dic₉⋊C₄	C₄	1	2	yes	—
16	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×#3663	C₄	1	4	yes	complemented
20	C₃×C₈×D₉	C₄	1	4	yes	complemented
21	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	1	no	—
22	C₂×C₃×D₃×C₈	—	3	1	no	—
23	C₃×#872	—	3	1	no	—
24	C₃²×#97	D₃	1	1	yes	characteristic
25	C₃×#815	—	3	1	no	—
26	C₃×C₈×Dic₃	—	3	1	no	—
27	C₃×C₄×C₃⋊C₈	—	3	1	no	—
28	C₂×#3663	C₂×C₃	1	1	yes	characteristic
29	C₂×C₈×D₉	C₂×C₃	1	1	yes	characteristic
30	#63434	C₂×C₃	1	1	yes	characteristic
31	C₉×#97	—	2	1	no	—
32	C₉×#97	C₂×C₃	1	1	yes	characteristic, complemented
33	#63375	C₂×C₃	1	1	yes	characteristic, complemented
34	C₈×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
35	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂²×C₃×D₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
40	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄×D₉	C₂×C₄	1	2	yes	—
42	C₃×C₄×D₉	C₂×C₄	1	2	yes	—
43	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
44	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
45	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	—
46	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	—
47	C₃×#405	—	9	1	no	—
48	#4708	—	3	1	no	—
49	C₂×C₃×C₄×D₃	—	3	1	no	—
50	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
52	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₃×(C₂×D₃)⋊C₄	—	3	2	no	—
54	C₃×Dic₃⋊C₄	—	3	2	no	—
55	C₃×C₄×Dic₃	—	3	1	no	—
56	C₂×C₃×C₃⋊C₈	—	3	1	no	—
57	C₂×C₃×C₃⋊C₈	—	3	1	no	—
58	C₃×ΓL(1,𝔽₂₅)	—	3	4	no	—
59	C₃×D₃×C₈	—	3	4	no	—
60	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₈×C₉	—	2	1	no	—
62	C₂×C₈×C₉	—	2	1	no	—
63	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
65	C₄²×C₉	—	2	1	no	—
66	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
67	(C₂×D₉)⋊C₄	C₃×C₄	1	2	yes	—
68	Dic₉⋊C₄	C₃×C₄	1	2	yes	—
69	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
72	#3663	C₃×C₄	1	4	yes	complemented
73	C₈×D₉	C₃×C₄	1	4	yes	complemented
74	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×D₉	—	2	2	no	—
76	C₂×C₃×D₉	C₄²	1	2	yes	—
77	C₃×C₄×C₉	—	2	2	no	—
78	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
80	C₃×Dic₉	—	2	2	no	—
81	C₃×Dic₉	C₄²	1	2	yes	—
82	C₂×C₃×M₁₆	—	9	1	no	—
83	C₂²×C₃×C₈	—	9	1	no	—
84	C₃×#117	—	9	1	no	—
85	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
86	C₂×D₃×C₈	—	3	1	no	—
87	#872	—	3	1	no	—
88	C₃×#97	—	2	1	no	—
89	C₃×#97	—	9	1	no	—
90	C₃×#97	D₉	1	1	yes	characteristic, complemented
91	C₃×#97	C₃×D₃	1	1	yes	characteristic
92	C₃×C₄×C₈	—	9	1	no	—
93	C₃×C₄×C₈	—	9	1	no	—
94	#815	—	3	1	no	—
95	C₈×Dic₃	—	3	1	no	—
96	C₄×C₃⋊C₈	—	3	1	no	—
97	C₂²×C₃×D₃	—	3	1	no	—
98	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
99	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₃×Dic₃	—	3	1	no	—
101	C₂×C₃×Dic₃	—	3	2	no	—
102	C₃×C₄×D₃	—	3	2	no	—
103	C₃×C₄×D₃	—	3	2	no	—
104	C₂²×D₉	C₂×C₃×C₄	1	1	yes	characteristic
105	C₃²×C₈	C₄×D₃	1	2	yes	—
106	C₃²×C₈	C₄×D₃	1	2	yes	—
107	C₃×C₃⋊C₈	—	3	2	no	—
108	C₃×C₃⋊C₈	—	3	2	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	1	no	—
111	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
112	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
113	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
114	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
115	C₄×D₉	C₂×C₃×C₄	1	2	yes	—
116	C₄×D₉	C₂×C₃×C₄	1	2	yes	—
117	C₈×C₉	—	2	2	no	—
118	C₈×C₉	—	2	2	no	—
119	C₈×C₉	C₂×C₃×C₄	1	2	yes	—
120	C₈×C₉	C₂×C₃×C₄	1	2	yes	—
121	C₉⋊C₈	C₂×C₃×C₄	1	2	yes	—
122	C₉⋊C₈	C₂×C₃×C₄	1	2	yes	—
123	#405	—	9	1	no	Sylow
124	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₃×C₉	#131	1	2	yes	—
126	C₃×D₉	—	2	2	no	—
127	C₂²×C₃×C₄	—	9	1	no	—
128	C₂×C₄×D₃	—	3	1	no	—
129	C₃×M₁₆	—	9	4	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	9	1	no	—
133	C₂×C₃×C₈	—	9	1	no	—
134	C₂×C₃×C₈	—	9	4	no	—
135	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	9	2	no	—
140	C₃×C₂²⋊C₄	—	9	2	no	—
141	C₃×C₄²	—	2	1	no	—
142	C₃×C₄²	—	9	1	no	—
143	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
144	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
145	(C₂×D₃)⋊C₄	—	3	2	no	—
146	Dic₃⋊C₄	—	3	2	no	—
147	C₄×Dic₃	—	3	1	no	—
148	C₂×C₃⋊C₈	—	3	1	no	—
149	C₂×C₃⋊C₈	—	3	1	no	—
150	ΓL(1,𝔽₂₅)	—	3	4	no	—
151	D₃×C₈	—	3	4	no	—
152	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₃×D₃	—	3	2	no	—
154	C₂×C₃×D₃	—	6	2	no	—
155	C₃²×C₄	—	2	2	no	—
156	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
157	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
158	C₃×Dic₃	—	3	2	no	—
159	C₃×Dic₃	—	6	2	no	—
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₂×D₉	—	2	2	no	—
163	C₂×D₉	C₃×C₄²	1	2	yes	—
164	C₄×C₉	—	2	2	no	—
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	—	2	1	no	—
167	C₄×C₉	—	4	2	no	—
168	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
170	Dic₉	—	2	2	no	—
171	Dic₉	C₃×C₄²	1	2	yes	—
172	C₂×M₁₆	—	9	1	no	—
173	C₂²×C₈	—	9	1	no	—
174	#117	—	9	1	no	—
175	#97	—	9	1	no	—
176	#97	C₃×D₉	1	1	yes	characteristic, complemented
177	C₄×C₈	—	9	1	no	—
178	C₄×C₈	—	9	1	no	—
179	C₃×C₉	#405	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	9	1	no	—
181	C₂²×D₃	—	3	1	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	9	1	no	—
185	C₂×C₃×C₄	—	9	2	no	—
186	C₂×C₃×C₄	—	9	2	no	—
187	C₂×C₃×C₄	—	9	2	no	—
188	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
189	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
190	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
191	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
192	C₂×Dic₃	—	3	1	no	—
193	C₂×Dic₃	—	3	2	no	—
194	C₄×D₃	—	3	2	no	—
195	C₄×D₃	—	3	2	no	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	9	2	no	—
199	C₃×C₈	—	9	2	no	—
200	C₃×C₈	C₄×D₉	1	2	yes	—
201	C₃×C₈	C₄×D₉	1	2	yes	—
202	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
203	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
204	C₃⋊C₈	—	3	2	no	—
205	C₃⋊C₈	—	3	2	no	—
206	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
207	C₂×C₃²	#907	1	2	yes	—
208	C₃×D₃	—	6	2	no	—
209	C₂×C₉	—	2	2	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
212	C₂×C₉	C₃×#131	1	2	yes	—
213	D₉	—	2	2	no	—
214	C₂²×C₄	—	9	1	no	—
215	M₁₆	—	9	4	no	—
216	C₂×C₈	—	9	1	no	—
217	C₂×C₈	—	9	1	no	—
218	C₂×C₈	—	9	4	no	—
219	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
220	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
221	C₄⋊C₄	—	9	2	no	—
222	C₂²⋊C₄	—	9	2	no	—
223	C₄²	—	9	1	no	—
224	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	9	2	no	—
227	C₂²×C₃	—	18	2	no	—
228	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
229	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
230	C₂×D₃	—	3	2	no	—
231	C₂×D₃	—	6	2	no	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	4	2	no	—
237	C₃×C₄	—	9	2	no	—
238	C₃×C₄	—	18	2	no	—
239	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
240	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
241	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
242	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
243	Dic₃	—	3	2	no	—
244	Dic₃	—	6	2	no	—
245	C₃²	#4708	1	1	yes	characteristic
246	C₉	—	2	1	no	—
247	C₉	C₃×#405	1	1	yes	characteristic, complemented
248	C₂³	—	9	1	no	—
249	C₂×C₄	—	9	1	no	—
250	C₂×C₄	—	9	2	no	—
251	C₂×C₄	—	9	2	no	—
252	C₂×C₄	—	9	2	no	—
253	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
254	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
255	C₈	—	9	2	no	—
256	C₈	—	9	2	no	—
257	C₈	C₃×C₄×D₉	1	2	yes	—
258	C₈	C₃×C₄×D₉	1	2	yes	—
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	18	2	no	—
262	C₂×C₃	C₄²×D₉	1	1	yes	characteristic
263	C₂×C₃	#63469	1	2	yes	—
264	C₂×C₃	C₃×C₄²×D₃	1	1	yes	characteristic
265	C₂×C₃	C₃×#907	1	2	yes	—
266	D₃	—	6	2	no	—
267	C₂²	—	9	2	no	—
268	C₂²	—	18	2	no	—
269	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	9	2	no	—
272	C₄	—	18	2	no	—
273	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
274	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
275	C₃	—	2	1	no	—
276	C₃	#10588389	1	1	yes	characteristic, direct product factor
277	C₃	C₃×#4708	1	1	yes	characteristic
278	C₂	—	18	2	no	—
279	C₂	C₃×C₄²×D₉	1	1	yes	characteristic
280	C₂	C₃×#63469	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator	C₂×C₉	C₂×C₃×C₄²
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#97	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃×D₉	C₂×C₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×#97	D₉
hypocenter	C₉	C₃×#405
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#97

Complements

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

N	H ≅ G/N	type
4. C₃×C₉×#97	C₂	semidirect
5. C₃×#63375	C₂	semidirect
6. C₃×C₈×Dic₉	C₂	semidirect
7. C₃×C₄×C₉⋊C₈	C₂	semidirect
9. #10588389	C₃	direct (N × H)
19. C₃×#3663	C₄	semidirect
20. C₃×C₈×D₉	C₄	semidirect
32. C₉×#97	C₂×C₃	semidirect
33. #63375	C₂×C₃	semidirect
34. C₈×Dic₉	C₂×C₃	semidirect
35. C₄×C₉⋊C₈	C₂×C₃	semidirect
72. #3663	C₃×C₄	semidirect
73. C₈×D₉	C₃×C₄	semidirect
90. C₃×#97	D₉	semidirect
176. #97	C₃×D₉	semidirect
179. C₃×C₉	#405	semidirect
247. C₉	C₃×#405	semidirect
276. C₃	#10588389	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×#3663	2
C₂×C₃×C₈×D₉	2
C₃×#63434	2
C₃×C₉×#97	2
C₃×#63375	2
C₃×C₈×Dic₉	2
C₃×C₄×C₉⋊C₈	2
C₃×#4708	3
#10588389	3

Subgroups — C3×#10588389 #13141502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588389 #13141502