Subgroups — C₃×#10588657 #13142551

Order 1728.

554 conjugacy classes of subgroups.

The subgroup lattice has 274 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	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉⋊#95	C₂	1	4	yes	maximal
5	C₃×#4977	—	3	1	no	maximal
6	#10588657	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	—	2	2	no	—
12	C₃×C₄⋊Dic₉	—	2	2	no	—
13	C₃×C₄×Dic₉	C₄	1	4	yes	complemented
14	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
15	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
16	C₂×C₃×C₄×Dic₃	—	3	1	no	—
17	C₃×C₃⋊#95	—	3	4	no	—
18	C₂×C₉×C₄⋊C₄	—	2	1	no	—
19	C₂×C₉×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄⋊Dic₉	C₂×C₃	1	1	yes	characteristic
21	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic
22	C₉⋊#95	C₂×C₃	1	4	yes	—
23	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	—	2	2	no	—
25	C₂×C₃×C₄×C₉	—	2	4	no	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
29	C₂×C₃×Dic₉	—	2	2	no	—
30	C₂×C₃×Dic₉	—	2	2	no	—
31	C₂×C₃×Dic₉	—	2	2	no	—
32	C₂×C₃×Dic₉	—	2	4	no	—
33	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
34	C₃×#382	—	9	1	no	—
35	#4977	—	3	1	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃×Dic₃	—	3	2	no	—
39	C₂²×C₃×Dic₃	—	3	2	no	—
40	C₃²×C₄⋊C₄	—	2	2	no	—
41	C₃×C₄⋊Dic₃	—	6	2	no	—
42	C₃×C₄×Dic₃	—	3	4	no	—
43	C₂²×C₄×C₉	—	2	1	no	—
44	C₂²×C₄×C₉	—	2	2	no	—
45	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
46	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
47	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
48	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
49	C₉×C₄⋊C₄	—	2	4	no	—
50	C₉×C₄⋊C₄	—	2	2	no	—
51	C₄⋊Dic₉	—	2	2	no	—
52	C₄×Dic₉	C₃×C₄	1	4	yes	complemented
53	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₃×C₉	#41	1	2	yes	—
57	C₂²×C₃×C₉	#41	1	2	yes	—
58	C₃×C₄×C₉	—	4	2	no	—
59	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	—
60	C₃×Dic₉	—	2	4	no	—
61	C₃×Dic₉	—	4	2	no	—
62	C₂×C₃×C₄⋊C₄	—	2	1	no	—
63	C₂×C₃×C₄⋊C₄	—	9	1	no	—
64	C₂×C₃×C₄⋊C₄	D₉	1	1	yes	characteristic
65	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
66	C₂×C₃×C₄²	—	9	1	no	—
67	C₂×C₄⋊Dic₃	—	3	1	no	—
68	C₂×C₄×Dic₃	—	3	1	no	—
69	C₃×#95	—	9	4	no	—
70	C₃⋊#95	—	3	4	no	—
71	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	—	2	4	no	—
74	C₂×C₃²×C₄	Dic₆	1	2	yes	—
75	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
76	C₂×C₃²×C₄	#67	1	2	yes	—
77	C₂×C₃×Dic₃	—	3	4	no	—
78	C₂×C₃×Dic₃	—	6	2	no	—
79	C₂×C₃×Dic₃	—	6	2	no	—
80	C₂×C₃×Dic₃	—	6	4	no	—
81	C₂×C₃×Dic₃	—	6	2	no	—
82	C₂³×C₉	—	2	1	no	—
83	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	4	no	—
87	C₂×C₄×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	4	4	no	—
91	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
93	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
94	C₂×Dic₉	—	2	2	no	—
95	C₂×Dic₉	—	2	2	no	—
96	C₂×Dic₉	—	2	2	no	—
97	C₂×Dic₉	—	2	4	no	—
98	C₂×Dic₉	C₂×C₃×C₄	1	4	yes	—
99	#382	—	9	1	no	Sylow
100	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₃×C₉	#117	1	2	yes	—
102	C₂×C₃×C₉	#124	1	1	yes	characteristic
103	C₂×C₃×C₉	#124	1	1	yes	characteristic
104	C₂×C₃×C₉	#125	1	2	yes	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	—	9	2	no	—
108	C₂²×C₃×C₄	—	9	2	no	—
109	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
110	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
112	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
113	C₂²×Dic₃	—	3	2	no	—
114	C₂²×Dic₃	—	3	2	no	—
115	C₃×C₄⋊C₄	—	2	4	no	—
116	C₃×C₄⋊C₄	—	2	2	no	—
117	C₃×C₄⋊C₄	—	2	2	no	—
118	C₃×C₄⋊C₄	—	18	2	no	—
119	C₃×C₄²	—	9	4	no	—
120	C₄⋊Dic₃	—	6	2	no	—
121	C₄×Dic₃	—	3	4	no	—
122	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
123	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
124	C₂²×C₃²	#237	1	2	yes	—
125	C₂²×C₃²	#239	1	2	yes	—
126	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
127	C₃²×C₄	—	4	2	no	—
128	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
129	C₃×Dic₃	—	6	4	no	—
130	C₃×Dic₃	—	12	2	no	—
131	C₂²×C₉	—	2	1	no	—
132	C₂²×C₉	—	2	1	no	—
133	C₂²×C₉	—	2	2	no	—
134	C₂²×C₉	—	2	2	no	—
135	C₂²×C₉	—	2	1	no	—
136	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
139	C₂²×C₉	C₃×#41	1	2	yes	—
140	C₂²×C₉	C₃×#41	1	2	yes	—
141	C₄×C₉	—	2	4	no	—
142	C₄×C₉	—	4	4	no	—
143	C₄×C₉	—	4	2	no	—
144	C₄×C₉	C₃×C₄⋊C₄	1	4	yes	—
145	Dic₉	—	2	4	no	—
146	Dic₉	—	4	2	no	—
147	C₂×C₄⋊C₄	—	9	1	no	—
148	C₂×C₄⋊C₄	C₃×D₉	1	1	yes	characteristic
149	C₂×C₄²	—	9	1	no	—
150	#95	—	9	4	no	—
151	C₃×C₉	#382	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₃	—	2	1	no	—
153	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
154	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	4	4	no	—
164	C₂×C₃×C₄	—	9	4	no	—
165	C₂×C₃×C₄	—	18	2	no	—
166	C₂×C₃×C₄	—	18	2	no	—
167	C₂×C₃×C₄	—	18	4	no	—
168	C₂×C₃×C₄	—	18	2	no	—
169	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
170	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
171	C₂×C₃×C₄	#611	1	2	yes	—
172	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
173	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
174	C₂×C₃×C₄	C₃×#67	1	2	yes	—
175	C₂×Dic₃	—	3	4	no	—
176	C₂×Dic₃	—	6	2	no	—
177	C₂×Dic₃	—	6	2	no	—
178	C₂×Dic₃	—	6	4	no	—
179	C₂×Dic₃	—	6	2	no	—
180	C₂×C₃²	#890	1	2	yes	—
181	C₂×C₃²	#892	1	2	yes	—
182	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
183	C₂×C₃²	#936	1	1	yes	characteristic
184	C₂×C₃²	#947	1	1	yes	characteristic
185	C₂×C₉	—	2	2	no	—
186	C₂×C₉	—	2	2	no	—
187	C₂×C₉	—	2	1	no	—
188	C₂×C₉	—	2	1	no	—
189	C₂×C₉	—	2	1	no	—
190	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₉	C₃×#117	1	2	yes	—
192	C₂×C₉	C₃×#124	1	1	yes	characteristic
193	C₂×C₉	C₃×#124	1	1	yes	characteristic
194	C₂×C₉	C₃×#125	1	2	yes	—
195	C₂²×C₄	—	9	2	no	—
196	C₂²×C₄	—	9	2	no	—
197	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
198	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	18	2	no	—
201	C₄²	—	9	4	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
208	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
209	C₂²×C₃	#3699	1	2	yes	—
210	C₂²×C₃	#3701	1	2	yes	—
211	C₂²×C₃	C₂×#611	1	1	yes	characteristic
212	C₂²×C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
213	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
214	C₂²×C₃	C₃×#237	1	2	yes	—
215	C₂²×C₃	C₃×#239	1	2	yes	—
216	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	4	4	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	—	18	4	no	—
222	C₃×C₄	—	36	2	no	—
223	C₃×C₄	Dic₉⋊C₄	1	4	yes	—
224	C₃×C₄	C₃×Dic₃⋊C₄	1	4	yes	—
225	Dic₃	—	6	4	no	—
226	Dic₃	—	12	2	no	—
227	C₃²	#4977	1	1	yes	characteristic
228	C₉	—	2	1	no	—
229	C₉	C₃×#382	1	1	yes	characteristic, complemented
230	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	9	4	no	—
234	C₂×C₄	—	18	2	no	—
235	C₂×C₄	—	18	2	no	—
236	C₂×C₄	—	18	4	no	—
237	C₂×C₄	—	18	2	no	—
238	C₂×C₄	C₃×Dic₁₈	1	2	yes	—
239	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
240	C₂×C₄	C₃×#611	1	2	yes	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	2	no	—
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	#63452	1	2	yes	—
247	C₂×C₃	#63454	1	2	yes	—
248	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
249	C₂×C₃	#63498	1	1	yes	characteristic
250	C₂×C₃	#63509	1	1	yes	characteristic
251	C₂×C₃	C₃×#890	1	2	yes	—
252	C₂×C₃	C₃×#892	1	2	yes	—
253	C₂×C₃	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
254	C₂×C₃	C₃×#936	1	1	yes	characteristic
255	C₂×C₃	C₃×#947	1	1	yes	characteristic
256	C₂²	C₂×C₃×Dic₁₈	1	1	yes	characteristic
257	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
258	C₂²	C₃×#3699	1	2	yes	—
259	C₂²	C₃×#3701	1	2	yes	—
260	C₂²	C₂×C₃×#611	1	1	yes	characteristic
261	C₄	—	4	2	no	—
262	C₄	—	18	4	no	—
263	C₄	—	36	2	no	—
264	C₄	C₃×Dic₉⋊C₄	1	4	yes	—
265	C₃	—	2	1	no	—
266	C₃	#10588657	1	1	yes	characteristic, direct product factor
267	C₃	C₃×#4977	1	1	yes	characteristic
268	C₂	C₃×#63452	1	2	yes	—
269	C₂	C₃×#63454	1	2	yes	—
270	C₂	C₂×C₃×Dic₉⋊C₄	1	1	yes	characteristic
271	C₂	C₃×#63498	1	1	yes	characteristic
272	C₂	C₃×#63509	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	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊C₄	D₉
hypocenter	C₉	C₃×#382
LCS	G ▷ C₂²×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
6. #10588657	C₃	direct (N × H)
13. C₃×C₄×Dic₉	C₄	semidirect
52. C₄×Dic₉	C₃×C₄	semidirect
151. C₃×C₉	#382	semidirect
229. C₉	C₃×#382	semidirect
266. C₃	#10588657	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
C₂×C₃×C₄×Dic₉	2
C₃×C₉⋊#95	2
C₃×#4977	3
#10588657	3

Subgroups — C3×#10588657 #13142551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588657 #13142551