Subgroups — C₂²×C₃×#470 #10679047

Order 768.

2878 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₂×C₃×#470	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×#103	C₂	1	2	yes	maximal, complemented
7	C₂²×#470	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	C₂²	1	2	yes	—
9	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃×Q₁₆	—	2	8	no	—
11	C₂²×C₃×Q₁₆	C₂²	1	12	yes	complemented
12	C₂²×C₃×M₁₆	—	2	8	no	—
13	C₂²×C₃×M₁₆	C₂²	1	12	yes	—
14	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₂²⋊Q₈	—	2	12	no	—
17	C₂×C₃×C₂²⋊Q₈	C₂²	1	24	yes	—
18	C₂²×C₃×C₄⋊C₄	—	2	2	no	—
19	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂²×C₃×C₂²⋊C₄	—	2	2	no	—
21	C₃×#470	C₂²	1	64	yes	direct product factor
22	C₂×C₃×#106	C₂²	1	12	yes	complemented
23	C₂×C₃×#103	C₂²	1	24	yes	complemented
24	C₂³×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂³×M₁₆	C₂×C₃	1	1	yes	characteristic
26	C₂²×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
27	C₂×#470	C₂×C₃	1	24	yes	direct product factor
28	C₂²×#106	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×#103	C₂×C₃	1	2	yes	complemented
30	C₂⁵×C₃	D₄	1	1	yes	characteristic
31	C₂²×C₃×Q₈	—	2	6	no	—
32	C₂²×C₃×Q₈	—	2	8	no	—
33	C₂²×C₃×Q₈	—	2	8	no	—
34	C₂²×C₃×Q₈	C₂³	1	12	yes	—
35	C₂³×C₃×C₄	—	2	4	no	—
36	C₂³×C₃×C₄	—	2	4	no	—
37	C₂³×C₃×C₄	—	2	2	no	—
38	C₂³×C₃×C₄	—	2	2	no	—
39	C₂³×C₃×C₄	D₄	1	6	yes	—
40	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
41	C₂³×C₃×C₄	C₂³	1	6	yes	—
42	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃×Q₁₆	—	2	48	no	—
44	C₂×C₃×Q₁₆	C₂³	1	16	yes	complemented
45	C₂×C₃×M₁₆	—	2	48	no	—
46	C₂×C₃×M₁₆	C₂³	1	16	yes	—
47	C₂²×C₃×C₈	—	2	4	no	—
48	C₂²×C₃×C₈	D₄	1	8	yes	—
49	C₂²×C₃×C₈	C₂³	1	6	yes	—
50	C₂²×C₃×C₈	C₂³	1	6	yes	—
51	C₃×C₂²⋊Q₈	—	2	48	no	—
52	C₃×C₂²⋊Q₈	C₂³	1	32	yes	—
53	C₂×C₃×C₄⋊C₄	—	2	24	no	—
54	C₂×C₃×C₄⋊C₄	—	2	6	no	—
55	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
56	C₂×C₃×C₂²⋊C₄	—	2	24	no	—
57	C₃×#106	C₂³	1	16	yes	complemented
58	C₃×#103	C₂³	1	32	yes	complemented
59	C₂³×Q₈	C₂²×C₃	1	2	yes	—
60	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂²×Q₁₆	—	2	8	no	—
62	C₂²×Q₁₆	C₂²×C₃	1	12	yes	complemented
63	C₂²×M₁₆	—	2	8	no	—
64	C₂²×M₁₆	C₂²×C₃	1	12	yes	—
65	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
66	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₂²⋊Q₈	—	2	12	no	—
68	C₂×C₂²⋊Q₈	C₂²×C₃	1	24	yes	—
69	C₂²×C₄⋊C₄	—	2	2	no	—
70	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂²×C₂²⋊C₄	—	2	2	no	—
72	#470	C₂²×C₃	1	64	yes	direct product factor
73	C₂×#106	C₂²×C₃	1	12	yes	complemented
74	C₂×#103	C₂²×C₃	1	24	yes	complemented
75	C₂⁴×C₃	—	2	4	no	—
76	C₂⁴×C₃	—	2	8	no	—
77	C₂⁴×C₃	C₂×D₄	1	6	yes	—
78	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×Q₈	—	2	12	no	—
80	C₂×C₃×Q₈	—	2	24	no	—
81	C₂×C₃×Q₈	—	2	24	no	—
82	C₂×C₃×Q₈	—	4	24	no	—
83	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
84	C₂²×C₃×C₄	—	2	4	no	—
85	C₂²×C₃×C₄	—	2	24	no	—
86	C₂²×C₃×C₄	—	2	12	no	—
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	—	4	8	no	—
91	C₂²×C₃×C₄	—	4	8	no	—
92	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
93	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
94	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
95	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
96	C₂²×C₃×C₄	#41	1	4	yes	—
97	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
98	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
99	C₃×Q₁₆	—	2	32	no	—
100	C₃×M₁₆	—	2	32	no	—
101	C₂×C₃×C₈	—	2	12	no	—
102	C₂×C₃×C₈	C₂×D₄	1	24	yes	—
103	C₂×C₃×C₈	C₂⁴	1	4	yes	—
104	C₂×C₃×C₈	C₂⁴	1	4	yes	—
105	C₃×C₄⋊C₄	—	2	32	no	—
106	C₃×C₄⋊C₄	—	2	12	no	—
107	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
108	C₃×C₂²⋊C₄	—	2	32	no	—
109	C₂⁵	C₃×D₄	1	1	yes	characteristic
110	C₂²×Q₈	—	2	6	no	—
111	C₂²×Q₈	—	2	8	no	—
112	C₂²×Q₈	—	2	8	no	—
113	C₂²×Q₈	C₂³×C₃	1	12	yes	—
114	C₂³×C₄	—	2	4	no	—
115	C₂³×C₄	—	2	4	no	—
116	C₂³×C₄	—	2	2	no	—
117	C₂³×C₄	—	2	2	no	—
118	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
119	C₂³×C₄	C₃×D₄	1	6	yes	—
120	C₂³×C₄	C₂³×C₃	1	6	yes	—
121	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂×Q₁₆	—	2	48	no	—
123	C₂×Q₁₆	C₂³×C₃	1	16	yes	complemented
124	C₂×M₁₆	—	2	48	no	—
125	C₂×M₁₆	C₂³×C₃	1	16	yes	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₃×D₄	1	8	yes	—
128	C₂²×C₈	C₂³×C₃	1	6	yes	—
129	C₂²×C₈	C₂³×C₃	1	6	yes	—
130	C₂²⋊Q₈	—	2	48	no	—
131	C₂²⋊Q₈	C₂³×C₃	1	32	yes	—
132	C₂×C₄⋊C₄	—	2	24	no	—
133	C₂×C₄⋊C₄	—	2	6	no	—
134	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
135	C₂×C₂²⋊C₄	—	2	24	no	—
136	#106	C₂³×C₃	1	16	yes	complemented
137	#103	C₂³×C₃	1	32	yes	complemented
138	C₂³×C₃	—	2	12	no	—
139	C₂³×C₃	—	2	24	no	—
140	C₂³×C₃	—	4	16	no	—
141	C₂³×C₃	#121	1	4	yes	—
142	C₂³×C₃	#137	1	8	yes	—
143	C₂³×C₃	C₂²×D₄	1	4	yes	—
144	C₂³×C₃	C₂²×D₄	1	3	yes	—
145	C₃×Q₈	—	2	8	no	—
146	C₃×Q₈	—	2	8	no	—
147	C₃×Q₈	—	4	24	no	—
148	C₂×C₃×C₄	—	2	12	no	—
149	C₂×C₃×C₄	—	2	24	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	8	no	—
153	C₂×C₃×C₄	—	2	8	no	—
154	C₂×C₃×C₄	—	4	24	no	—
155	C₂×C₃×C₄	—	4	24	no	—
156	C₂×C₃×C₄	#121	1	12	yes	—
157	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
158	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
159	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
160	C₂×C₃×C₄	C₂×#41	1	6	yes	—
161	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
162	C₃×C₈	—	2	4	no	—
163	C₃×C₈	C₂²×D₄	1	8	yes	—
164	C₂⁴	—	2	4	no	—
165	C₂⁴	—	2	8	no	—
166	C₂⁴	C₂×C₃×D₄	1	6	yes	—
167	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×Q₈	—	2	12	no	—
169	C₂×Q₈	—	2	24	no	—
170	C₂×Q₈	—	2	24	no	—
171	C₂×Q₈	—	4	24	no	—
172	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	24	no	—
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	—	2	12	no	—
177	C₂²×C₄	—	2	12	no	—
178	C₂²×C₄	—	2	12	no	—
179	C₂²×C₄	—	4	8	no	—
180	C₂²×C₄	—	4	8	no	—
181	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
182	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
183	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
184	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
185	C₂²×C₄	C₃×#41	1	4	yes	—
186	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
187	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
188	Q₁₆	—	2	32	no	—
189	M₁₆	—	2	32	no	—
190	C₂×C₈	—	2	12	no	—
191	C₂×C₈	C₂×C₃×D₄	1	24	yes	—
192	C₂×C₈	C₂⁴×C₃	1	4	yes	—
193	C₂×C₈	C₂⁴×C₃	1	4	yes	—
194	C₄⋊C₄	—	2	32	no	—
195	C₄⋊C₄	—	2	12	no	—
196	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
197	C₂²⋊C₄	—	2	32	no	—
198	C₂²×C₃	—	2	4	no	—
199	C₂²×C₃	—	2	8	no	—
200	C₂²×C₃	—	4	24	no	—
201	C₂²×C₃	#470	1	16	yes	direct product factor
202	C₂²×C₃	C₂×#121	1	6	yes	—
203	C₂²×C₃	C₂×#137	1	12	yes	—
204	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
205	C₃×C₄	—	2	4	no	—
206	C₃×C₄	—	4	8	no	—
207	C₃×C₄	—	4	8	no	—
208	C₃×C₄	C₂×#121	1	6	yes	—
209	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
210	C₃×C₄	C₂²×#41	1	1	yes	characteristic
211	C₂³	—	2	24	no	—
212	C₂³	—	2	12	no	—
213	C₂³	—	4	16	no	—
214	C₂³	C₃×#121	1	4	yes	—
215	C₂³	C₃×#137	1	8	yes	—
216	C₂³	C₂²×C₃×D₄	1	4	yes	—
217	C₂³	C₂²×C₃×D₄	1	3	yes	—
218	Q₈	—	2	8	no	—
219	Q₈	—	2	8	no	—
220	Q₈	—	4	24	no	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	24	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	4	24	no	—
228	C₂×C₄	—	4	24	no	—
229	C₂×C₄	C₃×#121	1	12	yes	—
230	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
231	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
232	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
233	C₂×C₄	C₂×C₃×#41	1	6	yes	—
234	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
235	C₈	—	2	4	no	—
236	C₈	C₂²×C₃×D₄	1	8	yes	—
237	C₂×C₃	—	4	4	no	—
238	C₂×C₃	C₂×#470	1	12	yes	direct product factor
239	C₂×C₃	C₂²×#121	1	1	yes	characteristic
240	C₂×C₃	C₂²×#137	1	2	yes	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	4	24	no	—
244	C₂²	C₃×#470	1	16	yes	direct product factor
245	C₂²	C₂×C₃×#121	1	6	yes	—
246	C₂²	C₂×C₃×#137	1	12	yes	—
247	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
248	C₄	—	2	4	no	—
249	C₄	—	4	8	no	—
250	C₄	—	4	8	no	—
251	C₄	C₂×C₃×#121	1	6	yes	—
252	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
253	C₄	C₂²×C₃×#41	1	1	yes	characteristic
254	C₃	C₂²×#470	1	1	yes	characteristic, direct product factor, Sylow
255	C₂	—	4	4	no	—
256	C₂	C₂×C₃×#470	1	12	yes	direct product factor
257	C₂	C₂²×C₃×#121	1	1	yes	characteristic
258	C₂	C₂²×C₃×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂⁵×C₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂⁴×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×Q₁₆	C₂	semidirect
4. C₂×C₃×#470	C₂	direct (N × H)
5. C₂²×C₃×#106	C₂	semidirect
6. C₂²×C₃×#103	C₂	semidirect
7. C₂²×#470	C₃	direct (N × H)
11. C₂²×C₃×Q₁₆	C₂²	semidirect
21. C₃×#470	C₂²	direct (N × H)
22. C₂×C₃×#106	C₂²	semidirect
23. C₂×C₃×#103	C₂²	semidirect
24. C₂³×Q₁₆	C₂×C₃	semidirect
27. C₂×#470	C₂×C₃	direct (N × H)
28. C₂²×#106	C₂×C₃	semidirect
29. C₂²×#103	C₂×C₃	semidirect
44. C₂×C₃×Q₁₆	C₂³	semidirect
57. C₃×#106	C₂³	semidirect
58. C₃×#103	C₂³	semidirect
62. C₂²×Q₁₆	C₂²×C₃	semidirect
72. #470	C₂²×C₃	direct (N × H)
73. C₂×#106	C₂²×C₃	semidirect
74. C₂×#103	C₂²×C₃	semidirect
123. C₂×Q₁₆	C₂³×C₃	semidirect
136. #106	C₂³×C₃	semidirect
137. #103	C₂³×C₃	semidirect
201. C₂²×C₃	#470	direct (N × H)
238. C₂×C₃	C₂×#470	direct (N × H)
244. C₂²	C₃×#470	direct (N × H)
254. C₃	C₂²×#470	direct (N × H)
256. C₂	C₂×C₃×#470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×Q₁₆	2
C₂³×C₃×M₁₆	2
C₂²×C₃×C₂²⋊Q₈	2
C₂×C₃×#470	2
C₂²×C₃×#106	2
C₂²×C₃×#103	2
C₂²×#470	3

Subgroups — C22×C3×#470 #10679047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#470 #10679047