Subgroups — C₃×#10588666 #13142562

Order 1728.

494 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉⋊#95	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉⋊#95	C₂	1	2	yes	maximal
5	C₃×C₉⋊#95	C₂	1	2	yes	maximal
6	C₃×#4986	—	3	1	no	maximal
7	#10588666	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
12	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	—	2	2	no	—
14	C₃×C₄⋊Dic₉	—	2	2	no	—
15	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
16	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
17	C₃×C₃⋊#95	—	3	2	no	—
18	C₃×C₃⋊#95	—	3	2	no	—
19	C₃×C₃⋊#95	—	3	1	no	—
20	C₂×C₉×C₄⋊C₄	—	2	1	no	—
21	C₂×C₉×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄⋊Dic₉	C₂×C₃	1	1	yes	characteristic
23	C₉⋊#95	C₂×C₃	1	1	yes	characteristic
24	C₉⋊#95	C₂×C₃	1	2	yes	—
25	C₉⋊#95	C₂×C₃	1	2	yes	—
26	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	—	2	1	no	—
28	C₂×C₃×C₄×C₉	—	2	1	no	—
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	C₂×C₃×Dic₉	—	2	4	no	—
36	C₂×C₃×Dic₉	—	2	2	no	—
37	C₂×C₃×Dic₉	—	2	2	no	—
38	C₂×C₃×Dic₉	—	2	2	no	—
39	C₃×#400	—	9	1	no	—
40	#4986	—	3	1	no	—
41	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×Dic₃	—	3	2	no	—
45	C₂²×C₃×Dic₃	—	3	2	no	—
46	C₃²×C₄⋊C₄	—	2	2	no	—
47	C₃×C₄⋊Dic₃	—	6	2	no	—
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	—	2	1	no	—
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
54	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
55	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
56	C₉×C₄⋊C₄	—	2	4	no	—
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₄⋊Dic₉	—	2	2	no	—
59	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₃×C₉	#41	1	1	yes	characteristic
62	C₂²×C₃×C₉	#41	1	2	yes	—
63	C₂²×C₃×C₉	#41	1	2	yes	—
64	C₃×C₄×C₉	—	2	2	no	—
65	C₃×C₄×C₉	—	2	2	no	—
66	C₃×C₄×C₉	—	4	1	no	—
67	C₃×Dic₉	—	4	2	no	—
68	C₃×Dic₉	—	4	2	no	—
69	C₂×C₃×C₄⋊C₄	—	2	1	no	—
70	C₂×C₃×C₄⋊C₄	—	9	1	no	—
71	C₂×C₃×C₄⋊C₄	D₉	1	1	yes	characteristic
72	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
73	C₂×C₄⋊Dic₃	—	3	1	no	—
74	C₃×#95	—	9	1	no	—
75	C₃×#95	—	9	2	no	—
76	C₃×#95	—	9	2	no	—
77	C₃⋊#95	—	3	1	no	—
78	C₃⋊#95	—	3	2	no	—
79	C₃⋊#95	—	3	2	no	—
80	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	—	2	1	no	—
83	C₂×C₃²×C₄	—	2	1	no	—
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	Dic₆	1	2	yes	—
87	C₂×C₃²×C₄	#67	1	2	yes	—
88	C₂×C₃×Dic₃	—	6	2	no	—
89	C₂×C₃×Dic₃	—	6	2	no	—
90	C₂×C₃×Dic₃	—	6	2	no	—
91	C₂×C₃×Dic₃	—	6	4	no	—
92	C₂×C₃×Dic₃	—	6	2	no	—
93	C₂³×C₉	—	2	1	no	—
94	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	4	2	no	—
106	C₂×C₄×C₉	—	4	2	no	—
107	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
108	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
109	C₂×Dic₉	—	2	2	no	—
110	C₂×Dic₉	—	2	2	no	—
111	C₂×Dic₉	—	2	2	no	—
112	C₂×Dic₉	—	2	4	no	—
113	C₂×Dic₉	—	2	2	no	—
114	#400	—	9	1	no	Sylow
115	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₃×C₉	#123	1	1	yes	characteristic
117	C₂×C₃×C₉	#124	1	1	yes	characteristic
118	C₂×C₃×C₉	#125	1	2	yes	—
119	C₂×C₃×C₉	#126	1	2	yes	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	2	1	no	—
122	C₂²×C₃×C₄	—	2	1	no	—
123	C₂²×C₃×C₄	—	9	2	no	—
124	C₂²×C₃×C₄	—	9	2	no	—
125	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
131	C₂²×Dic₃	—	3	2	no	—
132	C₂²×Dic₃	—	3	2	no	—
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	18	2	no	—
137	C₄⋊Dic₃	—	6	2	no	—
138	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
139	C₂²×C₃²	#235	1	1	yes	characteristic
140	C₂²×C₃²	#237	1	2	yes	—
141	C₂²×C₃²	#239	1	2	yes	—
142	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
143	C₃²×C₄	—	2	2	no	—
144	C₃²×C₄	—	2	2	no	—
145	C₃²×C₄	—	4	1	no	—
146	C₃×Dic₃	—	12	2	no	—
147	C₃×Dic₃	—	12	2	no	—
148	C₂²×C₉	—	2	1	no	—
149	C₂²×C₉	—	2	1	no	—
150	C₂²×C₉	—	2	2	no	—
151	C₂²×C₉	—	2	2	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
155	C₂²×C₉	C₃×#41	1	1	yes	characteristic
156	C₂²×C₉	C₃×#41	1	2	yes	—
157	C₂²×C₉	C₃×#41	1	2	yes	—
158	C₄×C₉	—	2	4	no	—
159	C₄×C₉	—	2	2	no	—
160	C₄×C₉	—	2	2	no	—
161	C₄×C₉	—	4	2	no	—
162	C₄×C₉	—	4	2	no	—
163	C₄×C₉	—	4	1	no	—
164	Dic₉	—	4	2	no	—
165	Dic₉	—	4	2	no	—
166	C₂×C₄⋊C₄	—	9	1	no	—
167	C₂×C₄⋊C₄	C₃×D₉	1	1	yes	characteristic
168	#95	—	9	1	no	—
169	#95	—	9	2	no	—
170	#95	—	9	2	no	—
171	C₃×C₉	#400	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
174	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	18	2	no	—
193	C₂×C₃×C₄	—	18	2	no	—
194	C₂×C₃×C₄	—	18	4	no	—
195	C₂×C₃×C₄	—	18	2	no	—
196	C₂×C₃×C₄	—	18	2	no	—
197	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
198	C₂×C₃×C₄	#611	1	2	yes	—
199	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
200	C₂×C₃×C₄	C₃×#67	1	2	yes	—
201	C₂×Dic₃	—	6	2	no	—
202	C₂×Dic₃	—	6	2	no	—
203	C₂×Dic₃	—	6	2	no	—
204	C₂×Dic₃	—	6	4	no	—
205	C₂×Dic₃	—	6	2	no	—
206	C₂×C₃²	#890	1	2	yes	—
207	C₂×C₃²	#898	1	2	yes	—
208	C₂×C₃²	#924	1	1	yes	characteristic
209	C₂×C₃²	#935	1	1	yes	characteristic
210	C₂×C₃²	#947	1	1	yes	characteristic
211	C₂×C₉	—	2	2	no	—
212	C₂×C₉	—	2	2	no	—
213	C₂×C₉	—	2	1	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	C₃×C₂²⋊Q₈	1	1	yes	characteristic
217	C₂×C₉	C₃×#123	1	1	yes	characteristic
218	C₂×C₉	C₃×#124	1	1	yes	characteristic
219	C₂×C₉	C₃×#125	1	2	yes	—
220	C₂×C₉	C₃×#126	1	2	yes	—
221	C₂²×C₄	—	9	2	no	—
222	C₂²×C₄	—	9	2	no	—
223	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	18	2	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
234	C₂²×C₃	#3697	1	1	yes	characteristic
235	C₂²×C₃	#3699	1	2	yes	—
236	C₂²×C₃	#3701	1	2	yes	—
237	C₂²×C₃	C₂×#611	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
239	C₂²×C₃	C₃×#235	1	1	yes	characteristic
240	C₂²×C₃	C₃×#237	1	2	yes	—
241	C₂²×C₃	C₃×#239	1	2	yes	—
242	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	4	2	no	—
249	C₃×C₄	—	4	2	no	—
250	C₃×C₄	—	4	1	no	—
251	C₃×C₄	—	4	1	no	—
252	C₃×C₄	—	36	2	no	—
253	C₃×C₄	—	36	2	no	—
254	Dic₃	—	12	2	no	—
255	Dic₃	—	12	2	no	—
256	C₃²	#4986	1	1	yes	characteristic
257	C₉	—	2	1	no	—
258	C₉	C₃×#400	1	1	yes	characteristic, complemented
259	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	18	2	no	—
266	C₂×C₄	—	18	2	no	—
267	C₂×C₄	—	18	4	no	—
268	C₂×C₄	—	18	2	no	—
269	C₂×C₄	—	18	2	no	—
270	C₂×C₄	C₃×Dic₁₈	1	2	yes	—
271	C₂×C₄	C₃×#611	1	2	yes	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	#63452	1	2	yes	—
278	C₂×C₃	#63460	1	2	yes	—
279	C₂×C₃	#63486	1	1	yes	characteristic
280	C₂×C₃	#63497	1	1	yes	characteristic
281	C₂×C₃	#63509	1	1	yes	characteristic
282	C₂×C₃	C₃×#890	1	2	yes	—
283	C₂×C₃	C₃×#898	1	2	yes	—
284	C₂×C₃	C₃×#924	1	1	yes	characteristic
285	C₂×C₃	C₃×#935	1	1	yes	characteristic
286	C₂×C₃	C₃×#947	1	1	yes	characteristic
287	C₂²	C₂×C₃×Dic₁₈	1	1	yes	characteristic
288	C₂²	C₃×#3697	1	1	yes	characteristic
289	C₂²	C₃×#3699	1	2	yes	—
290	C₂²	C₃×#3701	1	2	yes	—
291	C₂²	C₂×C₃×#611	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	1	no	—
295	C₄	—	36	2	no	—
296	C₄	—	36	2	no	—
297	C₃	—	2	1	no	—
298	C₃	#10588666	1	1	yes	characteristic, direct product factor
299	C₃	C₃×#4986	1	1	yes	characteristic
300	C₂	C₃×#63452	1	2	yes	—
301	C₂	C₃×#63460	1	2	yes	—
302	C₂	C₃×#63486	1	1	yes	characteristic
303	C₂	C₃×#63497	1	1	yes	characteristic
304	C₂	C₃×#63509	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator, cosocle	C₂³×C₉	C₂³×C₃
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
hypercenter	C₂×C₃×C₄⋊C₄	D₉
hypocenter	C₉	C₃×#400
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. #10588666	C₃	direct (N × H)
171. C₃×C₉	#400	semidirect
258. C₉	C₃×#400	semidirect
298. C₃	#10588666	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×C₉×C₄⋊C₄	2
C₂×C₃×C₄⋊Dic₉	2
C₃×C₉⋊#95	2
C₃×#4986	3
#10588666	3

Subgroups — C3×#10588666 #13142562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588666 #13142562