Subgroups — C₃×#10588380 #13141424

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#63467	C₂	1	1	yes	maximal, characteristic
2	C₃×#63464	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63372	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63364	C₂	1	2	yes	maximal, complemented
6	C₃×#4700	—	3	1	no	maximal
7	#10588380	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×#3697	C₂²	1	2	yes	—
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃×#3667	C₂²	1	2	yes	—
13	C₃×D₇₂	C₄	1	1	yes	characteristic, complemented
14	C₃×#3664	C₄	1	2	yes	complemented
15	C₃×#3663	—	2	2	no	—
16	C₃×C₈×D₉	—	2	2	no	—
17	C₃×Dic₃₆	C₄	1	1	yes	characteristic, complemented
18	C₃×#904	—	3	1	no	—
19	C₃×#901	—	3	2	no	—
20	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
21	C₃×#819	—	3	1	no	—
22	C₃×#805	—	3	2	no	—
23	#63467	C₂×C₃	1	1	yes	characteristic
24	#63464	C₂×C₃	1	2	yes	complemented
25	C₄×C₈×C₉	—	2	1	no	—
26	C₄×C₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
27	#63372	C₂×C₃	1	1	yes	characteristic, complemented
28	#63364	C₂×C₃	1	2	yes	complemented
29	C₂×C₃×C₄×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₃×#611	—	2	2	no	—
32	C₃×D₃₆	C₂×C₄	1	2	yes	—
33	C₃×C₄×D₉	—	2	2	no	—
34	C₃×Dic₁₈	C₂×C₄	1	2	yes	—
35	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
37	C₃×C₈×C₉	D₄	1	1	yes	characteristic
38	C₃×C₈×C₉	D₄	1	1	yes	characteristic
39	C₃×C₉⋊C₈	—	2	2	no	—
40	C₃×#443	—	9	1	no	—
41	#4700	—	3	1	no	—
42	C₃×#235	—	3	2	no	—
43	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
45	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₃×#208	—	3	2	no	—
47	C₃×Dic₁₂	—	3	1	no	—
48	C₃×D₂₄	—	3	1	no	—
49	C₃×#204	—	3	2	no	—
50	C₃×ΓL(1,𝔽₂₅)	—	6	2	no	—
51	C₃×D₃×C₈	—	6	2	no	—
52	#3697	C₂²×C₃	1	2	yes	—
53	C₂×C₈×C₉	—	2	1	no	—
54	C₂×C₈×C₉	—	2	1	no	—
55	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
57	C₄²×C₉	—	2	1	no	—
58	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
59	#3667	C₂²×C₃	1	2	yes	—
60	D₇₂	C₃×C₄	1	1	yes	characteristic, complemented
61	#3664	C₃×C₄	1	2	yes	complemented
62	#3663	—	2	2	no	—
63	C₈×D₉	—	2	2	no	—
64	Dic₃₆	C₃×C₄	1	1	yes	characteristic, complemented
65	C₂²×C₃×C₉	#41	1	1	yes	characteristic
66	C₂×C₃×D₉	—	2	2	no	—
67	C₃×C₄×C₉	—	2	2	no	—
68	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₃×Dic₉	—	2	2	no	—
71	C₃×#135	—	9	1	no	—
72	C₃×#131	—	9	2	no	—
73	#904	—	3	1	no	—
74	#901	—	3	2	no	—
75	C₃×#108	—	9	1	no	—
76	C₃×#104	—	9	2	no	—
77	C₃×C₄×C₈	—	2	1	no	—
78	C₃×C₄×C₈	D₉	1	1	yes	characteristic, complemented
79	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
80	#819	—	3	1	no	—
81	#805	—	3	2	no	—
82	C₂×C₃²×C₄	—	2	1	no	—
83	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₃×#67	—	6	2	no	—
85	C₃×D₁₂	—	3	2	no	—
86	C₃×C₄×D₃	—	6	2	no	—
87	C₃×Dic₆	—	3	2	no	—
88	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
89	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
90	C₃²×C₈	D₁₂	1	1	yes	characteristic
91	C₃²×C₈	D₁₂	1	1	yes	characteristic
92	C₃×C₃⋊C₈	—	6	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
97	#611	—	2	2	no	—
98	D₃₆	C₂×C₃×C₄	1	2	yes	—
99	C₄×D₉	—	2	2	no	—
100	Dic₁₈	C₂×C₃×C₄	1	2	yes	—
101	C₈×C₉	—	2	1	no	—
102	C₈×C₉	—	2	1	no	—
103	C₈×C₉	—	2	1	no	—
104	C₈×C₉	—	2	1	no	—
105	C₈×C₉	C₂×C₃×C₄	1	1	yes	characteristic
106	C₈×C₉	C₂×C₃×C₄	1	1	yes	characteristic
107	C₈×C₉	C₃×D₄	1	1	yes	characteristic
108	C₈×C₉	C₃×D₄	1	1	yes	characteristic
109	C₉⋊C₈	—	2	2	no	—
110	#443	—	9	1	no	Sylow
111	C₂×C₃×C₉	—	2	1	no	—
112	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
113	C₃×D₉	—	4	2	no	—
114	C₃×#41	—	9	2	no	—
115	#235	—	3	2	no	—
116	C₃×Q₁₆	—	9	1	no	—
117	C₃×SD₁₆	—	9	2	no	—
118	C₃×D₈	—	9	1	no	—
119	C₃×M₁₆	—	9	2	no	—
120	C₃×M₁₆	—	18	2	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	18	2	no	—
124	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
130	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
131	#208	—	3	2	no	—
132	Dic₁₂	—	3	1	no	—
133	D₂₄	—	3	1	no	—
134	#204	—	3	2	no	—
135	ΓL(1,𝔽₂₅)	—	6	2	no	—
136	D₃×C₈	—	6	2	no	—
137	C₂²×C₃²	#235	1	1	yes	characteristic
138	C₂×C₃×D₃	—	6	2	no	—
139	C₃²×C₄	—	2	2	no	—
140	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
141	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
142	C₃×Dic₃	—	6	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	C₃×#41	1	1	yes	characteristic
145	C₂×D₉	—	2	2	no	—
146	C₄×C₉	—	2	4	no	—
147	C₄×C₉	—	2	2	no	—
148	C₄×C₉	—	2	1	no	—
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
151	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
152	Dic₉	—	2	2	no	—
153	#135	—	9	1	no	—
154	#131	—	9	2	no	—
155	#108	—	9	1	no	—
156	#104	—	9	2	no	—
157	C₄×C₈	C₃×D₉	1	1	yes	characteristic, complemented
158	C₃×C₉	#443	1	1	yes	characteristic, complemented, Sylow
159	C₃×Q₈	—	9	2	no	—
160	C₃×D₄	—	9	2	no	—
161	C₃×D₄	—	18	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	18	2	no	—
167	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
169	#67	—	6	2	no	—
170	D₁₂	—	3	2	no	—
171	C₄×D₃	—	6	2	no	—
172	Dic₆	—	3	2	no	—
173	C₃×C₈	—	2	1	no	—
174	C₃×C₈	—	2	1	no	—
175	C₃×C₈	—	2	1	no	—
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	—	18	2	no	—
178	C₃×C₈	C₄×D₉	1	1	yes	characteristic
179	C₃×C₈	C₄×D₉	1	1	yes	characteristic
180	C₃×C₈	D₃₆	1	1	yes	characteristic
181	C₃×C₈	D₃₆	1	1	yes	characteristic
182	C₃×C₈	C₃×C₄×D₃	1	1	yes	characteristic
183	C₃×C₈	C₃×C₄×D₃	1	1	yes	characteristic
184	C₃×C₈	C₃×D₁₂	1	1	yes	characteristic
185	C₃×C₈	C₃×D₁₂	1	1	yes	characteristic
186	C₃⋊C₈	—	6	2	no	—
187	C₂×C₃²	—	2	1	no	—
188	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
189	C₃×D₃	—	12	2	no	—
190	C₂×C₉	—	2	2	no	—
191	C₂×C₉	—	2	1	no	—
192	C₂×C₉	—	2	1	no	—
193	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
194	D₉	—	4	2	no	—
195	#41	—	9	2	no	—
196	Q₁₆	—	9	1	no	—
197	SD₁₆	—	9	2	no	—
198	D₈	—	9	1	no	—
199	M₁₆	—	9	2	no	—
200	M₁₆	—	18	2	no	—
201	C₂×C₈	—	18	2	no	—
202	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
203	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
204	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	18	2	no	—
207	C₂²×C₃	#3697	1	1	yes	characteristic
208	C₂²×C₃	C₃×#235	1	1	yes	characteristic
209	C₂×D₃	—	6	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	18	2	no	—
216	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
217	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
218	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
219	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
220	Dic₃	—	6	2	no	—
221	C₃²	#4700	1	1	yes	characteristic
222	C₉	—	2	1	no	—
223	C₉	C₃×#443	1	1	yes	characteristic, complemented
224	Q₈	—	9	2	no	—
225	D₄	—	9	2	no	—
226	D₄	—	18	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	18	2	no	—
229	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
230	C₈	—	18	2	no	—
231	C₈	C₃×C₄×D₉	1	1	yes	characteristic
232	C₈	C₃×C₄×D₉	1	1	yes	characteristic
233	C₈	C₃×D₃₆	1	1	yes	characteristic
234	C₈	C₃×D₃₆	1	1	yes	characteristic
235	C₂×C₃	—	2	2	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	36	2	no	—
240	C₂×C₃	C₄×D₃₆	1	1	yes	characteristic
241	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
242	D₃	—	12	2	no	—
243	C₂²	—	18	2	no	—
244	C₂²	C₃×#3697	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	18	2	no	—
247	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
248	C₄	C₂×C₃×D₃₆	1	1	yes	characteristic
249	C₃	—	2	1	no	—
250	C₃	#10588380	1	1	yes	characteristic, direct product factor
251	C₃	C₃×#4700	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	36	2	no	—
254	C₂	C₃×C₄×D₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	D₃₆
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂×C₃²	C₄×D₁₂
prime radical	C₃×#63467	C₂
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄×C₈	D₉
hypocenter	C₉	C₃×#443
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×#63464	C₂	semidirect
3. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₃×#63372	C₂	semidirect
5. C₃×#63364	C₂	semidirect
7. #10588380	C₃	direct (N × H)
13. C₃×D₇₂	C₄	semidirect
14. C₃×#3664	C₄	semidirect
17. C₃×Dic₃₆	C₄	semidirect
24. #63464	C₂×C₃	semidirect
26. C₄×C₈×C₉	C₂×C₃	semidirect
27. #63372	C₂×C₃	semidirect
28. #63364	C₂×C₃	semidirect
60. D₇₂	C₃×C₄	semidirect
61. #3664	C₃×C₄	semidirect
64. Dic₃₆	C₃×C₄	semidirect
78. C₃×C₄×C₈	D₉	semidirect
157. C₄×C₈	C₃×D₉	semidirect
158. C₃×C₉	#443	semidirect
223. C₉	C₃×#443	semidirect
250. C₃	#10588380	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×#63467	2
C₃×#63464	2
C₃×C₄×C₈×C₉	2
C₃×#63372	2
C₃×#63364	2
C₃×#4700	3
#10588380	3

Subgroups — C3×#10588380 #13141424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588380 #13141424