Subgroups — C₃×#10588909 #13143589

Order 1728.

714 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	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉⋊#95	C₂	1	4	yes	maximal
5	C₃×#5229	—	3	1	no	maximal
6	#10588909	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×Q₈×C₉	—	2	2	no	—
8	C₂×C₃×Q₈×C₉	—	2	2	no	—
9	C₂×C₃×Q₈×C₉	C₄	1	4	yes	complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
13	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
14	C₃×C₄⋊Dic₉	—	2	2	no	—
15	C₃×C₄×Dic₉	—	2	2	no	—
16	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
17	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
18	C₂×C₃×C₄×Dic₃	—	3	1	no	—
19	C₂²×Q₈×C₉	—	2	1	no	—
20	C₂²×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
21	C₃×C₃⋊#95	—	3	4	no	—
22	C₂×C₄⋊Dic₉	C₂×C₃	1	1	yes	characteristic
23	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic
24	C₉⋊#95	C₂×C₃	1	4	yes	—
25	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
26	C₃×Q₈×C₉	—	2	8	no	—
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
33	C₂×C₃×Dic₉	—	2	2	no	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	C₂×C₃×Dic₉	—	2	2	no	—
36	C₂×C₃×Dic₉	—	2	4	no	—
37	C₂×C₃×Dic₉	Q₈	1	4	yes	complemented
38	C₃×#391	—	9	1	no	—
39	#5229	—	3	1	no	—
40	C₂×C₃²×Q₈	—	2	2	no	—
41	C₂×C₃²×Q₈	—	2	2	no	—
42	C₂×C₃²×Q₈	Dic₃	1	4	yes	—
43	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
44	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃×Dic₃	—	3	2	no	—
46	C₂²×C₃×Dic₃	—	3	2	no	—
47	C₃×C₄⋊Dic₃	—	6	2	no	—
48	C₃×C₄×Dic₃	—	6	2	no	—
49	C₂×Q₈×C₉	—	2	4	no	—
50	C₂×Q₈×C₉	—	2	4	no	—
51	C₂×Q₈×C₉	—	2	4	no	—
52	C₂×Q₈×C₉	—	2	2	no	—
53	C₂×Q₈×C₉	—	2	2	no	—
54	C₂×Q₈×C₉	C₃×C₄	1	4	yes	complemented
55	C₂²×C₄×C₉	—	2	2	no	—
56	C₂²×C₄×C₉	—	2	1	no	—
57	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
58	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
59	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
60	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
61	C₄⋊Dic₉	—	2	2	no	—
62	C₄×Dic₉	—	2	2	no	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
67	C₂²×C₃×C₉	#41	1	2	yes	—
68	C₃×C₄×C₉	—	2	4	no	—
69	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	—
70	C₃×Dic₉	—	2	4	no	—
71	C₃×Dic₉	—	4	2	no	—
72	C₂²×C₃×Q₈	—	2	1	no	—
73	C₂²×C₃×Q₈	D₉	1	1	yes	characteristic
74	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	—	9	1	no	—
76	C₂×C₃×C₄²	—	9	1	no	—
77	C₂×C₄⋊Dic₃	—	3	1	no	—
78	C₂×C₄×Dic₃	—	3	1	no	—
79	C₃×#95	—	9	4	no	—
80	C₃⋊#95	—	3	4	no	—
81	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
82	C₃²×Q₈	—	2	8	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
86	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
87	C₂×C₃²×C₄	#67	1	2	yes	—
88	C₂×C₃²×C₄	#67	1	2	yes	—
89	C₂×C₃×Dic₃	—	3	4	no	—
90	C₂×C₃×Dic₃	—	6	2	no	—
91	C₂×C₃×Dic₃	—	6	2	no	—
92	C₂×C₃×Dic₃	—	6	4	no	—
93	C₂×C₃×Dic₃	—	6	2	no	—
94	C₂³×C₉	—	2	1	no	—
95	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
96	Q₈×C₉	—	2	16	no	—
97	Q₈×C₉	—	2	8	no	—
98	C₂×C₄×C₉	—	2	4	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	4	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
107	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
108	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
109	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
110	C₂×Dic₉	—	2	2	no	—
111	C₂×Dic₉	—	2	2	no	—
112	C₂×Dic₉	—	2	2	no	—
113	C₂×Dic₉	—	2	4	no	—
114	C₂×Dic₉	C₃×Q₈	1	4	yes	complemented
115	#391	—	9	1	no	Sylow
116	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₃×C₉	C₄×Q₈	1	2	yes	—
118	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₃×C₉	#124	1	1	yes	characteristic
120	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₃×Q₈	—	2	4	no	—
122	C₂×C₃×Q₈	—	2	4	no	—
123	C₂×C₃×Q₈	—	2	2	no	—
124	C₂×C₃×Q₈	—	2	2	no	—
125	C₂×C₃×Q₈	—	2	2	no	—
126	C₂×C₃×Q₈	—	2	2	no	—
127	C₂×C₃×Q₈	—	2	4	no	—
128	C₂×C₃×Q₈	Dic₉	1	4	yes	complemented
129	C₂×C₃×Q₈	C₃×Dic₃	1	4	yes	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	—	2	1	no	—
132	C₂²×C₃×C₄	—	9	2	no	—
133	C₂²×C₃×C₄	—	9	2	no	—
134	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
135	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
137	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂²×Dic₃	—	3	2	no	—
139	C₂²×Dic₃	—	3	2	no	—
140	C₃×C₄⋊C₄	—	18	2	no	—
141	C₃×C₄²	—	18	2	no	—
142	C₄⋊Dic₃	—	6	2	no	—
143	C₄×Dic₃	—	6	2	no	—
144	C₂²×C₃²	D₃×Q₈	1	2	yes	—
145	C₂²×C₃²	#239	1	2	yes	—
146	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
147	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
148	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	#217	1	4	yes	—
151	C₃×Dic₃	—	6	4	no	—
152	C₃×Dic₃	—	12	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	2	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	2	1	no	—
158	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
159	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂²×C₉	C₂×C₃×Q₈	1	2	yes	—
162	C₂²×C₉	C₃×#41	1	2	yes	—
163	C₄×C₉	—	2	8	no	—
164	C₄×C₉	—	2	4	no	—
165	C₄×C₉	—	2	4	no	—
166	C₄×C₉	C₃×C₂²⋊C₄	1	4	yes	—
167	Dic₉	—	2	4	no	—
168	Dic₉	—	4	2	no	—
169	C₂²×Q₈	C₃×D₉	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	9	1	no	—
171	C₂×C₄²	—	9	1	no	—
172	#95	—	9	4	no	—
173	C₃×C₉	#391	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₃	—	2	1	no	—
175	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
176	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
177	C₃×Q₈	—	2	16	no	—
178	C₃×Q₈	—	2	8	no	—
179	C₃×Q₈	—	2	8	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	9	4	no	—
191	C₂×C₃×C₄	—	18	2	no	—
192	C₂×C₃×C₄	—	18	2	no	—
193	C₂×C₃×C₄	—	18	4	no	—
194	C₂×C₃×C₄	—	18	2	no	—
195	C₂×C₃×C₄	C₂×Dic₉	1	4	yes	—
196	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
197	C₂×C₃×C₄	#611	1	2	yes	—
198	C₂×C₃×C₄	#611	1	2	yes	—
199	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
200	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
201	C₂×C₃×C₄	C₃×#67	1	2	yes	—
202	C₂×C₃×C₄	C₃×#67	1	2	yes	—
203	C₂×Dic₃	—	3	4	no	—
204	C₂×Dic₃	—	6	2	no	—
205	C₂×Dic₃	—	6	2	no	—
206	C₂×Dic₃	—	6	4	no	—
207	C₂×Dic₃	—	6	2	no	—
208	C₂×C₃²	#944	1	1	yes	characteristic
209	C₂×C₃²	Q₈×Dic₃	1	2	yes	—
210	C₂×C₃²	#946	1	2	yes	—
211	C₂×C₃²	#947	1	1	yes	characteristic
212	C₂×C₃²	C₂×#217	1	1	yes	characteristic
213	C₂×C₉	—	2	1	no	—
214	C₂×C₉	—	2	2	no	—
215	C₂×C₉	—	2	2	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	—	2	1	no	—
218	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₉	C₃×C₄×Q₈	1	2	yes	—
220	C₂×C₉	C₃×C₂²⋊Q₈	1	2	yes	—
221	C₂×C₉	C₃×#124	1	1	yes	characteristic
222	C₂×C₉	C₃×C₄⋊Q₈	1	1	yes	characteristic
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	C₃×Dic₉	1	4	yes	complemented
226	C₂²×C₄	—	9	2	no	—
227	C₂²×C₄	—	9	2	no	—
228	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
229	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
230	C₄⋊C₄	—	18	2	no	—
231	C₄²	—	18	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	Q₈×D₉	1	2	yes	—
238	C₂²×C₃	#3701	1	2	yes	—
239	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
240	C₂²×C₃	C₂×#611	1	1	yes	characteristic
241	C₂²×C₃	C₂×#611	1	1	yes	characteristic
242	C₂²×C₃	C₃×D₃×Q₈	1	2	yes	—
243	C₂²×C₃	C₃×#239	1	2	yes	—
244	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
245	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
246	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
247	C₃×C₄	—	2	8	no	—
248	C₃×C₄	—	2	4	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	18	4	no	—
252	C₃×C₄	—	36	2	no	—
253	C₃×C₄	#3676	1	4	yes	—
254	C₃×C₄	C₃×#217	1	4	yes	—
255	Dic₃	—	6	4	no	—
256	Dic₃	—	12	2	no	—
257	C₃²	#5229	1	1	yes	characteristic
258	C₉	—	2	1	no	—
259	C₉	C₃×#391	1	1	yes	characteristic, complemented
260	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
261	Q₈	—	2	8	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	9	4	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	C₂×C₃×Dic₉	1	4	yes	—
270	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
271	C₂×C₄	C₃×#611	1	2	yes	—
272	C₂×C₄	C₃×#611	1	2	yes	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	#63506	1	1	yes	characteristic
279	C₂×C₃	Q₈×Dic₉	1	2	yes	—
280	C₂×C₃	#63508	1	2	yes	—
281	C₂×C₃	#63509	1	1	yes	characteristic
282	C₂×C₃	C₂×#3676	1	1	yes	characteristic
283	C₂×C₃	C₃×#944	1	1	yes	characteristic
284	C₂×C₃	C₃×Q₈×Dic₃	1	2	yes	—
285	C₂×C₃	C₃×#946	1	2	yes	—
286	C₂×C₃	C₃×#947	1	1	yes	characteristic
287	C₂×C₃	C₂×C₃×#217	1	1	yes	characteristic
288	C₂²	C₃×Q₈×D₉	1	2	yes	—
289	C₂²	C₃×#3701	1	2	yes	—
290	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
291	C₂²	C₂×C₃×#611	1	1	yes	characteristic
292	C₂²	C₂×C₃×#611	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	18	4	no	—
295	C₄	—	36	2	no	—
296	C₄	C₃×#3676	1	4	yes	—
297	C₃	—	2	1	no	—
298	C₃	#10588909	1	1	yes	characteristic, direct product factor
299	C₃	C₃×#5229	1	1	yes	characteristic
300	C₂	C₃×#63506	1	1	yes	characteristic
301	C₂	C₃×Q₈×Dic₉	1	2	yes	—
302	C₂	C₃×#63508	1	2	yes	—
303	C₂	C₃×#63509	1	1	yes	characteristic
304	C₂	C₂×C₃×#3676	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂²×C₃×Q₈	D₉
hypocenter	C₉	C₃×#391
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
6. #10588909	C₃	direct (N × H)
9. C₂×C₃×Q₈×C₉	C₄	semidirect
37. C₂×C₃×Dic₉	Q₈	semidirect
54. C₂×Q₈×C₉	C₃×C₄	semidirect
114. C₂×Dic₉	C₃×Q₈	semidirect
128. C₂×C₃×Q₈	Dic₉	semidirect
173. C₃×C₉	#391	semidirect
225. C₂×Q₈	C₃×Dic₉	semidirect
259. C₉	C₃×#391	semidirect
298. C₃	#10588909	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×Q₈×C₉	2
C₂×C₃×C₄⋊Dic₉	2
C₂×C₃×C₄×Dic₉	2
C₃×C₉⋊#95	2
C₃×#5229	3
#10588909	3

Subgroups — C3×#10588909 #13143589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588909 #13143589