Subgroups — C₃×#10588675 #13142664

Order 1728.

466 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#63362	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₉⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₃×#4997	—	3	1	no	maximal
7	#10588675	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
10	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	2	yes	—
12	C₃×#3667	C₄	1	4	yes	complemented
13	C₂×C₃×C₉⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
17	C₃²×#117	D₃	1	1	yes	characteristic
18	C₂×C₃×#208	—	3	1	no	—
19	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
20	C₃×#803	—	3	2	no	—
21	C₃×C₄×C₃⋊C₈	—	3	2	no	—
22	C₉×#117	—	2	1	no	—
23	C₉×#117	C₂×C₃	1	1	yes	characteristic
24	C₂×#3667	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic
26	#63362	C₂×C₃	1	2	yes	complemented
27	C₄×C₉⋊C₈	C₂×C₃	1	2	yes	complemented
28	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	—
35	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₃×C₉⋊C₈	C₂×C₄	1	4	yes	—
37	C₃×#405	—	9	1	no	—
38	#4997	—	3	1	no	—
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₃²×C₄⋊C₄	Dic₃	1	2	yes	—
41	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
42	C₃²×C₄²	C₂×D₃	1	2	yes	—
43	C₃×#208	—	3	4	no	—
44	C₂×C₃×C₃⋊C₈	—	3	4	no	—
45	C₂×C₃×C₃⋊C₈	—	3	1	no	—
46	C₂×C₃×C₃⋊C₈	—	3	1	no	—
47	C₂×C₃×C₃⋊C₈	—	3	2	no	—
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
50	C₉×C₄⋊C₄	—	2	2	no	—
51	C₉×C₄⋊C₄	C₃×C₄	1	2	yes	—
52	C₉×C₂²⋊C₄	—	2	2	no	—
53	C₉×C₂²⋊C₄	C₃×C₄	1	2	yes	—
54	C₄²×C₉	—	2	2	no	—
55	C₄²×C₉	C₂²×C₃	1	2	yes	—
56	#3667	C₃×C₄	1	4	yes	complemented
57	C₂×C₉⋊C₈	C₃×C₄	1	4	yes	complemented
58	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
60	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₃×C₉	—	2	2	no	—
62	C₂²×C₃×C₉	C₄²	1	2	yes	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	—	2	4	no	—
65	C₃×C₄×C₉	C₄²	1	2	yes	—
66	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×M₁₆	—	9	1	no	—
69	C₂²×C₃×C₈	—	9	1	no	—
70	C₃×#117	—	2	1	no	—
71	C₃×#117	D₉	1	1	yes	characteristic
72	C₃×#117	C₃×D₃	1	1	yes	characteristic
73	C₂×#208	—	3	1	no	—
74	C₂²×C₃⋊C₈	—	3	1	no	—
75	C₃×#97	—	9	2	no	—
76	C₃×C₄×C₈	—	9	2	no	—
77	#803	—	3	2	no	—
78	C₄×C₃⋊C₈	—	3	2	no	—
79	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
81	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
82	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
83	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
87	C₃×C₃⋊C₈	—	3	2	no	—
88	C₃×C₃⋊C₈	—	3	2	no	—
89	C₃×C₃⋊C₈	—	3	4	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	4	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
96	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
97	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
98	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
100	C₉⋊C₈	C₂×C₃×C₄	1	2	yes	—
101	C₉⋊C₈	C₂×C₃×C₄	1	2	yes	—
102	C₉⋊C₈	C₂×C₃×C₄	1	4	yes	—
103	#405	—	9	1	no	Sylow
104	C₂×C₃×C₉	—	2	2	no	—
105	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₃×C₉	#131	1	2	yes	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
110	C₃×M₁₆	—	9	4	no	—
111	C₂×C₃×C₈	—	9	4	no	—
112	C₂×C₃×C₈	—	9	1	no	—
113	C₂×C₃×C₈	—	9	2	no	—
114	C₂×C₃×C₈	—	9	1	no	—
115	C₃×C₄⋊C₄	—	2	2	no	—
116	C₃×C₄⋊C₄	Dic₉	1	2	yes	—
117	C₃×C₄⋊C₄	C₃×Dic₃	1	2	yes	—
118	C₃×C₂²⋊C₄	—	2	2	no	—
119	C₃×C₂²⋊C₄	Dic₉	1	2	yes	—
120	C₃×C₂²⋊C₄	C₃×Dic₃	1	2	yes	—
121	C₃×C₄²	—	2	2	no	—
122	C₃×C₄²	C₂×D₉	1	2	yes	—
123	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
124	#208	—	3	4	no	—
125	C₂×C₃⋊C₈	—	3	4	no	—
126	C₂×C₃⋊C₈	—	3	1	no	—
127	C₂×C₃⋊C₈	—	3	2	no	—
128	C₂×C₃⋊C₈	—	3	1	no	—
129	C₂²×C₃²	—	2	2	no	—
130	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
131	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
132	C₃²×C₄	—	2	4	no	—
133	C₃²×C₄	C₄×Dic₃	1	2	yes	—
134	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂²×C₉	—	2	2	no	—
137	C₂²×C₉	—	2	2	no	—
138	C₂²×C₉	—	2	1	no	—
139	C₂²×C₉	—	4	2	no	—
140	C₂²×C₉	C₃×C₄²	1	2	yes	—
141	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₄×C₉	—	2	4	no	—
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	4	4	no	—
147	C₄×C₉	C₃×C₄²	1	2	yes	—
148	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
150	C₂×M₁₆	—	9	1	no	—
151	C₂²×C₈	—	9	1	no	—
152	#117	C₃×D₉	1	1	yes	characteristic
153	#97	—	9	2	no	—
154	C₄×C₈	—	9	2	no	—
155	C₃×C₉	#405	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	2	1	no	—
157	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
158	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
165	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
166	C₂×C₃×C₄	C₂×Dic₉	1	4	yes	—
167	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
170	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
172	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
174	C₃×C₈	—	9	2	no	—
175	C₃×C₈	—	9	2	no	—
176	C₃×C₈	—	9	4	no	—
177	C₃⋊C₈	—	3	2	no	—
178	C₃⋊C₈	—	3	2	no	—
179	C₃⋊C₈	—	3	4	no	—
180	C₂×C₃²	—	2	2	no	—
181	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
182	C₂×C₃²	#948	1	2	yes	—
183	C₂×C₉	—	2	2	no	—
184	C₂×C₉	—	2	1	no	—
185	C₂×C₉	—	2	2	no	—
186	C₂×C₉	—	4	2	no	—
187	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
188	C₂×C₉	C₃×#131	1	2	yes	—
189	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
190	M₁₆	—	9	4	no	—
191	C₂×C₈	—	9	4	no	—
192	C₂×C₈	—	9	1	no	—
193	C₂×C₈	—	9	2	no	—
194	C₂×C₈	—	9	1	no	—
195	C₄⋊C₄	C₃×Dic₉	1	2	yes	—
196	C₂²⋊C₄	C₃×Dic₉	1	2	yes	—
197	C₄²	C₂×C₃×D₉	1	2	yes	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	4	2	no	—
203	C₂²×C₃	C₄×Dic₉	1	2	yes	—
204	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
205	C₂²×C₃	C₃×C₄×Dic₃	1	2	yes	—
206	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	4	4	no	—
213	C₃×C₄	C₄×Dic₉	1	2	yes	—
214	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
215	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
216	C₃×C₄	C₃×C₄×Dic₃	1	2	yes	—
217	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
218	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
219	C₃²	#4997	1	1	yes	characteristic
220	C₉	—	2	1	no	—
221	C₉	C₃×#405	1	1	yes	characteristic, complemented
222	C₂³	C₂×C₃×Dic₉	1	1	yes	characteristic
223	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
224	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
225	C₂×C₄	C₂×C₃×Dic₉	1	4	yes	—
226	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
228	C₈	—	9	2	no	—
229	C₈	—	9	2	no	—
230	C₈	—	9	4	no	—
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	—	2	2	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	2	no	—
235	C₂×C₃	—	4	2	no	—
236	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
237	C₂×C₃	#63510	1	2	yes	—
238	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
239	C₂×C₃	C₃×#948	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	C₃×C₄×Dic₉	1	2	yes	—
242	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	C₃×C₄×Dic₉	1	2	yes	—
245	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
246	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
247	C₃	—	2	1	no	—
248	C₃	#10588675	1	1	yes	characteristic, direct product factor
249	C₃	C₃×#4997	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	C₂×C₃×C₄×Dic₉	1	1	yes	characteristic
252	C₂	C₃×#63510	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×#117	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×#117	D₉
hypocenter	C₉	C₃×#405
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
4. C₃×#63362	C₂	semidirect
5. C₃×C₄×C₉⋊C₈	C₂	semidirect
7. #10588675	C₃	direct (N × H)
12. C₃×#3667	C₄	semidirect
13. C₂×C₃×C₉⋊C₈	C₄	semidirect
26. #63362	C₂×C₃	semidirect
27. C₄×C₉⋊C₈	C₂×C₃	semidirect
56. #3667	C₃×C₄	semidirect
57. C₂×C₉⋊C₈	C₃×C₄	semidirect
155. C₃×C₉	#405	semidirect
221. C₉	C₃×#405	semidirect
248. C₃	#10588675	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×#117	2
C₂×C₃×#3667	2
C₂²×C₃×C₉⋊C₈	2
C₃×#63362	2
C₃×C₄×C₉⋊C₈	2
C₃×#4997	3
#10588675	3

Subgroups — C3×#10588675 #13142664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588675 #13142664