Subgroups — C₃²×#4708 #13162145

Order 1728.

940 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×#872	C₂	1	1	yes	maximal, characteristic
4	C₃³×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#405	—	3	1	no	maximal
9	C₃×#4708	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃²×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
15	C₃²×Dic₃⋊C₄	C₄	1	2	yes	—
16	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃²×ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
20	C₃²×D₃×C₈	C₄	1	4	yes	complemented
21	C₂×C₃²×M₁₆	—	3	1	no	—
22	C₂²×C₃²×C₈	—	3	1	no	—
23	C₃²×#117	—	3	1	no	—
24	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	4	yes	—
25	C₂×C₃×D₃×C₈	C₂×C₃	1	4	yes	—
26	C₃×#872	C₂×C₃	1	4	yes	—
27	C₃²×#97	—	2	4	no	—
28	C₃²×#97	—	3	1	no	—
29	C₃²×#97	D₃	1	1	yes	characteristic, complemented
30	C₃²×#97	C₂×C₃	1	4	yes	complemented
31	C₃²×C₄×C₈	—	3	1	no	—
32	C₃²×C₄×C₈	—	3	1	no	—
33	C₃×#815	C₂×C₃	1	4	yes	complemented
34	C₃×C₈×Dic₃	C₂×C₃	1	4	yes	complemented
35	C₃×C₄×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
36	C₂²×C₃²×D₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
38	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
39	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
40	C₂×C₃²×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₃²×C₄×D₃	C₂×C₄	1	2	yes	—
42	C₃²×C₄×D₃	C₂×C₄	1	2	yes	—
43	C₃³×C₈	C₂×C₄	1	2	yes	—
44	C₃³×C₈	C₂×C₄	1	2	yes	—
45	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	—
46	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	—
47	C₃×#405	—	3	4	no	—
48	#4708	C₃²	1	1	yes	characteristic, direct product factor
49	C₂²×C₃²×C₄	—	3	1	no	—
50	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
51	C₃²×M₁₆	—	3	4	no	—
52	C₂×C₃²×C₈	—	2	4	no	—
53	C₂×C₃²×C₈	—	2	4	no	—
54	C₂×C₃²×C₈	—	3	1	no	—
55	C₂×C₃²×C₈	—	3	1	no	—
56	C₂×C₃²×C₈	—	3	4	no	—
57	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
60	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
61	C₃²×C₄⋊C₄	—	3	2	no	—
62	C₃²×C₂²⋊C₄	—	3	2	no	—
63	C₃²×C₄²	—	2	4	no	—
64	C₃²×C₄²	—	3	1	no	—
65	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
66	C₃²×C₄²	C₂²×C₃	1	4	yes	—
67	C₃×(C₂×D₃)⋊C₄	C₃×C₄	1	8	yes	—
68	C₃×Dic₃⋊C₄	C₃×C₄	1	8	yes	—
69	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
70	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
71	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
72	C₃×ΓL(1,𝔽₂₅)	C₃×C₄	1	16	yes	complemented
73	C₃×D₃×C₈	C₃×C₄	1	16	yes	complemented
74	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃²×D₃	—	2	2	no	—
76	C₂×C₃²×D₃	C₄²	1	2	yes	—
77	C₃³×C₄	—	2	2	no	—
78	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₃²×Dic₃	—	2	2	no	—
81	C₃²×Dic₃	C₄²	1	2	yes	—
82	C₂×C₃×M₁₆	—	3	4	no	—
83	C₂²×C₃×C₈	—	3	4	no	—
84	C₃×#117	—	3	4	no	—
85	C₂×ΓL(1,𝔽₂₅)	C₂×C₃²	1	1	yes	characteristic
86	C₂×D₃×C₈	C₂×C₃²	1	1	yes	characteristic
87	#872	C₂×C₃²	1	1	yes	characteristic
88	C₃×#97	—	2	4	no	—
89	C₃×#97	—	3	4	no	—
90	C₃×#97	C₃×D₃	1	4	yes	complemented
91	C₃×#97	C₂×C₃²	1	1	yes	characteristic, complemented
92	C₃×C₄×C₈	—	3	4	no	—
93	C₃×C₄×C₈	—	3	4	no	—
94	#815	C₂×C₃²	1	1	yes	characteristic, complemented
95	C₈×Dic₃	C₂×C₃²	1	1	yes	characteristic, complemented
96	C₄×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
97	C₂³×C₃²	—	3	1	no	—
98	C₂²×C₃×D₃	C₂×C₃×C₄	1	4	yes	—
99	C₂×C₃²×C₄	—	2	8	no	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	—	3	1	no	—
102	C₂×C₃²×C₄	—	3	2	no	—
103	C₂×C₃²×C₄	—	3	2	no	—
104	C₂×C₃²×C₄	—	3	2	no	—
105	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
106	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
107	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
109	C₂×C₃×Dic₃	C₂×C₃×C₄	1	8	yes	—
110	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
111	C₃×C₄×D₃	C₂×C₃×C₄	1	8	yes	—
112	C₃×C₄×D₃	C₂×C₃×C₄	1	8	yes	—
113	C₃²×C₈	—	2	8	no	—
114	C₃²×C₈	—	2	8	no	—
115	C₃²×C₈	—	3	2	no	—
116	C₃²×C₈	—	3	2	no	—
117	C₃²×C₈	C₄×D₃	1	2	yes	—
118	C₃²×C₈	C₄×D₃	1	2	yes	—
119	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
120	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
121	C₃×C₃⋊C₈	C₂×C₃×C₄	1	8	yes	—
122	C₃×C₃⋊C₈	C₂×C₃×C₄	1	8	yes	—
123	#405	—	3	1	no	Sylow
124	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₃³	#131	1	2	yes	—
126	C₃²×D₃	—	2	2	no	—
127	C₂²×C₃×C₄	—	3	4	no	—
128	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
129	C₃×M₁₆	—	3	16	no	—
130	C₂×C₃×C₈	—	2	4	no	—
131	C₂×C₃×C₈	—	2	4	no	—
132	C₂×C₃×C₈	—	3	4	no	—
133	C₂×C₃×C₈	—	3	4	no	—
134	C₂×C₃×C₈	—	3	16	no	—
135	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
136	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
137	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	3	8	no	—
140	C₃×C₂²⋊C₄	—	3	8	no	—
141	C₃×C₄²	—	2	4	no	—
142	C₃×C₄²	—	3	4	no	—
143	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
144	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
145	(C₂×D₃)⋊C₄	C₃²×C₄	1	2	yes	—
146	Dic₃⋊C₄	C₃²×C₄	1	2	yes	—
147	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
148	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
149	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
150	ΓL(1,𝔽₂₅)	C₃²×C₄	1	4	yes	complemented
151	D₃×C₈	C₃²×C₄	1	4	yes	complemented
152	C₂²×C₃²	—	2	4	no	—
153	C₂²×C₃²	—	3	2	no	—
154	C₂²×C₃²	—	6	2	no	—
155	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
157	C₂×C₃×D₃	—	2	8	no	—
158	C₂×C₃×D₃	C₃×C₄²	1	8	yes	—
159	C₃²×C₄	—	2	2	no	—
160	C₃²×C₄	—	2	4	no	—
161	C₃²×C₄	—	2	4	no	—
162	C₃²×C₄	—	2	8	no	—
163	C₃²×C₄	—	3	2	no	—
164	C₃²×C₄	—	4	8	no	—
165	C₃²×C₄	—	6	2	no	—
166	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
168	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
169	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
170	C₃×Dic₃	—	2	8	no	—
171	C₃×Dic₃	C₃×C₄²	1	8	yes	—
172	C₂×M₁₆	—	3	1	no	—
173	C₂²×C₈	—	3	1	no	—
174	#117	—	3	1	no	—
175	#97	—	3	1	no	—
176	#97	C₃²×D₃	1	1	yes	characteristic, complemented
177	C₄×C₈	—	3	1	no	—
178	C₄×C₈	—	3	1	no	—
179	C₃³	#405	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	3	4	no	—
181	C₂²×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	3	4	no	—
185	C₂×C₃×C₄	—	3	8	no	—
186	C₂×C₃×C₄	—	3	8	no	—
187	C₂×C₃×C₄	—	3	8	no	—
188	C₂×C₃×C₄	C₃×C₄×D₃	1	8	yes	—
189	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
190	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
191	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
192	C₂×Dic₃	C₂×C₃²×C₄	1	2	yes	—
193	C₂×Dic₃	C₂×C₃²×C₄	1	1	yes	characteristic
194	C₄×D₃	C₂×C₃²×C₄	1	2	yes	—
195	C₄×D₃	C₂×C₃²×C₄	1	2	yes	—
196	C₃×C₈	—	2	8	no	—
197	C₃×C₈	—	2	8	no	—
198	C₃×C₈	—	3	8	no	—
199	C₃×C₈	—	3	8	no	—
200	C₃×C₈	C₃×C₄×D₃	1	8	yes	—
201	C₃×C₈	C₃×C₄×D₃	1	8	yes	—
202	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
203	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
204	C₃⋊C₈	C₂×C₃²×C₄	1	2	yes	—
205	C₃⋊C₈	C₂×C₃²×C₄	1	2	yes	—
206	C₂×C₃²	—	2	8	no	—
207	C₂×C₃²	—	2	4	no	—
208	C₂×C₃²	—	6	2	no	—
209	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
210	C₂×C₃²	#907	1	2	yes	—
211	C₂×C₃²	C₂×C₃×C₄²	1	4	yes	—
212	C₂×C₃²	C₃×#131	1	8	yes	—
213	C₃×D₃	—	2	8	no	—
214	C₂²×C₄	—	3	1	no	—
215	M₁₆	—	3	4	no	—
216	C₂×C₈	—	3	1	no	—
217	C₂×C₈	—	3	1	no	—
218	C₂×C₈	—	3	4	no	—
219	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
220	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
221	C₄⋊C₄	—	3	2	no	—
222	C₂²⋊C₄	—	3	2	no	—
223	C₄²	—	3	1	no	—
224	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
225	C₂²×C₃	—	2	4	no	—
226	C₂²×C₃	—	3	8	no	—
227	C₂²×C₃	—	6	8	no	—
228	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
229	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
230	C₂×D₃	—	2	2	no	—
231	C₂×D₃	C₃²×C₄²	1	2	yes	—
232	C₃×C₄	—	2	8	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	4	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	3	8	no	—
237	C₃×C₄	—	4	8	no	—
238	C₃×C₄	—	6	8	no	—
239	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
240	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
241	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
242	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
243	Dic₃	—	2	2	no	—
244	Dic₃	C₃²×C₄²	1	2	yes	—
245	C₃²	—	2	4	no	—
246	C₃²	#4708	1	1	yes	characteristic, direct product factor
247	C₃²	C₃×#405	1	4	yes	complemented
248	C₂³	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	2	no	—
251	C₂×C₄	—	3	2	no	—
252	C₂×C₄	—	3	2	no	—
253	C₂×C₄	C₃²×C₄×D₃	1	2	yes	—
254	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
255	C₈	—	3	2	no	—
256	C₈	—	3	2	no	—
257	C₈	C₃²×C₄×D₃	1	2	yes	—
258	C₈	C₃²×C₄×D₃	1	2	yes	—
259	C₂×C₃	—	2	8	no	—
260	C₂×C₃	—	2	4	no	—
261	C₂×C₃	—	6	8	no	—
262	C₂×C₃	C₃×C₄²×D₃	1	4	yes	—
263	C₂×C₃	C₃×#907	1	8	yes	—
264	C₂×C₃	C₂×C₃²×C₄²	1	1	yes	characteristic
265	C₂×C₃	C₃²×#131	1	2	yes	—
266	D₃	—	2	2	no	—
267	C₂²	—	3	2	no	—
268	C₂²	—	6	2	no	—
269	C₂²	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	3	2	no	—
272	C₄	—	6	2	no	—
273	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
274	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
275	C₃	—	2	4	no	—
276	C₃	C₃×#4708	1	4	yes	direct product factor
277	C₃	C₃²×#405	1	1	yes	characteristic, complemented
278	C₂	—	6	2	no	—
279	C₂	C₃²×C₄²×D₃	1	1	yes	characteristic
280	C₂	C₃²×#907	1	2	yes	—

	subgroup	G/H
center	C₂×C₃²×C₄	C₂²×D₃
commutator	C₂×C₃	C₂×C₃²×C₄²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#97	C₂
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂²×C₃²×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×#97	D₃
hypocenter	C₃	C₃²×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×#97

Complements

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

N	H ≅ G/N	type
4. C₃³×#97	C₂	semidirect
5. C₃²×#815	C₂	semidirect
6. C₃²×C₈×Dic₃	C₂	semidirect
7. C₃²×C₄×C₃⋊C₈	C₂	semidirect
9. C₃×#4708	C₃	direct (N × H)
19. C₃²×ΓL(1,𝔽₂₅)	C₄	semidirect
20. C₃²×D₃×C₈	C₄	semidirect
29. C₃²×#97	D₃	semidirect
30. C₃²×#97	C₂×C₃	semidirect
33. C₃×#815	C₂×C₃	semidirect
34. C₃×C₈×Dic₃	C₂×C₃	semidirect
35. C₃×C₄×C₃⋊C₈	C₂×C₃	semidirect
48. #4708	C₃²	direct (N × H)
72. C₃×ΓL(1,𝔽₂₅)	C₃×C₄	semidirect
73. C₃×D₃×C₈	C₃×C₄	semidirect
90. C₃×#97	C₃×D₃	semidirect
91. C₃×#97	C₂×C₃²	semidirect
94. #815	C₂×C₃²	semidirect
95. C₈×Dic₃	C₂×C₃²	semidirect
96. C₄×C₃⋊C₈	C₂×C₃²	semidirect
150. ΓL(1,𝔽₂₅)	C₃²×C₄	semidirect
151. D₃×C₈	C₃²×C₄	semidirect
176. #97	C₃²×D₃	semidirect
179. C₃³	#405	semidirect
246. C₃²	#4708	direct (N × H)
247. C₃²	C₃×#405	semidirect
276. C₃	C₃×#4708	direct (N × H)
277. C₃	C₃²×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×ΓL(1,𝔽₂₅)	2
C₂×C₃²×D₃×C₈	2
C₃²×#872	2
C₃³×#97	2
C₃²×#815	2
C₃²×C₈×Dic₃	2
C₃²×C₄×C₃⋊C₈	2
C₃²×#405	3
C₃×#4708	3

Subgroups — C32×#4708 #13162145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4708 #13162145