Subgroups — C₃²×#4707 #13162137

Order 1728.

1012 conjugacy classes of subgroups.

The subgroup lattice has 228 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	2	yes	maximal
2	C₃²×C₄²×D₃	C₂	1	1	yes	maximal, characteristic
3	C₃³×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₈×Dic₃	C₂	1	2	yes	maximal, complemented
5	C₃²×#803	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₄×M₁₆	—	3	1	no	maximal
7	C₃×#4707	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×C₄×D₃	C₄	1	2	yes	—
9	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃³×C₈	C₂²	1	2	yes	—
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃²×C₄×Dic₃	C₄	1	2	yes	—
13	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃²×ΓL(1,𝔽₂₅)	C₄	1	8	yes	complemented
16	C₂×C₃²×M₁₆	—	3	2	no	—
17	C₂×C₃²×C₄²	—	3	1	no	—
18	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	8	yes	—
19	C₃×C₄²×D₃	C₂×C₃	1	4	yes	—
20	C₃²×#97	—	2	4	no	—
21	C₃²×#97	—	3	1	no	—
22	C₃²×#97	D₃	1	1	yes	characteristic, complemented
23	C₃²×#97	C₂×C₃	1	4	yes	complemented
24	C₃²×C₄×C₈	—	3	2	no	—
25	C₃×C₈×Dic₃	C₂×C₃	1	8	yes	complemented
26	C₃×#803	C₂×C₃	1	4	yes	complemented
27	C₂²×C₃²×D₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃²×Dic₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
32	C₃²×C₄×D₃	—	2	4	no	—
33	C₃²×C₄×D₃	C₂×C₄	1	4	yes	—
34	C₃³×C₈	C₂×C₄	1	4	yes	complemented
35	C₃²×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
36	C₃×C₄×M₁₆	—	3	4	no	—
37	#4707	C₃²	1	1	yes	characteristic, direct product factor
38	C₂²×C₃²×C₄	—	3	1	no	—
39	C₂²×C₃²×C₄	—	3	2	no	—
40	C₂×C₃×C₄×D₃	C₃×C₄	1	8	yes	—
41	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
42	C₃²×M₁₆	—	3	8	no	—
43	C₂×C₃²×C₈	—	2	8	no	—
44	C₂×C₃²×C₈	—	3	2	no	—
45	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
46	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
47	C₃²×C₄²	—	2	4	no	—
48	C₃²×C₄²	—	3	2	no	—
49	C₃²×C₄²	—	3	1	no	—
50	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
51	C₃²×C₄²	C₂²×C₃	1	4	yes	—
52	C₃×C₄×Dic₃	C₃×C₄	1	8	yes	—
53	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
54	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
55	C₃×ΓL(1,𝔽₂₅)	C₃×C₄	1	32	yes	complemented
56	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃²×D₃	—	2	2	no	—
58	C₂×C₃²×D₃	C₄²	1	2	yes	—
59	C₃³×C₄	—	2	2	no	—
60	C₃³×C₄	C₂²×C₄	1	2	yes	—
61	C₃²×Dic₃	C₄²	1	2	yes	—
62	C₃²×Dic₃	M₁₆	1	4	yes	complemented
63	C₂×C₃×M₁₆	—	3	8	no	—
64	C₂×C₃×C₄²	—	3	4	no	—
65	C₂×ΓL(1,𝔽₂₅)	C₂×C₃²	1	2	yes	—
66	C₄²×D₃	C₂×C₃²	1	1	yes	characteristic
67	C₃×#97	—	2	4	no	—
68	C₃×#97	—	3	4	no	—
69	C₃×#97	C₃×D₃	1	4	yes	complemented
70	C₃×#97	C₂×C₃²	1	1	yes	characteristic, complemented
71	C₃×C₄×C₈	—	3	8	no	—
72	C₈×Dic₃	C₂×C₃²	1	2	yes	complemented
73	#803	C₂×C₃²	1	1	yes	characteristic, complemented
74	C₂³×C₃²	—	3	1	no	—
75	C₂²×C₃×D₃	C₂×C₃×C₄	1	4	yes	—
76	C₂×C₃²×C₄	—	2	8	no	—
77	C₂×C₃²×C₄	—	2	4	no	—
78	C₂×C₃²×C₄	—	3	1	no	—
79	C₂×C₃²×C₄	—	3	2	no	—
80	C₂×C₃²×C₄	—	3	4	no	—
81	C₂×C₃²×C₄	—	6	4	no	—
82	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
83	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
86	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
87	C₂×C₃×Dic₃	C₂×C₃×C₄	1	8	yes	—
88	C₃×C₄×D₃	—	2	16	no	—
89	C₃×C₄×D₃	C₂×C₃×C₄	1	16	yes	—
90	C₃²×C₈	—	2	16	no	—
91	C₃²×C₈	—	3	4	no	—
92	C₃²×C₈	C₄×D₃	1	4	yes	complemented
93	C₃²×C₈	C₂×C₃×C₄	1	16	yes	complemented
94	C₃×C₃⋊C₈	C₂×C₃×C₄	1	16	yes	complemented
95	C₄×M₁₆	—	3	1	no	Sylow
96	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₃³	C₂×M₁₆	1	2	yes	—
98	C₃²×D₃	—	2	2	no	—
99	C₂²×C₃×C₄	—	3	4	no	—
100	C₂²×C₃×C₄	—	3	8	no	—
101	C₂×C₄×D₃	C₃²×C₄	1	2	yes	—
102	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
103	C₃×M₁₆	—	3	32	no	—
104	C₂×C₃×C₈	—	2	8	no	—
105	C₂×C₃×C₈	—	3	8	no	—
106	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
107	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
108	C₃×C₄²	—	2	4	no	—
109	C₃×C₄²	—	3	8	no	—
110	C₃×C₄²	—	3	4	no	—
111	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
112	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
113	C₄×Dic₃	C₃²×C₄	1	2	yes	—
114	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
115	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
116	ΓL(1,𝔽₂₅)	C₃²×C₄	1	8	yes	complemented
117	C₂²×C₃²	—	2	4	no	—
118	C₂²×C₃²	—	3	2	no	—
119	C₂²×C₃²	—	6	2	no	—
120	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
121	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
122	C₂×C₃×D₃	—	2	8	no	—
123	C₂×C₃×D₃	C₃×C₄²	1	8	yes	—
124	C₃²×C₄	—	2	2	no	—
125	C₃²×C₄	—	2	8	no	—
126	C₃²×C₄	—	2	8	no	—
127	C₃²×C₄	—	3	2	no	—
128	C₃²×C₄	—	3	4	no	—
129	C₃²×C₄	—	4	8	no	—
130	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
131	C₃²×C₄	C₂²×C₃×C₄	1	8	yes	—
132	C₃×Dic₃	C₃×C₄²	1	8	yes	—
133	C₃×Dic₃	C₃×M₁₆	1	16	yes	complemented
134	C₂×M₁₆	—	3	2	no	—
135	C₂×C₄²	—	3	1	no	—
136	#97	—	3	1	no	—
137	#97	C₃²×D₃	1	1	yes	characteristic, complemented
138	C₄×C₈	—	3	2	no	—
139	C₃³	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₃	—	3	4	no	—
141	C₂²×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	8	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	3	8	no	—
145	C₂×C₃×C₄	—	3	4	no	—
146	C₂×C₃×C₄	—	3	16	no	—
147	C₂×C₃×C₄	—	6	16	no	—
148	C₂×C₃×C₄	C₃×C₄×D₃	1	8	yes	—
149	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
150	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
151	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
152	C₂×Dic₃	C₂×C₃²×C₄	1	1	yes	characteristic
153	C₂×Dic₃	C₂×C₃²×C₄	1	2	yes	—
154	C₄×D₃	—	2	4	no	—
155	C₄×D₃	C₂×C₃²×C₄	1	4	yes	—
156	C₃×C₈	—	2	16	no	—
157	C₃×C₈	—	3	16	no	—
158	C₃×C₈	C₃×C₄×D₃	1	16	yes	complemented
159	C₃×C₈	C₂×C₃²×C₄	1	4	yes	complemented
160	C₃⋊C₈	C₂×C₃²×C₄	1	4	yes	complemented
161	C₂×C₃²	—	2	8	no	—
162	C₂×C₃²	—	2	4	no	—
163	C₂×C₃²	—	6	2	no	—
164	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
165	C₂×C₃²	D₃×M₁₆	1	2	yes	—
166	C₂×C₃²	C₂×C₃×C₄²	1	4	yes	—
167	C₂×C₃²	C₂×C₃×M₁₆	1	8	yes	—
168	C₃×D₃	—	2	8	no	—
169	C₂²×C₄	—	3	1	no	—
170	C₂²×C₄	—	3	2	no	—
171	M₁₆	—	3	8	no	—
172	C₂×C₈	—	3	2	no	—
173	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
174	C₄²	—	3	2	no	—
175	C₄²	—	3	1	no	—
176	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	3	8	no	—
179	C₂²×C₃	—	6	8	no	—
180	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
181	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
182	C₂×D₃	—	2	2	no	—
183	C₂×D₃	C₃²×C₄²	1	2	yes	—
184	C₃×C₄	—	2	8	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	8	no	—
187	C₃×C₄	—	3	16	no	—
188	C₃×C₄	—	3	8	no	—
189	C₃×C₄	—	4	8	no	—
190	C₃×C₄	C₂×C₃×C₄×D₃	1	8	yes	—
191	C₃×C₄	C₂²×C₃²×C₄	1	2	yes	—
192	Dic₃	C₃²×C₄²	1	2	yes	—
193	Dic₃	C₃²×M₁₆	1	4	yes	complemented
194	C₃²	—	2	4	no	—
195	C₃²	#4707	1	1	yes	characteristic, direct product factor
196	C₃²	C₃×C₄×M₁₆	1	4	yes	complemented
197	C₂³	—	3	1	no	—
198	C₂×C₄	—	3	1	no	—
199	C₂×C₄	—	3	2	no	—
200	C₂×C₄	—	3	4	no	—
201	C₂×C₄	—	6	4	no	—
202	C₂×C₄	C₃²×C₄×D₃	1	2	yes	—
203	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
204	C₈	—	3	4	no	—
205	C₈	C₃²×C₄×D₃	1	4	yes	complemented
206	C₂×C₃	—	2	4	no	—
207	C₂×C₃	—	2	8	no	—
208	C₂×C₃	—	6	8	no	—
209	C₂×C₃	C₃×C₄²×D₃	1	4	yes	—
210	C₂×C₃	C₃×D₃×M₁₆	1	8	yes	—
211	C₂×C₃	C₂×C₃²×C₄²	1	1	yes	characteristic
212	C₂×C₃	C₂×C₃²×M₁₆	1	2	yes	—
213	D₃	—	2	2	no	—
214	C₂²	—	3	2	no	—
215	C₂²	—	6	2	no	—
216	C₂²	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	3	2	no	—
219	C₄	—	3	4	no	—
220	C₄	C₂×C₃²×C₄×D₃	1	2	yes	—
221	C₃	—	2	4	no	—
222	C₃	C₃×#4707	1	4	yes	direct product factor
223	C₃	C₃²×C₄×M₁₆	1	1	yes	characteristic, complemented
224	C₂	—	6	2	no	—
225	C₂	C₃²×C₄²×D₃	1	1	yes	characteristic
226	C₂	C₃²×D₃×M₁₆	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×C₄×M₁₆
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
3. C₃³×#97	C₂	semidirect
4. C₃²×C₈×Dic₃	C₂	semidirect
5. C₃²×#803	C₂	semidirect
7. C₃×#4707	C₃	direct (N × H)
15. C₃²×ΓL(1,𝔽₂₅)	C₄	semidirect
22. C₃²×#97	D₃	semidirect
23. C₃²×#97	C₂×C₃	semidirect
25. C₃×C₈×Dic₃	C₂×C₃	semidirect
26. C₃×#803	C₂×C₃	semidirect
34. C₃³×C₈	C₂×C₄	semidirect
35. C₃²×C₃⋊C₈	C₂×C₄	semidirect
37. #4707	C₃²	direct (N × H)
55. C₃×ΓL(1,𝔽₂₅)	C₃×C₄	semidirect
62. C₃²×Dic₃	M₁₆	semidirect
69. C₃×#97	C₃×D₃	semidirect
70. C₃×#97	C₂×C₃²	semidirect
72. C₈×Dic₃	C₂×C₃²	semidirect
73. #803	C₂×C₃²	semidirect
92. C₃²×C₈	C₄×D₃	semidirect
93. C₃²×C₈	C₂×C₃×C₄	semidirect
94. C₃×C₃⋊C₈	C₂×C₃×C₄	semidirect
116. ΓL(1,𝔽₂₅)	C₃²×C₄	semidirect
133. C₃×Dic₃	C₃×M₁₆	semidirect
137. #97	C₃²×D₃	semidirect
139. C₃³	C₄×M₁₆	semidirect
158. C₃×C₈	C₃×C₄×D₃	semidirect
159. C₃×C₈	C₂×C₃²×C₄	semidirect
160. C₃⋊C₈	C₂×C₃²×C₄	semidirect
193. Dic₃	C₃²×M₁₆	semidirect
195. C₃²	#4707	direct (N × H)
196. C₃²	C₃×C₄×M₁₆	semidirect
205. C₈	C₃²×C₄×D₃	semidirect
222. C₃	C₃×#4707	direct (N × H)
223. C₃	C₃²×C₄×M₁₆	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
C₃²×C₄²×D₃	2
C₃³×#97	2
C₃²×C₈×Dic₃	2
C₃²×#803	2
C₃²×C₄×M₁₆	3
C₃×#4707	3

Subgroups — C32×#4707 #13162137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4707 #13162137