Subgroups — C₃×#10588367 #13141411

Order 1728.

398 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×C₄²×D₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×(C₂×D₉)⋊C₈	C₂	1	2	yes	maximal
4	C₃×Dic₉⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4685	—	3	1	no	maximal
7	#10588367	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×C₄×D₉	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×D₉	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄×Dic₉	C₄	1	1	yes	characteristic
14	C₃×C₄×Dic₉	C₄	1	1	yes	characteristic
15	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
17	C₃×C₄²×D₃	—	3	1	no	—
18	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
19	C₃×(C₂×D₃)⋊C₈	—	3	2	no	—
20	C₃×Dic₃⋊C₈	—	3	2	no	—
21	C₃×C₄×C₃⋊C₈	—	3	1	no	—
22	C₄²×D₉	C₂×C₃	1	1	yes	characteristic
23	C₄×C₈×C₉	—	2	1	no	—
24	C₄×C₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
25	(C₂×D₉)⋊C₈	C₂×C₃	1	2	yes	—
26	Dic₉⋊C₈	C₂×C₃	1	2	yes	complemented
27	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×C₃×D₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×D₉	—	2	2	no	—
36	C₃×C₄×D₉	—	2	2	no	—
37	C₃×C₄×D₉	C₈	1	4	yes	complemented
38	C₃×C₈×C₉	—	2	2	no	—
39	C₃×C₉⋊C₈	—	2	2	no	—
40	C₃×#431	—	9	1	no	—
41	#4685	—	3	1	no	—
42	C₂×C₃×C₄×D₃	—	3	1	no	—
43	C₂×C₃×C₄×D₃	—	3	1	no	—
44	C₂×C₃×C₄×D₃	—	3	1	no	—
45	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
46	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
47	C₃×C₄×Dic₃	—	3	1	no	—
48	C₃×C₄×Dic₃	—	3	1	no	—
49	C₃×C₄×Dic₃	—	3	1	no	—
50	C₂×C₃×C₃⋊C₈	—	3	2	no	—
51	C₂×C₄×D₉	C₃×C₄	1	1	yes	characteristic
52	C₂×C₄×D₉	C₃×C₄	1	1	yes	characteristic
53	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₈×C₉	—	2	2	no	—
55	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
56	C₄²×C₉	—	2	1	no	—
57	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
58	C₄×Dic₉	C₃×C₄	1	1	yes	characteristic
59	C₄×Dic₉	C₃×C₄	1	1	yes	characteristic
60	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
62	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×D₉	—	2	1	no	—
64	C₂×C₃×D₉	—	2	1	no	—
65	C₂×C₃×D₉	C₂×C₈	1	2	yes	—
66	C₃×C₄×C₉	C₂×C₈	1	2	yes	—
67	C₃×C₄×C₉	M₁₆	1	2	yes	—
68	C₃×C₄×C₉	#41	1	2	yes	—
69	C₃×Dic₉	—	2	1	no	—
70	C₃×Dic₉	—	2	1	no	—
71	C₃×Dic₉	C₂×C₈	1	2	yes	—
72	C₂×C₃×C₄²	—	9	1	no	—
73	C₄²×D₃	—	3	1	no	—
74	C₃×C₄⋊C₈	—	9	2	no	—
75	C₃×C₂²⋊C₈	—	9	2	no	—
76	C₃×C₄×C₈	—	2	1	no	—
77	C₃×C₄×C₈	—	9	1	no	—
78	C₃×C₄×C₈	D₉	1	1	yes	characteristic, complemented
79	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
80	(C₂×D₃)⋊C₈	—	3	2	no	—
81	Dic₃⋊C₈	—	3	2	no	—
82	C₄×C₃⋊C₈	—	3	1	no	—
83	C₂²×C₃×D₃	—	3	1	no	—
84	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃×Dic₃	—	3	1	no	—
88	C₂×C₃×Dic₃	—	3	1	no	—
89	C₂×C₃×Dic₃	—	3	1	no	—
90	C₃×C₄×D₃	—	3	4	no	—
91	C₃×C₄×D₃	—	6	2	no	—
92	C₃×C₄×D₃	—	6	2	no	—
93	C₂²×D₉	C₂×C₃×C₄	1	1	yes	characteristic
94	C₃²×C₈	—	2	2	no	—
95	C₃×C₃⋊C₈	—	6	2	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
102	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
105	C₄×D₉	—	2	2	no	—
106	C₄×D₉	—	2	2	no	—
107	C₄×D₉	C₃×C₈	1	4	yes	complemented
108	C₈×C₉	—	2	4	no	—
109	C₈×C₉	—	2	2	no	—
110	C₉⋊C₈	—	2	2	no	—
111	#431	—	9	1	no	Sylow
112	C₂×C₃×C₉	#117	1	1	yes	characteristic
113	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
115	C₃×D₉	—	2	2	no	—
116	C₂²×C₃×C₄	—	9	1	no	—
117	C₂²×C₃×C₄	—	9	1	no	—
118	C₂²×C₃×C₄	—	9	1	no	—
119	C₂×C₄×D₃	—	3	1	no	—
120	C₂×C₄×D₃	—	3	1	no	—
121	C₂×C₄×D₃	—	3	1	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	9	2	no	—
124	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
125	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	—	9	1	no	—
128	C₃×C₄²	—	9	1	no	—
129	C₃×C₄²	—	9	1	no	—
130	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
131	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
132	C₄×Dic₃	—	3	1	no	—
133	C₄×Dic₃	—	3	1	no	—
134	C₄×Dic₃	—	3	1	no	—
135	C₂×C₃⋊C₈	—	3	2	no	—
136	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂×C₃×D₃	—	3	2	no	—
138	C₂×C₃×D₃	—	6	1	no	—
139	C₂×C₃×D₃	—	6	1	no	—
140	C₃²×C₄	D₃×C₈	1	2	yes	—
141	C₃²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
142	C₃²×C₄	#235	1	2	yes	—
143	C₃×Dic₃	—	3	2	no	—
144	C₃×Dic₃	—	6	1	no	—
145	C₃×Dic₃	—	6	1	no	—
146	C₂²×C₉	—	2	1	no	—
147	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₂×D₉	—	2	1	no	—
149	C₂×D₉	—	2	1	no	—
150	C₂×D₉	C₂×C₃×C₈	1	2	yes	—
151	C₄×C₉	—	2	2	no	—
152	C₄×C₉	—	2	2	no	—
153	C₄×C₉	—	2	2	no	—
154	C₄×C₉	C₂×C₃×C₈	1	2	yes	—
155	C₄×C₉	C₃×M₁₆	1	2	yes	—
156	C₄×C₉	C₃×#41	1	2	yes	—
157	Dic₉	—	2	1	no	—
158	Dic₉	—	2	1	no	—
159	Dic₉	C₂×C₃×C₈	1	2	yes	—
160	C₂×C₄²	—	9	1	no	—
161	C₄⋊C₈	—	9	2	no	—
162	C₂²⋊C₈	—	9	2	no	—
163	C₄×C₈	—	9	1	no	—
164	C₄×C₈	C₃×D₉	1	1	yes	characteristic, complemented
165	C₃×C₉	#431	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₃	—	9	1	no	—
167	C₂²×D₃	—	3	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	9	1	no	—
172	C₂×C₃×C₄	—	9	4	no	—
173	C₂×C₃×C₄	—	9	1	no	—
174	C₂×C₃×C₄	—	9	1	no	—
175	C₂×C₃×C₄	—	18	2	no	—
176	C₂×C₃×C₄	—	18	2	no	—
177	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
178	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
180	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
183	C₂×Dic₃	—	3	1	no	—
184	C₂×Dic₃	—	3	1	no	—
185	C₂×Dic₃	—	3	1	no	—
186	C₄×D₃	—	3	4	no	—
187	C₄×D₃	—	6	2	no	—
188	C₄×D₃	—	6	2	no	—
189	C₃×C₈	—	2	4	no	—
190	C₃×C₈	—	2	2	no	—
191	C₃×C₈	—	2	2	no	—
192	C₃×C₈	—	18	2	no	—
193	C₃⋊C₈	—	6	2	no	—
194	C₂×C₃²	#872	1	1	yes	characteristic
195	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
196	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
197	C₃×D₃	—	6	2	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	2	1	no	—
200	C₂×C₉	—	2	1	no	—
201	C₂×C₉	C₃×#117	1	1	yes	characteristic
202	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
203	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
204	D₉	—	2	2	no	—
205	C₂²×C₄	—	9	1	no	—
206	C₂²×C₄	—	9	1	no	—
207	C₂²×C₄	—	9	1	no	—
208	C₂×C₈	—	9	2	no	—
209	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
210	C₄²	—	9	1	no	—
211	C₄²	—	9	1	no	—
212	C₄²	—	9	1	no	—
213	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	9	2	no	—
216	C₂²×C₃	—	18	1	no	—
217	C₂²×C₃	—	18	1	no	—
218	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
220	C₂×D₃	—	3	2	no	—
221	C₂×D₃	—	6	1	no	—
222	C₂×D₃	—	6	1	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	9	2	no	—
227	C₃×C₄	—	18	1	no	—
228	C₃×C₄	—	18	1	no	—
229	C₃×C₄	C₈×D₉	1	2	yes	—
230	C₃×C₄	#3663	1	2	yes	—
231	C₃×C₄	#3697	1	2	yes	—
232	C₃×C₄	C₃×D₃×C₈	1	2	yes	—
233	C₃×C₄	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
234	C₃×C₄	C₃×#235	1	2	yes	—
235	Dic₃	—	3	2	no	—
236	Dic₃	—	6	1	no	—
237	Dic₃	—	6	1	no	—
238	C₃²	#4685	1	1	yes	characteristic
239	C₉	—	2	1	no	—
240	C₉	C₃×#431	1	1	yes	characteristic, complemented
241	C₂³	—	9	1	no	—
242	C₂×C₄	—	9	1	no	—
243	C₂×C₄	—	9	4	no	—
244	C₂×C₄	—	9	1	no	—
245	C₂×C₄	—	9	1	no	—
246	C₂×C₄	—	18	2	no	—
247	C₂×C₄	—	18	2	no	—
248	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
249	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	18	2	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	18	2	no	—
257	C₂×C₃	#63434	1	1	yes	characteristic
258	C₂×C₃	C₂×C₈×D₉	1	1	yes	characteristic
259	C₂×C₃	C₂×#3663	1	1	yes	characteristic
260	C₂×C₃	C₃×#872	1	1	yes	characteristic
261	C₂×C₃	C₂×C₃×D₃×C₈	1	1	yes	characteristic
262	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
263	D₃	—	6	2	no	—
264	C₂²	—	9	2	no	—
265	C₂²	—	18	1	no	—
266	C₂²	—	18	1	no	—
267	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
268	C₄	—	9	2	no	—
269	C₄	—	18	1	no	—
270	C₄	—	18	1	no	—
271	C₄	C₃×C₈×D₉	1	2	yes	—
272	C₄	C₃×#3663	1	2	yes	—
273	C₄	C₃×#3697	1	2	yes	—
274	C₃	—	2	1	no	—
275	C₃	#10588367	1	1	yes	characteristic, direct product factor
276	C₃	C₃×#4685	1	1	yes	characteristic
277	C₂	—	18	2	no	—
278	C₂	C₃×#63434	1	1	yes	characteristic
279	C₂	C₂×C₃×C₈×D₉	1	1	yes	characteristic
280	C₂	C₂×C₃×#3663	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄²	C₂×D₉
commutator	C₂×C₉	C₂²×C₃×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃×D₉	C₂×C₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄×C₈	D₉
hypocenter	C₉	C₃×#431
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₃×C₄² ◁ C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₃×Dic₉⋊C₈	C₂	semidirect
5. C₃×C₄×C₉⋊C₈	C₂	semidirect
7. #10588367	C₃	direct (N × H)
24. C₄×C₈×C₉	C₂×C₃	semidirect
26. Dic₉⋊C₈	C₂×C₃	semidirect
27. C₄×C₉⋊C₈	C₂×C₃	semidirect
37. C₃×C₄×D₉	C₈	semidirect
78. C₃×C₄×C₈	D₉	semidirect
107. C₄×D₉	C₃×C₈	semidirect
164. C₄×C₈	C₃×D₉	semidirect
165. C₃×C₉	#431	semidirect
240. C₉	C₃×#431	semidirect
275. C₃	#10588367	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×C₄²×D₉	2
C₃×C₄×C₈×C₉	2
C₃×(C₂×D₉)⋊C₈	2
C₃×Dic₉⋊C₈	2
C₃×C₄×C₉⋊C₈	2
C₃×#4685	3
#10588367	3

Subgroups — C3×#10588367 #13141411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588367 #13141411