Subgroups — C₂×C₃×#4677 #11808478

Order 1152.

1981 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×(C₂×D₃)⋊C₄	C₂	1	3	yes	maximal
2	C₂²×C₃×C₄⋊Dic₃	C₂	1	3	yes	maximal, complemented
3	C₂×C₃²×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4677	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#399	—	3	1	no	maximal
6	C₂×#4677	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂⁴×C₃×D₃	C₂²	1	1	yes	characteristic
8	C₂³×C₃²×C₄	C₂²	1	3	yes	—
9	C₂³×C₃×Dic₃	C₂²	1	3	yes	—
10	C₂×C₃×(C₂×D₃)⋊C₄	—	2	12	no	—
11	C₂×C₃×(C₂×D₃)⋊C₄	C₂²	1	12	yes	—
12	C₂×C₃×C₄⋊Dic₃	—	2	12	no	—
13	C₂×C₃×C₄⋊Dic₃	C₂²	1	12	yes	complemented
14	C₃²×#95	C₂²	1	4	yes	complemented
15	C₂²×C₃×C₄⋊C₄	—	3	3	no	—
16	C₂²×C₃×C₂²⋊C₄	—	3	3	no	—
17	C₂²×(C₂×D₃)⋊C₄	C₂×C₃	1	3	yes	—
18	C₂²×C₄⋊Dic₃	C₂×C₃	1	3	yes	complemented
19	C₃×#399	—	3	8	no	—
20	C₂×C₃×#95	—	2	1	no	—
21	C₂×C₃×#95	D₃	1	1	yes	characteristic, complemented
22	C₂×C₃×#95	C₂×C₃	1	1	yes	characteristic, complemented
23	#4677	C₂×C₃	1	8	yes	direct product factor
24	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
25	C₂³×C₃×D₃	—	2	6	no	—
26	C₂³×C₃×D₃	—	2	6	no	—
27	C₂³×C₃×D₃	Q₈	1	4	yes	—
28	C₂³×C₃×D₃	C₂³	1	2	yes	—
29	C₂²×C₃²×C₄	—	2	12	no	—
30	C₂²×C₃²×C₄	D₄	1	12	yes	—
31	C₂²×C₃²×C₄	C₂³	1	6	yes	—
32	C₂²×C₃×Dic₃	—	2	12	no	—
33	C₂²×C₃×Dic₃	—	2	6	no	—
34	C₂²×C₃×Dic₃	C₂³	1	6	yes	—
35	C₃×(C₂×D₃)⋊C₄	—	2	24	no	—
36	C₃×C₄⋊Dic₃	—	2	24	no	—
37	C₂×#399	—	3	1	no	Sylow
38	C₂⁵×C₃	—	3	1	no	—
39	C₂⁴×D₃	C₂²×C₃	1	1	yes	characteristic
40	C₂³×C₃×C₄	—	2	3	no	—
41	C₂³×C₃×C₄	—	3	3	no	—
42	C₂³×C₃×C₄	C₂×D₃	1	3	yes	—
43	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
44	C₂³×Dic₃	C₂²×C₃	1	3	yes	—
45	C₂×C₃×C₄⋊C₄	—	3	12	no	—
46	C₂×C₃×C₄⋊C₄	—	6	12	no	—
47	C₂×C₃×C₂²⋊C₄	—	3	12	no	—
48	C₂×C₃×C₂²⋊C₄	—	6	12	no	—
49	C₂×(C₂×D₃)⋊C₄	—	2	12	no	—
50	C₂×(C₂×D₃)⋊C₄	C₂²×C₃	1	12	yes	—
51	C₂×C₄⋊Dic₃	—	2	12	no	—
52	C₂×C₄⋊Dic₃	C₂²×C₃	1	12	yes	complemented
53	C₃×#95	—	2	4	no	—
54	C₃×#95	C₂×D₃	1	4	yes	complemented
55	C₃×#95	C₂²×C₃	1	4	yes	complemented
56	C₂³×C₃²	C₂×D₄	1	6	yes	—
57	C₂³×C₃²	C₂×Q₈	1	2	yes	—
58	C₂³×C₃²	#41	1	6	yes	—
59	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
60	C₂²×C₃×D₃	—	2	24	no	—
61	C₂²×C₃×D₃	—	2	6	no	—
62	C₂²×C₃×D₃	—	2	6	no	—
63	C₂²×C₃×D₃	—	4	12	no	—
64	C₂²×C₃×D₃	—	4	4	no	—
65	C₂²×C₃×D₃	C₂×Q₈	1	4	yes	—
66	C₂×C₃²×C₄	—	2	24	no	—
67	C₂×C₃²×C₄	—	2	12	no	—
68	C₂×C₃²×C₄	C₂×D₄	1	12	yes	—
69	C₂×C₃×Dic₃	—	2	12	no	—
70	C₂×C₃×Dic₃	—	2	6	no	—
71	C₂×C₃×Dic₃	—	4	12	no	—
72	C₂²×C₄⋊C₄	—	3	3	no	—
73	C₂²×C₂²⋊C₄	—	3	3	no	—
74	#399	—	3	8	no	—
75	C₂×#95	C₃×D₃	1	1	yes	characteristic, complemented
76	C₂⁴×C₃	—	2	1	no	—
77	C₂⁴×C₃	—	3	4	no	—
78	C₂⁴×C₃	—	3	2	no	—
79	C₂⁴×C₃	—	6	6	no	—
80	C₂⁴×C₃	—	6	6	no	—
81	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
82	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
83	C₂³×D₃	—	2	6	no	—
84	C₂³×D₃	—	2	6	no	—
85	C₂³×D₃	C₃×Q₈	1	4	yes	—
86	C₂³×D₃	C₂³×C₃	1	2	yes	—
87	C₂²×C₃×C₄	—	2	12	no	—
88	C₂²×C₃×C₄	—	2	12	no	—
89	C₂²×C₃×C₄	—	2	12	no	—
90	C₂²×C₃×C₄	—	2	6	no	—
91	C₂²×C₃×C₄	—	3	6	no	—
92	C₂²×C₃×C₄	—	4	12	no	—
93	C₂²×C₃×C₄	—	6	6	no	—
94	C₂²×C₃×C₄	—	6	12	no	—
95	C₂²×C₃×C₄	D₁₂	1	12	yes	—
96	C₂²×C₃×C₄	C₃×D₄	1	12	yes	—
97	C₂²×C₃×C₄	C₂²×D₃	1	6	yes	—
98	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
99	C₂²×Dic₃	—	2	12	no	—
100	C₂²×Dic₃	—	2	6	no	—
101	C₂²×Dic₃	C₂³×C₃	1	6	yes	—
102	C₃×C₄⋊C₄	—	6	24	no	—
103	C₃×C₂²⋊C₄	—	6	24	no	—
104	(C₂×D₃)⋊C₄	—	2	24	no	—
105	C₄⋊Dic₃	—	2	24	no	—
106	C₂²×C₃²	C₂²⋊Q₈	1	12	yes	—
107	C₂²×C₃²	#123	1	12	yes	—
108	C₂²×C₃²	#127	1	4	yes	—
109	C₂²×C₃²	C₂²×D₄	1	3	yes	—
110	C₂²×C₃²	C₂²×Q₈	1	1	yes	characteristic
111	C₂²×C₃²	C₂×#41	1	3	yes	—
112	C₂×C₃×D₃	—	2	12	no	—
113	C₂×C₃×D₃	—	4	6	no	—
114	C₂×C₃×D₃	—	4	2	no	—
115	C₂×C₃×D₃	—	4	16	no	—
116	C₃²×C₄	—	2	12	no	—
117	C₃×Dic₃	—	4	6	no	—
118	C₂⁵	—	3	1	no	—
119	C₂³×C₄	—	3	3	no	—
120	C₂³×C₄	C₂×C₃×D₃	1	3	yes	—
121	C₂×C₄⋊C₄	—	3	12	no	—
122	C₂×C₄⋊C₄	—	6	12	no	—
123	C₂×C₂²⋊C₄	—	3	12	no	—
124	C₂×C₂²⋊C₄	—	6	12	no	—
125	#95	C₂×C₃×D₃	1	4	yes	complemented
126	C₂³×C₃	—	2	1	no	—
127	C₂³×C₃	—	2	6	no	—
128	C₂³×C₃	—	2	6	no	—
129	C₂³×C₃	—	2	2	no	—
130	C₂³×C₃	—	3	4	no	—
131	C₂³×C₃	—	6	24	no	—
132	C₂³×C₃	—	6	6	no	—
133	C₂³×C₃	—	6	6	no	—
134	C₂³×C₃	—	12	4	no	—
135	C₂³×C₃	—	12	12	no	—
136	C₂³×C₃	C₂×D₁₂	1	6	yes	—
137	C₂³×C₃	#237	1	6	yes	—
138	C₂³×C₃	D₃×Q₈	1	2	yes	—
139	C₂³×C₃	C₂×C₃×D₄	1	6	yes	—
140	C₂³×C₃	C₂×C₃×Q₈	1	2	yes	—
141	C₂³×C₃	C₃×#41	1	6	yes	—
142	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
143	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
144	C₂²×D₃	—	2	24	no	—
145	C₂²×D₃	—	2	6	no	—
146	C₂²×D₃	—	2	6	no	—
147	C₂²×D₃	—	4	12	no	—
148	C₂²×D₃	—	4	4	no	—
149	C₂²×D₃	C₂×C₃×Q₈	1	4	yes	—
150	C₂×C₃×C₄	—	2	24	no	—
151	C₂×C₃×C₄	—	2	24	no	—
152	C₂×C₃×C₄	—	2	12	no	—
153	C₂×C₃×C₄	—	2	12	no	—
154	C₂×C₃×C₄	—	2	12	no	—
155	C₂×C₃×C₄	—	4	24	no	—
156	C₂×C₃×C₄	—	4	12	no	—
157	C₂×C₃×C₄	—	6	6	no	—
158	C₂×C₃×C₄	—	6	12	no	—
159	C₂×C₃×C₄	—	12	12	no	—
160	C₂×C₃×C₄	C₂×D₁₂	1	12	yes	—
161	C₂×C₃×C₄	C₂×C₃×D₄	1	12	yes	—
162	C₂×Dic₃	—	2	12	no	—
163	C₂×Dic₃	—	2	6	no	—
164	C₂×Dic₃	—	4	12	no	—
165	C₂×C₃²	#399	1	8	yes	complemented
166	C₂×C₃²	C₂×C₂²⋊Q₈	1	3	yes	—
167	C₂×C₃²	C₂×#123	1	3	yes	—
168	C₂×C₃²	C₂×#127	1	1	yes	characteristic
169	C₃×D₃	—	4	4	no	—
170	C₂⁴	—	3	2	no	—
171	C₂⁴	—	3	4	no	—
172	C₂⁴	—	6	6	no	—
173	C₂⁴	—	6	6	no	—
174	C₂⁴	C₂²×C₃×D₃	1	1	yes	characteristic
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	—	3	6	no	—
177	C₂²×C₄	—	6	6	no	—
178	C₂²×C₄	—	6	12	no	—
179	C₂²×C₄	C₃×D₁₂	1	12	yes	—
180	C₂²×C₄	C₂²×C₃×D₃	1	6	yes	—
181	C₄⋊C₄	—	6	24	no	—
182	C₂²⋊C₄	—	6	24	no	—
183	C₂²×C₃	—	2	12	no	—
184	C₂²×C₃	—	2	12	no	—
185	C₂²×C₃	—	2	3	no	—
186	C₂²×C₃	—	2	3	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	4	no	—
189	C₂²×C₃	—	6	12	no	—
190	C₂²×C₃	—	12	16	no	—
191	C₂²×C₃	—	12	2	no	—
192	C₂²×C₃	—	12	6	no	—
193	C₂²×C₃	#874	1	4	yes	—
194	C₂²×C₃	#886	1	12	yes	—
195	C₂²×C₃	#897	1	12	yes	—
196	C₂²×C₃	C₃×C₂²⋊Q₈	1	12	yes	—
197	C₂²×C₃	C₃×#123	1	12	yes	—
198	C₂²×C₃	C₃×#127	1	4	yes	—
199	C₂²×C₃	C₂²×D₁₂	1	3	yes	—
200	C₂²×C₃	C₂×#237	1	3	yes	—
201	C₂²×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
202	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
203	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
204	C₂²×C₃	C₂×C₃×#41	1	3	yes	—
205	C₂×D₃	—	2	12	no	—
206	C₂×D₃	—	4	16	no	—
207	C₂×D₃	—	4	6	no	—
208	C₂×D₃	—	4	2	no	—
209	C₃×C₄	—	2	12	no	—
210	C₃×C₄	—	2	12	no	—
211	C₃×C₄	—	4	12	no	—
212	C₃×C₄	—	12	6	no	—
213	Dic₃	—	4	6	no	—
214	C₃²	C₂×#399	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	3	4	no	—
216	C₂³	—	6	24	no	—
217	C₂³	—	6	6	no	—
218	C₂³	—	6	6	no	—
219	C₂³	—	12	12	no	—
220	C₂³	—	12	4	no	—
221	C₂³	C₂×C₃×D₁₂	1	6	yes	—
222	C₂³	C₃×#237	1	6	yes	—
223	C₂³	C₃×D₃×Q₈	1	2	yes	—
224	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
225	C₂×C₄	—	2	24	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	6	6	no	—
228	C₂×C₄	—	6	12	no	—
229	C₂×C₄	—	12	12	no	—
230	C₂×C₄	C₂×C₃×D₁₂	1	12	yes	—
231	C₂×C₃	—	2	3	no	—
232	C₂×C₃	—	2	3	no	—
233	C₂×C₃	—	2	8	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	12	4	no	—
236	C₂×C₃	#4677	1	8	yes	direct product factor
237	C₂×C₃	C₃×#399	1	8	yes	complemented
238	C₂×C₃	C₂×#874	1	1	yes	characteristic
239	C₂×C₃	C₂×#886	1	3	yes	—
240	C₂×C₃	C₂×#897	1	3	yes	—
241	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	3	yes	—
242	C₂×C₃	C₂×C₃×#123	1	3	yes	—
243	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
244	D₃	—	4	4	no	—
245	C₂²	—	6	12	no	—
246	C₂²	—	12	6	no	—
247	C₂²	—	12	16	no	—
248	C₂²	—	12	2	no	—
249	C₂²	C₃×#874	1	4	yes	—
250	C₂²	C₃×#886	1	12	yes	—
251	C₂²	C₃×#897	1	12	yes	—
252	C₂²	C₂²×C₃×D₁₂	1	3	yes	—
253	C₂²	C₂×C₃×#237	1	3	yes	—
254	C₂²	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
255	C₄	—	2	12	no	—
256	C₄	—	12	6	no	—
257	C₃	—	2	1	no	—
258	C₃	C₂×#4677	1	1	yes	characteristic, direct product factor
259	C₃	C₂×C₃×#399	1	1	yes	characteristic, complemented
260	C₂	—	12	4	no	—
261	C₂	C₃×#4677	1	8	yes	direct product factor
262	C₂	C₂×C₃×#874	1	1	yes	characteristic
263	C₂	C₂×C₃×#886	1	3	yes	—
264	C₂	C₂×C₃×#897	1	3	yes	—

	subgroup	G/H
center	C₂⁴×C₃	C₂²×D₃
commutator, cosocle	C₂³×C₃	C₂⁴×C₃
Frattini	C₂³	C₂³×C₃×D₃
Fitting	C₂×C₃²×#95	C₂
socle	C₂⁴×C₃²	C₂³
prime radical	C₂⁴×C₃×D₃	C₂²
hypercenter	C₂×C₃×#95	D₃
hypocenter	C₃	C₂×C₃×#399
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂×C₃×#95

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₄⋊Dic₃	C₂	semidirect
3. C₂×C₃²×#95	C₂	semidirect
4. C₃×#4677	C₂	direct (N × H)
6. C₂×#4677	C₃	direct (N × H)
13. C₂×C₃×C₄⋊Dic₃	C₂²	semidirect
14. C₃²×#95	C₂²	semidirect
18. C₂²×C₄⋊Dic₃	C₂×C₃	semidirect
21. C₂×C₃×#95	D₃	semidirect
22. C₂×C₃×#95	C₂×C₃	semidirect
23. #4677	C₂×C₃	direct (N × H)
52. C₂×C₄⋊Dic₃	C₂²×C₃	semidirect
54. C₃×#95	C₂×D₃	semidirect
55. C₃×#95	C₂²×C₃	semidirect
75. C₂×#95	C₃×D₃	semidirect
125. #95	C₂×C₃×D₃	semidirect
165. C₂×C₃²	#399	semidirect
214. C₃²	C₂×#399	semidirect
236. C₂×C₃	#4677	direct (N × H)
237. C₂×C₃	C₃×#399	semidirect
258. C₃	C₂×#4677	direct (N × H)
259. C₃	C₂×C₃×#399	semidirect
261. C₂	C₃×#4677	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×(C₂×D₃)⋊C₄	2
C₂²×C₃×C₄⋊Dic₃	2
C₂×C₃²×#95	2
C₃×#4677	2
C₂×C₃×#399	3
C₂×#4677	3

Subgroups — C2×C3×#4677 #11808478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4677 #11808478