Subgroups — C₃²×#5720 #13179097

Order 1728.

1944 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#939	C₂	1	6	yes	maximal, complemented
3	C₃²×#937	C₂	1	3	yes	maximal, complemented
4	C₃²×#935	C₂	1	3	yes	maximal, complemented
5	C₃²×#876	C₂	1	2	yes	maximal, complemented
6	C₃²×#560	—	3	1	no	maximal
7	C₃×#5720	C₃	1	4	yes	maximal, direct product factor
8	C₂³×C₃²×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃³×D₄	C₂²	1	3	yes	—
10	C₂×C₃³×D₄	C₂²	1	3	yes	complemented
11	C₂×C₃²×#67	C₂²	1	6	yes	—
12	C₂²×C₃²×Dic₃	C₂²	1	3	yes	complemented
13	C₃³×C₄²	C₂²	1	1	yes	characteristic, complemented
14	C₃²×#217	C₂²	1	3	yes	—
15	C₃²×#217	C₂²	1	3	yes	complemented
16	C₃²×(C₂×D₃)⋊C₄	C₂²	1	6	yes	—
17	C₃²×Dic₃⋊C₄	C₂²	1	6	yes	complemented
18	C₃²×#127	—	2	4	no	—
19	C₃²×#127	D₃	1	1	yes	characteristic, complemented
20	C₃²×#127	C₂×C₃	1	4	yes	complemented
21	C₃²×#126	—	3	2	no	—
22	C₃²×#123	—	3	3	no	—
23	C₃²×#121	—	3	6	no	—
24	C₃²×#120	—	3	3	no	—
25	C₃×#939	C₂×C₃	1	24	yes	complemented
26	C₃×#937	C₂×C₃	1	12	yes	complemented
27	C₃×#935	C₂×C₃	1	12	yes	complemented
28	C₃×#876	C₂×C₃	1	8	yes	complemented
29	C₂³×C₃³	C₂³	1	1	yes	characteristic
30	C₂³×C₃³	C₂³	1	3	yes	—
31	C₂²×C₃²×D₃	—	4	3	no	—
32	C₂²×C₃²×D₃	C₂³	1	2	yes	—
33	C₃³×D₄	—	4	3	no	—
34	C₃³×D₄	—	4	3	no	—
35	C₂×C₃³×C₄	C₂³	1	3	yes	—
36	C₃²×#67	—	4	6	no	—
37	C₂×C₃²×Dic₃	—	4	3	no	—
38	C₂×C₃²×Dic₃	C₂³	1	6	yes	—
39	C₃×#560	—	3	4	no	—
40	#5720	C₃²	1	1	yes	characteristic, direct product factor
41	C₂⁴×C₃²	—	3	1	no	—
42	C₂³×C₃×D₃	C₂²×C₃	1	4	yes	—
43	C₂×C₃²×D₄	—	2	12	no	—
44	C₂×C₃²×D₄	—	2	12	no	—
45	C₂×C₃²×D₄	—	3	6	no	—
46	C₂×C₃²×D₄	C₂×D₃	1	3	yes	—
47	C₂×C₃²×D₄	C₂×D₃	1	3	yes	complemented
48	C₂×C₃²×D₄	C₂²×C₃	1	12	yes	—
49	C₂×C₃²×D₄	C₂²×C₃	1	12	yes	complemented
50	C₂²×C₃²×C₄	—	3	3	no	—
51	C₂×C₃×#67	C₂²×C₃	1	24	yes	—
52	C₂²×C₃×Dic₃	C₂²×C₃	1	12	yes	complemented
53	C₃²×C₄⋊C₄	—	3	6	no	—
54	C₃²×C₂²⋊C₄	—	3	3	no	—
55	C₃²×C₂²⋊C₄	—	3	3	no	—
56	C₃²×C₂²⋊C₄	—	3	6	no	—
57	C₃²×C₄²	—	2	4	no	—
58	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic, complemented
59	C₃²×C₄²	C₂²×C₃	1	4	yes	complemented
60	C₃×#217	C₂²×C₃	1	12	yes	—
61	C₃×#217	C₂²×C₃	1	12	yes	complemented
62	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	24	yes	—
63	C₃×Dic₃⋊C₄	C₂²×C₃	1	24	yes	complemented
64	C₂²×C₃³	—	2	3	no	—
65	C₂²×C₃³	—	2	6	no	—
66	C₂²×C₃³	—	2	3	no	—
67	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
68	C₂×C₃²×D₃	—	2	6	no	—
69	C₂×C₃²×D₃	—	4	4	no	—
70	C₃³×C₄	—	2	3	no	—
71	C₃²×Dic₃	—	2	6	no	—
72	C₃×#127	—	2	4	no	—
73	C₃×#127	C₃×D₃	1	4	yes	complemented
74	C₃×#127	C₂×C₃²	1	1	yes	characteristic, complemented
75	C₃×#126	—	3	8	no	—
76	C₃×#123	—	3	12	no	—
77	C₃×#121	—	3	24	no	—
78	C₃×#120	—	3	12	no	—
79	#939	C₂×C₃²	1	6	yes	complemented
80	#937	C₂×C₃²	1	3	yes	complemented
81	#935	C₂×C₃²	1	3	yes	complemented
82	#876	C₂×C₃²	1	2	yes	complemented
83	C₂³×C₃²	—	2	4	no	—
84	C₂³×C₃²	—	2	12	no	—
85	C₂³×C₃²	—	3	2	no	—
86	C₂³×C₃²	—	12	3	no	—
87	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
88	C₂³×C₃²	C₂²×D₃	1	3	yes	—
89	C₂³×C₃²	C₂³×C₃	1	4	yes	—
90	C₂³×C₃²	C₂³×C₃	1	12	yes	—
91	C₂²×C₃×D₃	—	4	12	no	—
92	C₂²×C₃×D₃	C₂³×C₃	1	8	yes	—
93	C₃²×D₄	—	4	3	no	—
94	C₃²×D₄	—	4	3	no	—
95	C₃²×D₄	—	4	12	no	—
96	C₃²×D₄	—	4	12	no	—
97	C₃²×D₄	—	4	24	no	—
98	C₃²×D₄	—	4	24	no	—
99	C₃²×D₄	—	12	6	no	—
100	C₂×C₃²×C₄	—	2	12	no	—
101	C₂×C₃²×C₄	—	3	6	no	—
102	C₂×C₃²×C₄	—	12	3	no	—
103	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
104	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
105	C₃×#67	—	4	24	no	—
106	C₂×C₃×Dic₃	—	4	12	no	—
107	C₂×C₃×Dic₃	C₂³×C₃	1	24	yes	—
108	#560	—	3	1	no	Sylow
109	C₂×C₃³	—	4	1	no	—
110	C₂×C₃³	—	4	3	no	—
111	C₂×C₃³	H(5,𝔽₂)	1	3	yes	—
112	C₃²×D₃	—	4	2	no	—
113	C₂⁴×C₃	—	3	4	no	—
114	C₂³×D₃	C₂²×C₃²	1	1	yes	characteristic
115	C₂×C₃×D₄	—	2	12	no	—
116	C₂×C₃×D₄	—	2	12	no	—
117	C₂×C₃×D₄	—	3	24	no	—
118	C₂×C₃×D₄	C₂×C₃×D₃	1	12	yes	—
119	C₂×C₃×D₄	C₂×C₃×D₃	1	12	yes	complemented
120	C₂×C₃×D₄	C₂²×C₃²	1	3	yes	—
121	C₂×C₃×D₄	C₂²×C₃²	1	3	yes	complemented
122	C₂²×C₃×C₄	—	3	12	no	—
123	C₂×#67	C₂²×C₃²	1	6	yes	—
124	C₂²×Dic₃	C₂²×C₃²	1	3	yes	complemented
125	C₃×C₄⋊C₄	—	3	24	no	—
126	C₃×C₂²⋊C₄	—	3	12	no	—
127	C₃×C₂²⋊C₄	—	3	12	no	—
128	C₃×C₂²⋊C₄	—	3	24	no	—
129	C₃×C₄²	—	2	4	no	—
130	C₃×C₄²	C₂×C₃×D₃	1	4	yes	complemented
131	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic, complemented
132	#217	C₂²×C₃²	1	3	yes	—
133	#217	C₂²×C₃²	1	3	yes	complemented
134	(C₂×D₃)⋊C₄	C₂²×C₃²	1	6	yes	—
135	Dic₃⋊C₄	C₂²×C₃²	1	6	yes	complemented
136	C₂²×C₃²	—	2	3	no	—
137	C₂²×C₃²	—	2	6	no	—
138	C₂²×C₃²	—	2	3	no	—
139	C₂²×C₃²	—	2	4	no	—
140	C₂²×C₃²	—	2	12	no	—
141	C₂²×C₃²	—	2	24	no	—
142	C₂²×C₃²	—	2	12	no	—
143	C₂²×C₃²	—	4	12	no	—
144	C₂²×C₃²	—	4	24	no	—
145	C₂²×C₃²	—	4	12	no	—
146	C₂²×C₃²	—	6	6	no	—
147	C₂²×C₃²	—	12	4	no	—
148	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
149	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
150	C₂×C₃×D₃	—	2	24	no	—
151	C₂×C₃×D₃	—	4	16	no	—
152	C₃²×C₄	—	2	24	no	—
153	C₃²×C₄	—	2	3	no	—
154	C₃²×C₄	—	2	12	no	—
155	C₃²×C₄	—	6	6	no	—
156	C₃×Dic₃	—	2	24	no	—
157	#127	C₃²×D₃	1	1	yes	characteristic, complemented
158	#126	—	3	2	no	—
159	#123	—	3	3	no	—
160	#121	—	3	6	no	—
161	#120	—	3	3	no	—
162	C₃³	#560	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₃	—	2	4	no	—
164	C₂³×C₃	—	2	12	no	—
165	C₂³×C₃	—	3	8	no	—
166	C₂³×C₃	—	12	12	no	—
167	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
168	C₂³×C₃	C₂²×C₃×D₃	1	12	yes	—
169	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
170	C₂³×C₃	C₂³×C₃²	1	3	yes	—
171	C₂²×D₃	—	4	3	no	—
172	C₂²×D₃	C₂³×C₃²	1	2	yes	—
173	C₃×D₄	—	4	12	no	—
174	C₃×D₄	—	4	12	no	—
175	C₃×D₄	—	4	3	no	—
176	C₃×D₄	—	4	3	no	—
177	C₃×D₄	—	4	24	no	—
178	C₃×D₄	—	4	24	no	—
179	C₃×D₄	—	12	24	no	—
180	C₂×C₃×C₄	—	2	12	no	—
181	C₂×C₃×C₄	—	3	24	no	—
182	C₂×C₃×C₄	—	12	12	no	—
183	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
184	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
185	#67	—	4	6	no	—
186	C₂×Dic₃	—	4	3	no	—
187	C₂×Dic₃	C₂³×C₃²	1	6	yes	—
188	C₂×C₃²	—	2	12	no	—
189	C₂×C₃²	—	4	3	no	—
190	C₂×C₃²	—	4	1	no	—
191	C₂×C₃²	—	4	4	no	—
192	C₂×C₃²	—	4	12	no	—
193	C₂×C₃²	—	8	4	no	—
194	C₂×C₃²	—	8	12	no	—
195	C₂×C₃²	—	12	2	no	—
196	C₂×C₃²	#1004	1	3	yes	—
197	C₂×C₃²	C₃×H(5,𝔽₂)	1	12	yes	—
198	C₃×D₃	—	4	8	no	—
199	C₂⁴	—	3	1	no	—
200	C₂×D₄	—	3	6	no	—
201	C₂×D₄	C₂×C₃²×D₃	1	3	yes	—
202	C₂×D₄	C₂×C₃²×D₃	1	3	yes	complemented
203	C₂²×C₄	—	3	3	no	—
204	C₄⋊C₄	—	3	6	no	—
205	C₂²⋊C₄	—	3	3	no	—
206	C₂²⋊C₄	—	3	3	no	—
207	C₂²⋊C₄	—	3	6	no	—
208	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
209	C₂²×C₃	—	2	12	no	—
210	C₂²×C₃	—	2	24	no	—
211	C₂²×C₃	—	2	12	no	—
212	C₂²×C₃	—	2	3	no	—
213	C₂²×C₃	—	2	6	no	—
214	C₂²×C₃	—	2	3	no	—
215	C₂²×C₃	—	2	4	no	—
216	C₂²×C₃	—	4	12	no	—
217	C₂²×C₃	—	4	24	no	—
218	C₂²×C₃	—	4	12	no	—
219	C₂²×C₃	—	6	24	no	—
220	C₂²×C₃	—	12	16	no	—
221	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
222	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
223	C₂×D₃	—	2	6	no	—
224	C₂×D₃	—	4	4	no	—
225	C₃×C₄	—	2	24	no	—
226	C₃×C₄	—	2	12	no	—
227	C₃×C₄	—	2	3	no	—
228	C₃×C₄	—	6	24	no	—
229	Dic₃	—	2	6	no	—
230	C₃²	—	2	4	no	—
231	C₃²	#5720	1	1	yes	characteristic, direct product factor
232	C₃²	C₃×#560	1	4	yes	complemented
233	C₂³	—	3	2	no	—
234	C₂³	—	12	3	no	—
235	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic
236	C₂³	C₂²×C₃²×D₃	1	3	yes	—
237	D₄	—	4	3	no	—
238	D₄	—	4	3	no	—
239	D₄	—	12	6	no	—
240	C₂×C₄	—	3	6	no	—
241	C₂×C₄	—	12	3	no	—
242	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
243	C₂×C₃	—	2	12	no	—
244	C₂×C₃	—	4	1	no	—
245	C₂×C₃	—	4	3	no	—
246	C₂×C₃	—	4	4	no	—
247	C₂×C₃	—	4	12	no	—
248	C₂×C₃	—	8	4	no	—
249	C₂×C₃	—	8	12	no	—
250	C₂×C₃	—	12	8	no	—
251	C₂×C₃	C₃×#1004	1	12	yes	—
252	C₂×C₃	C₃²×H(5,𝔽₂)	1	3	yes	—
253	D₃	—	4	2	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	2	6	no	—
256	C₂²	—	2	3	no	—
257	C₂²	—	6	6	no	—
258	C₂²	—	12	4	no	—
259	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
260	C₄	—	2	3	no	—
261	C₄	—	6	6	no	—
262	C₃	—	2	4	no	—
263	C₃	C₃×#5720	1	4	yes	direct product factor
264	C₃	C₃²×#560	1	1	yes	characteristic, complemented
265	C₂	—	4	1	no	—
266	C₂	—	4	3	no	—
267	C₂	—	12	2	no	—
268	C₂	C₃²×#1004	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×#127	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃²×#127	D₃
hypocenter	C₃	C₃²×#560
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#127

Complements

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

N	H ≅ G/N	type
1. C₃³×#127	C₂	semidirect
2. C₃²×#939	C₂	semidirect
3. C₃²×#937	C₂	semidirect
4. C₃²×#935	C₂	semidirect
5. C₃²×#876	C₂	semidirect
7. C₃×#5720	C₃	direct (N × H)
10. C₂×C₃³×D₄	C₂²	semidirect
12. C₂²×C₃²×Dic₃	C₂²	semidirect
13. C₃³×C₄²	C₂²	semidirect
15. C₃²×#217	C₂²	semidirect
17. C₃²×Dic₃⋊C₄	C₂²	semidirect
19. C₃²×#127	D₃	semidirect
20. C₃²×#127	C₂×C₃	semidirect
25. C₃×#939	C₂×C₃	semidirect
26. C₃×#937	C₂×C₃	semidirect
27. C₃×#935	C₂×C₃	semidirect
28. C₃×#876	C₂×C₃	semidirect
40. #5720	C₃²	direct (N × H)
47. C₂×C₃²×D₄	C₂×D₃	semidirect
49. C₂×C₃²×D₄	C₂²×C₃	semidirect
52. C₂²×C₃×Dic₃	C₂²×C₃	semidirect
58. C₃²×C₄²	C₂×D₃	semidirect
59. C₃²×C₄²	C₂²×C₃	semidirect
61. C₃×#217	C₂²×C₃	semidirect
63. C₃×Dic₃⋊C₄	C₂²×C₃	semidirect
73. C₃×#127	C₃×D₃	semidirect
74. C₃×#127	C₂×C₃²	semidirect
79. #939	C₂×C₃²	semidirect
80. #937	C₂×C₃²	semidirect
81. #935	C₂×C₃²	semidirect
82. #876	C₂×C₃²	semidirect
119. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
121. C₂×C₃×D₄	C₂²×C₃²	semidirect
124. C₂²×Dic₃	C₂²×C₃²	semidirect
130. C₃×C₄²	C₂×C₃×D₃	semidirect
131. C₃×C₄²	C₂²×C₃²	semidirect
133. #217	C₂²×C₃²	semidirect
135. Dic₃⋊C₄	C₂²×C₃²	semidirect
157. #127	C₃²×D₃	semidirect
162. C₃³	#560	semidirect
202. C₂×D₄	C₂×C₃²×D₃	semidirect
208. C₄²	C₂×C₃²×D₃	semidirect
231. C₃²	#5720	direct (N × H)
232. C₃²	C₃×#560	semidirect
263. C₃	C₃×#5720	direct (N × H)
264. C₃	C₃²×#560	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#127	2
C₃²×#939	2
C₃²×#937	2
C₃²×#935	2
C₃²×#876	2
C₃²×#560	3
C₃×#5720	3

Subgroups — C32×#5720 #13179097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5720 #13179097