Subgroups — C₃×#10588522 #13142018

Order 1728.

404 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3663	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×D₃₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×(C₂×D₉)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4838	—	3	1	no	maximal
7	#10588522	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₃₆	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×D₉	C₂²	1	2	yes	—
10	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃×(C₂×D₉)⋊C₄	C₄	1	2	yes	—
13	C₃×C₄⋊Dic₉	C₄	1	1	yes	characteristic
14	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×#3663	—	2	4	no	—
17	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	2	no	—
18	C₃×C₄×D₁₂	—	3	1	no	—
19	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
20	C₃×(C₂×D₃)⋊C₈	—	3	2	no	—
21	C₃×C₄×C₃⋊C₈	—	3	1	no	—
22	C₂×#3663	C₂×C₃	1	2	yes	complemented
23	C₄×D₃₆	C₂×C₃	1	1	yes	characteristic
24	C₉×C₄⋊C₈	—	2	1	no	—
25	C₉×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
26	(C₂×D₉)⋊C₈	C₂×C₃	1	2	yes	complemented
27	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×C₃×D₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
33	C₃×D₃₆	—	2	2	no	—
34	C₃×C₄×D₉	—	2	2	no	—
35	C₃×C₄×D₉	—	2	2	no	—
36	C₃×C₈×C₉	—	2	2	no	—
37	C₃×C₉⋊C₈	—	2	1	no	—
38	C₃×C₉⋊C₈	D₄	1	2	yes	complemented
39	C₃×#436	—	9	1	no	—
40	#4838	—	3	1	no	—
41	C₂×C₃×D₁₂	—	3	1	no	—
42	C₂×C₃×C₄×D₃	—	3	2	no	—
43	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
44	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
45	C₃×(C₂×D₃)⋊C₄	—	3	2	no	—
46	C₃×C₄⋊Dic₃	—	3	1	no	—
47	C₂×C₃×C₃⋊C₈	—	3	1	no	—
48	C₂×C₃×C₃⋊C₈	—	3	1	no	—
49	C₃×ΓL(1,𝔽₂₅)	—	6	4	no	—
50	C₂×D₃₆	C₃×C₄	1	1	yes	characteristic
51	C₂×C₄×D₉	C₂²×C₃	1	2	yes	—
52	C₂×C₈×C₉	—	2	2	no	—
53	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
56	(C₂×D₉)⋊C₄	C₃×C₄	1	2	yes	—
57	C₄⋊Dic₉	C₃×C₄	1	1	yes	characteristic
58	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
60	#3663	—	2	4	no	—
61	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×D₉	—	2	2	no	—
63	C₂×C₃×D₉	—	2	2	no	—
64	C₂×C₃×D₉	—	2	2	no	—
65	C₃×C₄×C₉	—	2	1	no	—
66	C₃×C₄×C₉	M₁₆	1	2	yes	complemented
67	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₃×C₄×C₉	#41	1	1	yes	characteristic
69	C₃×Dic₉	—	2	2	no	—
70	C₂×C₃×M₁₆	—	9	2	no	—
71	C₃×C₄×D₄	—	9	1	no	—
72	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
73	C₄×D₁₂	—	3	1	no	—
74	C₃×C₄⋊C₈	—	2	1	no	—
75	C₃×C₄⋊C₈	D₉	1	1	yes	characteristic, complemented
76	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
77	C₃×C₂²⋊C₈	—	9	2	no	—
78	C₃×C₄×C₈	—	9	1	no	—
79	(C₂×D₃)⋊C₈	—	3	2	no	—
80	C₄×C₃⋊C₈	—	3	1	no	—
81	C₂²×C₃×D₃	—	3	2	no	—
82	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₃×Dic₃	—	3	2	no	—
86	C₃×D₁₂	—	6	2	no	—
87	C₃×C₄×D₃	—	6	2	no	—
88	C₃×C₄×D₃	—	6	2	no	—
89	C₂²×D₉	C₂×C₃×C₄	1	2	yes	—
90	C₃²×C₈	—	2	2	no	—
91	C₃×C₃⋊C₈	—	3	2	no	—
92	C₃×C₃⋊C₈	—	6	1	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
99	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
100	D₃₆	—	2	2	no	—
101	C₄×D₉	—	2	2	no	—
102	C₄×D₉	—	2	2	no	—
103	C₈×C₉	—	2	2	no	—
104	C₈×C₉	—	4	2	no	—
105	C₉⋊C₈	—	2	1	no	—
106	C₉⋊C₈	C₃×D₄	1	2	yes	complemented
107	#436	—	9	1	no	Sylow
108	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₃×C₉	#131	1	1	yes	characteristic
111	C₃×D₉	—	4	2	no	—
112	C₂×C₃×D₄	—	9	1	no	—
113	C₂²×C₃×C₄	—	9	2	no	—
114	C₂×D₁₂	—	3	1	no	—
115	C₂×C₄×D₃	—	3	2	no	—
116	C₃×M₁₆	—	18	4	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	—	9	1	no	—
119	C₂×C₃×C₈	—	9	1	no	—
120	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
121	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
122	C₃×C₄⋊C₄	—	9	1	no	—
123	C₃×C₂²⋊C₄	—	9	2	no	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
126	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
127	(C₂×D₃)⋊C₄	—	3	2	no	—
128	C₄⋊Dic₃	—	3	1	no	—
129	C₂×C₃⋊C₈	—	3	1	no	—
130	C₂×C₃⋊C₈	—	3	1	no	—
131	ΓL(1,𝔽₂₅)	—	6	4	no	—
132	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
133	C₂×C₃×D₃	—	6	2	no	—
134	C₂×C₃×D₃	—	6	2	no	—
135	C₂×C₃×D₃	—	6	2	no	—
136	C₃²×C₄	—	2	1	no	—
137	C₃²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
138	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
139	C₃²×C₄	#239	1	1	yes	characteristic
140	C₃×Dic₃	—	6	2	no	—
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
143	C₂×D₉	—	2	2	no	—
144	C₂×D₉	—	2	2	no	—
145	C₂×D₉	—	2	2	no	—
146	C₄×C₉	—	2	1	no	—
147	C₄×C₉	—	2	2	no	—
148	C₄×C₉	—	2	1	no	—
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	—	4	1	no	—
151	C₄×C₉	C₃×M₁₆	1	2	yes	complemented
152	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
153	C₄×C₉	C₃×#41	1	1	yes	characteristic
154	Dic₉	—	2	2	no	—
155	C₂×M₁₆	—	9	2	no	—
156	C₄×D₄	—	9	1	no	—
157	C₄⋊C₈	C₃×D₉	1	1	yes	characteristic, complemented
158	C₂²⋊C₈	—	9	2	no	—
159	C₄×C₈	—	9	1	no	—
160	C₃×C₉	#436	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₃	—	9	2	no	—
162	C₂²×D₃	—	3	2	no	—
163	C₃×D₄	—	18	2	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	9	2	no	—
168	C₂×C₃×C₄	—	18	2	no	—
169	C₂×C₃×C₄	—	18	2	no	—
170	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
171	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
173	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
176	C₂×Dic₃	—	3	2	no	—
177	D₁₂	—	6	2	no	—
178	C₄×D₃	—	6	2	no	—
179	C₄×D₃	—	6	2	no	—
180	C₃×C₈	—	2	2	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃×C₈	—	4	2	no	—
183	C₃×C₈	—	9	2	no	—
184	C₃×C₈	—	18	1	no	—
185	C₃⋊C₈	—	3	2	no	—
186	C₃⋊C₈	—	6	1	no	—
187	C₂×C₃²	#893	1	1	yes	characteristic
188	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
189	C₂×C₃²	#907	1	1	yes	characteristic
190	C₃×D₃	—	12	2	no	—
191	C₂×C₉	—	2	1	no	—
192	C₂×C₉	—	2	1	no	—
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
195	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
196	C₂×C₉	C₃×#131	1	1	yes	characteristic
197	D₉	—	4	2	no	—
198	C₂×D₄	—	9	1	no	—
199	C₂²×C₄	—	9	2	no	—
200	M₁₆	—	18	4	no	—
201	C₂×C₈	—	9	1	no	—
202	C₂×C₈	—	9	1	no	—
203	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
204	C₄⋊C₄	—	9	1	no	—
205	C₂²⋊C₄	—	9	2	no	—
206	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	18	2	no	—
209	C₂²×C₃	—	18	2	no	—
210	C₂²×C₃	—	18	2	no	—
211	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
212	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
213	C₂×D₃	—	6	2	no	—
214	C₂×D₃	—	6	2	no	—
215	C₂×D₃	—	6	2	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	4	1	no	—
222	C₃×C₄	—	18	2	no	—
223	C₃×C₄	#3663	1	2	yes	complemented
224	C₃×C₄	D₄×D₉	1	1	yes	characteristic
225	C₃×C₄	#3701	1	1	yes	characteristic
226	C₃×C₄	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
227	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
228	C₃×C₄	C₃×#239	1	1	yes	characteristic
229	Dic₃	—	6	2	no	—
230	C₃²	#4838	1	1	yes	characteristic
231	C₉	—	2	1	no	—
232	C₉	C₃×#436	1	1	yes	characteristic, complemented
233	C₂³	—	9	2	no	—
234	D₄	—	18	2	no	—
235	C₂×C₄	—	9	2	no	—
236	C₂×C₄	—	18	2	no	—
237	C₂×C₄	—	18	2	no	—
238	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
239	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	9	2	no	—
243	C₈	—	18	1	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	36	2	no	—
248	C₂×C₃	#63455	1	1	yes	characteristic
249	C₂×C₃	C₂×#3663	1	1	yes	characteristic
250	C₂×C₃	#63469	1	1	yes	characteristic
251	C₂×C₃	C₃×#893	1	1	yes	characteristic
252	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
253	C₂×C₃	C₃×#907	1	1	yes	characteristic
254	D₃	—	12	2	no	—
255	C₂²	—	18	2	no	—
256	C₂²	—	18	2	no	—
257	C₂²	—	18	2	no	—
258	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	18	2	no	—
261	C₄	C₃×#3663	1	2	yes	complemented
262	C₄	C₃×D₄×D₉	1	1	yes	characteristic
263	C₄	C₃×#3701	1	1	yes	characteristic
264	C₃	—	2	1	no	—
265	C₃	#10588522	1	1	yes	characteristic, direct product factor
266	C₃	C₃×#4838	1	1	yes	characteristic
267	C₂	—	36	2	no	—
268	C₂	C₃×#63455	1	1	yes	characteristic
269	C₂	C₂×C₃×#3663	1	1	yes	characteristic
270	C₂	C₃×#63469	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂×C₃×D₃₆	C₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	D₉
hypocenter	C₉	C₃×#436
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3663	C₂	semidirect
3. C₃×C₉×C₄⋊C₈	C₂	semidirect
4. C₃×(C₂×D₉)⋊C₈	C₂	semidirect
5. C₃×C₄×C₉⋊C₈	C₂	semidirect
7. #10588522	C₃	direct (N × H)
22. C₂×#3663	C₂×C₃	semidirect
25. C₉×C₄⋊C₈	C₂×C₃	semidirect
26. (C₂×D₉)⋊C₈	C₂×C₃	semidirect
27. C₄×C₉⋊C₈	C₂×C₃	semidirect
38. C₃×C₉⋊C₈	D₄	semidirect
66. C₃×C₄×C₉	M₁₆	semidirect
75. C₃×C₄⋊C₈	D₉	semidirect
106. C₉⋊C₈	C₃×D₄	semidirect
151. C₄×C₉	C₃×M₁₆	semidirect
157. C₄⋊C₈	C₃×D₉	semidirect
160. C₃×C₉	#436	semidirect
223. C₃×C₄	#3663	semidirect
232. C₉	C₃×#436	semidirect
261. C₄	C₃×#3663	semidirect
265. C₃	#10588522	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×#3663	2
C₃×C₄×D₃₆	2
C₃×C₉×C₄⋊C₈	2
C₃×(C₂×D₉)⋊C₈	2
C₃×C₄×C₉⋊C₈	2
C₃×#4838	3
#10588522	3

Subgroups — C3×#10588522 #13142018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588522 #13142018