Subgroups — C₃×#10588560 #13142188

Order 1728.

378 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×D₉	C₂	1	1	yes	maximal, characteristic
2	C₃×#63454	C₂	1	2	yes	maximal
3	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63378	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63367	C₂	1	2	yes	maximal, complemented
6	C₃×#4880	—	3	1	no	maximal
7	#10588560	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
11	C₃×(C₂×D₉)⋊C₄	—	2	2	no	—
12	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₃×C₄×Dic₉	—	2	2	no	—
14	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₈×D₉	C₄	1	4	yes	complemented
16	C₂×C₃×D₃×C₈	—	3	1	no	—
17	C₃×#892	—	3	2	no	—
18	C₃²×#107	D₃	1	1	yes	characteristic
19	C₃×#818	—	3	1	no	—
20	C₃×#808	—	3	2	no	—
21	C₂×C₈×D₉	C₂×C₃	1	1	yes	characteristic
22	#63454	C₂×C₃	1	2	yes	—
23	C₉×#107	—	2	1	no	—
24	C₉×#107	C₂×C₃	1	1	yes	characteristic, complemented
25	#63378	C₂×C₃	1	1	yes	characteristic, complemented
26	#63367	C₂×C₃	1	2	yes	complemented
27	C₂²×C₃×D₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₃×Dic₉	—	2	2	no	—
31	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
32	C₃×C₄×D₉	C₂×C₄	1	2	yes	—
33	C₃×C₄×D₉	Q₈	1	2	yes	—
34	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
35	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₃×#427	—	9	1	no	—
37	#4880	—	3	1	no	—
38	C₂×C₃×C₄×D₃	—	3	1	no	—
39	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
40	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
41	C₃×(C₂×D₃)⋊C₄	—	6	2	no	—
42	C₃×C₄⋊Dic₃	—	3	2	no	—
43	C₃×C₄×Dic₃	—	6	2	no	—
44	C₂×C₃×C₃⋊C₈	—	3	1	no	—
45	C₃×D₃×C₈	—	3	4	no	—
46	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₈×C₉	—	2	1	no	—
48	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
49	C₉×C₄⋊C₄	—	2	2	no	—
50	C₉×C₄⋊C₄	C₂²×C₃	1	2	yes	—
51	(C₂×D₉)⋊C₄	—	2	2	no	—
52	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
53	C₄×Dic₉	—	2	2	no	—
54	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
55	C₈×D₉	C₃×C₄	1	4	yes	complemented
56	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×D₉	—	2	1	no	—
58	C₂×C₃×D₉	—	2	1	no	—
59	C₂×C₃×D₉	C₄⋊C₄	1	2	yes	—
60	C₃×C₄×C₉	—	4	2	no	—
61	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
63	C₃×Dic₉	—	4	2	no	—
64	C₃×Dic₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₃×C₈	—	9	1	no	—
66	C₃×#117	—	9	2	no	—
67	C₂×D₃×C₈	—	3	1	no	—
68	#892	—	3	2	no	—
69	C₃×#107	—	2	1	no	—
70	C₃×#107	—	9	1	no	—
71	C₃×#107	D₉	1	1	yes	characteristic, complemented
72	C₃×#107	C₃×D₃	1	1	yes	characteristic
73	C₃×#106	—	9	2	no	—
74	#818	—	3	1	no	—
75	#808	—	3	2	no	—
76	C₂²×C₃×D₃	—	3	1	no	—
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
79	C₂×C₃×Dic₃	—	3	1	no	—
80	C₂×C₃×Dic₃	—	6	2	no	—
81	C₃×C₄×D₃	—	3	2	no	—
82	C₃×C₄×D₃	—	3	2	no	—
83	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
84	C₃²×C₈	C₄×D₃	1	2	yes	—
85	C₃×C₃⋊C₈	—	3	2	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	4	2	no	—
89	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
90	C₂×Dic₉	—	2	2	no	—
91	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
92	C₄×D₉	C₂×C₃×C₄	1	2	yes	—
93	C₄×D₉	C₃×Q₈	1	2	yes	—
94	C₈×C₉	—	2	2	no	—
95	C₈×C₉	C₂×C₃×C₄	1	2	yes	—
96	C₉⋊C₈	C₂×C₃×C₄	1	2	yes	—
97	#427	—	9	1	no	Sylow
98	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₃×C₉	#135	1	1	yes	characteristic
100	C₂×C₃×C₉	#135	1	1	yes	characteristic
101	C₃×D₉	—	2	2	no	—
102	C₂²×C₃×C₄	—	9	1	no	—
103	C₂×C₄×D₃	—	3	1	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	9	1	no	—
106	C₂×C₃×C₈	—	9	4	no	—
107	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	—	9	2	no	—
111	C₃×C₄⋊C₄	C₂×D₉	1	2	yes	—
112	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
113	C₃×C₂²⋊C₄	—	18	2	no	—
114	C₃×C₄²	—	18	2	no	—
115	(C₂×D₃)⋊C₄	—	6	2	no	—
116	C₄⋊Dic₃	—	3	2	no	—
117	C₄×Dic₃	—	6	2	no	—
118	C₂×C₃⋊C₈	—	3	1	no	—
119	D₃×C₈	—	3	4	no	—
120	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
121	C₂×C₃×D₃	—	3	2	no	—
122	C₂×C₃×D₃	—	6	1	no	—
123	C₂×C₃×D₃	—	6	1	no	—
124	C₃²×C₄	—	4	2	no	—
125	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
126	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
127	C₃×Dic₃	—	3	2	no	—
128	C₃×Dic₃	—	12	2	no	—
129	C₂²×C₉	—	2	1	no	—
130	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂×D₉	—	2	1	no	—
132	C₂×D₉	—	2	1	no	—
133	C₂×D₉	C₃×C₄⋊C₄	1	2	yes	—
134	C₄×C₉	—	2	1	no	—
135	C₄×C₉	—	2	1	no	—
136	C₄×C₉	—	4	2	no	—
137	C₄×C₉	—	8	2	no	—
138	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
139	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
140	Dic₉	—	4	2	no	—
141	Dic₉	C₃×C₄⋊C₄	1	2	yes	—
142	C₂²×C₈	—	9	1	no	—
143	#117	—	9	2	no	—
144	#107	—	9	1	no	—
145	#107	C₃×D₉	1	1	yes	characteristic, complemented
146	#106	—	9	2	no	—
147	C₃×C₉	#427	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₃	—	9	1	no	—
149	C₂²×D₃	—	3	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	9	1	no	—
155	C₂×C₃×C₄	—	9	2	no	—
156	C₂×C₃×C₄	—	9	2	no	—
157	C₂×C₃×C₄	—	18	2	no	—
158	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
160	C₂×Dic₃	—	3	1	no	—
161	C₂×Dic₃	—	6	2	no	—
162	C₄×D₃	—	3	2	no	—
163	C₄×D₃	—	3	2	no	—
164	C₃×C₈	—	2	2	no	—
165	C₃×C₈	—	9	2	no	—
166	C₃×C₈	C₄×D₉	1	2	yes	—
167	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
168	C₃⋊C₈	—	3	2	no	—
169	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₃²	#912	1	1	yes	characteristic
171	C₂×C₃²	#919	1	1	yes	characteristic
172	C₃×D₃	—	6	2	no	—
173	C₂×C₉	—	2	1	no	—
174	C₂×C₉	—	2	1	no	—
175	C₂×C₉	—	2	1	no	—
176	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₉	C₃×#135	1	1	yes	characteristic
178	C₂×C₉	C₃×#135	1	1	yes	characteristic
179	D₉	—	2	2	no	—
180	C₂²×C₄	—	9	1	no	—
181	C₂×C₈	—	9	1	no	—
182	C₂×C₈	—	9	4	no	—
183	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
184	C₄⋊C₄	—	9	2	no	—
185	C₄⋊C₄	C₂×C₃×D₉	1	2	yes	—
186	C₂²⋊C₄	—	18	2	no	—
187	C₄²	—	18	2	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	9	2	no	—
190	C₂²×C₃	—	18	1	no	—
191	C₂²×C₃	—	18	1	no	—
192	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
193	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
194	C₂×D₃	—	3	2	no	—
195	C₂×D₃	—	6	1	no	—
196	C₂×D₃	—	6	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	4	2	no	—
200	C₃×C₄	—	4	2	no	—
201	C₃×C₄	—	8	2	no	—
202	C₃×C₄	—	9	2	no	—
203	C₃×C₄	—	36	2	no	—
204	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
205	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
206	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
207	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
208	Dic₃	—	3	2	no	—
209	Dic₃	—	12	2	no	—
210	C₃²	#4880	1	1	yes	characteristic
211	C₉	—	2	1	no	—
212	C₉	C₃×#427	1	1	yes	characteristic, complemented
213	C₂³	—	9	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	9	1	no	—
216	C₂×C₄	—	9	2	no	—
217	C₂×C₄	—	9	2	no	—
218	C₂×C₄	—	18	2	no	—
219	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
220	C₈	—	9	2	no	—
221	C₈	C₃×C₄×D₉	1	2	yes	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	18	2	no	—
226	C₂×C₃	C₄⋊C₄×D₉	1	1	yes	characteristic
227	C₂×C₃	#63474	1	1	yes	characteristic
228	C₂×C₃	#63481	1	1	yes	characteristic
229	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
230	C₂×C₃	C₃×#912	1	1	yes	characteristic
231	C₂×C₃	C₃×#919	1	1	yes	characteristic
232	D₃	—	6	2	no	—
233	C₂²	—	9	2	no	—
234	C₂²	—	18	1	no	—
235	C₂²	—	18	1	no	—
236	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
237	C₄	—	4	2	no	—
238	C₄	—	9	2	no	—
239	C₄	—	36	2	no	—
240	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
241	C₄	C₃×Q₈×D₉	1	1	yes	characteristic
242	C₃	—	2	1	no	—
243	C₃	#10588560	1	1	yes	characteristic, direct product factor
244	C₃	C₃×#4880	1	1	yes	characteristic
245	C₂	—	18	2	no	—
246	C₂	C₃×C₄⋊C₄×D₉	1	1	yes	characteristic
247	C₂	C₃×#63474	1	1	yes	characteristic
248	C₂	C₃×#63481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×D₉
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#107	C₂
socle	C₂²×C₃²	D₃×D₄
prime radical	C₂²×C₃×D₉	D₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×#107	D₉
hypocenter	C₉	C₃×#427
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#107

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×#107	C₂	semidirect
4. C₃×#63378	C₂	semidirect
5. C₃×#63367	C₂	semidirect
7. #10588560	C₃	direct (N × H)
15. C₃×C₈×D₉	C₄	semidirect
24. C₉×#107	C₂×C₃	semidirect
25. #63378	C₂×C₃	semidirect
26. #63367	C₂×C₃	semidirect
55. C₈×D₉	C₃×C₄	semidirect
71. C₃×#107	D₉	semidirect
145. #107	C₃×D₉	semidirect
147. C₃×C₉	#427	semidirect
212. C₉	C₃×#427	semidirect
243. C₃	#10588560	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×C₈×D₉	2
C₃×#63454	2
C₃×C₉×#107	2
C₃×#63378	2
C₃×#63367	2
C₃×#4880	3
#10588560	3

Subgroups — C3×#10588560 #13142188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588560 #13142188