Subgroups — C₃²×C₄×#208 #13162358

Order 1728.

1284 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₃²×C₄²	C₂×D₃
commutator	C₂×C₃	C₂×C₃²×C₄²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₂×C₃³×C₄²	C₂
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₂×C₃²×C₄²	D₃
hypocenter	C₃	C₃²×C₄×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄² ◁ C₂×C₃²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₃²×#803	C₂	semidirect
4. C₃²×C₄×C₃⋊C₈	C₂	semidirect
6. C₃×C₄×#208	C₃	direct (N × H)
12. C₃²×#208	C₄	direct (N × H)
21. C₃×#803	C₂×C₃	semidirect
22. C₃×C₄×C₃⋊C₈	C₂×C₃	semidirect
30. C₃²×C₃⋊C₈	C₂×C₄	semidirect
32. C₄×#208	C₃²	direct (N × H)
50. C₃×#208	C₃×C₄	direct (N × H)
57. C₃³×C₄	M₁₆	semidirect
66. #803	C₂×C₃²	semidirect
67. C₄×C₃⋊C₈	C₂×C₃²	semidirect
90. C₃×C₃⋊C₈	C₂×C₃×C₄	semidirect
112. #208	C₃²×C₄	direct (N × H)
129. C₃²×C₄	#208	direct (N × H)
132. C₃²×C₄	C₃×M₁₆	semidirect
139. C₃³	C₄×M₁₆	semidirect
162. C₃⋊C₈	C₂×C₃²×C₄	semidirect
194. C₃×C₄	C₃×#208	direct (N × H)
197. C₃×C₄	C₃²×M₁₆	semidirect
201. C₃²	C₄×#208	direct (N × H)
202. C₃²	C₃×C₄×M₁₆	semidirect
224. C₄	C₃²×#208	direct (N × H)
228. C₃	C₃×C₄×#208	direct (N × H)
229. 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×C₃³×C₄²	2
C₂×C₃²×#208	2
C₃²×#803	2
C₃²×C₄×C₃⋊C₈	2
C₃²×C₄×M₁₆	3
C₃×C₄×#208	3

Subgroups — C32×C4×#208 #13162358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×#208 #13162358