Subgroups — C₃×C₄×#874 #11802526

Order 1152.

1977 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#874	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₁₂	C₂	1	6	yes	maximal, complemented
3	C₃²×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4939	C₂	1	6	yes	maximal, complemented
5	C₃×#4935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×#127	—	3	1	no	maximal
7	C₄×#874	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×D₁₂	C₂²	1	6	yes	—
9	C₂²×C₃×C₄×D₃	C₂²	1	4	yes	—
10	C₂×C₃²×C₄²	C₂²	1	3	yes	—
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄²	C₂²	1	3	yes	—
13	C₂×C₃×(C₂×D₃)⋊C₄	C₂²	1	12	yes	—
14	C₂×C₃×C₄⋊Dic₃	C₂²	1	6	yes	—
15	C₃×#874	C₄	1	8	yes	direct product factor
16	C₃×C₄×D₁₂	—	2	24	no	—
17	C₂×C₃×#127	—	3	1	no	—
18	C₂×C₃×C₄×D₄	—	3	6	no	—
19	C₂×#874	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄×D₁₂	C₂×C₃	1	6	yes	complemented
21	C₃×#390	—	3	6	no	—
22	C₃×#384	—	3	1	no	—
23	C₃×C₄³	—	2	1	no	—
24	C₃×C₄³	D₃	1	1	yes	characteristic, complemented
25	C₃×C₄³	C₂×C₃	1	1	yes	characteristic, complemented
26	#4939	C₂×C₃	1	6	yes	complemented
27	#4935	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×C₃×D₃	C₂³	1	4	yes	—
29	C₂²×C₃²×C₄	C₂³	1	3	yes	—
30	C₂²×C₃²×C₄	C₂³	1	3	yes	—
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	C₂³	1	4	yes	—
33	C₂×C₃×D₁₂	—	2	12	no	—
34	C₂×C₃×D₁₂	—	2	12	no	—
35	C₂×C₃×D₁₂	C₂×C₄	1	24	yes	complemented
36	C₂×C₃×C₄×D₃	—	2	24	no	—
37	C₃²×C₄²	—	2	6	no	—
38	C₃²×C₄²	C₂×C₄	1	4	yes	complemented
39	C₃²×C₄²	D₄	1	12	yes	complemented
40	C₃×(C₂×D₃)⋊C₄	—	2	24	no	—
41	C₃×C₄⋊Dic₃	—	2	12	no	—
42	C₄×#127	—	3	1	no	Sylow
43	C₂²×C₃×D₄	—	3	6	no	—
44	C₂³×C₃×C₄	—	3	4	no	—
45	C₂²×D₁₂	C₂²×C₃	1	6	yes	—
46	C₂²×C₄×D₃	C₂²×C₃	1	4	yes	—
47	C₃×#127	—	3	8	no	—
48	C₃×C₄×D₄	—	6	24	no	—
49	C₂×C₃×C₄⋊C₄	—	3	6	no	—
50	C₂×C₃×C₂²⋊C₄	—	3	12	no	—
51	C₂×C₃×C₄²	—	2	3	no	—
52	C₂×C₃×C₄²	—	2	1	no	—
53	C₂×C₃×C₄²	—	2	3	no	—
54	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
57	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
58	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
60	C₂×(C₂×D₃)⋊C₄	C₂²×C₃	1	12	yes	—
61	C₂×C₄⋊Dic₃	C₂²×C₃	1	6	yes	—
62	#874	C₃×C₄	1	8	yes	direct product factor
63	C₄×D₁₂	—	2	24	no	—
64	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
65	C₂²×C₃×D₃	—	2	12	no	—
66	C₂²×C₃×D₃	—	2	12	no	—
67	C₂²×C₃×D₃	C₂²×C₄	1	8	yes	—
68	C₂×C₃²×C₄	—	2	6	no	—
69	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
70	C₂×C₃²×C₄	C₂×D₄	1	6	yes	—
71	C₂×C₃²×C₄	C₂×D₄	1	6	yes	—
72	C₂×C₃²×C₄	C₂×D₄	1	6	yes	—
73	C₂×C₃²×C₄	#41	1	6	yes	—
74	C₂×C₃×Dic₃	—	2	12	no	—
75	C₃×D₁₂	—	2	48	no	—
76	C₃×C₄×D₃	—	4	16	no	—
77	C₂×#127	—	3	1	no	—
78	C₂×C₄×D₄	—	3	6	no	—
79	#390	—	3	6	no	—
80	#384	—	3	1	no	—
81	C₄³	C₃×D₃	1	1	yes	characteristic, complemented
82	C₂⁴×C₃	—	3	4	no	—
83	C₂³×D₃	C₂³×C₃	1	4	yes	—
84	C₂×C₃×D₄	—	3	24	no	—
85	C₂×C₃×D₄	—	6	12	no	—
86	C₂×C₃×D₄	—	6	12	no	—
87	C₂²×C₃×C₄	—	2	3	no	—
88	C₂²×C₃×C₄	—	2	3	no	—
89	C₂²×C₃×C₄	—	2	1	no	—
90	C₂²×C₃×C₄	—	3	4	no	—
91	C₂²×C₃×C₄	—	6	24	no	—
92	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
93	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
96	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
97	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂²×Dic₃	C₂³×C₃	1	4	yes	—
99	C₂×D₁₂	—	2	12	no	—
100	C₂×D₁₂	—	2	12	no	—
101	C₂×D₁₂	C₂×C₃×C₄	1	24	yes	complemented
102	C₂×C₄×D₃	—	2	24	no	—
103	C₃×C₄⋊C₄	—	6	12	no	—
104	C₃×C₂²⋊C₄	—	6	24	no	—
105	C₃×C₄²	—	2	12	no	—
106	C₃×C₄²	—	2	6	no	—
107	C₃×C₄²	—	2	6	no	—
108	C₃×C₄²	—	2	4	no	—
109	C₃×C₄²	—	2	12	no	—
110	C₃×C₄²	C₄×D₃	1	4	yes	complemented
111	C₃×C₄²	D₁₂	1	12	yes	complemented
112	C₃×C₄²	C₂×C₃×C₄	1	4	yes	complemented
113	C₃×C₄²	C₃×D₄	1	12	yes	complemented
114	(C₂×D₃)⋊C₄	—	2	24	no	—
115	C₄⋊Dic₃	—	2	12	no	—
116	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
117	C₂²×C₃²	C₂²×D₄	1	3	yes	—
118	C₂²×C₃²	C₂×#41	1	3	yes	—
119	C₂×C₃×D₃	—	2	24	no	—
120	C₂×C₃×D₃	—	4	4	no	—
121	C₂×C₃×D₃	—	4	12	no	—
122	C₃²×C₄	—	2	6	no	—
123	C₃²×C₄	C₄×D₄	1	12	yes	complemented
124	C₃²×C₄	#127	1	4	yes	complemented
125	C₃×Dic₃	—	4	4	no	—
126	C₂²×D₄	—	3	6	no	—
127	C₂³×C₄	—	3	4	no	—
128	#127	—	3	8	no	—
129	C₄×D₄	—	6	24	no	—
130	C₂×C₄⋊C₄	—	3	6	no	—
131	C₂×C₂²⋊C₄	—	3	12	no	—
132	C₂×C₄²	C₂×C₃×D₃	1	3	yes	—
133	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
134	C₂×C₄²	C₂×C₃×D₃	1	3	yes	—
135	C₂³×C₃	—	2	1	no	—
136	C₂³×C₃	—	3	8	no	—
137	C₂³×C₃	—	6	12	no	—
138	C₂³×C₃	—	6	12	no	—
139	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
140	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
141	C₂²×D₃	—	2	12	no	—
142	C₂²×D₃	—	2	12	no	—
143	C₂²×D₃	C₂²×C₃×C₄	1	8	yes	—
144	C₃×D₄	—	6	48	no	—
145	C₂×C₃×C₄	—	2	12	no	—
146	C₂×C₃×C₄	—	2	6	no	—
147	C₂×C₃×C₄	—	2	6	no	—
148	C₂×C₃×C₄	—	2	6	no	—
149	C₂×C₃×C₄	—	2	6	no	—
150	C₂×C₃×C₄	—	2	6	no	—
151	C₂×C₃×C₄	—	2	6	no	—
152	C₂×C₃×C₄	—	2	6	no	—
153	C₂×C₃×C₄	—	6	12	no	—
154	C₂×C₃×C₄	—	12	16	no	—
155	C₂×C₃×C₄	C₂×C₄×D₃	1	6	yes	—
156	C₂×C₃×C₄	C₂×D₁₂	1	6	yes	—
157	C₂×C₃×C₄	C₂×D₁₂	1	6	yes	—
158	C₂×C₃×C₄	C₂×D₁₂	1	6	yes	—
159	C₂×C₃×C₄	#235	1	6	yes	—
160	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
161	C₂×C₃×C₄	C₂×C₃×D₄	1	6	yes	—
162	C₂×C₃×C₄	C₂×C₃×D₄	1	6	yes	—
163	C₂×C₃×C₄	C₂×C₃×D₄	1	6	yes	—
164	C₂×C₃×C₄	C₃×#41	1	6	yes	—
165	C₂×Dic₃	—	2	12	no	—
166	D₁₂	—	2	48	no	—
167	C₄×D₃	—	4	16	no	—
168	C₂×C₃²	C₂×C₄×D₄	1	3	yes	—
169	C₂×C₃²	C₂×#127	1	1	yes	characteristic
170	C₂×C₃²	#532	1	3	yes	—
171	C₃×D₃	—	4	8	no	—
172	C₂⁴	—	3	4	no	—
173	C₂×D₄	—	3	24	no	—
174	C₂×D₄	—	6	12	no	—
175	C₂×D₄	—	6	12	no	—
176	C₂²×C₄	—	3	4	no	—
177	C₂²×C₄	—	6	24	no	—
178	C₂²×C₄	C₂²×C₃×D₃	1	3	yes	—
179	C₂²×C₄	C₂²×C₃×D₃	1	3	yes	—
180	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
181	C₄⋊C₄	—	6	12	no	—
182	C₂²⋊C₄	—	6	24	no	—
183	C₄²	—	2	6	no	—
184	C₄²	C₃×C₄×D₃	1	4	yes	complemented
185	C₄²	C₃×D₁₂	1	12	yes	complemented
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	3	no	—
188	C₂²×C₃	—	2	3	no	—
189	C₂²×C₃	—	6	24	no	—
190	C₂²×C₃	—	12	4	no	—
191	C₂²×C₃	—	12	12	no	—
192	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
193	C₂²×C₃	C₂²×D₁₂	1	3	yes	—
194	C₂²×C₃	C₂×#235	1	3	yes	—
195	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
196	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
197	C₂²×C₃	C₂×C₃×#41	1	3	yes	—
198	C₂×D₃	—	2	24	no	—
199	C₂×D₃	—	4	4	no	—
200	C₂×D₃	—	4	12	no	—
201	C₃×C₄	—	2	12	no	—
202	C₃×C₄	—	2	6	no	—
203	C₃×C₄	—	2	6	no	—
204	C₃×C₄	—	2	12	no	—
205	C₃×C₄	—	2	4	no	—
206	C₃×C₄	—	12	4	no	—
207	C₃×C₄	C₄×D₁₂	1	12	yes	complemented
208	C₃×C₄	#874	1	4	yes	direct product factor
209	C₃×C₄	C₃×C₄×D₄	1	12	yes	complemented
210	C₃×C₄	C₃×#127	1	4	yes	complemented
211	Dic₃	—	4	4	no	—
212	C₃²	C₄×#127	1	1	yes	characteristic, complemented, Sylow
213	C₂³	—	3	8	no	—
214	C₂³	—	6	12	no	—
215	C₂³	—	6	12	no	—
216	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
217	D₄	—	6	48	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	6	12	no	—
220	C₂×C₄	—	12	16	no	—
221	C₂×C₄	C₂×C₃×C₄×D₃	1	6	yes	—
222	C₂×C₄	C₂×C₃×D₁₂	1	6	yes	—
223	C₂×C₄	C₂×C₃×D₁₂	1	6	yes	—
224	C₂×C₄	C₂×C₃×D₁₂	1	6	yes	—
225	C₂×C₄	C₃×#235	1	6	yes	—
226	C₂×C₃	—	2	3	no	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	3	no	—
229	C₂×C₃	—	12	8	no	—
230	C₂×C₃	C₂×C₄×D₁₂	1	3	yes	—
231	C₂×C₃	C₂×#874	1	1	yes	characteristic
232	C₂×C₃	#5477	1	3	yes	—
233	C₂×C₃	C₂×C₃×C₄×D₄	1	3	yes	—
234	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
235	C₂×C₃	C₃×#532	1	3	yes	—
236	D₃	—	4	8	no	—
237	C₂²	—	6	24	no	—
238	C₂²	—	12	4	no	—
239	C₂²	—	12	12	no	—
240	C₂²	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
241	C₂²	C₂²×C₃×D₁₂	1	3	yes	—
242	C₂²	C₂×C₃×#235	1	3	yes	—
243	C₄	—	2	6	no	—
244	C₄	—	12	4	no	—
245	C₄	C₃×C₄×D₁₂	1	12	yes	complemented
246	C₄	C₃×#874	1	4	yes	direct product factor
247	C₃	—	2	1	no	—
248	C₃	C₄×#874	1	1	yes	characteristic, direct product factor
249	C₃	C₃×C₄×#127	1	1	yes	characteristic, complemented
250	C₂	—	12	8	no	—
251	C₂	C₂×C₃×C₄×D₁₂	1	3	yes	—
252	C₂	C₂×C₃×#874	1	1	yes	characteristic
253	C₂	C₃×#5477	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₃
Fitting	C₃²×C₄³	C₂
socle	C₂³×C₃²	C₂⁴
prime radical	C₂×C₃×#874	C₂
cosocle	C₂³×C₃	C₂⁴×C₃
hypercenter	C₃×C₄³	D₃
hypocenter	C₃	C₃×C₄×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×C₄³

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×D₁₂	C₂	semidirect
3. C₃²×C₄³	C₂	semidirect
4. C₃×#4939	C₂	semidirect
5. C₃×#4935	C₂	semidirect
7. C₄×#874	C₃	direct (N × H)
15. C₃×#874	C₄	direct (N × H)
20. C₂×C₄×D₁₂	C₂×C₃	semidirect
24. C₃×C₄³	D₃	semidirect
25. C₃×C₄³	C₂×C₃	semidirect
26. #4939	C₂×C₃	semidirect
27. #4935	C₂×C₃	semidirect
35. C₂×C₃×D₁₂	C₂×C₄	semidirect
38. C₃²×C₄²	C₂×C₄	semidirect
39. C₃²×C₄²	D₄	semidirect
62. #874	C₃×C₄	direct (N × H)
81. C₄³	C₃×D₃	semidirect
101. C₂×D₁₂	C₂×C₃×C₄	semidirect
110. C₃×C₄²	C₄×D₃	semidirect
111. C₃×C₄²	D₁₂	semidirect
112. C₃×C₄²	C₂×C₃×C₄	semidirect
113. C₃×C₄²	C₃×D₄	semidirect
123. C₃²×C₄	C₄×D₄	semidirect
124. C₃²×C₄	#127	semidirect
184. C₄²	C₃×C₄×D₃	semidirect
185. C₄²	C₃×D₁₂	semidirect
207. C₃×C₄	C₄×D₁₂	semidirect
208. C₃×C₄	#874	direct (N × H)
209. C₃×C₄	C₃×C₄×D₄	semidirect
210. C₃×C₄	C₃×#127	semidirect
212. C₃²	C₄×#127	semidirect
245. C₄	C₃×C₄×D₁₂	semidirect
246. C₄	C₃×#874	direct (N × H)
248. C₃	C₄×#874	direct (N × H)
249. C₃	C₃×C₄×#127	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#874	2
C₂×C₃×C₄×D₁₂	2
C₃²×C₄³	2
C₃×#4939	2
C₃×#4935	2
C₃×C₄×#127	3
C₄×#874	3

Subgroups — C3×C4×#874 #11802526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#874 #11802526