Subgroups — C₃×#10589815 #13153256

Order 1728.

816 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₃×#63498	C₂	1	4	yes	maximal, complemented
3	C₃×#63497	C₂	1	4	yes	maximal, complemented
4	C₃×D₄×Dic₉	C₂	1	4	yes	maximal, complemented
5	C₃×C₄×Dic₁₈	C₂	1	2	yes	maximal, complemented
6	C₃×#5713	—	3	1	no	maximal
7	#10589815	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₄×C₉	C₂²	1	4	yes	—
9	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₉	C₂²	1	4	yes	—
11	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₃×#3676	C₂²	1	8	yes	—
14	C₃×#3676	C₂²	1	4	yes	—
15	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
16	C₃×Dic₉⋊C₄	C₂²	1	4	yes	—
17	C₃×C₄×Dic₉	C₂²	1	4	yes	—
18	C₃²×#127	D₃	1	1	yes	characteristic
19	C₃×#936	—	3	4	no	—
20	C₃×#935	—	3	4	no	—
21	C₃×D₄×Dic₃	—	3	4	no	—
22	C₃×C₄×Dic₆	—	3	2	no	—
23	C₉×#127	—	2	1	no	—
24	C₉×#127	C₂×C₃	1	1	yes	characteristic
25	#63498	C₂×C₃	1	4	yes	complemented
26	#63497	C₂×C₃	1	4	yes	complemented
27	D₄×Dic₉	C₂×C₃	1	4	yes	complemented
28	C₄×Dic₁₈	C₂×C₃	1	2	yes	complemented
29	C₂³×C₃×C₉	C₂³	1	4	yes	—
30	C₃×D₄×C₉	—	2	8	no	—
31	C₃×D₄×C₉	—	4	2	no	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
34	C₂×C₃×Dic₉	—	4	4	no	—
35	C₂×C₃×Dic₉	C₂³	1	4	yes	—
36	C₂×C₃×Dic₉	C₂³	1	4	yes	—
37	C₃×Dic₁₈	—	2	4	no	—
38	C₃×#559	—	9	1	no	—
39	#5713	—	3	1	no	—
40	C₂×C₃²×D₄	C₂×D₃	1	4	yes	—
41	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
42	C₂²×C₃×Dic₃	—	3	4	no	—
43	C₂×C₃×Dic₆	—	3	2	no	—
44	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
45	C₃×#217	—	3	8	no	—
46	C₃×#217	—	3	4	no	—
47	C₃×C₄⋊Dic₃	—	3	2	no	—
48	C₃×Dic₃⋊C₄	—	3	4	no	—
49	C₃×C₄×Dic₃	—	3	4	no	—
50	C₂×D₄×C₉	—	2	4	no	—
51	C₂×D₄×C₉	—	2	2	no	—
52	C₂×D₄×C₉	C₂²×C₃	1	4	yes	—
53	C₂×D₄×C₉	C₂²×C₃	1	2	yes	—
54	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
55	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
56	C₄²×C₉	—	2	1	no	—
57	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
58	#3676	C₂²×C₃	1	8	yes	—
59	#3676	C₂²×C₃	1	4	yes	—
60	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
61	Dic₉⋊C₄	C₂²×C₃	1	4	yes	—
62	C₄×Dic₉	C₂²×C₃	1	4	yes	—
63	C₂²×C₃×C₉	—	2	8	no	—
64	C₂²×C₃×C₉	—	2	4	no	—
65	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
66	C₃×C₄×C₉	—	2	1	no	—
67	C₃×C₄×C₉	#41	1	4	yes	—
68	C₃×Dic₉	—	2	4	no	—
69	C₃×Dic₉	—	2	4	no	—
70	C₃×#127	—	2	1	no	—
71	C₃×#127	D₉	1	1	yes	characteristic
72	C₃×#127	C₃×D₃	1	1	yes	characteristic
73	C₃×#124	—	9	4	no	—
74	C₃×#123	—	9	4	no	—
75	C₃×C₄×Q₈	—	9	2	no	—
76	C₃×C₄×D₄	—	9	4	no	—
77	#936	—	3	4	no	—
78	#935	—	3	4	no	—
79	D₄×Dic₃	—	3	4	no	—
80	C₄×Dic₆	—	3	2	no	—
81	C₂³×C₃²	C₂²×D₃	1	4	yes	—
82	C₃²×D₄	—	2	8	no	—
83	C₃²×D₄	—	4	2	no	—
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
86	C₂×C₃×Dic₃	—	3	4	no	—
87	C₂×C₃×Dic₃	—	3	4	no	—
88	C₂×C₃×Dic₃	—	12	4	no	—
89	C₃×Dic₆	—	6	4	no	—
90	C₂³×C₉	—	2	4	no	—
91	C₂³×C₉	C₂³×C₃	1	4	yes	—
92	D₄×C₉	—	2	8	no	—
93	D₄×C₉	—	4	8	no	—
94	D₄×C₉	—	4	2	no	—
95	D₄×C₉	—	4	4	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
100	C₂×Dic₉	—	4	4	no	—
101	C₂×Dic₉	C₂³×C₃	1	4	yes	—
102	C₂×Dic₉	C₂³×C₃	1	4	yes	—
103	Dic₁₈	—	2	4	no	—
104	#559	—	9	1	no	Sylow
105	C₂×C₃×C₉	—	4	4	no	—
106	C₂×C₃×C₉	C₂×#41	1	2	yes	—
107	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂×C₃×Q₈	—	9	2	no	—
109	C₂×C₃×D₄	—	2	4	no	—
110	C₂×C₃×D₄	—	2	2	no	—
111	C₂×C₃×D₄	C₂×D₉	1	4	yes	—
112	C₂×C₃×D₄	C₂×D₉	1	2	yes	—
113	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	—
114	C₂×C₃×D₄	C₂×C₃×D₃	1	2	yes	—
115	C₂²×C₃×C₄	—	9	4	no	—
116	C₂²×Dic₃	—	3	4	no	—
117	C₂×Dic₆	—	3	2	no	—
118	C₃×C₄⋊C₄	—	9	4	no	—
119	C₃×C₄⋊C₄	—	9	2	no	—
120	C₃×C₂²⋊C₄	—	9	8	no	—
121	C₃×C₂²⋊C₄	—	9	4	no	—
122	C₃×C₄²	—	2	1	no	—
123	C₃×C₄²	—	9	4	no	—
124	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
125	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
126	#217	—	3	8	no	—
127	#217	—	3	4	no	—
128	C₄⋊Dic₃	—	3	2	no	—
129	Dic₃⋊C₄	—	3	4	no	—
130	C₄×Dic₃	—	3	4	no	—
131	C₂²×C₃²	—	2	8	no	—
132	C₂²×C₃²	—	2	4	no	—
133	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
134	C₃²×C₄	—	2	1	no	—
135	C₃²×C₄	#237	1	4	yes	—
136	C₃×Dic₃	—	6	4	no	—
137	C₃×Dic₃	—	6	4	no	—
138	C₂²×C₉	—	2	8	no	—
139	C₂²×C₉	—	2	4	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	—	4	8	no	—
142	C₂²×C₉	—	4	4	no	—
143	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
144	C₄×C₉	—	2	2	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	2	4	no	—
147	C₄×C₉	C₃×#41	1	4	yes	—
148	Dic₉	—	2	4	no	—
149	Dic₉	—	2	4	no	—
150	#127	C₃×D₉	1	1	yes	characteristic
151	#124	—	9	4	no	—
152	#123	—	9	4	no	—
153	C₄×Q₈	—	9	2	no	—
154	C₄×D₄	—	9	4	no	—
155	C₃×C₉	#559	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	2	4	no	—
157	C₂³×C₃	C₂²×D₉	1	4	yes	—
158	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
159	C₃×Q₈	—	18	4	no	—
160	C₃×D₄	—	2	8	no	—
161	C₃×D₄	—	2	8	no	—
162	C₃×D₄	—	4	8	no	—
163	C₃×D₄	—	4	2	no	—
164	C₃×D₄	—	4	2	no	—
165	C₃×D₄	—	4	4	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	9	4	no	—
169	C₂×C₃×C₄	—	9	4	no	—
170	C₂×C₃×C₄	—	36	4	no	—
171	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
173	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
175	C₂×Dic₃	—	3	4	no	—
176	C₂×Dic₃	—	3	4	no	—
177	C₂×Dic₃	—	12	4	no	—
178	Dic₆	—	6	4	no	—
179	C₂×C₃²	—	4	4	no	—
180	C₂×C₃²	C₂×#237	1	2	yes	—
181	C₂×C₃²	#1004	1	1	yes	characteristic
182	C₂×C₉	—	2	2	no	—
183	C₂×C₉	—	2	1	no	—
184	C₂×C₉	—	4	4	no	—
185	C₂×C₉	—	8	4	no	—
186	C₂×C₉	C₂×C₃×#41	1	2	yes	—
187	C₂×C₉	C₃×H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×Q₈	—	9	2	no	—
189	C₂×D₄	C₂×C₃×D₉	1	4	yes	—
190	C₂×D₄	C₂×C₃×D₉	1	2	yes	—
191	C₂²×C₄	—	9	4	no	—
192	C₄⋊C₄	—	9	2	no	—
193	C₄⋊C₄	—	9	4	no	—
194	C₂²⋊C₄	—	9	8	no	—
195	C₂²⋊C₄	—	9	4	no	—
196	C₄²	—	9	4	no	—
197	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
198	C₂²×C₃	—	2	8	no	—
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	8	no	—
201	C₂²×C₃	—	2	4	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	4	8	no	—
204	C₂²×C₃	—	4	4	no	—
205	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
206	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	18	4	no	—
212	C₃×C₄	—	18	4	no	—
213	C₃×C₄	#3699	1	4	yes	—
214	C₃×C₄	C₃×#237	1	4	yes	—
215	Dic₃	—	6	4	no	—
216	Dic₃	—	6	4	no	—
217	C₃²	#5713	1	1	yes	characteristic
218	C₉	—	2	1	no	—
219	C₉	C₃×#559	1	1	yes	characteristic, complemented
220	C₂³	C₂²×C₃×D₉	1	4	yes	—
221	Q₈	—	18	4	no	—
222	D₄	—	2	8	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	9	4	no	—
225	C₂×C₄	—	9	4	no	—
226	C₂×C₄	—	36	4	no	—
227	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	4	4	no	—
232	C₂×C₃	—	4	4	no	—
233	C₂×C₃	—	8	4	no	—
234	C₂×C₃	C₂×#3699	1	2	yes	—
235	C₂×C₃	#63710	1	1	yes	characteristic
236	C₂×C₃	C₂×C₃×#237	1	2	yes	—
237	C₂×C₃	C₃×#1004	1	1	yes	characteristic
238	C₂²	—	2	8	no	—
239	C₂²	—	2	4	no	—
240	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	18	4	no	—
243	C₄	—	18	4	no	—
244	C₄	C₃×#3699	1	4	yes	—
245	C₃	—	2	1	no	—
246	C₃	#10589815	1	1	yes	characteristic, direct product factor
247	C₃	C₃×#5713	1	1	yes	characteristic
248	C₂	—	4	4	no	—
249	C₂	C₂×C₃×#3699	1	2	yes	—
250	C₂	C₃×#63710	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#127	C₂
socle	C₂²×C₃²	C₂³×D₃
prime radical	C₃²×#127	D₃
hypercenter	C₃×#127	D₉
hypocenter	C₉	C₃×#559
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₃×#63498	C₂	semidirect
3. C₃×#63497	C₂	semidirect
4. C₃×D₄×Dic₉	C₂	semidirect
5. C₃×C₄×Dic₁₈	C₂	semidirect
7. #10589815	C₃	direct (N × H)
25. #63498	C₂×C₃	semidirect
26. #63497	C₂×C₃	semidirect
27. D₄×Dic₉	C₂×C₃	semidirect
28. C₄×Dic₁₈	C₂×C₃	semidirect
155. C₃×C₉	#559	semidirect
219. C₉	C₃×#559	semidirect
246. C₃	#10589815	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×#127	2
C₃×#63498	2
C₃×#63497	2
C₃×D₄×Dic₉	2
C₃×C₄×Dic₁₈	2
C₃×#5713	3
#10589815	3

Subgroups — C3×#10589815 #13153256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589815 #13153256