Subgroups — C₃²×#4700 #13162136

Order 1728.

812 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#904	C₂	1	1	yes	maximal, characteristic
2	C₃²×#901	C₂	1	2	yes	maximal, complemented
3	C₃³×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#819	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#805	C₂	1	2	yes	maximal, complemented
6	C₃²×#443	—	3	1	no	maximal
7	C₃×#4700	C₃	1	4	yes	maximal, direct product factor
8	C₃²×#235	C₂²	1	2	yes	—
9	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃²×#208	C₂²	1	2	yes	—
13	C₃²×Dic₁₂	C₄	1	1	yes	characteristic, complemented
14	C₃²×D₂₄	C₄	1	1	yes	characteristic, complemented
15	C₃²×#204	C₄	1	2	yes	complemented
16	C₃²×ΓL(1,𝔽₂₅)	—	2	2	no	—
17	C₃²×D₃×C₈	—	2	2	no	—
18	C₃²×#135	—	3	1	no	—
19	C₃²×#131	—	3	2	no	—
20	C₃×#904	C₂×C₃	1	4	yes	—
21	C₃×#901	C₂×C₃	1	8	yes	complemented
22	C₃²×#108	—	3	1	no	—
23	C₃²×#104	—	3	2	no	—
24	C₃²×C₄×C₈	—	2	4	no	—
25	C₃²×C₄×C₈	D₃	1	1	yes	characteristic, complemented
26	C₃²×C₄×C₈	C₂×C₃	1	4	yes	complemented
27	C₃×#819	C₂×C₃	1	4	yes	complemented
28	C₃×#805	C₂×C₃	1	8	yes	complemented
29	C₂×C₃³×C₄	—	2	1	no	—
30	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
31	C₃²×#67	—	2	2	no	—
32	C₃²×D₁₂	C₂×C₄	1	2	yes	—
33	C₃²×C₄×D₃	—	2	2	no	—
34	C₃²×Dic₆	C₂×C₄	1	2	yes	—
35	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃³×C₈	D₄	1	1	yes	characteristic
38	C₃³×C₈	D₄	1	1	yes	characteristic
39	C₃²×C₃⋊C₈	—	2	2	no	—
40	C₃×#443	—	3	4	no	—
41	#4700	C₃²	1	1	yes	characteristic, direct product factor
42	C₃²×#41	—	3	2	no	—
43	C₃×#235	C₂²×C₃	1	8	yes	—
44	C₃²×Q₁₆	—	3	1	no	—
45	C₃²×SD₁₆	—	3	2	no	—
46	C₃²×D₈	—	3	1	no	—
47	C₃²×M₁₆	—	3	2	no	—
48	C₃²×M₁₆	—	6	2	no	—
49	C₂×C₃²×C₈	—	2	4	no	—
50	C₂×C₃²×C₈	—	2	4	no	—
51	C₂×C₃²×C₈	—	6	2	no	—
52	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
55	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
56	C₃²×C₄²	—	2	4	no	—
57	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₃²×C₄²	C₂²×C₃	1	4	yes	—
59	C₃×#208	C₂²×C₃	1	8	yes	—
60	C₃×Dic₁₂	C₃×C₄	1	4	yes	complemented
61	C₃×D₂₄	C₃×C₄	1	4	yes	complemented
62	C₃×#204	C₃×C₄	1	8	yes	complemented
63	C₃×ΓL(1,𝔽₂₅)	—	2	8	no	—
64	C₃×D₃×C₈	—	2	8	no	—
65	C₂²×C₃³	#41	1	1	yes	characteristic
66	C₂×C₃²×D₃	—	2	2	no	—
67	C₃³×C₄	—	2	2	no	—
68	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
70	C₃²×Dic₃	—	2	2	no	—
71	C₃×#135	—	3	4	no	—
72	C₃×#131	—	3	8	no	—
73	#904	C₂×C₃²	1	1	yes	characteristic
74	#901	C₂×C₃²	1	2	yes	complemented
75	C₃×#108	—	3	4	no	—
76	C₃×#104	—	3	8	no	—
77	C₃×C₄×C₈	—	2	4	no	—
78	C₃×C₄×C₈	C₃×D₃	1	4	yes	complemented
79	C₃×C₄×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
80	#819	C₂×C₃²	1	1	yes	characteristic, complemented
81	#805	C₂×C₃²	1	2	yes	complemented
82	C₃²×Q₈	—	3	2	no	—
83	C₃²×D₄	—	3	2	no	—
84	C₃²×D₄	—	6	2	no	—
85	C₂×C₃²×C₄	—	2	8	no	—
86	C₂×C₃²×C₄	—	2	1	no	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	—	2	4	no	—
89	C₂×C₃²×C₄	—	6	2	no	—
90	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
92	C₃×#67	—	2	8	no	—
93	C₃×D₁₂	C₂×C₃×C₄	1	8	yes	—
94	C₃×C₄×D₃	—	2	8	no	—
95	C₃×Dic₆	C₂×C₃×C₄	1	8	yes	—
96	C₃²×C₈	—	2	4	no	—
97	C₃²×C₈	—	2	4	no	—
98	C₃²×C₈	—	2	4	no	—
99	C₃²×C₈	—	2	4	no	—
100	C₃²×C₈	—	6	2	no	—
101	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
102	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
103	C₃²×C₈	D₁₂	1	1	yes	characteristic
104	C₃²×C₈	D₁₂	1	1	yes	characteristic
105	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
106	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
107	C₃²×C₈	C₃×D₄	1	4	yes	—
108	C₃²×C₈	C₃×D₄	1	4	yes	—
109	C₃×C₃⋊C₈	—	2	8	no	—
110	#443	—	3	1	no	Sylow
111	C₂×C₃³	—	2	1	no	—
112	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
113	C₃²×D₃	—	4	2	no	—
114	C₃×#41	—	3	8	no	—
115	#235	C₂²×C₃²	1	2	yes	—
116	C₃×Q₁₆	—	3	4	no	—
117	C₃×SD₁₆	—	3	8	no	—
118	C₃×D₈	—	3	4	no	—
119	C₃×M₁₆	—	3	8	no	—
120	C₃×M₁₆	—	6	8	no	—
121	C₂×C₃×C₈	—	2	4	no	—
122	C₂×C₃×C₈	—	2	4	no	—
123	C₂×C₃×C₈	—	6	8	no	—
124	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
125	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
126	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
128	C₃×C₄²	—	2	4	no	—
129	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
130	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
131	#208	C₂²×C₃²	1	2	yes	—
132	Dic₁₂	C₃²×C₄	1	1	yes	characteristic, complemented
133	D₂₄	C₃²×C₄	1	1	yes	characteristic, complemented
134	#204	C₃²×C₄	1	2	yes	complemented
135	ΓL(1,𝔽₂₅)	—	2	2	no	—
136	D₃×C₈	—	2	2	no	—
137	C₂²×C₃²	—	2	4	no	—
138	C₂²×C₃²	—	6	2	no	—
139	C₂²×C₃²	#235	1	1	yes	characteristic
140	C₂²×C₃²	C₃×#41	1	4	yes	—
141	C₂×C₃×D₃	—	2	8	no	—
142	C₃²×C₄	—	2	16	no	—
143	C₃²×C₄	—	2	2	no	—
144	C₃²×C₄	—	2	8	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	—	6	2	no	—
148	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
149	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
150	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
151	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
152	C₃×Dic₃	—	2	8	no	—
153	#135	—	3	1	no	—
154	#131	—	3	2	no	—
155	#108	—	3	1	no	—
156	#104	—	3	2	no	—
157	C₄×C₈	C₃²×D₃	1	1	yes	characteristic, complemented
158	C₃³	#443	1	1	yes	characteristic, complemented, Sylow
159	C₃×Q₈	—	3	8	no	—
160	C₃×D₄	—	3	8	no	—
161	C₃×D₄	—	6	8	no	—
162	C₂×C₃×C₄	—	2	8	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	6	8	no	—
167	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
168	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
169	#67	—	2	2	no	—
170	D₁₂	C₂×C₃²×C₄	1	2	yes	—
171	C₄×D₃	—	2	2	no	—
172	Dic₆	C₂×C₃²×C₄	1	2	yes	—
173	C₃×C₈	—	2	4	no	—
174	C₃×C₈	—	2	4	no	—
175	C₃×C₈	—	2	4	no	—
176	C₃×C₈	—	2	4	no	—
177	C₃×C₈	—	6	8	no	—
178	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
179	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
180	C₃×C₈	C₃×D₁₂	1	4	yes	—
181	C₃×C₈	C₃×D₁₂	1	4	yes	—
182	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
183	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
184	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
185	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
186	C₃⋊C₈	—	2	2	no	—
187	C₂×C₃²	—	2	8	no	—
188	C₂×C₃²	—	2	1	no	—
189	C₂×C₃²	—	2	4	no	—
190	C₂×C₃²	—	2	4	no	—
191	C₂×C₃²	—	12	2	no	—
192	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
193	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
194	C₃×D₃	—	4	8	no	—
195	#41	—	3	2	no	—
196	Q₁₆	—	3	1	no	—
197	SD₁₆	—	3	2	no	—
198	D₈	—	3	1	no	—
199	M₁₆	—	3	2	no	—
200	M₁₆	—	6	2	no	—
201	C₂×C₈	—	6	2	no	—
202	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
203	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
204	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	6	8	no	—
207	C₂²×C₃	C₃×#235	1	4	yes	—
208	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
209	C₂×D₃	—	2	2	no	—
210	C₃×C₄	—	2	16	no	—
211	C₃×C₄	—	2	8	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	6	8	no	—
216	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
217	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
218	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
219	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
220	Dic₃	—	2	2	no	—
221	C₃²	—	2	4	no	—
222	C₃²	#4700	1	1	yes	characteristic, direct product factor
223	C₃²	C₃×#443	1	4	yes	complemented
224	Q₈	—	3	2	no	—
225	D₄	—	3	2	no	—
226	D₄	—	6	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	6	2	no	—
229	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
230	C₈	—	6	2	no	—
231	C₈	C₃²×C₄×D₃	1	1	yes	characteristic
232	C₈	C₃²×C₄×D₃	1	1	yes	characteristic
233	C₈	C₃²×D₁₂	1	1	yes	characteristic
234	C₈	C₃²×D₁₂	1	1	yes	characteristic
235	C₂×C₃	—	2	8	no	—
236	C₂×C₃	—	2	4	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	—	12	8	no	—
240	C₂×C₃	C₃×C₄×D₁₂	1	4	yes	—
241	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
242	D₃	—	4	2	no	—
243	C₂²	—	6	2	no	—
244	C₂²	C₃²×#235	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	6	2	no	—
247	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
248	C₄	C₂×C₃²×D₁₂	1	1	yes	characteristic
249	C₃	—	2	4	no	—
250	C₃	C₃×#4700	1	4	yes	direct product factor
251	C₃	C₃²×#443	1	1	yes	characteristic, complemented
252	C₂	—	2	1	no	—
253	C₂	—	12	2	no	—
254	C₂	C₃²×C₄×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₈	D₁₂
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄×C₈	C₂
socle	C₂×C₃³	C₄×D₄
prime radical	C₃²×#904	C₂
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×C₄×C₈	D₃
hypocenter	C₃	C₃²×#443
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₈ ◁ C₂×C₃²×C₈ ◁ C₃²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₃²×#901	C₂	semidirect
3. C₃³×C₄×C₈	C₂	semidirect
4. C₃²×#819	C₂	semidirect
5. C₃²×#805	C₂	semidirect
7. C₃×#4700	C₃	direct (N × H)
13. C₃²×Dic₁₂	C₄	semidirect
14. C₃²×D₂₄	C₄	semidirect
15. C₃²×#204	C₄	semidirect
21. C₃×#901	C₂×C₃	semidirect
25. C₃²×C₄×C₈	D₃	semidirect
26. C₃²×C₄×C₈	C₂×C₃	semidirect
27. C₃×#819	C₂×C₃	semidirect
28. C₃×#805	C₂×C₃	semidirect
41. #4700	C₃²	direct (N × H)
60. C₃×Dic₁₂	C₃×C₄	semidirect
61. C₃×D₂₄	C₃×C₄	semidirect
62. C₃×#204	C₃×C₄	semidirect
74. #901	C₂×C₃²	semidirect
78. C₃×C₄×C₈	C₃×D₃	semidirect
79. C₃×C₄×C₈	C₂×C₃²	semidirect
80. #819	C₂×C₃²	semidirect
81. #805	C₂×C₃²	semidirect
132. Dic₁₂	C₃²×C₄	semidirect
133. D₂₄	C₃²×C₄	semidirect
134. #204	C₃²×C₄	semidirect
157. C₄×C₈	C₃²×D₃	semidirect
158. C₃³	#443	semidirect
222. C₃²	#4700	direct (N × H)
223. C₃²	C₃×#443	semidirect
250. C₃	C₃×#4700	direct (N × H)
251. C₃	C₃²×#443	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#904	2
C₃²×#901	2
C₃³×C₄×C₈	2
C₃²×#819	2
C₃²×#805	2
C₃²×#443	3
C₃×#4700	3

Subgroups — C32×#4700 #13162136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4700 #13162136