Subgroups — C₃×C₄×#812 #11795854

Order 1152.

473 conjugacy classes of subgroups.

The subgroup lattice has 243 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#812	C₂	1	2	yes	maximal
3	C₃×#4461	C₂	1	2	yes	maximal, complemented
4	C₃×C₄×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×M₃₂	—	3	1	no	maximal
6	C₄×#812	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
10	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×#812	C₄	1	8	yes	direct product factor
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	4	yes	—
16	C₂×C₃×M₃₂	—	3	2	no	—
17	C₂×C₃×C₄×C₈	—	2	1	no	—
18	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×#812	C₂×C₃	1	2	yes	—
21	C₃×#346	—	3	2	no	—
22	C₃×C₄×C₁₆	—	3	2	no	—
23	#4461	C₂×C₃	1	2	yes	complemented
24	C₄×C₃⋊C₁₆	C₂×C₃	1	2	yes	complemented
25	C₂²×C₃²×C₄	C₈	1	2	yes	—
26	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₈	1	4	yes	—
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
30	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
33	C₃²×C₄²	C₈	1	2	yes	—
34	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₃×C₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
37	C₄×M₃₂	—	3	1	no	Sylow
38	C₂²×C₃×C₈	—	2	1	no	—
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	Dic₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₃×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃×C₄²	—	2	1	no	—
45	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
47	C₃×M₃₂	—	3	8	no	—
48	C₂×C₃×C₁₆	—	3	4	no	—
49	C₃×C₄×C₈	—	2	1	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	Dic₃	1	1	yes	characteristic
54	C₃×C₄×C₈	Dic₃	1	1	yes	characteristic
55	C₃×C₄×C₈	C₃×C₄	1	1	yes	characteristic
56	C₃×C₄×C₈	C₃×C₄	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	#812	C₃×C₄	1	8	yes	direct product factor
62	C₂×C₃⋊C₁₆	C₂²×C₃	1	4	yes	—
63	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
64	C₂×C₃²×C₄	—	2	4	no	—
65	C₂×C₃²×C₄	C₄²	1	2	yes	—
66	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
67	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
69	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
70	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₃²×C₈	C₄²	1	2	yes	—
72	C₃²×C₈	C₂×C₈	1	2	yes	—
73	C₃²×C₈	C₂×C₈	1	2	yes	—
74	C₃²×C₈	C₂²×C₄	1	2	yes	—
75	C₂×M₃₂	—	3	2	no	—
76	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
77	#346	—	3	2	no	—
78	C₄×C₁₆	—	3	2	no	—
79	C₂²×C₃×C₄	—	2	2	no	—
80	C₂²×C₃×C₄	—	2	1	no	—
81	C₂²×C₃×C₄	C₃⋊C₈	1	2	yes	—
82	C₂²×C₃×C₄	C₃×C₈	1	2	yes	—
83	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×C₃×C₈	—	2	4	no	—
86	C₂×C₃×C₈	—	2	4	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	C₃⋊C₈	1	4	yes	—
92	C₂×C₃×C₈	C₃×C₈	1	4	yes	—
93	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
94	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
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	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
103	C₃×C₄²	—	2	2	no	—
104	C₃×C₄²	—	2	1	no	—
105	C₃×C₄²	—	2	1	no	—
106	C₃×C₄²	C₃⋊C₈	1	2	yes	—
107	C₃×C₄²	C₃×C₈	1	2	yes	—
108	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
109	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
110	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
111	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
112	C₃×C₁₆	—	3	8	no	—
113	C₃⋊C₁₆	C₂×C₃×C₄	1	8	yes	complemented
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	C₄×C₈	1	2	yes	—
116	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
117	C₃²×C₄	—	2	2	no	—
118	C₃²×C₄	C₄×C₈	1	2	yes	—
119	C₃²×C₄	M₃₂	1	4	yes	complemented
120	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
121	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
122	C₂²×C₈	C₃×Dic₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
124	C₂×C₄²	C₃×Dic₃	1	1	yes	characteristic
125	M₃₂	—	3	8	no	—
126	C₂×C₁₆	—	3	4	no	—
127	C₄×C₈	C₃×Dic₃	1	1	yes	characteristic
128	C₄×C₈	C₃×Dic₃	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	C₂³×C₃	—	2	1	no	—
132	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
133	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
134	C₂×C₃×C₄	—	2	8	no	—
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
144	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
147	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
148	C₂×C₃×C₄	C₃×C₄²	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
150	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
152	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
153	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₃×C₈	—	2	2	no	—
156	C₃×C₈	—	2	2	no	—
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	C₂×C₃⋊C₈	1	2	yes	—
160	C₃×C₈	C₂×C₃⋊C₈	1	2	yes	—
161	C₃×C₈	C₄×Dic₃	1	2	yes	—
162	C₃×C₈	C₃×C₄²	1	2	yes	—
163	C₃×C₈	C₂×C₃×C₈	1	2	yes	—
164	C₃×C₈	C₂×C₃×C₈	1	2	yes	—
165	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
166	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
167	C₂×C₃²	—	2	2	no	—
168	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
169	C₂×C₃²	C₂×M₃₂	1	2	yes	—
170	C₂²×C₄	C₃×C₃⋊C₈	1	2	yes	—
171	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
172	C₂×C₈	C₃×C₃⋊C₈	1	4	yes	—
173	C₂×C₈	C₃×C₄×D₃	1	4	yes	—
174	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
175	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
176	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
178	C₄²	C₃×C₃⋊C₈	1	2	yes	—
179	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
180	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
181	C₁₆	—	3	8	no	—
182	C₂²×C₃	—	2	4	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	2	1	no	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	C₄×C₃⋊C₈	1	2	yes	—
188	C₂²×C₃	C₃×C₄×C₈	1	2	yes	—
189	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
190	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
191	C₃×C₄	—	2	4	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	C₄×C₃⋊C₈	1	2	yes	—
199	C₃×C₄	#812	1	4	yes	direct product factor
200	C₃×C₄	C₃×C₄×C₈	1	2	yes	—
201	C₃×C₄	C₃×M₃₂	1	4	yes	complemented
202	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
203	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
204	C₃×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
205	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
206	C₃²	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
207	C₂³	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
210	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
211	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
212	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
213	C₂×C₄	C₃×C₄×Dic₃	1	2	yes	—
214	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
215	C₈	C₂×C₃×C₃⋊C₈	1	2	yes	—
216	C₈	C₂×C₃×C₃⋊C₈	1	2	yes	—
217	C₈	C₃×C₄×Dic₃	1	2	yes	—
218	C₈	C₂×C₃×C₄×D₃	1	2	yes	—
219	C₂×C₃	—	2	4	no	—
220	C₂×C₃	—	2	2	no	—
221	C₂×C₃	—	2	2	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	2	no	—
224	C₂×C₃	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
225	C₂×C₃	C₂×#812	1	2	yes	—
226	C₂×C₃	C₂×C₃×C₄×C₈	1	1	yes	characteristic
227	C₂×C₃	C₂×C₃×M₃₂	1	2	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	C₃×C₄×C₃⋊C₈	1	2	yes	—
230	C₂²	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	C₃×C₄×C₃⋊C₈	1	2	yes	—
233	C₄	C₃×#812	1	4	yes	direct product factor
234	C₄	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
235	C₄	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
236	C₃	—	2	1	no	—
237	C₃	C₄×#812	1	1	yes	characteristic, direct product factor
238	C₃	C₃×C₄×M₃₂	1	1	yes	characteristic, complemented
239	C₂	—	2	2	no	—
240	C₂	C₂×C₃×C₄×C₃⋊C₈	1	1	yes	characteristic
241	C₂	C₂×C₃×#812	1	2	yes	—

	subgroup	G/H
center	C₃×C₄×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₃×C₄×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×C₄×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄×C₈ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#4461	C₂	semidirect
4. C₃×C₄×C₃⋊C₁₆	C₂	semidirect
6. C₄×#812	C₃	direct (N × H)
14. C₃×#812	C₄	direct (N × H)
23. #4461	C₂×C₃	semidirect
24. C₄×C₃⋊C₁₆	C₂×C₃	semidirect
36. C₃×C₃⋊C₁₆	C₂×C₄	semidirect
61. #812	C₃×C₄	direct (N × H)
113. C₃⋊C₁₆	C₂×C₃×C₄	semidirect
119. C₃²×C₄	M₃₂	semidirect
199. C₃×C₄	#812	direct (N × H)
201. C₃×C₄	C₃×M₃₂	semidirect
206. C₃²	C₄×M₃₂	semidirect
233. C₄	C₃×#812	direct (N × H)
237. C₃	C₄×#812	direct (N × H)
238. C₃	C₃×C₄×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×C₃×#812	2
C₃×#4461	2
C₃×C₄×C₃⋊C₁₆	2
C₃×C₄×M₃₂	3
C₄×#812	3

Subgroups — C3×C4×#812 #11795854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#812 #11795854