Subgroups — C₂²×C₃×#431 #10678721

Order 768.

3086 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#431	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃×C₂²⋊C₈	C₂	1	2	yes	maximal
5	C₂²×C₃×C₄×C₈	C₂	1	2	yes	maximal, complemented
6	C₂²×#431	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁴×C₃×C₄	C₄	1	1	yes	characteristic
8	C₂⁴×C₃×C₄	C₄	1	1	yes	characteristic
9	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₈	C₂²	1	4	yes	—
11	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
12	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
13	C₂²×C₃×C₄²	C₄	1	12	yes	—
14	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄²	C₂²	1	12	yes	—
16	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₃×#431	C₂²	1	64	yes	direct product factor
18	C₂×C₃×C₄⋊C₈	C₂²	1	24	yes	complemented
19	C₂×C₃×C₂²⋊C₈	C₂²	1	24	yes	—
20	C₂×C₃×C₄×C₈	C₂²	1	24	yes	complemented
21	C₂³×C₄²	C₂×C₃	1	1	yes	characteristic
22	C₂×#431	C₂×C₃	1	24	yes	direct product factor
23	C₂²×C₄⋊C₈	C₂×C₃	1	2	yes	complemented
24	C₂²×C₂²⋊C₈	C₂×C₃	1	2	yes	—
25	C₂²×C₄×C₈	C₂×C₃	1	2	yes	complemented
26	C₂⁵×C₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₃×C₄	—	2	8	no	—
28	C₂³×C₃×C₄	—	2	8	no	—
29	C₂³×C₃×C₄	C₈	1	16	yes	complemented
30	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₃×C₄	C₂×C₄	1	12	yes	—
32	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₃×C₄	C₂×C₄	1	6	yes	—
34	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₃×C₄	C₂×C₄	1	12	yes	—
37	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₃×C₄	C₂³	1	6	yes	—
39	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₈	—	2	16	no	—
41	C₂²×C₃×C₈	C₂³	1	24	yes	—
42	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
43	C₂×C₃×C₄²	C₂×C₄	1	48	yes	—
44	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
45	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
46	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
47	C₂×C₃×C₄²	C₂³	1	6	yes	—
48	C₂×C₃×C₄²	C₂³	1	6	yes	—
49	C₂×C₃×C₄²	C₂³	1	16	yes	—
50	C₃×C₄⋊C₈	C₂³	1	32	yes	complemented
51	C₃×C₂²⋊C₈	C₂³	1	32	yes	—
52	C₃×C₄×C₈	C₂³	1	32	yes	complemented
53	C₂⁴×C₄	C₃×C₄	1	1	yes	characteristic
54	C₂⁴×C₄	C₃×C₄	1	1	yes	characteristic
55	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
56	C₂³×C₈	C₂²×C₃	1	4	yes	—
57	C₂²×C₄²	C₃×C₄	1	1	yes	characteristic
58	C₂²×C₄²	C₃×C₄	1	12	yes	—
59	C₂²×C₄²	C₃×C₄	1	1	yes	characteristic
60	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₄²	C₂²×C₃	1	12	yes	—
63	#431	C₂²×C₃	1	64	yes	direct product factor
64	C₂×C₄⋊C₈	C₂²×C₃	1	24	yes	complemented
65	C₂×C₂²⋊C₈	C₂²×C₃	1	24	yes	—
66	C₂×C₄×C₈	C₂²×C₃	1	24	yes	complemented
67	C₂⁴×C₃	—	2	4	no	—
68	C₂⁴×C₃	—	2	4	no	—
69	C₂⁴×C₃	C₂×C₈	1	8	yes	—
70	C₂⁴×C₃	C₂²×C₄	1	6	yes	—
71	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	—	2	4	no	—
73	C₂²×C₃×C₄	—	2	4	no	—
74	C₂²×C₃×C₄	—	2	24	no	—
75	C₂²×C₃×C₄	—	2	48	no	—
76	C₂²×C₃×C₄	—	2	24	no	—
77	C₂²×C₃×C₄	C₂×C₈	1	8	yes	—
78	C₂²×C₃×C₄	C₂×C₈	1	8	yes	—
79	C₂²×C₃×C₄	C₂×C₈	1	96	yes	complemented
80	C₂²×C₃×C₄	M₁₆	1	8	yes	—
81	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
82	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
83	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
84	C₂²×C₃×C₄	C₂²×C₄	1	16	yes	—
85	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
86	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
87	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
88	C₂²×C₃×C₄	C₂²×C₄	1	16	yes	—
89	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
90	C₂²×C₃×C₄	#41	1	8	yes	—
91	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
92	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
93	C₂×C₃×C₈	—	2	48	no	—
94	C₂×C₃×C₈	C₂⁴	1	16	yes	—
95	C₃×C₄²	C₂²×C₄	1	16	yes	—
96	C₃×C₄²	C₂²×C₄	1	16	yes	—
97	C₃×C₄²	C₂²×C₄	1	12	yes	—
98	C₃×C₄²	C₂²×C₄	1	12	yes	—
99	C₃×C₄²	C₂⁴	1	4	yes	—
100	C₃×C₄²	C₂⁴	1	4	yes	—
101	C₂⁵	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂³×C₄	—	2	8	no	—
103	C₂³×C₄	—	2	8	no	—
104	C₂³×C₄	C₃×C₈	1	16	yes	complemented
105	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂³×C₄	C₂×C₃×C₄	1	12	yes	—
107	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂³×C₄	C₂×C₃×C₄	1	6	yes	—
109	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂³×C₄	C₂×C₃×C₄	1	12	yes	—
112	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂³×C₄	C₂³×C₃	1	6	yes	—
114	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₂²×C₈	—	2	16	no	—
116	C₂²×C₈	C₂³×C₃	1	24	yes	—
117	C₂×C₄²	C₂×C₃×C₄	1	12	yes	—
118	C₂×C₄²	C₂×C₃×C₄	1	48	yes	—
119	C₂×C₄²	C₂×C₃×C₄	1	12	yes	—
120	C₂×C₄²	C₂×C₃×C₄	1	6	yes	—
121	C₂×C₄²	C₂×C₃×C₄	1	6	yes	—
122	C₂×C₄²	C₂³×C₃	1	6	yes	—
123	C₂×C₄²	C₂³×C₃	1	6	yes	—
124	C₂×C₄²	C₂³×C₃	1	16	yes	—
125	C₄⋊C₈	C₂³×C₃	1	32	yes	complemented
126	C₂²⋊C₈	C₂³×C₃	1	32	yes	—
127	C₄×C₈	C₂³×C₃	1	32	yes	complemented
128	C₂³×C₃	—	2	12	no	—
129	C₂³×C₃	—	2	12	no	—
130	C₂³×C₃	—	2	32	no	—
131	C₂³×C₃	#117	1	4	yes	—
132	C₂³×C₃	C₂²×C₈	1	24	yes	—
133	C₂³×C₃	C₂²×C₈	1	4	yes	—
134	C₂³×C₃	C₂×M₁₆	1	4	yes	—
135	C₂³×C₃	C₂³×C₄	1	4	yes	—
136	C₂³×C₃	C₂³×C₄	1	3	yes	—
137	C₂×C₃×C₄	—	2	12	no	—
138	C₂×C₃×C₄	—	2	12	no	—
139	C₂×C₃×C₄	—	2	24	no	—
140	C₂×C₃×C₄	—	2	32	no	—
141	C₂×C₃×C₄	—	2	8	no	—
142	C₂×C₃×C₄	#117	1	12	yes	—
143	C₂×C₃×C₄	C₂²×C₈	1	24	yes	—
144	C₂×C₃×C₄	C₂²×C₈	1	24	yes	—
145	C₂×C₃×C₄	C₂²×C₈	1	64	yes	complemented
146	C₂×C₃×C₄	C₂×M₁₆	1	24	yes	—
147	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
148	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
149	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
150	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
151	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
152	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
153	C₂×C₃×C₄	C₂×#41	1	12	yes	—
154	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
155	C₃×C₈	—	2	16	no	—
156	C₂⁴	—	2	4	no	—
157	C₂⁴	—	2	4	no	—
158	C₂⁴	C₂×C₃×C₈	1	8	yes	—
159	C₂⁴	C₂²×C₃×C₄	1	6	yes	—
160	C₂⁴	C₂²×C₃×C₄	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	24	no	—
164	C₂²×C₄	—	2	48	no	—
165	C₂²×C₄	—	2	24	no	—
166	C₂²×C₄	C₂×C₃×C₈	1	8	yes	—
167	C₂²×C₄	C₂×C₃×C₈	1	8	yes	—
168	C₂²×C₄	C₂×C₃×C₈	1	96	yes	complemented
169	C₂²×C₄	C₃×M₁₆	1	8	yes	—
170	C₂²×C₄	C₂²×C₃×C₄	1	6	yes	—
171	C₂²×C₄	C₂²×C₃×C₄	1	12	yes	—
172	C₂²×C₄	C₂²×C₃×C₄	1	6	yes	—
173	C₂²×C₄	C₂²×C₃×C₄	1	16	yes	—
174	C₂²×C₄	C₂²×C₃×C₄	1	3	yes	—
175	C₂²×C₄	C₂²×C₃×C₄	1	6	yes	—
176	C₂²×C₄	C₂²×C₃×C₄	1	6	yes	—
177	C₂²×C₄	C₂²×C₃×C₄	1	16	yes	—
178	C₂²×C₄	C₂²×C₃×C₄	1	6	yes	—
179	C₂²×C₄	C₃×#41	1	8	yes	—
180	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
181	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
182	C₂×C₈	—	2	48	no	—
183	C₂×C₈	C₂⁴×C₃	1	16	yes	—
184	C₄²	C₂²×C₃×C₄	1	16	yes	—
185	C₄²	C₂²×C₃×C₄	1	16	yes	—
186	C₄²	C₂²×C₃×C₄	1	12	yes	—
187	C₄²	C₂²×C₃×C₄	1	12	yes	—
188	C₄²	C₂⁴×C₃	1	4	yes	—
189	C₄²	C₂⁴×C₃	1	4	yes	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	—	2	48	no	—
192	C₂²×C₃	—	2	4	no	—
193	C₂²×C₃	#431	1	16	yes	direct product factor
194	C₂²×C₃	C₂×#117	1	6	yes	—
195	C₂²×C₃	C₂³×C₈	1	8	yes	—
196	C₂²×C₃	C₂³×C₈	1	6	yes	—
197	C₂²×C₃	C₂²×M₁₆	1	6	yes	—
198	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	C₂×#117	1	6	yes	—
202	C₃×C₄	C₂³×C₈	1	8	yes	—
203	C₃×C₄	C₂³×C₈	1	8	yes	—
204	C₃×C₄	C₂²×M₁₆	1	8	yes	—
205	C₃×C₄	C₂²×#41	1	2	yes	—
206	C₂³	—	2	12	no	—
207	C₂³	—	2	32	no	—
208	C₂³	—	2	12	no	—
209	C₂³	C₃×#117	1	4	yes	—
210	C₂³	C₂²×C₃×C₈	1	24	yes	—
211	C₂³	C₂²×C₃×C₈	1	4	yes	—
212	C₂³	C₂×C₃×M₁₆	1	4	yes	—
213	C₂³	C₂³×C₃×C₄	1	4	yes	—
214	C₂³	C₂³×C₃×C₄	1	3	yes	—
215	C₂×C₄	—	2	12	no	—
216	C₂×C₄	—	2	12	no	—
217	C₂×C₄	—	2	32	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	24	no	—
220	C₂×C₄	C₃×#117	1	12	yes	—
221	C₂×C₄	C₂²×C₃×C₈	1	24	yes	—
222	C₂×C₄	C₂²×C₃×C₈	1	24	yes	—
223	C₂×C₄	C₂²×C₃×C₈	1	64	yes	complemented
224	C₂×C₄	C₂×C₃×M₁₆	1	24	yes	—
225	C₂×C₄	C₂³×C₃×C₄	1	4	yes	—
226	C₂×C₄	C₂³×C₃×C₄	1	4	yes	—
227	C₂×C₄	C₂³×C₃×C₄	1	4	yes	—
228	C₂×C₄	C₂³×C₃×C₄	1	4	yes	—
229	C₂×C₄	C₂³×C₃×C₄	1	4	yes	—
230	C₂×C₄	C₂³×C₃×C₄	1	3	yes	—
231	C₂×C₄	C₂×C₃×#41	1	12	yes	—
232	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
233	C₈	—	2	16	no	—
234	C₂×C₃	—	2	8	no	—
235	C₂×C₃	C₂×#431	1	12	yes	direct product factor
236	C₂×C₃	C₂²×#117	1	1	yes	characteristic
237	C₂×C₃	C₂⁴×C₈	1	1	yes	characteristic
238	C₂×C₃	C₂³×M₁₆	1	1	yes	characteristic
239	C₂²	—	2	4	no	—
240	C₂²	—	2	48	no	—
241	C₂²	—	2	4	no	—
242	C₂²	C₃×#431	1	16	yes	direct product factor
243	C₂²	C₂×C₃×#117	1	6	yes	—
244	C₂²	C₂³×C₃×C₈	1	8	yes	—
245	C₂²	C₂³×C₃×C₈	1	6	yes	—
246	C₂²	C₂²×C₃×M₁₆	1	6	yes	—
247	C₂²	C₂⁴×C₃×C₄	1	1	yes	characteristic
248	C₄	—	2	4	no	—
249	C₄	—	2	4	no	—
250	C₄	C₂×C₃×#117	1	6	yes	—
251	C₄	C₂³×C₃×C₈	1	8	yes	—
252	C₄	C₂³×C₃×C₈	1	8	yes	—
253	C₄	C₂²×C₃×M₁₆	1	8	yes	—
254	C₄	C₂²×C₃×#41	1	2	yes	—
255	C₃	C₂²×#431	1	1	yes	characteristic, direct product factor, Sylow
256	C₂	—	2	8	no	—
257	C₂	C₂×C₃×#431	1	12	yes	direct product factor
258	C₂	C₂²×C₃×#117	1	1	yes	characteristic
259	C₂	C₂⁴×C₃×C₈	1	1	yes	characteristic
260	C₂	C₂³×C₃×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄²	C₂²
commutator	C₂	C₂⁴×C₃×C₈
Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂⁴×C₃	C₂²×C₄
prime radical	C₂⁵×C₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#431	C₂	direct (N × H)
3. C₂²×C₃×C₄⋊C₈	C₂	semidirect
5. C₂²×C₃×C₄×C₈	C₂	semidirect
6. C₂²×#431	C₃	direct (N × H)
17. C₃×#431	C₂²	direct (N × H)
18. C₂×C₃×C₄⋊C₈	C₂²	semidirect
20. C₂×C₃×C₄×C₈	C₂²	semidirect
22. C₂×#431	C₂×C₃	direct (N × H)
23. C₂²×C₄⋊C₈	C₂×C₃	semidirect
25. C₂²×C₄×C₈	C₂×C₃	semidirect
29. C₂³×C₃×C₄	C₈	semidirect
50. C₃×C₄⋊C₈	C₂³	semidirect
52. C₃×C₄×C₈	C₂³	semidirect
63. #431	C₂²×C₃	direct (N × H)
64. C₂×C₄⋊C₈	C₂²×C₃	semidirect
66. C₂×C₄×C₈	C₂²×C₃	semidirect
79. C₂²×C₃×C₄	C₂×C₈	semidirect
104. C₂³×C₄	C₃×C₈	semidirect
125. C₄⋊C₈	C₂³×C₃	semidirect
127. C₄×C₈	C₂³×C₃	semidirect
145. C₂×C₃×C₄	C₂²×C₈	semidirect
168. C₂²×C₄	C₂×C₃×C₈	semidirect
193. C₂²×C₃	#431	direct (N × H)
223. C₂×C₄	C₂²×C₃×C₈	semidirect
235. C₂×C₃	C₂×#431	direct (N × H)
242. C₂²	C₃×#431	direct (N × H)
255. C₃	C₂²×#431	direct (N × H)
257. C₂	C₂×C₃×#431	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×C₄²	2
C₂×C₃×#431	2
C₂²×C₃×C₄⋊C₈	2
C₂²×C₃×C₂²⋊C₈	2
C₂²×C₃×C₄×C₈	2
C₂²×#431	3

Subgroups — C22×C3×#431 #10678721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#431 #10678721