Subgroups — C₃²×#5753 #13179130

Order 1728.

1736 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#1010	C₂	1	2	yes	maximal, complemented
2	C₃³×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#909	C₂	1	6	yes	maximal, complemented
4	C₂×C₃²×Dic₁₂	C₂	1	3	yes	maximal, complemented
5	C₃²×#904	C₂	1	3	yes	maximal, complemented
6	C₃²×#578	—	3	1	no	maximal
7	C₃×#5753	C₃	1	4	yes	maximal, direct product factor
8	C₃³×#41	C₂²	1	1	yes	characteristic
9	C₃²×D₃×Q₈	—	2	2	no	—
10	C₃²×#237	—	2	6	no	—
11	C₃²×#235	C₂²	1	6	yes	—
12	C₂×C₃²×Dic₆	C₂²	1	6	yes	—
13	C₃³×M₁₆	C₂²	1	3	yes	—
14	C₂×C₃³×C₈	C₂²	1	3	yes	—
15	C₃²×Dic₁₂	C₂²	1	6	yes	—
16	C₃²×Dic₁₂	C₂²	1	3	yes	—
17	C₃²×D₂₄	C₂²	1	1	yes	characteristic
18	C₃²×#204	C₂²	1	6	yes	—
19	C₃²×2₋¹⁺⁴	—	3	2	no	—
20	C₃×#1010	C₂×C₃	1	8	yes	complemented
21	C₃²×#137	—	3	6	no	—
22	C₃²×#135	—	3	3	no	—
23	C₂×C₃²×Q₁₆	—	3	3	no	—
24	C₃²×#131	—	2	4	no	—
25	C₃²×#131	D₃	1	1	yes	characteristic, complemented
26	C₃²×#131	C₂×C₃	1	4	yes	complemented
27	C₃×#909	C₂×C₃	1	24	yes	complemented
28	C₂×C₃×Dic₁₂	C₂×C₃	1	12	yes	complemented
29	C₃×#904	C₂×C₃	1	12	yes	complemented
30	C₃³×Q₈	D₄	1	1	yes	characteristic
31	C₃³×D₄	D₄	1	3	yes	—
32	C₂×C₃³×C₄	C₂³	1	3	yes	—
33	C₃²×#67	—	2	6	no	—
34	C₂×C₃²×Dic₃	—	2	6	no	—
35	C₃²×D₁₂	C₂³	1	2	yes	—
36	C₃²×C₄×D₃	—	2	6	no	—
37	C₃²×Dic₆	—	2	6	no	—
38	C₃²×Dic₆	C₂³	1	6	yes	—
39	C₃³×C₈	C₂³	1	1	yes	characteristic
40	C₃³×C₈	C₂³	1	3	yes	—
41	C₃×#578	—	3	4	no	—
42	#5753	C₃²	1	1	yes	characteristic, direct product factor
43	C₃²×#41	—	2	4	no	—
44	C₃²×#41	—	3	6	no	—
45	C₃²×#41	—	6	6	no	—
46	C₃²×#41	C₂×D₃	1	1	yes	characteristic
47	C₃²×#41	C₂²×C₃	1	4	yes	—
48	C₂×C₃²×Q₈	—	3	6	no	—
49	C₂×C₃²×Q₈	—	6	2	no	—
50	C₃×D₃×Q₈	—	2	8	no	—
51	C₃×#237	—	2	24	no	—
52	C₃×#235	C₂²×C₃	1	24	yes	—
53	C₂×C₃×Dic₆	C₂²×C₃	1	24	yes	—
54	C₃²×Q₁₆	—	3	3	no	—
55	C₃²×Q₁₆	—	3	6	no	—
56	C₃²×SD₁₆	—	3	6	no	—
57	C₃²×D₈	—	3	1	no	—
58	C₃²×M₁₆	—	2	12	no	—
59	C₃²×M₁₆	C₂×D₃	1	3	yes	—
60	C₃²×M₁₆	C₂²×C₃	1	12	yes	—
61	C₂×C₃²×C₈	—	2	12	no	—
62	C₂×C₃²×C₈	C₂×D₃	1	3	yes	—
63	C₂×C₃²×C₈	C₂²×C₃	1	12	yes	—
64	C₃×Dic₁₂	C₂²×C₃	1	12	yes	—
65	C₃×Dic₁₂	C₂²×C₃	1	24	yes	—
66	C₃×D₂₄	C₂²×C₃	1	4	yes	—
67	C₃×#204	C₂²×C₃	1	24	yes	—
68	C₂²×C₃³	C₂×D₄	1	3	yes	—
69	C₂×C₃²×D₃	—	2	2	no	—
70	C₃³×C₄	C₂×D₄	1	3	yes	—
71	C₃³×C₄	C₂⁴	1	1	yes	characteristic
72	C₃²×Dic₃	—	2	6	no	—
73	C₃×2₋¹⁺⁴	—	3	8	no	—
74	#1010	C₂×C₃²	1	2	yes	complemented
75	C₃×#137	—	3	24	no	—
76	C₃×#135	—	3	12	no	—
77	C₂×C₃×Q₁₆	—	3	12	no	—
78	C₃×#131	—	2	4	no	—
79	C₃×#131	C₃×D₃	1	4	yes	complemented
80	C₃×#131	C₂×C₃²	1	1	yes	characteristic, complemented
81	#909	C₂×C₃²	1	6	yes	complemented
82	C₂×Dic₁₂	C₂×C₃²	1	3	yes	complemented
83	#904	C₂×C₃²	1	3	yes	complemented
84	C₃²×Q₈	—	2	4	no	—
85	C₃²×Q₈	—	3	6	no	—
86	C₃²×Q₈	—	6	6	no	—
87	C₃²×Q₈	D₁₂	1	1	yes	characteristic
88	C₃²×Q₈	C₃×D₄	1	4	yes	—
89	C₃²×D₄	—	2	12	no	—
90	C₃²×D₄	—	3	2	no	—
91	C₃²×D₄	—	6	6	no	—
92	C₃²×D₄	D₁₂	1	3	yes	—
93	C₃²×D₄	C₃×D₄	1	12	yes	—
94	C₂×C₃²×C₄	—	2	12	no	—
95	C₂×C₃²×C₄	—	6	6	no	—
96	C₂×C₃²×C₄	—	6	6	no	—
97	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
98	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
99	C₃×#67	—	2	24	no	—
100	C₂×C₃×Dic₃	—	2	24	no	—
101	C₃×D₁₂	C₂³×C₃	1	8	yes	—
102	C₃×C₄×D₃	—	2	24	no	—
103	C₃×Dic₆	—	2	24	no	—
104	C₃×Dic₆	C₂³×C₃	1	24	yes	—
105	C₃²×C₈	—	2	4	no	—
106	C₃²×C₈	—	2	12	no	—
107	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
108	C₃²×C₈	C₂²×D₃	1	3	yes	—
109	C₃²×C₈	C₂³×C₃	1	12	yes	—
110	C₃²×C₈	C₂³×C₃	1	4	yes	—
111	#578	—	3	1	no	Sylow
112	C₂×C₃³	—	2	3	no	—
113	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
114	C₃²×D₃	—	4	2	no	—
115	C₃×#41	—	2	4	no	—
116	C₃×#41	—	3	24	no	—
117	C₃×#41	—	6	24	no	—
118	C₃×#41	C₂×C₃×D₃	1	4	yes	—
119	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
120	C₂×C₃×Q₈	—	3	24	no	—
121	C₂×C₃×Q₈	—	6	8	no	—
122	D₃×Q₈	—	2	2	no	—
123	#237	—	2	6	no	—
124	#235	C₂²×C₃²	1	6	yes	—
125	C₂×Dic₆	C₂²×C₃²	1	6	yes	—
126	C₃×Q₁₆	—	3	12	no	—
127	C₃×Q₁₆	—	3	24	no	—
128	C₃×SD₁₆	—	3	24	no	—
129	C₃×D₈	—	3	4	no	—
130	C₃×M₁₆	—	2	12	no	—
131	C₃×M₁₆	C₂×C₃×D₃	1	12	yes	—
132	C₃×M₁₆	C₂²×C₃²	1	3	yes	—
133	C₂×C₃×C₈	—	2	12	no	—
134	C₂×C₃×C₈	C₂×C₃×D₃	1	12	yes	—
135	C₂×C₃×C₈	C₂²×C₃²	1	3	yes	—
136	Dic₁₂	C₂²×C₃²	1	3	yes	—
137	Dic₁₂	C₂²×C₃²	1	6	yes	—
138	D₂₄	C₂²×C₃²	1	1	yes	characteristic
139	#204	C₂²×C₃²	1	6	yes	—
140	C₂²×C₃²	—	2	12	no	—
141	C₂²×C₃²	—	6	2	no	—
142	C₂²×C₃²	C₂×D₁₂	1	3	yes	—
143	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
144	C₂×C₃×D₃	—	2	8	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	—	2	12	no	—
147	C₃²×C₄	—	6	6	no	—
148	C₃²×C₄	C₂×D₁₂	1	3	yes	—
149	C₃²×C₄	C₂×C₃×D₄	1	12	yes	—
150	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
151	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
152	C₃×Dic₃	—	2	24	no	—
153	2₋¹⁺⁴	—	3	2	no	—
154	#137	—	3	6	no	—
155	#135	—	3	3	no	—
156	C₂×Q₁₆	—	3	3	no	—
157	#131	C₃²×D₃	1	1	yes	characteristic, complemented
158	C₃³	#578	1	1	yes	characteristic, complemented, Sylow
159	C₃×Q₈	—	2	4	no	—
160	C₃×Q₈	—	3	24	no	—
161	C₃×Q₈	—	6	24	no	—
162	C₃×Q₈	C₃×D₁₂	1	4	yes	—
163	C₃×Q₈	C₃²×D₄	1	1	yes	characteristic
164	C₃×D₄	—	2	12	no	—
165	C₃×D₄	—	3	8	no	—
166	C₃×D₄	—	6	24	no	—
167	C₃×D₄	C₃×D₁₂	1	12	yes	—
168	C₃×D₄	C₃²×D₄	1	3	yes	—
169	C₂×C₃×C₄	—	2	12	no	—
170	C₂×C₃×C₄	—	6	24	no	—
171	C₂×C₃×C₄	—	6	24	no	—
172	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
173	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
174	#67	—	2	6	no	—
175	C₂×Dic₃	—	2	6	no	—
176	D₁₂	C₂³×C₃²	1	2	yes	—
177	C₄×D₃	—	2	6	no	—
178	Dic₆	—	2	6	no	—
179	Dic₆	C₂³×C₃²	1	6	yes	—
180	C₃×C₈	—	2	4	no	—
181	C₃×C₈	—	2	12	no	—
182	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
183	C₃×C₈	C₂²×C₃×D₃	1	12	yes	—
184	C₃×C₈	C₂³×C₃²	1	3	yes	—
185	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
186	C₂×C₃²	—	2	3	no	—
187	C₂×C₃²	—	2	4	no	—
188	C₂×C₃²	—	2	12	no	—
189	C₂×C₃²	—	4	12	no	—
190	C₂×C₃²	—	12	2	no	—
191	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
192	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
193	C₃×D₃	—	4	8	no	—
194	#41	—	3	6	no	—
195	#41	—	6	6	no	—
196	#41	C₂×C₃²×D₃	1	1	yes	characteristic
197	C₂×Q₈	—	3	6	no	—
198	C₂×Q₈	—	6	2	no	—
199	Q₁₆	—	3	3	no	—
200	Q₁₆	—	3	6	no	—
201	SD₁₆	—	3	6	no	—
202	D₈	—	3	1	no	—
203	M₁₆	C₂×C₃²×D₃	1	3	yes	—
204	C₂×C₈	C₂×C₃²×D₃	1	3	yes	—
205	C₂²×C₃	—	2	12	no	—
206	C₂²×C₃	—	6	8	no	—
207	C₂²×C₃	C₂×C₃×D₁₂	1	12	yes	—
208	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
209	C₂×D₃	—	2	2	no	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	2	12	no	—
212	C₃×C₄	—	6	24	no	—
213	C₃×C₄	C₂×C₃×D₁₂	1	12	yes	—
214	C₃×C₄	C₂×C₃²×D₄	1	3	yes	—
215	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
216	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
217	Dic₃	—	2	6	no	—
218	C₃²	—	2	4	no	—
219	C₃²	#5753	1	1	yes	characteristic, direct product factor
220	C₃²	C₃×#578	1	4	yes	complemented
221	Q₈	—	3	6	no	—
222	Q₈	—	6	6	no	—
223	Q₈	C₃²×D₁₂	1	1	yes	characteristic
224	D₄	—	3	2	no	—
225	D₄	—	6	6	no	—
226	D₄	C₃²×D₁₂	1	3	yes	—
227	C₂×C₄	—	6	6	no	—
228	C₂×C₄	—	6	6	no	—
229	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
230	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
231	C₈	C₂²×C₃²×D₃	1	3	yes	—
232	C₂×C₃	—	2	12	no	—
233	C₂×C₃	—	2	3	no	—
234	C₂×C₃	—	2	4	no	—
235	C₂×C₃	—	4	12	no	—
236	C₂×C₃	—	12	8	no	—
237	C₂×C₃	C₂²×C₃×D₁₂	1	4	yes	—
238	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
239	D₃	—	4	2	no	—
240	C₂²	—	6	2	no	—
241	C₂²	C₂×C₃²×D₁₂	1	3	yes	—
242	C₄	—	6	6	no	—
243	C₄	C₂×C₃²×D₁₂	1	3	yes	—
244	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
245	C₃	—	2	4	no	—
246	C₃	C₃×#5753	1	4	yes	direct product factor
247	C₃	C₃²×#578	1	1	yes	characteristic, complemented
248	C₂	—	2	3	no	—
249	C₂	—	12	2	no	—
250	C₂	C₂²×C₃²×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	C₂²×D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₃²
Frattini	C₄	C₂³×C₃²×D₃
Fitting	C₃³×#131	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₃²×#131	D₃
hypocenter	C₃	C₃²×#578
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×#41 ◁ C₃²×#131

Complements

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

N	H ≅ G/N	type
1. C₃²×#1010	C₂	semidirect
2. C₃³×#131	C₂	semidirect
3. C₃²×#909	C₂	semidirect
4. C₂×C₃²×Dic₁₂	C₂	semidirect
5. C₃²×#904	C₂	semidirect
7. C₃×#5753	C₃	direct (N × H)
20. C₃×#1010	C₂×C₃	semidirect
25. C₃²×#131	D₃	semidirect
26. C₃²×#131	C₂×C₃	semidirect
27. C₃×#909	C₂×C₃	semidirect
28. C₂×C₃×Dic₁₂	C₂×C₃	semidirect
29. C₃×#904	C₂×C₃	semidirect
42. #5753	C₃²	direct (N × H)
74. #1010	C₂×C₃²	semidirect
79. C₃×#131	C₃×D₃	semidirect
80. C₃×#131	C₂×C₃²	semidirect
81. #909	C₂×C₃²	semidirect
82. C₂×Dic₁₂	C₂×C₃²	semidirect
83. #904	C₂×C₃²	semidirect
157. #131	C₃²×D₃	semidirect
158. C₃³	#578	semidirect
219. C₃²	#5753	direct (N × H)
220. C₃²	C₃×#578	semidirect
246. C₃	C₃×#5753	direct (N × H)
247. C₃	C₃²×#578	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#1010	2
C₃³×#131	2
C₃²×#909	2
C₂×C₃²×Dic₁₂	2
C₃²×#904	2
C₃²×#578	3
C₃×#5753	3

Subgroups — C32×#5753 #13179130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5753 #13179130