Subgroups — C₃×#10588209 #13135572

Order 1728.

346 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	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	C₃×#4533	—	3	1	no	maximal
5	#10588209	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₉×C₄⋊C₄	—	2	1	no	—
8	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
9	C₃×C₄²×C₉	—	2	1	no	—
10	C₃×#3667	—	2	1	no	—
11	C₃×#3667	—	2	1	no	—
12	C₃×#3667	—	2	1	no	—
13	C₃×#3667	C₄	1	2	yes	complemented
14	C₂×C₃×C₉⋊C₈	—	2	1	no	—
15	C₂×C₃×C₉⋊C₈	C₄	1	2	yes	complemented
16	C₃²×#117	D₃	1	1	yes	characteristic
17	C₂×C₃×#208	—	3	1	no	—
18	C₂×C₃×#208	—	3	1	no	—
19	C₉×#117	—	2	1	no	—
20	C₉×#117	C₂×C₃	1	1	yes	characteristic
21	C₂×#3667	C₂×C₃	1	1	yes	characteristic
22	C₂×#3667	C₂×C₃	1	1	yes	characteristic
23	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	—	2	2	no	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	Q₈	1	1	yes	characteristic
31	C₃×C₉⋊C₈	—	2	2	no	—
32	C₃×C₉⋊C₈	—	2	2	no	—
33	C₃×#344	—	9	1	no	—
34	#4533	—	3	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₃²×C₄⋊C₄	—	2	1	no	—
37	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
38	C₃²×C₄²	—	2	1	no	—
39	C₃×#208	—	3	2	no	—
40	C₃×#208	—	6	1	no	—
41	C₃×#208	—	6	1	no	—
42	C₃×#208	—	6	1	no	—
43	C₂×C₃×C₃⋊C₈	—	3	2	no	—
44	C₂×C₃×C₃⋊C₈	—	6	1	no	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	—	2	1	no	—
49	C₉×C₂²⋊C₄	—	2	2	no	—
50	C₉×C₂²⋊C₄	C₃×C₄	1	2	yes	—
51	C₄²×C₉	—	2	2	no	—
52	C₄²×C₉	—	2	1	no	—
53	#3667	—	2	1	no	—
54	#3667	—	2	1	no	—
55	#3667	—	2	1	no	—
56	#3667	C₃×C₄	1	2	yes	complemented
57	C₂×C₉⋊C₈	—	2	1	no	—
58	C₂×C₉⋊C₈	C₃×C₄	1	2	yes	complemented
59	C₂²×C₃×C₉	—	4	1	no	—
60	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	—	4	2	no	—
64	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
67	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃×M₁₆	—	9	1	no	—
69	C₂×C₃×M₁₆	—	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	C₂×#208	—	3	1	no	—
75	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
81	C₂×C₃²×C₄	#67	1	1	yes	characteristic
82	C₂×C₃²×C₄	#67	1	1	yes	characteristic
83	C₂³×C₉	—	2	1	no	—
84	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
85	C₃×C₃⋊C₈	—	6	2	no	—
86	C₃×C₃⋊C₈	—	6	2	no	—
87	C₂×C₄×C₉	—	2	4	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
100	C₂×C₄×C₉	C₃×Q₈	1	1	yes	characteristic
101	C₉⋊C₈	—	2	2	no	—
102	C₉⋊C₈	—	2	2	no	—
103	#344	—	9	1	no	Sylow
104	C₂×C₃×C₉	—	2	1	no	—
105	C₂×C₃×C₉	—	2	1	no	—
106	C₂×C₃×C₉	—	2	1	no	—
107	C₂×C₃×C₉	#95	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
111	C₃×M₁₆	—	9	2	no	—
112	C₃×M₁₆	—	18	1	no	—
113	C₃×M₁₆	—	18	1	no	—
114	C₃×M₁₆	—	18	1	no	—
115	C₂×C₃×C₈	—	9	2	no	—
116	C₂×C₃×C₈	—	18	1	no	—
117	C₃×C₄⋊C₄	—	2	2	no	—
118	C₃×C₄⋊C₄	—	2	1	no	—
119	C₃×C₄⋊C₄	—	2	1	no	—
120	C₃×C₂²⋊C₄	—	2	2	no	—
121	C₃×C₂²⋊C₄	Dic₉	1	2	yes	—
122	C₃×C₂²⋊C₄	C₃×Dic₃	1	2	yes	—
123	C₃×C₄²	—	2	2	no	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	—	2	1	no	—
126	#208	—	3	2	no	—
127	#208	—	6	1	no	—
128	#208	—	6	1	no	—
129	#208	—	6	1	no	—
130	C₂×C₃⋊C₈	—	3	2	no	—
131	C₂×C₃⋊C₈	—	6	1	no	—
132	C₂²×C₃²	—	4	1	no	—
133	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
134	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
135	C₂²×C₃²	#217	1	1	yes	characteristic
136	C₃²×C₄	—	4	2	no	—
137	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
138	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
139	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
140	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	—	2	1	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	4	1	no	—
145	C₂²×C₉	—	4	2	no	—
146	C₂²×C₉	C₃×C₄²	1	1	yes	characteristic
147	C₂²×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₉	C₃×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	2	1	no	—
153	C₄×C₉	—	4	4	no	—
154	C₄×C₉	—	4	2	no	—
155	C₄×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₉	C₃×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₉	C₃×C₄⋊C₄	1	1	yes	characteristic
159	C₂×M₁₆	—	9	1	no	—
160	C₂×M₁₆	—	9	1	no	—
161	#117	C₃×D₉	1	1	yes	characteristic
162	C₃×C₉	#344	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₃	—	2	1	no	—
164	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
165	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	Dic₁₈	1	1	yes	characteristic
176	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
177	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
178	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
179	C₂×C₃×C₄	#611	1	1	yes	characteristic
180	C₂×C₃×C₄	#611	1	1	yes	characteristic
181	C₂×C₃×C₄	C₃×Dic₆	1	1	yes	characteristic
182	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
183	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
185	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
186	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
187	C₃×C₈	—	18	2	no	—
188	C₃×C₈	—	18	2	no	—
189	C₃⋊C₈	—	6	2	no	—
190	C₃⋊C₈	—	6	2	no	—
191	C₂×C₃²	—	2	1	no	—
192	C₂×C₃²	—	2	1	no	—
193	C₂×C₃²	—	2	1	no	—
194	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
195	C₂×C₉	—	2	2	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	—	2	1	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	2	1	no	—
200	C₂×C₉	—	4	1	no	—
201	C₂×C₉	—	4	1	no	—
202	C₂×C₉	C₃×#95	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
204	M₁₆	—	9	2	no	—
205	M₁₆	—	18	1	no	—
206	M₁₆	—	18	1	no	—
207	M₁₆	—	18	1	no	—
208	C₂×C₈	—	9	2	no	—
209	C₂×C₈	—	18	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	C₃×Dic₉	1	2	yes	—
212	C₄²	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	4	1	no	—
217	C₂²×C₃	—	4	1	no	—
218	C₂²×C₃	—	4	2	no	—
219	C₂²×C₃	C₄×Dic₉	1	1	yes	characteristic
220	C₂²×C₃	C₄⋊Dic₉	1	1	yes	characteristic
221	C₂²×C₃	#3676	1	1	yes	characteristic
222	C₂²×C₃	C₃×C₄×Dic₃	1	1	yes	characteristic
223	C₂²×C₃	C₃×C₄⋊Dic₃	1	1	yes	characteristic
224	C₂²×C₃	C₃×#217	1	1	yes	characteristic
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	4	2	no	—
230	C₃×C₄	—	4	4	no	—
231	C₃×C₄	—	4	2	no	—
232	C₃×C₄	Dic₉⋊C₄	1	1	yes	characteristic
233	C₃×C₄	Dic₉⋊C₄	1	1	yes	characteristic
234	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
235	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
236	C₃×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
237	C₃×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
238	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
239	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
240	C₃²	#4533	1	1	yes	characteristic
241	C₉	—	2	1	no	—
242	C₉	C₃×#344	1	1	yes	characteristic, complemented
243	C₂³	C₂×C₃×Dic₉	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	C₃×Dic₁₈	1	1	yes	characteristic
246	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
247	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
248	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
249	C₂×C₄	C₃×#611	1	1	yes	characteristic
250	C₂×C₄	C₃×#611	1	1	yes	characteristic
251	C₈	—	18	2	no	—
252	C₈	—	18	2	no	—
253	C₂×C₃	—	2	2	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	4	1	no	—
262	C₂×C₃	—	4	1	no	—
263	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
264	C₂×C₃	C₃×C₃⋊#95	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	C₃×C₄×Dic₉	1	1	yes	characteristic
267	C₂²	C₃×C₄⋊Dic₉	1	1	yes	characteristic
268	C₂²	C₃×#3676	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	C₃×Dic₉⋊C₄	1	1	yes	characteristic
271	C₄	C₃×Dic₉⋊C₄	1	1	yes	characteristic
272	C₄	C₃×(C₂×D₉)⋊C₄	1	1	yes	characteristic
273	C₄	C₃×(C₂×D₉)⋊C₄	1	1	yes	characteristic
274	C₃	—	2	1	no	—
275	C₃	#10588209	1	1	yes	characteristic, direct product factor
276	C₃	C₃×#4533	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	C₃×C₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	(C₂×D₉)⋊C₄
commutator	C₂²×C₉	C₃×C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃×D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂×C₃²	C₃⋊#95
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂²×C₄×C₉	C₂²×C₃
hypercenter	C₃×#117	D₉
hypocenter	C₉	C₃×#344
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
5. #10588209	C₃	direct (N × H)
13. C₃×#3667	C₄	semidirect
15. C₂×C₃×C₉⋊C₈	C₄	semidirect
56. #3667	C₃×C₄	semidirect
58. C₂×C₉⋊C₈	C₃×C₄	semidirect
162. C₃×C₉	#344	semidirect
242. C₉	C₃×#344	semidirect
275. C₃	#10588209	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
C₃×#4533	3
#10588209	3

Subgroups — C3×#10588209 #13135572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588209 #13135572