Subgroups — C₃²×#4901 #13162337

Order 1728.

616 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#901	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#798	C₂	1	2	yes	maximal, complemented
5	C₃²×D₃×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₃²×#504	—	3	1	no	maximal
7	C₃×#4901	C₃	1	4	yes	maximal, direct product factor
8	C₃²×#235	C₄	1	1	yes	characteristic
9	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
10	C₃²×#208	C₄	1	1	yes	characteristic
11	C₃²×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
12	C₃²×D₃×C₈	C₂²	1	2	yes	—
13	C₃³×C₁₆	C₂²	1	1	yes	characteristic
14	C₃³×C₁₆	C₂²	1	1	yes	characteristic
15	C₃²×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₃²×#131	—	3	1	no	—
17	C₃×#901	C₂×C₃	1	4	yes	—
18	C₃²×M₃₂	—	3	1	no	—
19	C₃²×M₃₂	—	3	2	no	—
20	C₂×C₃²×C₁₆	—	2	4	no	—
21	C₂×C₃²×C₁₆	—	3	2	no	—
22	C₂×C₃²×C₁₆	D₃	1	1	yes	characteristic, complemented
23	C₂×C₃²×C₁₆	C₂×C₃	1	4	yes	complemented
24	C₃×#812	C₂×C₃	1	4	yes	complemented
25	C₃×#798	C₂×C₃	1	8	yes	complemented
26	C₃×D₃×C₁₆	C₂×C₃	1	8	yes	complemented
27	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₃²×#67	C₈	1	2	yes	—
29	C₃²×D₁₂	C₈	1	1	yes	characteristic
30	C₃²×C₄×D₃	C₂×C₄	1	2	yes	—
31	C₃²×Dic₆	C₈	1	1	yes	characteristic
32	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃³×C₈	C₂³	1	1	yes	characteristic
34	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₃×#504	—	3	4	no	—
36	#4901	C₃²	1	1	yes	characteristic, direct product factor
37	C₃²×#41	—	3	1	no	—
38	C₃×#235	C₃×C₄	1	4	yes	—
39	C₃²×M₁₆	—	3	1	no	—
40	C₃²×M₁₆	—	3	2	no	—
41	C₂×C₃²×C₈	—	2	4	no	—
42	C₂×C₃²×C₈	—	3	2	no	—
43	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
45	C₃×#208	C₃×C₄	1	4	yes	—
46	C₃×ΓL(1,𝔽₂₅)	C₃×C₄	1	8	yes	—
47	C₃×D₃×C₈	C₂²×C₃	1	8	yes	—
48	C₃²×C₁₆	—	2	4	no	—
49	C₃²×C₁₆	—	2	4	no	—
50	C₃²×C₁₆	—	3	2	no	—
51	C₃²×C₁₆	C₂×D₃	1	1	yes	characteristic
52	C₃²×C₁₆	C₂×D₃	1	1	yes	characteristic
53	C₃²×C₁₆	C₂²×C₃	1	4	yes	—
54	C₃²×C₁₆	C₂²×C₃	1	4	yes	—
55	C₃×C₃⋊C₁₆	C₂²×C₃	1	8	yes	—
56	C₂²×C₃³	C₂×C₈	1	1	yes	characteristic
57	C₂×C₃²×D₃	C₂×C₈	1	2	yes	—
58	C₃³×C₄	C₂×C₈	1	1	yes	characteristic
59	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₃²×Dic₃	C₂×C₈	1	2	yes	—
61	C₃×#131	—	3	4	no	—
62	#901	C₂×C₃²	1	1	yes	characteristic
63	C₃×M₃₂	—	3	4	no	—
64	C₃×M₃₂	—	3	8	no	—
65	C₂×C₃×C₁₆	—	2	4	no	—
66	C₂×C₃×C₁₆	—	3	8	no	—
67	C₂×C₃×C₁₆	C₃×D₃	1	4	yes	complemented
68	C₂×C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
69	#812	C₂×C₃²	1	1	yes	characteristic, complemented
70	#798	C₂×C₃²	1	2	yes	complemented
71	D₃×C₁₆	C₂×C₃²	1	2	yes	complemented
72	C₃²×Q₈	—	3	1	no	—
73	C₃²×D₄	—	3	2	no	—
74	C₃²×D₄	—	3	1	no	—
75	C₂×C₃²×C₄	—	2	4	no	—
76	C₂×C₃²×C₄	—	3	2	no	—
77	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
79	C₃×#67	C₃×C₈	1	8	yes	—
80	C₃×D₁₂	C₃×C₈	1	4	yes	—
81	C₃×C₄×D₃	C₂×C₃×C₄	1	8	yes	—
82	C₃×Dic₆	C₃×C₈	1	4	yes	—
83	C₃²×C₈	—	2	4	no	—
84	C₃²×C₈	—	2	4	no	—
85	C₃²×C₈	—	3	2	no	—
86	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
87	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
88	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
89	C₃²×C₈	C₂³×C₃	1	4	yes	—
90	C₃×C₃⋊C₈	C₂×C₃×C₄	1	8	yes	—
91	#504	—	3	1	no	Sylow
92	C₂×C₃³	—	2	1	no	—
93	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
94	C₃²×D₃	—	2	2	no	—
95	C₃×#41	—	3	4	no	—
96	#235	C₃²×C₄	1	1	yes	characteristic
97	C₃×M₁₆	—	3	4	no	—
98	C₃×M₁₆	—	3	8	no	—
99	C₂×C₃×C₈	—	2	4	no	—
100	C₂×C₃×C₈	—	3	8	no	—
101	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
102	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
103	#208	C₃²×C₄	1	1	yes	characteristic
104	ΓL(1,𝔽₂₅)	C₃²×C₄	1	2	yes	—
105	D₃×C₈	C₂²×C₃²	1	2	yes	—
106	C₃×C₁₆	—	2	4	no	—
107	C₃×C₁₆	—	2	4	no	—
108	C₃×C₁₆	—	3	8	no	—
109	C₃×C₁₆	C₂×C₃×D₃	1	4	yes	—
110	C₃×C₁₆	C₂×C₃×D₃	1	4	yes	—
111	C₃×C₁₆	C₂²×C₃²	1	1	yes	characteristic
112	C₃×C₁₆	C₂²×C₃²	1	1	yes	characteristic
113	C₃⋊C₁₆	C₂²×C₃²	1	2	yes	—
114	C₂²×C₃²	—	2	4	no	—
115	C₂²×C₃²	—	3	2	no	—
116	C₂²×C₃²	D₃×C₈	1	1	yes	characteristic
117	C₂²×C₃²	C₂×C₃×C₈	1	4	yes	—
118	C₂×C₃×D₃	C₂×C₃×C₈	1	8	yes	—
119	C₃²×C₄	—	2	4	no	—
120	C₃²×C₄	—	2	4	no	—
121	C₃²×C₄	—	3	2	no	—
122	C₃²×C₄	D₃×C₈	1	1	yes	characteristic
123	C₃²×C₄	C₂×C₃×C₈	1	4	yes	—
124	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
125	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
126	C₃×Dic₃	C₂×C₃×C₈	1	8	yes	—
127	#131	—	3	1	no	—
128	M₃₂	—	3	1	no	—
129	M₃₂	—	3	2	no	—
130	C₂×C₁₆	—	3	2	no	—
131	C₂×C₁₆	C₃²×D₃	1	1	yes	characteristic, complemented
132	C₃³	#504	1	1	yes	characteristic, complemented, Sylow
133	C₃×Q₈	—	3	4	no	—
134	C₃×D₄	—	3	8	no	—
135	C₃×D₄	—	3	4	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	3	8	no	—
138	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
139	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
140	#67	C₃²×C₈	1	2	yes	—
141	D₁₂	C₃²×C₈	1	1	yes	characteristic
142	C₄×D₃	C₂×C₃²×C₄	1	2	yes	—
143	Dic₆	C₃²×C₈	1	1	yes	characteristic
144	C₃×C₈	—	2	4	no	—
145	C₃×C₈	—	2	4	no	—
146	C₃×C₈	—	3	8	no	—
147	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
148	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
149	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
150	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
151	C₃⋊C₈	C₂×C₃²×C₄	1	2	yes	—
152	C₂×C₃²	—	2	8	no	—
153	C₂×C₃²	—	2	4	no	—
154	C₂×C₃²	—	2	1	no	—
155	C₂×C₃²	—	2	4	no	—
156	C₂×C₃²	—	6	2	no	—
157	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
158	C₂×C₃²	C₂²×C₃×C₈	1	4	yes	—
159	C₃×D₃	—	2	8	no	—
160	#41	—	3	1	no	—
161	M₁₆	—	3	1	no	—
162	M₁₆	—	3	2	no	—
163	C₂×C₈	—	3	2	no	—
164	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
165	C₁₆	—	3	2	no	—
166	C₁₆	C₂×C₃²×D₃	1	1	yes	characteristic
167	C₁₆	C₂×C₃²×D₃	1	1	yes	characteristic
168	C₂²×C₃	—	2	4	no	—
169	C₂²×C₃	—	3	8	no	—
170	C₂²×C₃	C₃×D₃×C₈	1	4	yes	—
171	C₂²×C₃	C₂×C₃²×C₈	1	1	yes	characteristic
172	C₂×D₃	C₂×C₃²×C₈	1	2	yes	—
173	C₃×C₄	—	2	4	no	—
174	C₃×C₄	—	2	4	no	—
175	C₃×C₄	—	3	8	no	—
176	C₃×C₄	C₃×D₃×C₈	1	4	yes	—
177	C₃×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
178	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
179	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
180	Dic₃	C₂×C₃²×C₈	1	2	yes	—
181	C₃²	—	2	4	no	—
182	C₃²	#4901	1	1	yes	characteristic, direct product factor
183	C₃²	C₃×#504	1	4	yes	complemented
184	Q₈	—	3	1	no	—
185	D₄	—	3	2	no	—
186	D₄	—	3	1	no	—
187	C₂×C₄	—	3	2	no	—
188	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
189	C₈	—	3	2	no	—
190	C₈	C₃²×C₄×D₃	1	1	yes	characteristic
191	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
192	C₂×C₃	—	2	8	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	—	2	4	no	—
195	C₂×C₃	—	2	4	no	—
196	C₂×C₃	—	6	8	no	—
197	C₂×C₃	C₂×C₃×D₃×C₈	1	4	yes	—
198	C₂×C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
199	D₃	—	2	2	no	—
200	C₂²	—	3	2	no	—
201	C₂²	C₃²×D₃×C₈	1	1	yes	characteristic
202	C₄	—	3	2	no	—
203	C₄	C₃²×D₃×C₈	1	1	yes	characteristic
204	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
205	C₃	—	2	4	no	—
206	C₃	C₃×#4901	1	4	yes	direct product factor
207	C₃	C₃²×#504	1	1	yes	characteristic, complemented
208	C₂	—	2	1	no	—
209	C₂	—	6	2	no	—
210	C₂	C₂×C₃²×D₃×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₁₆	C₂×D₃
commutator	C₂×C₃	C₂²×C₃²×C₈
Frattini	C₈	C₂²×C₃²×D₃
Fitting	C₂×C₃³×C₁₆	C₂
socle	C₂×C₃³	C₂²×C₈
prime radical	C₃²×#235	C₄
cosocle	C₃×C₈	C₂³×C₃²
hypercenter	C₂×C₃²×C₁₆	D₃
hypocenter	C₃	C₃²×#504
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₁₆ ◁ C₂×C₃²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×C₁₆	C₂	semidirect
3. C₃²×#812	C₂	semidirect
4. C₃²×#798	C₂	semidirect
5. C₃²×D₃×C₁₆	C₂	semidirect
7. C₃×#4901	C₃	direct (N × H)
22. C₂×C₃²×C₁₆	D₃	semidirect
23. C₂×C₃²×C₁₆	C₂×C₃	semidirect
24. C₃×#812	C₂×C₃	semidirect
25. C₃×#798	C₂×C₃	semidirect
26. C₃×D₃×C₁₆	C₂×C₃	semidirect
36. #4901	C₃²	direct (N × H)
67. C₂×C₃×C₁₆	C₃×D₃	semidirect
68. C₂×C₃×C₁₆	C₂×C₃²	semidirect
69. #812	C₂×C₃²	semidirect
70. #798	C₂×C₃²	semidirect
71. D₃×C₁₆	C₂×C₃²	semidirect
131. C₂×C₁₆	C₃²×D₃	semidirect
132. C₃³	#504	semidirect
182. C₃²	#4901	direct (N × H)
183. C₃²	C₃×#504	semidirect
206. C₃	C₃×#4901	direct (N × H)
207. C₃	C₃²×#504	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#901	2
C₂×C₃³×C₁₆	2
C₃²×#812	2
C₃²×#798	2
C₃²×D₃×C₁₆	2
C₃²×#504	3
C₃×#4901	3

Subgroups — C32×#4901 #13162337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4901 #13162337