Subgroups — C₃²×C₄×Dic₁₂ #13162134

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×Dic₆	C₂	1	2	yes	maximal
3	C₃³×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃²×#818	C₂	1	1	yes	maximal, characteristic
5	C₃²×#816	C₂	1	2	yes	maximal
6	C₃²×C₄×Q₁₆	—	3	1	no	maximal
7	C₃×C₄×Dic₁₂	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×Dic₆	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×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₃²×Dic₃⋊C₄	—	2	2	no	—
14	C₃²×C₄×Dic₃	—	2	2	no	—
15	C₃²×Dic₁₂	C₄	1	4	yes	direct product factor
16	C₂×C₃²×Q₁₆	—	3	1	no	—
17	C₃²×C₄×Q₈	—	3	2	no	—
18	C₂×C₃×Dic₁₂	C₂×C₃	1	4	yes	—
19	C₃×C₄×Dic₆	C₂×C₃	1	8	yes	—
20	C₃²×#107	—	3	1	no	—
21	C₃²×#103	—	3	2	no	—
22	C₃²×C₄×C₈	—	2	4	no	—
23	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
24	C₃²×C₄×C₈	C₂×C₃	1	4	yes	—
25	C₃×#818	C₂×C₃	1	4	yes	—
26	C₃×#816	C₂×C₃	1	8	yes	—
27	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃²×Dic₃	—	2	2	no	—
31	C₂×C₃²×Dic₃	—	2	2	no	—
32	C₃²×Dic₆	—	2	2	no	—
33	C₃²×Dic₆	C₂×C₄	1	4	yes	—
34	C₃³×C₈	—	2	1	no	—
35	C₃³×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄×Q₁₆	—	3	4	no	—
37	C₄×Dic₁₂	C₃²	1	1	yes	characteristic, direct product factor
38	C₂×C₃²×Q₈	—	3	2	no	—
39	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
40	C₃²×Q₁₆	—	3	4	no	—
41	C₂×C₃²×C₈	—	2	4	no	—
42	C₂×C₃²×C₈	—	2	4	no	—
43	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
46	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
47	C₃²×C₄⋊C₄	—	3	2	no	—
48	C₃²×C₄⋊C₄	—	6	2	no	—
49	C₃²×C₄²	—	2	4	no	—
50	C₃²×C₄²	—	6	2	no	—
51	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₃²×C₄²	C₂²×C₃	1	4	yes	—
53	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
54	C₃×Dic₃⋊C₄	—	2	8	no	—
55	C₃×C₄×Dic₃	—	2	8	no	—
56	C₃×Dic₁₂	C₃×C₄	1	16	yes	direct product factor
57	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
58	C₃³×C₄	—	2	1	no	—
59	C₃³×C₄	Q₁₆	1	2	yes	complemented
60	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₃³×C₄	#41	1	1	yes	characteristic
62	C₃²×Dic₃	—	2	4	no	—
63	C₃²×Dic₃	—	4	2	no	—
64	C₂×C₃×Q₁₆	—	3	4	no	—
65	C₃×C₄×Q₈	—	3	8	no	—
66	C₂×Dic₁₂	C₂×C₃²	1	1	yes	characteristic
67	C₄×Dic₆	C₂×C₃²	1	2	yes	—
68	C₃×#107	—	3	4	no	—
69	C₃×#103	—	3	8	no	—
70	C₃×C₄×C₈	—	2	4	no	—
71	C₃×C₄×C₈	C₃×D₃	1	4	yes	—
72	C₃×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
73	#818	C₂×C₃²	1	1	yes	characteristic
74	#816	C₂×C₃²	1	2	yes	—
75	C₃²×Q₈	—	3	4	no	—
76	C₃²×Q₈	—	6	2	no	—
77	C₂×C₃²×C₄	—	2	4	no	—
78	C₂×C₃²×C₄	—	2	4	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	6	2	no	—
81	C₂×C₃²×C₄	—	6	2	no	—
82	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
83	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
85	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
88	C₂×C₃×Dic₃	—	2	8	no	—
89	C₂×C₃×Dic₃	—	2	8	no	—
90	C₃×Dic₆	—	2	8	no	—
91	C₃×Dic₆	C₂×C₃×C₄	1	16	yes	—
92	C₃²×C₈	—	2	8	no	—
93	C₃²×C₈	—	2	1	no	—
94	C₃²×C₈	—	2	4	no	—
95	C₃²×C₈	—	2	8	no	—
96	C₃²×C₈	C₄×D₃	1	2	yes	—
97	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
98	C₄×Q₁₆	—	3	1	no	Sylow
99	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
100	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₃³	#135	1	1	yes	characteristic
102	C₂×C₃×Q₈	—	3	8	no	—
103	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
104	C₃×Q₁₆	—	3	16	no	—
105	C₂×C₃×C₈	—	2	4	no	—
106	C₂×C₃×C₈	—	2	4	no	—
107	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
108	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
109	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	3	8	no	—
112	C₃×C₄⋊C₄	—	6	8	no	—
113	C₃×C₄²	—	2	4	no	—
114	C₃×C₄²	—	6	8	no	—
115	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
116	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
117	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
118	Dic₃⋊C₄	—	2	2	no	—
119	C₄×Dic₃	—	2	2	no	—
120	Dic₁₂	C₃²×C₄	1	4	yes	direct product factor
121	C₂²×C₃²	—	2	4	no	—
122	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
123	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
124	C₃²×C₄	—	2	8	no	—
125	C₃²×C₄	—	2	1	no	—
126	C₃²×C₄	—	2	4	no	—
127	C₃²×C₄	—	2	8	no	—
128	C₃²×C₄	—	2	4	no	—
129	C₃²×C₄	—	2	4	no	—
130	C₃²×C₄	—	6	4	no	—
131	C₃²×C₄	—	12	2	no	—
132	C₃²×C₄	Dic₁₂	1	2	yes	direct product factor
133	C₃²×C₄	C₃×Q₁₆	1	8	yes	complemented
134	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₃²×C₄	#235	1	1	yes	characteristic
136	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
137	C₃²×C₄	C₃×#41	1	4	yes	—
138	C₃×Dic₃	—	2	16	no	—
139	C₃×Dic₃	—	4	8	no	—
140	C₂×Q₁₆	—	3	1	no	—
141	C₄×Q₈	—	3	2	no	—
142	#107	—	3	1	no	—
143	#103	—	3	2	no	—
144	C₄×C₈	C₃²×D₃	1	1	yes	characteristic
145	C₃³	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
146	C₃×Q₈	—	3	16	no	—
147	C₃×Q₈	—	6	8	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	6	8	no	—
152	C₂×C₃×C₄	—	6	8	no	—
153	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
154	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
155	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
158	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
159	C₂×Dic₃	—	2	2	no	—
160	C₂×Dic₃	—	2	2	no	—
161	Dic₆	—	2	2	no	—
162	Dic₆	C₂×C₃²×C₄	1	4	yes	—
163	C₃×C₈	—	2	8	no	—
164	C₃×C₈	—	2	4	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	2	8	no	—
167	C₃×C₈	C₃×C₄×D₃	1	8	yes	—
168	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
169	C₂×C₃²	—	2	4	no	—
170	C₂×C₃²	—	2	4	no	—
171	C₂×C₃²	—	2	4	no	—
172	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
173	C₂×C₃²	#904	1	1	yes	characteristic
174	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
175	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
176	C₂×C₃²	C₂×C₃×Q₁₆	1	4	yes	—
177	C₂×C₃²	C₃×#135	1	4	yes	—
178	C₂×Q₈	—	3	2	no	—
179	Q₁₆	—	3	4	no	—
180	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
181	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
182	C₄⋊C₄	—	3	2	no	—
183	C₄⋊C₄	—	6	2	no	—
184	C₄²	—	6	2	no	—
185	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
186	C₂²×C₃	—	2	4	no	—
187	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
188	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
189	C₃×C₄	—	2	8	no	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	2	8	no	—
195	C₃×C₄	—	6	16	no	—
196	C₃×C₄	—	12	8	no	—
197	C₃×C₄	C₃×Dic₁₂	1	8	yes	direct product factor
198	C₃×C₄	C₃²×Q₁₆	1	2	yes	complemented
199	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
200	C₃×C₄	C₃×#235	1	4	yes	—
201	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
202	C₃×C₄	C₃²×#41	1	1	yes	characteristic
203	Dic₃	—	2	4	no	—
204	Dic₃	—	4	2	no	—
205	C₃²	—	2	4	no	—
206	C₃²	C₄×Dic₁₂	1	1	yes	characteristic, direct product factor
207	C₃²	C₃×C₄×Q₁₆	1	4	yes	complemented
208	Q₈	—	3	4	no	—
209	Q₈	—	6	2	no	—
210	C₂×C₄	—	6	2	no	—
211	C₂×C₄	—	6	2	no	—
212	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
213	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	C₃²×C₄×D₃	1	2	yes	—
217	C₂×C₃	—	2	4	no	—
218	C₂×C₃	—	2	4	no	—
219	C₂×C₃	—	2	4	no	—
220	C₂×C₃	C₃×C₄×D₁₂	1	4	yes	—
221	C₂×C₃	C₃×#904	1	4	yes	—
222	C₂×C₃	C₂×C₃×Dic₁₂	1	4	yes	—
223	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
224	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
225	C₂×C₃	C₃²×#135	1	1	yes	characteristic
226	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	6	4	no	—
229	C₄	—	12	2	no	—
230	C₄	C₃²×Dic₁₂	1	2	yes	direct product factor
231	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
232	C₄	C₃²×#235	1	1	yes	characteristic
233	C₃	—	2	4	no	—
234	C₃	C₃×C₄×Dic₁₂	1	4	yes	direct product factor
235	C₃	C₃²×C₄×Q₁₆	1	1	yes	characteristic, complemented
236	C₂	C₃²×C₄×D₁₂	1	1	yes	characteristic
237	C₂	C₃²×#904	1	1	yes	characteristic
238	C₂	C₂×C₃²×Dic₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	D₁₂
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄×C₈	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×C₄×C₈	D₃
hypocenter	C₃	C₃²×C₄×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄² ◁ C₃²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. C₃×C₄×Dic₁₂	C₃	direct (N × H)
15. C₃²×Dic₁₂	C₄	direct (N × H)
37. C₄×Dic₁₂	C₃²	direct (N × H)
56. C₃×Dic₁₂	C₃×C₄	direct (N × H)
59. C₃³×C₄	Q₁₆	semidirect
120. Dic₁₂	C₃²×C₄	direct (N × H)
132. C₃²×C₄	Dic₁₂	direct (N × H)
133. C₃²×C₄	C₃×Q₁₆	semidirect
145. C₃³	C₄×Q₁₆	semidirect
197. C₃×C₄	C₃×Dic₁₂	direct (N × H)
198. C₃×C₄	C₃²×Q₁₆	semidirect
206. C₃²	C₄×Dic₁₂	direct (N × H)
207. C₃²	C₃×C₄×Q₁₆	semidirect
230. C₄	C₃²×Dic₁₂	direct (N × H)
234. C₃	C₃×C₄×Dic₁₂	direct (N × H)
235. C₃	C₃²×C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×Dic₁₂	2
C₃²×C₄×Dic₆	2
C₃³×C₄×C₈	2
C₃²×#818	2
C₃²×#816	2
C₃²×C₄×Q₁₆	3
C₃×C₄×Dic₁₂	3

Subgroups — C32×C4×Dic12 #13162134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×Dic₁₂ #13162134