Subgroups — C₃×C₈×Dic₁₂ #11777228

Order 1152.

365 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	C₃×C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×Dic₆	C₂	1	2	yes	maximal
3	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic
4	C₃×#4458	C₂	1	1	yes	maximal, characteristic
5	C₃×#4456	C₂	1	2	yes	maximal
6	C₃×C₈×Q₁₆	—	3	1	no	maximal
7	C₈×Dic₁₂	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×Dic₁₂	C₄	1	1	yes	characteristic
9	C₃×C₄×Dic₆	C₂²	1	2	yes	—
10	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×#818	C₄	1	1	yes	characteristic
14	C₃×#816	C₄	1	2	yes	—
15	C₃×Dic₃⋊C₈	—	2	2	no	—
16	C₃×C₈×Dic₃	—	2	2	no	—
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃×C₈×Q₈	—	3	2	no	—
19	C₃×C₄×Q₁₆	—	3	1	no	—
20	C₄×Dic₁₂	C₂×C₃	1	1	yes	characteristic
21	C₈×Dic₆	C₂×C₃	1	2	yes	—
22	C₃×#335	—	3	1	no	—
23	C₃×Q₈⋊C₈	—	3	2	no	—
24	C₃×C₈²	—	2	1	no	—
25	C₃×C₈²	D₃	1	1	yes	characteristic
26	C₃×C₈²	C₂×C₃	1	1	yes	characteristic
27	#4458	C₂×C₃	1	1	yes	characteristic
28	#4456	C₂×C₃	1	2	yes	—
29	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	—	2	1	no	—
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
35	C₃²×C₄²	C₂³	1	1	yes	characteristic
36	C₃×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
37	C₃×Dic₃⋊C₄	—	2	2	no	—
38	C₃×C₄×Dic₃	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₃×Dic₁₂	C₈	1	4	yes	direct product factor
41	C₈×Q₁₆	—	3	1	no	Sylow
42	C₂×C₃×Q₁₆	—	3	1	no	—
43	C₃×C₄×Q₈	—	3	2	no	—
44	C₂×Dic₁₂	C₃×C₄	1	1	yes	characteristic
45	C₄×Dic₆	C₂²×C₃	1	2	yes	—
46	C₃×#107	—	3	1	no	—
47	C₃×C₄⋊C₈	—	3	2	no	—
48	C₃×C₄⋊C₈	—	6	2	no	—
49	C₃×#103	—	3	2	no	—
50	C₃×C₄×C₈	—	2	1	no	—
51	C₃×C₄×C₈	—	2	1	no	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×C₄×C₈	—	6	2	no	—
54	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	#818	C₃×C₄	1	1	yes	characteristic
61	#816	C₃×C₄	1	2	yes	—
62	Dic₃⋊C₈	—	2	2	no	—
63	C₈×Dic₃	—	2	2	no	—
64	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
65	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	#41	1	1	yes	characteristic
68	C₂×C₃×Dic₃	—	2	2	no	—
69	C₂×C₃×Dic₃	—	2	2	no	—
70	C₃×Dic₆	—	2	2	no	—
71	C₃×Dic₆	C₂×C₈	1	4	yes	—
72	C₃²×C₈	—	2	2	no	—
73	C₃²×C₈	—	2	1	no	—
74	C₃²×C₈	—	2	1	no	—
75	C₃²×C₈	C₂×C₈	1	2	yes	—
76	C₃²×C₈	Q₁₆	1	2	yes	complemented
77	C₃×C₃⋊C₈	—	4	2	no	—
78	C₈×Q₈	—	3	2	no	—
79	C₄×Q₁₆	—	3	1	no	—
80	#335	—	3	1	no	—
81	Q₈⋊C₈	—	3	2	no	—
82	C₈²	C₃×D₃	1	1	yes	characteristic
83	C₂×C₃×Q₈	—	3	2	no	—
84	C₂×Dic₆	C₂×C₃×C₄	1	2	yes	—
85	C₃×Q₁₆	—	3	4	no	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	6	2	no	—
94	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
96	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
97	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
102	C₃×C₄⋊C₄	—	3	2	no	—
103	C₃×C₄⋊C₄	—	6	2	no	—
104	C₃×C₄²	—	2	1	no	—
105	C₃×C₄²	—	6	2	no	—
106	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
107	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
108	C₄⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
109	Dic₃⋊C₄	—	2	2	no	—
110	C₄×Dic₃	—	2	2	no	—
111	C₂×C₃⋊C₈	—	2	2	no	—
112	Dic₁₂	C₃×C₈	1	4	yes	direct product factor
113	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
114	C₃²×C₄	—	2	1	no	—
115	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
116	C₃²×C₄	#131	1	1	yes	characteristic
117	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
118	C₃²×C₄	#135	1	1	yes	characteristic
119	C₃×Dic₃	—	2	4	no	—
120	C₃×Dic₃	—	4	2	no	—
121	C₂×Q₁₆	—	3	1	no	—
122	C₄×Q₈	—	3	2	no	—
123	#107	—	3	1	no	—
124	C₄⋊C₈	—	3	2	no	—
125	C₄⋊C₈	—	6	2	no	—
126	#103	—	3	2	no	—
127	C₄×C₈	—	6	2	no	—
128	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
129	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
130	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
131	C₃×Q₈	—	3	4	no	—
132	C₃×Q₈	—	6	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	6	2	no	—
137	C₂×C₃×C₄	—	6	2	no	—
138	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
140	C₂×C₃×C₄	#235	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
144	C₂×Dic₃	—	2	2	no	—
145	C₂×Dic₃	—	2	2	no	—
146	Dic₆	—	2	2	no	—
147	Dic₆	C₂×C₃×C₈	1	4	yes	—
148	C₃×C₈	—	2	4	no	—
149	C₃×C₈	—	2	2	no	—
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	—	2	2	no	—
153	C₃×C₈	—	2	1	no	—
154	C₃×C₈	—	2	1	no	—
155	C₃×C₈	—	2	1	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	12	2	no	—
160	C₃×C₈	D₃×C₈	1	2	yes	—
161	C₃×C₈	Dic₁₂	1	2	yes	direct product factor
162	C₃×C₈	C₂×C₃×C₈	1	2	yes	—
163	C₃×C₈	C₃×Q₁₆	1	2	yes	complemented
164	C₃⋊C₈	—	4	2	no	—
165	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
166	C₂×C₃²	C₄×Q₁₆	1	1	yes	characteristic
167	C₂×C₃²	#443	1	1	yes	characteristic
168	C₂×Q₈	—	3	2	no	—
169	Q₁₆	—	3	4	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	6	2	no	—
172	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
173	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
174	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
175	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
176	C₄⋊C₄	—	3	2	no	—
177	C₄⋊C₄	—	6	2	no	—
178	C₄²	—	6	2	no	—
179	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
182	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	6	4	no	—
191	C₃×C₄	—	12	2	no	—
192	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
193	C₃×C₄	#901	1	1	yes	characteristic
194	C₃×C₄	#904	1	1	yes	characteristic
195	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
196	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
197	C₃×C₄	C₃×#131	1	1	yes	characteristic
198	C₃×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
199	C₃×C₄	C₃×#135	1	1	yes	characteristic
200	Dic₃	—	2	4	no	—
201	Dic₃	—	4	2	no	—
202	C₃²	C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
203	Q₈	—	3	4	no	—
204	Q₈	—	6	2	no	—
205	C₂×C₄	—	6	2	no	—
206	C₂×C₄	—	6	2	no	—
207	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
208	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
209	C₂×C₄	C₃×#235	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	12	2	no	—
214	C₈	C₃×D₃×C₈	1	2	yes	—
215	C₈	C₃×Dic₁₂	1	2	yes	direct product factor
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
220	C₂×C₃	C₄×Dic₁₂	1	1	yes	characteristic
221	C₂×C₃	#4700	1	1	yes	characteristic
222	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
223	C₂×C₃	C₃×C₄×Q₁₆	1	1	yes	characteristic
224	C₂×C₃	C₃×#443	1	1	yes	characteristic
225	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	6	4	no	—
228	C₄	—	12	2	no	—
229	C₄	C₂×C₃×D₃×C₈	1	1	yes	characteristic
230	C₄	C₃×#901	1	1	yes	characteristic
231	C₄	C₃×#904	1	1	yes	characteristic
232	C₄	C₂×C₃×Dic₁₂	1	1	yes	characteristic
233	C₃	—	2	1	no	—
234	C₃	C₈×Dic₁₂	1	1	yes	characteristic, direct product factor
235	C₃	C₃×C₈×Q₁₆	1	1	yes	characteristic, complemented
236	C₂	C₃×C₈×D₁₂	1	1	yes	characteristic
237	C₂	C₃×C₄×Dic₁₂	1	1	yes	characteristic
238	C₂	C₃×#4700	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	D₁₂
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×C₈²	C₂
socle, prime radical	C₂²×C₃²	C₄×D₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×C₈²	D₃
hypocenter	C₃	C₃×C₈×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₈ ◁ C₃×C₈²

Complements

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

N	H ≅ G/N	type
7. C₈×Dic₁₂	C₃	direct (N × H)
40. C₃×Dic₁₂	C₈	direct (N × H)
76. C₃²×C₈	Q₁₆	semidirect
112. Dic₁₂	C₃×C₈	direct (N × H)
161. C₃×C₈	Dic₁₂	direct (N × H)
163. C₃×C₈	C₃×Q₁₆	semidirect
202. C₃²	C₈×Q₁₆	semidirect
215. C₈	C₃×Dic₁₂	direct (N × H)
234. C₃	C₈×Dic₁₂	direct (N × H)
235. C₃	C₃×C₈×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Dic₁₂	2
C₃×C₈×Dic₆	2
C₃²×C₈²	2
C₃×#4458	2
C₃×#4456	2
C₃×C₈×Q₁₆	3
C₈×Dic₁₂	3

Subgroups — C3×C8×Dic12 #11777228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×Dic₁₂ #11777228