Subgroups — C₂²×C₃×#466 #10679042

Order 768.

3694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#121	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×#466	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×#102	C₂	1	2	yes	maximal, complemented
7	C₂²×#466	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×D₄	—	2	2	no	—
9	C₂³×C₃×D₄	C₂²	1	2	yes	—
10	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×D₈	—	2	8	no	—
12	C₂²×C₃×D₈	C₂²	1	12	yes	complemented
13	C₂³×C₃×C₈	—	2	8	no	—
14	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₈	C₂²	1	12	yes	complemented
17	C₂×C₃×#121	—	2	12	no	—
18	C₂×C₃×#121	C₂²	1	24	yes	complemented
19	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂²×C₃×C₂²⋊C₄	—	2	2	no	—
21	C₃×#466	C₂²	1	64	yes	direct product factor
22	C₂×C₃×#107	C₂²	1	12	yes	complemented
23	C₂×C₃×#102	C₂²	1	24	yes	complemented
24	C₂³×D₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂⁴×C₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²×#121	C₂×C₃	1	2	yes	complemented
27	C₂×#466	C₂×C₃	1	24	yes	direct product factor
28	C₂²×#107	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×#102	C₂×C₃	1	2	yes	complemented
30	C₂⁵×C₃	—	2	2	no	—
31	C₂⁵×C₃	D₄	1	1	yes	characteristic
32	C₂²×C₃×D₄	—	2	24	no	—
33	C₂²×C₃×D₄	—	2	8	no	—
34	C₂²×C₃×D₄	—	2	6	no	—
35	C₂²×C₃×D₄	—	2	8	no	—
36	C₂²×C₃×D₄	—	4	16	no	—
37	C₂²×C₃×D₄	C₂³	1	12	yes	—
38	C₂³×C₃×C₄	—	2	4	no	—
39	C₂³×C₃×C₄	—	2	4	no	—
40	C₂³×C₃×C₄	—	2	2	no	—
41	C₂³×C₃×C₄	D₄	1	6	yes	—
42	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
43	C₂³×C₃×C₄	C₂³	1	6	yes	—
44	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₃×D₈	—	2	48	no	—
46	C₂×C₃×D₈	C₂³	1	16	yes	complemented
47	C₂²×C₃×C₈	—	2	4	no	—
48	C₂²×C₃×C₈	—	2	48	no	—
49	C₂²×C₃×C₈	D₄	1	8	yes	complemented
50	C₂²×C₃×C₈	C₂³	1	6	yes	—
51	C₂²×C₃×C₈	C₂³	1	6	yes	—
52	C₂²×C₃×C₈	C₂³	1	16	yes	complemented
53	C₃×#121	—	2	48	no	—
54	C₃×#121	C₂³	1	32	yes	complemented
55	C₂×C₃×C₄⋊C₄	—	2	6	no	—
56	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
57	C₂×C₃×C₂²⋊C₄	—	2	24	no	—
58	C₃×#107	C₂³	1	16	yes	complemented
59	C₃×#102	C₂³	1	32	yes	complemented
60	C₂³×D₄	—	2	2	no	—
61	C₂³×D₄	C₂²×C₃	1	2	yes	—
62	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂²×D₈	—	2	8	no	—
64	C₂²×D₈	C₂²×C₃	1	12	yes	complemented
65	C₂³×C₈	—	2	8	no	—
66	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
67	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
68	C₂³×C₈	C₂²×C₃	1	12	yes	complemented
69	C₂×#121	—	2	12	no	—
70	C₂×#121	C₂²×C₃	1	24	yes	complemented
71	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂²×C₂²⋊C₄	—	2	2	no	—
73	#466	C₂²×C₃	1	64	yes	direct product factor
74	C₂×#107	C₂²×C₃	1	12	yes	complemented
75	C₂×#102	C₂²×C₃	1	24	yes	complemented
76	C₂⁴×C₃	—	2	4	no	—
77	C₂⁴×C₃	—	2	4	no	—
78	C₂⁴×C₃	—	2	12	no	—
79	C₂⁴×C₃	—	4	8	no	—
80	C₂⁴×C₃	—	4	8	no	—
81	C₂⁴×C₃	—	4	8	no	—
82	C₂⁴×C₃	D₈	1	8	yes	complemented
83	C₂⁴×C₃	C₂×D₄	1	6	yes	—
84	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×D₄	—	2	32	no	—
86	C₂×C₃×D₄	—	2	24	no	—
87	C₂×C₃×D₄	—	2	12	no	—
88	C₂×C₃×D₄	—	2	24	no	—
89	C₂×C₃×D₄	—	4	96	no	—
90	C₂×C₃×D₄	—	4	24	no	—
91	C₂×C₃×D₄	C₂⁴	1	8	yes	—
92	C₂²×C₃×C₄	—	2	4	no	—
93	C₂²×C₃×C₄	—	2	24	no	—
94	C₂²×C₃×C₄	—	2	12	no	—
95	C₂²×C₃×C₄	—	2	12	no	—
96	C₂²×C₃×C₄	—	2	12	no	—
97	C₂²×C₃×C₄	—	4	8	no	—
98	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
99	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
100	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
101	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
102	C₂²×C₃×C₄	#41	1	4	yes	—
103	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
104	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
105	C₃×D₈	—	2	32	no	—
106	C₂×C₃×C₈	—	2	12	no	—
107	C₂×C₃×C₈	—	2	32	no	—
108	C₂×C₃×C₈	C₂×D₄	1	24	yes	complemented
109	C₂×C₃×C₈	C₂⁴	1	4	yes	—
110	C₂×C₃×C₈	C₂⁴	1	4	yes	—
111	C₃×C₄⋊C₄	—	2	12	no	—
112	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
113	C₃×C₂²⋊C₄	—	2	32	no	—
114	C₂⁵	—	2	2	no	—
115	C₂⁵	C₃×D₄	1	1	yes	characteristic
116	C₂²×D₄	—	2	24	no	—
117	C₂²×D₄	—	2	8	no	—
118	C₂²×D₄	—	2	6	no	—
119	C₂²×D₄	—	2	8	no	—
120	C₂²×D₄	—	4	16	no	—
121	C₂²×D₄	C₂³×C₃	1	12	yes	—
122	C₂³×C₄	—	2	4	no	—
123	C₂³×C₄	—	2	4	no	—
124	C₂³×C₄	—	2	2	no	—
125	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
126	C₂³×C₄	C₃×D₄	1	6	yes	—
127	C₂³×C₄	C₂³×C₃	1	6	yes	—
128	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₂×D₈	—	2	48	no	—
130	C₂×D₈	C₂³×C₃	1	16	yes	complemented
131	C₂²×C₈	—	2	4	no	—
132	C₂²×C₈	—	2	48	no	—
133	C₂²×C₈	C₃×D₄	1	8	yes	complemented
134	C₂²×C₈	C₂³×C₃	1	6	yes	—
135	C₂²×C₈	C₂³×C₃	1	6	yes	—
136	C₂²×C₈	C₂³×C₃	1	16	yes	complemented
137	#121	—	2	48	no	—
138	#121	C₂³×C₃	1	32	yes	complemented
139	C₂×C₄⋊C₄	—	2	6	no	—
140	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
141	C₂×C₂²⋊C₄	—	2	24	no	—
142	#107	C₂³×C₃	1	16	yes	complemented
143	#102	C₂³×C₃	1	32	yes	complemented
144	C₂³×C₃	—	2	12	no	—
145	C₂³×C₃	—	2	12	no	—
146	C₂³×C₃	—	2	32	no	—
147	C₂³×C₃	—	2	8	no	—
148	C₂³×C₃	—	4	24	no	—
149	C₂³×C₃	—	4	24	no	—
150	C₂³×C₃	—	4	24	no	—
151	C₂³×C₃	—	8	32	no	—
152	C₂³×C₃	#121	1	4	yes	—
153	C₂³×C₃	C₂×D₈	1	4	yes	—
154	C₂³×C₃	C₂×D₈	1	24	yes	complemented
155	C₂³×C₃	#135	1	4	yes	—
156	C₂³×C₃	C₂²×D₄	1	4	yes	—
157	C₂³×C₃	C₂²×D₄	1	3	yes	—
158	C₃×D₄	—	2	8	no	—
159	C₃×D₄	—	2	8	no	—
160	C₃×D₄	—	4	64	no	—
161	C₃×D₄	—	4	24	no	—
162	C₂×C₃×C₄	—	2	12	no	—
163	C₂×C₃×C₄	—	2	24	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	—	4	24	no	—
168	C₂×C₃×C₄	#121	1	12	yes	—
169	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
170	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
171	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
172	C₂×C₃×C₄	C₂×#41	1	6	yes	—
173	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
174	C₃×C₈	—	2	4	no	—
175	C₃×C₈	C₂²×D₄	1	8	yes	complemented
176	C₂⁴	—	2	4	no	—
177	C₂⁴	—	2	4	no	—
178	C₂⁴	—	2	12	no	—
179	C₂⁴	—	4	8	no	—
180	C₂⁴	—	4	8	no	—
181	C₂⁴	—	4	8	no	—
182	C₂⁴	C₃×D₈	1	8	yes	complemented
183	C₂⁴	C₂×C₃×D₄	1	6	yes	—
184	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×D₄	—	2	32	no	—
186	C₂×D₄	—	2	24	no	—
187	C₂×D₄	—	2	12	no	—
188	C₂×D₄	—	2	24	no	—
189	C₂×D₄	—	4	96	no	—
190	C₂×D₄	—	4	24	no	—
191	C₂×D₄	C₂⁴×C₃	1	8	yes	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	2	12	no	—
195	C₂²×C₄	—	2	12	no	—
196	C₂²×C₄	—	2	12	no	—
197	C₂²×C₄	—	4	8	no	—
198	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
199	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
200	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
201	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
202	C₂²×C₄	C₃×#41	1	4	yes	—
203	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
204	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
205	D₈	—	2	32	no	—
206	C₂×C₈	—	2	12	no	—
207	C₂×C₈	—	2	32	no	—
208	C₂×C₈	C₂×C₃×D₄	1	24	yes	complemented
209	C₂×C₈	C₂⁴×C₃	1	4	yes	—
210	C₂×C₈	C₂⁴×C₃	1	4	yes	—
211	C₄⋊C₄	—	2	12	no	—
212	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
213	C₂²⋊C₄	—	2	32	no	—
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	2	48	no	—
216	C₂²×C₃	—	2	4	no	—
217	C₂²×C₃	—	4	8	no	—
218	C₂²×C₃	—	4	8	no	—
219	C₂²×C₃	—	4	8	no	—
220	C₂²×C₃	—	8	48	no	—
221	C₂²×C₃	#466	1	16	yes	direct product factor
222	C₂²×C₃	C₂×#121	1	6	yes	—
223	C₂²×C₃	C₂²×D₈	1	6	yes	—
224	C₂²×C₃	C₂²×D₈	1	8	yes	complemented
225	C₂²×C₃	C₂×#135	1	6	yes	—
226	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
227	C₃×C₄	—	2	4	no	—
228	C₃×C₄	—	4	8	no	—
229	C₃×C₄	C₂×#121	1	6	yes	—
230	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
231	C₃×C₄	C₂²×#41	1	1	yes	characteristic
232	C₂³	—	2	12	no	—
233	C₂³	—	2	32	no	—
234	C₂³	—	2	12	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	4	24	no	—
237	C₂³	—	4	24	no	—
238	C₂³	—	4	24	no	—
239	C₂³	—	8	32	no	—
240	C₂³	C₃×#121	1	4	yes	—
241	C₂³	C₂×C₃×D₈	1	4	yes	—
242	C₂³	C₂×C₃×D₈	1	24	yes	complemented
243	C₂³	C₃×#135	1	4	yes	—
244	C₂³	C₂²×C₃×D₄	1	4	yes	—
245	C₂³	C₂²×C₃×D₄	1	3	yes	—
246	D₄	—	2	8	no	—
247	D₄	—	2	8	no	—
248	D₄	—	4	64	no	—
249	D₄	—	4	24	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	24	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	24	no	—
256	C₂×C₄	C₃×#121	1	12	yes	—
257	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
258	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
259	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
260	C₂×C₄	C₂×C₃×#41	1	6	yes	—
261	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	C₂²×C₃×D₄	1	8	yes	complemented
264	C₂×C₃	—	2	8	no	—
265	C₂×C₃	—	8	8	no	—
266	C₂×C₃	C₂×#466	1	12	yes	direct product factor
267	C₂×C₃	C₂²×#121	1	1	yes	characteristic
268	C₂×C₃	C₂³×D₈	1	1	yes	characteristic
269	C₂×C₃	C₂²×#135	1	1	yes	characteristic
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	48	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	4	8	no	—
275	C₂²	—	4	8	no	—
276	C₂²	—	8	48	no	—
277	C₂²	C₃×#466	1	16	yes	direct product factor
278	C₂²	C₂×C₃×#121	1	6	yes	—
279	C₂²	C₂²×C₃×D₈	1	6	yes	—
280	C₂²	C₂²×C₃×D₈	1	8	yes	complemented
281	C₂²	C₂×C₃×#135	1	6	yes	—
282	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	4	8	no	—
285	C₄	C₂×C₃×#121	1	6	yes	—
286	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
287	C₄	C₂²×C₃×#41	1	1	yes	characteristic
288	C₃	C₂²×#466	1	1	yes	characteristic, direct product factor, Sylow
289	C₂	—	2	8	no	—
290	C₂	—	8	8	no	—
291	C₂	C₂×C₃×#466	1	12	yes	direct product factor
292	C₂	C₂²×C₃×#121	1	1	yes	characteristic
293	C₂	C₂³×C₃×D₈	1	1	yes	characteristic
294	C₂	C₂²×C₃×#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
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×D₈	C₂	semidirect
2. C₂⁴×C₃×C₈	C₂	semidirect
3. C₂²×C₃×#121	C₂	semidirect
4. C₂×C₃×#466	C₂	direct (N × H)
5. C₂²×C₃×#107	C₂	semidirect
6. C₂²×C₃×#102	C₂	semidirect
7. C₂²×#466	C₃	direct (N × H)
12. C₂²×C₃×D₈	C₂²	semidirect
16. C₂³×C₃×C₈	C₂²	semidirect
18. C₂×C₃×#121	C₂²	semidirect
21. C₃×#466	C₂²	direct (N × H)
22. C₂×C₃×#107	C₂²	semidirect
23. C₂×C₃×#102	C₂²	semidirect
24. C₂³×D₈	C₂×C₃	semidirect
25. C₂⁴×C₈	C₂×C₃	semidirect
26. C₂²×#121	C₂×C₃	semidirect
27. C₂×#466	C₂×C₃	direct (N × H)
28. C₂²×#107	C₂×C₃	semidirect
29. C₂²×#102	C₂×C₃	semidirect
46. C₂×C₃×D₈	C₂³	semidirect
49. C₂²×C₃×C₈	D₄	semidirect
52. C₂²×C₃×C₈	C₂³	semidirect
54. C₃×#121	C₂³	semidirect
58. C₃×#107	C₂³	semidirect
59. C₃×#102	C₂³	semidirect
64. C₂²×D₈	C₂²×C₃	semidirect
68. C₂³×C₈	C₂²×C₃	semidirect
70. C₂×#121	C₂²×C₃	semidirect
73. #466	C₂²×C₃	direct (N × H)
74. C₂×#107	C₂²×C₃	semidirect
75. C₂×#102	C₂²×C₃	semidirect
82. C₂⁴×C₃	D₈	semidirect
108. C₂×C₃×C₈	C₂×D₄	semidirect
130. C₂×D₈	C₂³×C₃	semidirect
133. C₂²×C₈	C₃×D₄	semidirect
136. C₂²×C₈	C₂³×C₃	semidirect
138. #121	C₂³×C₃	semidirect
142. #107	C₂³×C₃	semidirect
143. #102	C₂³×C₃	semidirect
154. C₂³×C₃	C₂×D₈	semidirect
175. C₃×C₈	C₂²×D₄	semidirect
182. C₂⁴	C₃×D₈	semidirect
208. C₂×C₈	C₂×C₃×D₄	semidirect
221. C₂²×C₃	#466	direct (N × H)
224. C₂²×C₃	C₂²×D₈	semidirect
242. C₂³	C₂×C₃×D₈	semidirect
263. C₈	C₂²×C₃×D₄	semidirect
266. C₂×C₃	C₂×#466	direct (N × H)
277. C₂²	C₃×#466	direct (N × H)
280. C₂²	C₂²×C₃×D₈	semidirect
288. C₃	C₂²×#466	direct (N × H)
291. C₂	C₂×C₃×#466	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×D₈	2
C₂⁴×C₃×C₈	2
C₂²×C₃×#121	2
C₂×C₃×#466	2
C₂²×C₃×#107	2
C₂²×C₃×#102	2
C₂²×#466	3

Subgroups — C22×C3×#466 #10679042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#466 #10679042