Subgroups — C₂×C₃×#4695 #11874183

Order 1152.

1601 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	C₂²×C₃×D₂₄	C₂	1	4	yes	maximal, complemented
2	C₂×C₃×#874	C₂	1	2	yes	maximal, complemented
3	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4695	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#493	—	3	1	no	maximal
6	C₂×#4695	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×D₁₂	—	2	4	no	—
8	C₂²×C₃×D₁₂	C₂²	1	4	yes	—
9	C₂²×C₃²×C₈	C₂²	1	2	yes	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₂₄	—	2	16	no	—
12	C₂×C₃×D₂₄	C₂²	1	16	yes	complemented
13	C₃×#874	—	2	4	no	—
14	C₃×#874	C₂²	1	8	yes	complemented
15	C₃²×C₄×C₈	C₂²	1	4	yes	complemented
16	C₂²×C₃×D₈	—	3	4	no	—
17	C₂×C₃×#127	—	3	2	no	—
18	C₂²×D₂₄	C₂×C₃	1	4	yes	complemented
19	C₂×#874	C₂×C₃	1	2	yes	complemented
20	C₃×#493	—	3	8	no	—
21	C₂×C₃×C₄×C₈	—	2	1	no	—
22	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic, complemented
23	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
24	#4695	C₂×C₃	1	8	yes	direct product factor
25	C₂³×C₃×D₃	—	2	4	no	—
26	C₂²×C₃²×C₄	D₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂×C₃×D₁₂	—	2	16	no	—
29	C₂×C₃×D₁₂	—	2	16	no	—
30	C₂×C₃×D₁₂	—	2	4	no	—
31	C₂×C₃×D₁₂	—	4	16	no	—
32	C₂×C₃×D₁₂	C₂³	1	8	yes	—
33	C₂×C₃²×C₈	D₄	1	8	yes	complemented
34	C₂×C₃²×C₈	C₂³	1	4	yes	—
35	C₃²×C₄²	D₄	1	2	yes	—
36	C₃²×C₄²	C₂³	1	2	yes	—
37	C₃×D₂₄	—	2	32	no	—
38	C₂×#493	—	3	1	no	Sylow
39	C₂²×C₃×D₄	—	3	4	no	—
40	C₂²×C₃×D₄	—	6	4	no	—
41	C₂²×D₁₂	—	2	4	no	—
42	C₂²×D₁₂	C₂²×C₃	1	4	yes	—
43	C₂×C₃×D₈	—	3	16	no	—
44	C₂×C₃×D₈	—	6	16	no	—
45	C₂²×C₃×C₈	—	2	2	no	—
46	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
47	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
48	C₃×#127	—	3	8	no	—
49	C₃×#127	—	6	4	no	—
50	C₂×C₃×C₄²	—	2	1	no	—
51	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₂×D₂₄	—	2	16	no	—
54	C₂×D₂₄	C₂²×C₃	1	16	yes	complemented
55	#874	—	2	4	no	—
56	#874	C₂²×C₃	1	8	yes	complemented
57	C₃×C₄×C₈	—	2	4	no	—
58	C₃×C₄×C₈	C₂×D₃	1	4	yes	complemented
59	C₃×C₄×C₈	C₂²×C₃	1	4	yes	complemented
60	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₃×D₃	—	2	8	no	—
62	C₂²×C₃×D₃	—	4	8	no	—
63	C₂²×C₃×D₃	—	4	16	no	—
64	C₂×C₃²×C₄	D₈	1	8	yes	complemented
65	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
67	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
68	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
69	C₃×D₁₂	—	2	16	no	—
70	C₃×D₁₂	—	4	8	no	—
71	C₃×D₁₂	—	4	32	no	—
72	C₃²×C₈	C₂×D₄	1	8	yes	complemented
73	C₂²×D₈	—	3	4	no	—
74	C₂×#127	—	3	2	no	—
75	#493	—	3	8	no	—
76	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic, complemented
77	C₂⁴×C₃	—	6	4	no	—
78	C₂³×D₃	—	2	4	no	—
79	C₂×C₃×D₄	—	3	8	no	—
80	C₂×C₃×D₄	—	6	16	no	—
81	C₂×C₃×D₄	—	6	16	no	—
82	C₂×C₃×D₄	—	6	4	no	—
83	C₂×C₃×D₄	—	12	16	no	—
84	C₂²×C₃×C₄	—	2	2	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	D₁₂	1	2	yes	—
87	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
88	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₂×D₁₂	—	2	16	no	—
91	C₂×D₁₂	—	2	4	no	—
92	C₂×D₁₂	—	2	16	no	—
93	C₂×D₁₂	—	4	16	no	—
94	C₂×D₁₂	C₂³×C₃	1	8	yes	—
95	C₃×D₈	—	6	32	no	—
96	C₂×C₃×C₈	—	2	8	no	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	D₁₂	1	8	yes	complemented
99	C₂×C₃×C₈	C₃×D₄	1	8	yes	complemented
100	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
101	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
102	C₃×C₄²	—	2	2	no	—
103	C₃×C₄²	—	2	2	no	—
104	C₃×C₄²	D₁₂	1	2	yes	—
105	C₃×C₄²	C₃×D₄	1	2	yes	—
106	C₃×C₄²	C₂²×D₃	1	2	yes	—
107	C₃×C₄²	C₂³×C₃	1	2	yes	—
108	D₂₄	—	2	32	no	—
109	C₂²×C₃²	#127	1	2	yes	—
110	C₂²×C₃²	C₂×D₈	1	4	yes	—
111	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₃×D₃	—	4	8	no	—
113	C₂×C₃×D₃	—	4	16	no	—
114	C₂×C₃×D₃	—	8	16	no	—
115	C₃²×C₄	#127	1	2	yes	—
116	C₃²×C₄	C₂×D₈	1	8	yes	complemented
117	C₃²×C₄	C₂²×D₄	1	2	yes	—
118	C₂²×D₄	—	3	4	no	—
119	C₂²×D₄	—	6	4	no	—
120	C₂×D₈	—	3	16	no	—
121	C₂×D₈	—	6	16	no	—
122	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
123	#127	—	3	8	no	—
124	#127	—	6	4	no	—
125	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
126	C₄×C₈	C₂×C₃×D₃	1	4	yes	complemented
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	—	6	8	no	—
129	C₂³×C₃	—	12	8	no	—
130	C₂³×C₃	—	12	16	no	—
131	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
132	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
133	C₂²×D₃	—	2	8	no	—
134	C₂²×D₃	—	4	8	no	—
135	C₂²×D₃	—	4	16	no	—
136	C₃×D₄	—	6	16	no	—
137	C₃×D₄	—	12	8	no	—
138	C₃×D₄	—	12	32	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	8	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	D₂₄	1	8	yes	complemented
145	C₂×C₃×C₄	C₃×D₈	1	8	yes	complemented
146	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
148	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
149	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
151	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
152	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
154	D₁₂	—	2	16	no	—
155	D₁₂	—	4	8	no	—
156	D₁₂	—	4	32	no	—
157	C₃×C₈	—	2	8	no	—
158	C₃×C₈	C₂×D₁₂	1	8	yes	complemented
159	C₃×C₈	C₂×C₃×D₄	1	8	yes	complemented
160	C₂×C₃²	#493	1	4	yes	complemented
161	C₂×C₃²	C₂×#127	1	1	yes	characteristic
162	C₂×C₃²	C₂²×D₈	1	2	yes	—
163	C₃×D₃	—	8	8	no	—
164	C₂⁴	—	6	4	no	—
165	C₂×D₄	—	3	8	no	—
166	C₂×D₄	—	6	16	no	—
167	C₂×D₄	—	6	16	no	—
168	C₂×D₄	—	6	4	no	—
169	C₂×D₄	—	12	16	no	—
170	C₂²×C₄	C₃×D₁₂	1	2	yes	—
171	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
172	D₈	—	6	32	no	—
173	C₂×C₈	C₃×D₁₂	1	8	yes	complemented
174	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
175	C₄²	C₃×D₁₂	1	2	yes	—
176	C₄²	C₂²×C₃×D₃	1	2	yes	—
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	12	8	no	—
181	C₂²×C₃	—	12	16	no	—
182	C₂²×C₃	—	24	16	no	—
183	C₂²×C₃	#874	1	2	yes	—
184	C₂²×C₃	C₂×D₂₄	1	4	yes	—
185	C₂²×C₃	C₃×#127	1	2	yes	—
186	C₂²×C₃	C₂×C₃×D₈	1	4	yes	—
187	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
188	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
189	C₂×D₃	—	4	16	no	—
190	C₂×D₃	—	4	8	no	—
191	C₂×D₃	—	8	16	no	—
192	C₃×C₄	—	2	8	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	#874	1	2	yes	—
196	C₃×C₄	C₂×D₂₄	1	8	yes	complemented
197	C₃×C₄	C₃×#127	1	2	yes	—
198	C₃×C₄	C₂×C₃×D₈	1	8	yes	complemented
199	C₃×C₄	C₂²×D₁₂	1	2	yes	—
200	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
201	C₃²	C₂×#493	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	6	8	no	—
203	C₂³	—	12	16	no	—
204	C₂³	—	12	8	no	—
205	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
206	D₄	—	6	16	no	—
207	D₄	—	12	32	no	—
208	D₄	—	12	8	no	—
209	C₂×C₄	C₃×D₂₄	1	8	yes	complemented
210	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
211	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
212	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
213	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
214	C₈	C₂×C₃×D₁₂	1	8	yes	complemented
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	—	24	8	no	—
219	C₂×C₃	#4695	1	4	yes	direct product factor
220	C₂×C₃	C₃×#493	1	4	yes	complemented
221	C₂×C₃	C₂×#874	1	1	yes	characteristic
222	C₂×C₃	C₂²×D₂₄	1	2	yes	—
223	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
224	C₂×C₃	C₂²×C₃×D₈	1	2	yes	—
225	D₃	—	8	8	no	—
226	C₂²	—	12	8	no	—
227	C₂²	—	12	16	no	—
228	C₂²	—	24	16	no	—
229	C₂²	C₃×#874	1	2	yes	—
230	C₂²	C₂×C₃×D₂₄	1	4	yes	—
231	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
232	C₄	C₃×#874	1	2	yes	—
233	C₄	C₂×C₃×D₂₄	1	8	yes	complemented
234	C₄	C₂²×C₃×D₁₂	1	2	yes	—
235	C₃	—	2	1	no	—
236	C₃	C₂×#4695	1	1	yes	characteristic, direct product factor
237	C₃	C₂×C₃×#493	1	1	yes	characteristic, complemented
238	C₂	—	24	8	no	—
239	C₂	C₃×#4695	1	4	yes	direct product factor
240	C₂	C₂×C₃×#874	1	1	yes	characteristic
241	C₂	C₂²×C₃×D₂₄	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂³×C₃²	C₂×D₄
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₂×C₃×#493
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₂₄	C₂	semidirect
2. C₂×C₃×#874	C₂	semidirect
3. C₂×C₃²×C₄×C₈	C₂	semidirect
4. C₃×#4695	C₂	direct (N × H)
6. C₂×#4695	C₃	direct (N × H)
12. C₂×C₃×D₂₄	C₂²	semidirect
14. C₃×#874	C₂²	semidirect
15. C₃²×C₄×C₈	C₂²	semidirect
18. C₂²×D₂₄	C₂×C₃	semidirect
19. C₂×#874	C₂×C₃	semidirect
22. C₂×C₃×C₄×C₈	D₃	semidirect
23. C₂×C₃×C₄×C₈	C₂×C₃	semidirect
24. #4695	C₂×C₃	direct (N × H)
33. C₂×C₃²×C₈	D₄	semidirect
54. C₂×D₂₄	C₂²×C₃	semidirect
56. #874	C₂²×C₃	semidirect
58. C₃×C₄×C₈	C₂×D₃	semidirect
59. C₃×C₄×C₈	C₂²×C₃	semidirect
64. C₂×C₃²×C₄	D₈	semidirect
72. C₃²×C₈	C₂×D₄	semidirect
76. C₂×C₄×C₈	C₃×D₃	semidirect
98. C₂×C₃×C₈	D₁₂	semidirect
99. C₂×C₃×C₈	C₃×D₄	semidirect
116. C₃²×C₄	C₂×D₈	semidirect
126. C₄×C₈	C₂×C₃×D₃	semidirect
144. C₂×C₃×C₄	D₂₄	semidirect
145. C₂×C₃×C₄	C₃×D₈	semidirect
158. C₃×C₈	C₂×D₁₂	semidirect
159. C₃×C₈	C₂×C₃×D₄	semidirect
160. C₂×C₃²	#493	semidirect
173. C₂×C₈	C₃×D₁₂	semidirect
196. C₃×C₄	C₂×D₂₄	semidirect
198. C₃×C₄	C₂×C₃×D₈	semidirect
201. C₃²	C₂×#493	semidirect
209. C₂×C₄	C₃×D₂₄	semidirect
214. C₈	C₂×C₃×D₁₂	semidirect
219. C₂×C₃	#4695	direct (N × H)
220. C₂×C₃	C₃×#493	semidirect
233. C₄	C₂×C₃×D₂₄	semidirect
236. C₃	C₂×#4695	direct (N × H)
237. C₃	C₂×C₃×#493	semidirect
239. C₂	C₃×#4695	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×D₂₄	2
C₂×C₃×#874	2
C₂×C₃²×C₄×C₈	2
C₃×#4695	2
C₂×C₃×#493	3
C₂×#4695	3

Subgroups — C2×C3×#4695 #11874183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4695 #11874183