Subgroups — C₃²×#5966 #13181725

Order 1728.

5524 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	C₃³×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#1010	C₂	1	6	yes	maximal, complemented
3	C₃²×D₃×#41	C₂	1	6	yes	maximal, complemented
4	C₃²×#1004	C₂	1	9	yes	maximal, complemented
5	C₂×C₃²×#237	C₂	1	9	yes	maximal, complemented
6	C₃²×#585	—	3	1	no	maximal
7	C₃×#5966	C₃	1	4	yes	maximal, direct product factor
8	C₃³×#41	C₂²	1	6	yes	complemented
9	C₂×C₃³×D₄	C₂²	1	9	yes	complemented
10	C₂×C₃²×#67	C₂²	1	18	yes	complemented
11	C₂²×C₃²×Dic₃	C₂²	1	6	yes	complemented
12	C₃²×#239	C₂²	1	2	yes	complemented
13	C₃²×D₃×Q₈	C₂²	1	6	yes	complemented
14	C₃²×#237	C₂²	1	18	yes	complemented
15	C₃²×#237	C₂²	1	36	yes	complemented
16	C₃²×D₃×D₄	C₂²	1	18	yes	complemented
17	C₃²×#235	C₂²	1	18	yes	complemented
18	C₂×C₃²×C₄×D₃	C₂²	1	9	yes	complemented
19	C₂×C₃²×Dic₆	C₂²	1	9	yes	complemented
20	C₃²×2₋¹⁺⁴	—	3	6	no	—
21	C₃²×H(5,𝔽₂)	—	2	4	no	—
22	C₃²×H(5,𝔽₂)	—	3	9	no	—
23	C₃²×H(5,𝔽₂)	D₃	1	1	yes	characteristic, complemented
24	C₃²×H(5,𝔽₂)	C₂×C₃	1	4	yes	complemented
25	C₂×C₃²×#41	—	3	6	no	—
26	C₂×C₃²×#41	—	3	9	no	—
27	C₃×#1010	C₂×C₃	1	24	yes	complemented
28	C₃×D₃×#41	C₂×C₃	1	24	yes	complemented
29	C₃×#1004	C₂×C₃	1	36	yes	complemented
30	C₂×C₃×#237	C₂×C₃	1	36	yes	complemented
31	C₂³×C₃³	C₂³	1	6	yes	—
32	C₂²×C₃²×D₃	C₂³	1	9	yes	—
33	C₃³×Q₈	C₂³	1	2	yes	—
34	C₃³×D₄	C₂³	1	18	yes	—
35	C₂×C₃³×C₄	C₂³	1	9	yes	—
36	C₃²×#67	C₂³	1	36	yes	—
37	C₂×C₃²×Dic₃	C₂³	1	18	yes	—
38	C₂×C₃²×Dic₃	C₂³	1	9	yes	—
39	C₃²×D₁₂	C₂³	1	6	yes	—
40	C₃²×C₄×D₃	C₂³	1	6	yes	—
41	C₃²×C₄×D₃	C₂³	1	18	yes	—
42	C₃²×Dic₆	C₂³	1	18	yes	—
43	C₃×#585	—	3	4	no	—
44	#5966	C₃²	1	1	yes	characteristic, direct product factor
45	C₃²×#41	—	2	24	no	—
46	C₃²×#41	—	3	2	no	—
47	C₃²×#41	—	3	18	no	—
48	C₃²×#41	—	3	36	no	—
49	C₃²×#41	—	3	18	no	—
50	C₃²×#41	C₂×D₃	1	6	yes	complemented
51	C₃²×#41	C₂²×C₃	1	24	yes	complemented
52	C₂×C₃²×Q₈	—	3	6	no	—
53	C₂×C₃²×Q₈	—	3	9	no	—
54	C₂×C₃²×D₄	—	2	36	no	—
55	C₂×C₃²×D₄	—	3	18	no	—
56	C₂×C₃²×D₄	—	3	18	no	—
57	C₂×C₃²×D₄	C₂×D₃	1	9	yes	complemented
58	C₂×C₃²×D₄	C₂²×C₃	1	36	yes	complemented
59	C₂²×C₃²×C₄	—	3	6	no	—
60	C₂²×C₃²×C₄	—	3	9	no	—
61	C₂×C₃×#67	C₂²×C₃	1	72	yes	complemented
62	C₂²×C₃×Dic₃	C₂²×C₃	1	24	yes	complemented
63	C₃×#239	C₂²×C₃	1	8	yes	complemented
64	C₃×D₃×Q₈	C₂²×C₃	1	24	yes	complemented
65	C₃×#237	C₂²×C₃	1	72	yes	complemented
66	C₃×#237	C₂²×C₃	1	144	yes	complemented
67	C₃×D₃×D₄	C₂²×C₃	1	72	yes	complemented
68	C₃×#235	C₂²×C₃	1	72	yes	complemented
69	C₂×C₃×C₄×D₃	C₂²×C₃	1	36	yes	complemented
70	C₂×C₃×Dic₆	C₂²×C₃	1	36	yes	complemented
71	C₂²×C₃³	—	4	6	no	—
72	C₂²×C₃³	C₂⁴	1	9	yes	—
73	C₂×C₃²×D₃	—	4	9	no	—
74	C₂×C₃²×D₃	C₂⁴	1	6	yes	—
75	C₃³×C₄	C₂⁴	1	6	yes	—
76	C₃²×Dic₃	C₂⁴	1	9	yes	—
77	C₃²×Dic₃	C₂⁴	1	1	yes	characteristic
78	C₃×2₋¹⁺⁴	—	3	24	no	—
79	C₃×H(5,𝔽₂)	—	2	4	no	—
80	C₃×H(5,𝔽₂)	—	3	36	no	—
81	C₃×H(5,𝔽₂)	C₃×D₃	1	4	yes	complemented
82	C₃×H(5,𝔽₂)	C₂×C₃²	1	1	yes	characteristic, complemented
83	C₂×C₃×#41	—	3	24	no	—
84	C₂×C₃×#41	—	3	36	no	—
85	#1010	C₂×C₃²	1	6	yes	complemented
86	D₃×#41	C₂×C₃²	1	6	yes	complemented
87	#1004	C₂×C₃²	1	9	yes	complemented
88	C₂×#237	C₂×C₃²	1	9	yes	complemented
89	C₂³×C₃²	—	2	24	no	—
90	C₂³×C₃²	—	3	9	no	—
91	C₂³×C₃²	C₂²×D₃	1	6	yes	—
92	C₂³×C₃²	C₂³×C₃	1	24	yes	—
93	C₂²×C₃×D₃	C₂³×C₃	1	36	yes	—
94	C₃²×Q₈	—	2	8	no	—
95	C₃²×Q₈	—	3	18	no	—
96	C₃²×Q₈	C₂²×D₃	1	2	yes	—
97	C₃²×Q₈	C₂³×C₃	1	8	yes	—
98	C₃²×D₄	—	2	72	no	—
99	C₃²×D₄	—	3	36	no	—
100	C₃²×D₄	—	3	6	no	—
101	C₃²×D₄	C₂²×D₃	1	18	yes	—
102	C₃²×D₄	C₂³×C₃	1	72	yes	—
103	C₂×C₃²×C₄	—	2	36	no	—
104	C₂×C₃²×C₄	—	3	9	no	—
105	C₂×C₃²×C₄	—	3	18	no	—
106	C₂×C₃²×C₄	—	3	6	no	—
107	C₂×C₃²×C₄	—	3	18	no	—
108	C₂×C₃²×C₄	C₂²×D₃	1	9	yes	—
109	C₂×C₃²×C₄	C₂³×C₃	1	36	yes	—
110	C₃×#67	C₂³×C₃	1	144	yes	—
111	C₂×C₃×Dic₃	C₂³×C₃	1	72	yes	—
112	C₂×C₃×Dic₃	C₂³×C₃	1	36	yes	—
113	C₃×D₁₂	C₂³×C₃	1	24	yes	—
114	C₃×C₄×D₃	C₂³×C₃	1	24	yes	—
115	C₃×C₄×D₃	C₂³×C₃	1	72	yes	—
116	C₃×Dic₆	C₂³×C₃	1	72	yes	—
117	#585	—	3	1	no	Sylow
118	C₂×C₃³	—	2	9	no	—
119	C₂×C₃³	C₂⁵	1	1	yes	characteristic
120	C₃²×D₃	—	2	6	no	—
121	C₃×#41	—	2	24	no	—
122	C₃×#41	—	3	8	no	—
123	C₃×#41	—	3	72	no	—
124	C₃×#41	—	3	144	no	—
125	C₃×#41	—	3	72	no	—
126	C₃×#41	C₂×C₃×D₃	1	24	yes	complemented
127	C₃×#41	C₂²×C₃²	1	6	yes	complemented
128	C₂×C₃×Q₈	—	3	24	no	—
129	C₂×C₃×Q₈	—	3	36	no	—
130	C₂×C₃×D₄	—	2	36	no	—
131	C₂×C₃×D₄	—	3	72	no	—
132	C₂×C₃×D₄	—	3	72	no	—
133	C₂×C₃×D₄	C₂×C₃×D₃	1	36	yes	complemented
134	C₂×C₃×D₄	C₂²×C₃²	1	9	yes	complemented
135	C₂²×C₃×C₄	—	3	24	no	—
136	C₂²×C₃×C₄	—	3	36	no	—
137	C₂×#67	C₂²×C₃²	1	18	yes	complemented
138	C₂²×Dic₃	C₂²×C₃²	1	6	yes	complemented
139	#239	C₂²×C₃²	1	2	yes	complemented
140	D₃×Q₈	C₂²×C₃²	1	6	yes	complemented
141	#237	C₂²×C₃²	1	18	yes	complemented
142	#237	C₂²×C₃²	1	36	yes	complemented
143	D₃×D₄	C₂²×C₃²	1	18	yes	complemented
144	#235	C₂²×C₃²	1	18	yes	complemented
145	C₂×C₄×D₃	C₂²×C₃²	1	9	yes	complemented
146	C₂×Dic₆	C₂²×C₃²	1	9	yes	complemented
147	C₂²×C₃²	—	2	36	no	—
148	C₂²×C₃²	—	3	6	no	—
149	C₂²×C₃²	—	4	6	no	—
150	C₂²×C₃²	—	4	24	no	—
151	C₂²×C₃²	—	8	24	no	—
152	C₂²×C₃²	—	12	9	no	—
153	C₂²×C₃²	C₂³×D₃	1	9	yes	—
154	C₂²×C₃²	C₂⁴×C₃	1	36	yes	—
155	C₂×C₃×D₃	—	4	36	no	—
156	C₂×C₃×D₃	C₂⁴×C₃	1	24	yes	—
157	C₃²×C₄	—	2	24	no	—
158	C₃²×C₄	—	3	1	no	—
159	C₃²×C₄	—	3	9	no	—
160	C₃²×C₄	C₂³×D₃	1	6	yes	—
161	C₃²×C₄	C₂⁴×C₃	1	24	yes	—
162	C₃×Dic₃	C₂⁴×C₃	1	36	yes	—
163	C₃×Dic₃	C₂⁴×C₃	1	4	yes	—
164	2₋¹⁺⁴	—	3	6	no	—
165	H(5,𝔽₂)	—	3	9	no	—
166	H(5,𝔽₂)	C₃²×D₃	1	1	yes	characteristic, complemented
167	C₂×#41	—	3	6	no	—
168	C₂×#41	—	3	9	no	—
169	C₃³	#585	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	2	24	no	—
171	C₂³×C₃	—	3	36	no	—
172	C₂³×C₃	C₂²×C₃×D₃	1	24	yes	—
173	C₂³×C₃	C₂³×C₃²	1	6	yes	—
174	C₂²×D₃	C₂³×C₃²	1	9	yes	—
175	C₃×Q₈	—	2	8	no	—
176	C₃×Q₈	—	3	72	no	—
177	C₃×Q₈	C₂²×C₃×D₃	1	8	yes	—
178	C₃×Q₈	C₂³×C₃²	1	2	yes	—
179	C₃×D₄	—	2	72	no	—
180	C₃×D₄	—	3	144	no	—
181	C₃×D₄	—	3	24	no	—
182	C₃×D₄	C₂²×C₃×D₃	1	72	yes	—
183	C₃×D₄	C₂³×C₃²	1	18	yes	—
184	C₂×C₃×C₄	—	2	36	no	—
185	C₂×C₃×C₄	—	3	36	no	—
186	C₂×C₃×C₄	—	3	72	no	—
187	C₂×C₃×C₄	—	3	24	no	—
188	C₂×C₃×C₄	—	3	72	no	—
189	C₂×C₃×C₄	C₂²×C₃×D₃	1	36	yes	—
190	C₂×C₃×C₄	C₂³×C₃²	1	9	yes	—
191	#67	C₂³×C₃²	1	36	yes	—
192	C₂×Dic₃	C₂³×C₃²	1	18	yes	—
193	C₂×Dic₃	C₂³×C₃²	1	9	yes	—
194	D₁₂	C₂³×C₃²	1	6	yes	—
195	C₄×D₃	C₂³×C₃²	1	6	yes	—
196	C₄×D₃	C₂³×C₃²	1	18	yes	—
197	Dic₆	C₂³×C₃²	1	18	yes	—
198	C₂×C₃²	—	2	9	no	—
199	C₂×C₃²	—	2	4	no	—
200	C₂×C₃²	—	2	36	no	—
201	C₂×C₃²	—	4	36	no	—
202	C₂×C₃²	—	6	6	no	—
203	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
204	C₂×C₃²	C₂⁵×C₃	1	4	yes	—
205	C₃×D₃	—	2	24	no	—
206	#41	—	3	2	no	—
207	#41	—	3	18	no	—
208	#41	—	3	36	no	—
209	#41	—	3	18	no	—
210	#41	C₂×C₃²×D₃	1	6	yes	complemented
211	C₂×Q₈	—	3	6	no	—
212	C₂×Q₈	—	3	9	no	—
213	C₂×D₄	—	3	18	no	—
214	C₂×D₄	—	3	18	no	—
215	C₂×D₄	C₂×C₃²×D₃	1	9	yes	complemented
216	C₂²×C₄	—	3	6	no	—
217	C₂²×C₄	—	3	9	no	—
218	C₂²×C₃	—	2	36	no	—
219	C₂²×C₃	—	3	24	no	—
220	C₂²×C₃	—	4	24	no	—
221	C₂²×C₃	—	4	6	no	—
222	C₂²×C₃	—	8	24	no	—
223	C₂²×C₃	—	12	36	no	—
224	C₂²×C₃	C₂³×C₃×D₃	1	36	yes	—
225	C₂²×C₃	C₂⁴×C₃²	1	9	yes	—
226	C₂×D₃	—	4	9	no	—
227	C₂×D₃	C₂⁴×C₃²	1	6	yes	—
228	C₃×C₄	—	2	24	no	—
229	C₃×C₄	—	3	4	no	—
230	C₃×C₄	—	3	36	no	—
231	C₃×C₄	C₂³×C₃×D₃	1	24	yes	—
232	C₃×C₄	C₂⁴×C₃²	1	6	yes	—
233	Dic₃	C₂⁴×C₃²	1	9	yes	—
234	Dic₃	C₂⁴×C₃²	1	1	yes	characteristic
235	C₃²	—	2	4	no	—
236	C₃²	#5966	1	1	yes	characteristic, direct product factor
237	C₃²	C₃×#585	1	4	yes	complemented
238	C₂³	—	3	9	no	—
239	C₂³	C₂²×C₃²×D₃	1	6	yes	—
240	Q₈	—	3	18	no	—
241	Q₈	C₂²×C₃²×D₃	1	2	yes	—
242	D₄	—	3	36	no	—
243	D₄	—	3	6	no	—
244	D₄	C₂²×C₃²×D₃	1	18	yes	—
245	C₂×C₄	—	3	9	no	—
246	C₂×C₄	—	3	18	no	—
247	C₂×C₄	—	3	6	no	—
248	C₂×C₄	—	3	18	no	—
249	C₂×C₄	C₂²×C₃²×D₃	1	9	yes	—
250	C₂×C₃	—	2	36	no	—
251	C₂×C₃	—	2	9	no	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	—	4	36	no	—
254	C₂×C₃	—	6	24	no	—
255	C₂×C₃	C₂⁴×C₃×D₃	1	4	yes	—
256	C₂×C₃	C₂⁵×C₃²	1	1	yes	characteristic
257	D₃	—	2	6	no	—
258	C₂²	—	3	6	no	—
259	C₂²	—	4	6	no	—
260	C₂²	—	12	9	no	—
261	C₂²	C₂³×C₃²×D₃	1	9	yes	—
262	C₄	—	3	1	no	—
263	C₄	—	3	9	no	—
264	C₄	C₂³×C₃²×D₃	1	6	yes	—
265	C₃	—	2	4	no	—
266	C₃	C₃×#5966	1	4	yes	direct product factor
267	C₃	C₃²×#585	1	1	yes	characteristic, complemented
268	C₂	—	2	9	no	—
269	C₂	—	6	6	no	—
270	C₂	C₂⁴×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	C₂⁴×D₃
commutator, cosocle	C₂×C₃	C₂⁵×C₃²
Frattini	C₂	C₂⁴×C₃²×D₃
Fitting	C₃³×H(5,𝔽₂)	C₂
socle	C₂×C₃³	C₂⁵
hypercenter	C₃²×H(5,𝔽₂)	D₃
hypocenter	C₃	C₃²×#585
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₃³×H(5,𝔽₂)	C₂	semidirect
2. C₃²×#1010	C₂	semidirect
3. C₃²×D₃×#41	C₂	semidirect
4. C₃²×#1004	C₂	semidirect
5. C₂×C₃²×#237	C₂	semidirect
7. C₃×#5966	C₃	direct (N × H)
8. C₃³×#41	C₂²	semidirect
9. C₂×C₃³×D₄	C₂²	semidirect
10. C₂×C₃²×#67	C₂²	semidirect
11. C₂²×C₃²×Dic₃	C₂²	semidirect
12. C₃²×#239	C₂²	semidirect
13. C₃²×D₃×Q₈	C₂²	semidirect
14. C₃²×#237	C₂²	semidirect
15. C₃²×#237	C₂²	semidirect
16. C₃²×D₃×D₄	C₂²	semidirect
17. C₃²×#235	C₂²	semidirect
18. C₂×C₃²×C₄×D₃	C₂²	semidirect
19. C₂×C₃²×Dic₆	C₂²	semidirect
23. C₃²×H(5,𝔽₂)	D₃	semidirect
24. C₃²×H(5,𝔽₂)	C₂×C₃	semidirect
27. C₃×#1010	C₂×C₃	semidirect
28. C₃×D₃×#41	C₂×C₃	semidirect
29. C₃×#1004	C₂×C₃	semidirect
30. C₂×C₃×#237	C₂×C₃	semidirect
44. #5966	C₃²	direct (N × H)
50. C₃²×#41	C₂×D₃	semidirect
51. C₃²×#41	C₂²×C₃	semidirect
57. C₂×C₃²×D₄	C₂×D₃	semidirect
58. C₂×C₃²×D₄	C₂²×C₃	semidirect
61. C₂×C₃×#67	C₂²×C₃	semidirect
62. C₂²×C₃×Dic₃	C₂²×C₃	semidirect
63. C₃×#239	C₂²×C₃	semidirect
64. C₃×D₃×Q₈	C₂²×C₃	semidirect
65. C₃×#237	C₂²×C₃	semidirect
66. C₃×#237	C₂²×C₃	semidirect
67. C₃×D₃×D₄	C₂²×C₃	semidirect
68. C₃×#235	C₂²×C₃	semidirect
69. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
70. C₂×C₃×Dic₆	C₂²×C₃	semidirect
81. C₃×H(5,𝔽₂)	C₃×D₃	semidirect
82. C₃×H(5,𝔽₂)	C₂×C₃²	semidirect
85. #1010	C₂×C₃²	semidirect
86. D₃×#41	C₂×C₃²	semidirect
87. #1004	C₂×C₃²	semidirect
88. C₂×#237	C₂×C₃²	semidirect
126. C₃×#41	C₂×C₃×D₃	semidirect
127. C₃×#41	C₂²×C₃²	semidirect
133. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
134. C₂×C₃×D₄	C₂²×C₃²	semidirect
137. C₂×#67	C₂²×C₃²	semidirect
138. C₂²×Dic₃	C₂²×C₃²	semidirect
139. #239	C₂²×C₃²	semidirect
140. D₃×Q₈	C₂²×C₃²	semidirect
141. #237	C₂²×C₃²	semidirect
142. #237	C₂²×C₃²	semidirect
143. D₃×D₄	C₂²×C₃²	semidirect
144. #235	C₂²×C₃²	semidirect
145. C₂×C₄×D₃	C₂²×C₃²	semidirect
146. C₂×Dic₆	C₂²×C₃²	semidirect
166. H(5,𝔽₂)	C₃²×D₃	semidirect
169. C₃³	#585	semidirect
210. #41	C₂×C₃²×D₃	semidirect
215. C₂×D₄	C₂×C₃²×D₃	semidirect
236. C₃²	#5966	direct (N × H)
237. C₃²	C₃×#585	semidirect
266. C₃	C₃×#5966	direct (N × H)
267. C₃	C₃²×#585	semidirect

Maximal subgroups

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

subgroup	index
C₃³×H(5,𝔽₂)	2
C₃²×#1010	2
C₃²×D₃×#41	2
C₃²×#1004	2
C₂×C₃²×#237	2
C₃²×#585	3
C₃×#5966	3

Subgroups — C32×#5966 #13181725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5966 #13181725