Subgroups — C₂²×C₃×#467 #10679043

Order 768.

3006 conjugacy classes of subgroups.

The subgroup lattice has 272 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	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₂×C₃×#467	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×#103	C₂	1	2	yes	maximal, complemented
7	C₂²×#467	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	C₂³×C₃×C₈	—	2	8	no	—
13	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₈	C₂²	1	12	yes	—
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×#467	C₂²	1	64	yes	direct product factor
22	C₂×C₃×#107	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	C₂⁴×C₈	C₂×C₃	1	1	yes	characteristic
26	C₂²×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
27	C₂×#467	C₂×C₃	1	24	yes	direct product factor
28	C₂²×#107	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	C₂²×C₃×C₈	—	2	4	no	—
46	C₂²×C₃×C₈	—	2	48	no	—
47	C₂²×C₃×C₈	D₄	1	8	yes	—
48	C₂²×C₃×C₈	C₂³	1	6	yes	—
49	C₂²×C₃×C₈	C₂³	1	16	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×#107	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	C₂³×C₈	—	2	8	no	—
64	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₂³×C₈	C₂²×C₃	1	12	yes	—
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	#467	C₂²×C₃	1	64	yes	direct product factor
73	C₂×#107	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	4	no	—
77	C₂⁴×C₃	Q₁₆	1	8	yes	complemented
78	C₂⁴×C₃	C₂×D₄	1	6	yes	—
79	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×Q₈	—	2	12	no	—
81	C₂×C₃×Q₈	—	2	24	no	—
82	C₂×C₃×Q₈	—	2	24	no	—
83	C₂×C₃×Q₈	—	4	24	no	—
84	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
85	C₂²×C₃×C₄	—	2	4	no	—
86	C₂²×C₃×C₄	—	2	24	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	—	2	12	no	—
91	C₂²×C₃×C₄	—	4	8	no	—
92	C₂²×C₃×C₄	—	4	8	no	—
93	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
94	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
95	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
96	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
97	C₂²×C₃×C₄	#41	1	4	yes	—
98	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
99	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
100	C₃×Q₁₆	—	2	32	no	—
101	C₂×C₃×C₈	—	2	12	no	—
102	C₂×C₃×C₈	—	2	32	no	—
103	C₂×C₃×C₈	C₂×D₄	1	24	yes	—
104	C₂×C₃×C₈	C₂⁴	1	4	yes	—
105	C₂×C₃×C₈	C₂⁴	1	4	yes	—
106	C₃×C₄⋊C₄	—	2	32	no	—
107	C₃×C₄⋊C₄	—	2	12	no	—
108	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
109	C₃×C₂²⋊C₄	—	2	32	no	—
110	C₂⁵	C₃×D₄	1	1	yes	characteristic
111	C₂²×Q₈	—	2	6	no	—
112	C₂²×Q₈	—	2	8	no	—
113	C₂²×Q₈	—	2	8	no	—
114	C₂²×Q₈	C₂³×C₃	1	12	yes	—
115	C₂³×C₄	—	2	4	no	—
116	C₂³×C₄	—	2	4	no	—
117	C₂³×C₄	—	2	2	no	—
118	C₂³×C₄	—	2	2	no	—
119	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
120	C₂³×C₄	C₃×D₄	1	6	yes	—
121	C₂³×C₄	C₂³×C₃	1	6	yes	—
122	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
123	C₂×Q₁₆	—	2	48	no	—
124	C₂×Q₁₆	C₂³×C₃	1	16	yes	complemented
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	48	no	—
127	C₂²×C₈	C₃×D₄	1	8	yes	—
128	C₂²×C₈	C₂³×C₃	1	6	yes	—
129	C₂²×C₈	C₂³×C₃	1	16	yes	—
130	C₂²×C₈	C₂³×C₃	1	6	yes	—
131	C₂²⋊Q₈	—	2	48	no	—
132	C₂²⋊Q₈	C₂³×C₃	1	32	yes	—
133	C₂×C₄⋊C₄	—	2	24	no	—
134	C₂×C₄⋊C₄	—	2	6	no	—
135	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
136	C₂×C₂²⋊C₄	—	2	24	no	—
137	#107	C₂³×C₃	1	16	yes	complemented
138	#103	C₂³×C₃	1	32	yes	complemented
139	C₂³×C₃	—	2	12	no	—
140	C₂³×C₃	—	2	12	no	—
141	C₂³×C₃	—	2	32	no	—
142	C₂³×C₃	#121	1	4	yes	—
143	C₂³×C₃	C₂×Q₁₆	1	4	yes	—
144	C₂³×C₃	C₂×Q₁₆	1	24	yes	complemented
145	C₂³×C₃	#135	1	4	yes	—
146	C₂³×C₃	C₂²×D₄	1	4	yes	—
147	C₂³×C₃	C₂²×D₄	1	3	yes	—
148	C₃×Q₈	—	2	8	no	—
149	C₃×Q₈	—	2	8	no	—
150	C₃×Q₈	—	4	24	no	—
151	C₂×C₃×C₄	—	2	12	no	—
152	C₂×C₃×C₄	—	2	24	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	8	no	—
157	C₂×C₃×C₄	—	4	24	no	—
158	C₂×C₃×C₄	—	4	24	no	—
159	C₂×C₃×C₄	#121	1	12	yes	—
160	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
161	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
162	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
163	C₂×C₃×C₄	C₂×#41	1	6	yes	—
164	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
165	C₃×C₈	—	2	4	no	—
166	C₃×C₈	C₂²×D₄	1	8	yes	—
167	C₂⁴	—	2	4	no	—
168	C₂⁴	—	2	4	no	—
169	C₂⁴	C₃×Q₁₆	1	8	yes	complemented
170	C₂⁴	C₂×C₃×D₄	1	6	yes	—
171	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×Q₈	—	2	12	no	—
173	C₂×Q₈	—	2	24	no	—
174	C₂×Q₈	—	2	24	no	—
175	C₂×Q₈	—	4	24	no	—
176	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	24	no	—
179	C₂²×C₄	—	2	12	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	2	12	no	—
183	C₂²×C₄	—	4	8	no	—
184	C₂²×C₄	—	4	8	no	—
185	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
186	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
187	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
188	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
189	C₂²×C₄	C₃×#41	1	4	yes	—
190	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
191	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
192	Q₁₆	—	2	32	no	—
193	C₂×C₈	—	2	12	no	—
194	C₂×C₈	—	2	32	no	—
195	C₂×C₈	C₂×C₃×D₄	1	24	yes	—
196	C₂×C₈	C₂⁴×C₃	1	4	yes	—
197	C₂×C₈	C₂⁴×C₃	1	4	yes	—
198	C₄⋊C₄	—	2	32	no	—
199	C₄⋊C₄	—	2	12	no	—
200	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
201	C₂²⋊C₄	—	2	32	no	—
202	C₂²×C₃	—	2	4	no	—
203	C₂²×C₃	—	2	48	no	—
204	C₂²×C₃	—	2	4	no	—
205	C₂²×C₃	#467	1	16	yes	direct product factor
206	C₂²×C₃	C₂×#121	1	6	yes	—
207	C₂²×C₃	C₂²×Q₁₆	1	6	yes	—
208	C₂²×C₃	C₂²×Q₁₆	1	8	yes	complemented
209	C₂²×C₃	C₂×#135	1	6	yes	—
210	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	4	8	no	—
213	C₃×C₄	—	4	8	no	—
214	C₃×C₄	C₂×#121	1	6	yes	—
215	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
216	C₃×C₄	C₂²×#41	1	1	yes	characteristic
217	C₂³	—	2	12	no	—
218	C₂³	—	2	12	no	—
219	C₂³	—	2	32	no	—
220	C₂³	C₃×#121	1	4	yes	—
221	C₂³	C₂×C₃×Q₁₆	1	4	yes	—
222	C₂³	C₂×C₃×Q₁₆	1	24	yes	complemented
223	C₂³	C₃×#135	1	4	yes	—
224	C₂³	C₂²×C₃×D₄	1	4	yes	—
225	C₂³	C₂²×C₃×D₄	1	3	yes	—
226	Q₈	—	2	8	no	—
227	Q₈	—	2	8	no	—
228	Q₈	—	4	24	no	—
229	C₂×C₄	—	2	12	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	24	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	24	no	—
236	C₂×C₄	—	4	24	no	—
237	C₂×C₄	C₃×#121	1	12	yes	—
238	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
239	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
240	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
241	C₂×C₄	C₂×C₃×#41	1	6	yes	—
242	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	C₂²×C₃×D₄	1	8	yes	—
245	C₂×C₃	—	2	8	no	—
246	C₂×C₃	C₂×#467	1	12	yes	direct product factor
247	C₂×C₃	C₂²×#121	1	1	yes	characteristic
248	C₂×C₃	C₂³×Q₁₆	1	1	yes	characteristic
249	C₂×C₃	C₂²×#135	1	1	yes	characteristic
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	48	no	—
253	C₂²	C₃×#467	1	16	yes	direct product factor
254	C₂²	C₂×C₃×#121	1	6	yes	—
255	C₂²	C₂²×C₃×Q₁₆	1	6	yes	—
256	C₂²	C₂²×C₃×Q₁₆	1	8	yes	complemented
257	C₂²	C₂×C₃×#135	1	6	yes	—
258	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
259	C₄	—	2	4	no	—
260	C₄	—	4	8	no	—
261	C₄	—	4	8	no	—
262	C₄	C₂×C₃×#121	1	6	yes	—
263	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
264	C₄	C₂²×C₃×#41	1	1	yes	characteristic
265	C₃	C₂²×#467	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	—	2	8	no	—
267	C₂	C₂×C₃×#467	1	12	yes	direct product factor
268	C₂	C₂²×C₃×#121	1	1	yes	characteristic
269	C₂	C₂³×C₃×Q₁₆	1	1	yes	characteristic
270	C₂	C₂²×C₃×#135	1	1	yes	characteristic

	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×#467	C₂	direct (N × H)
5. C₂²×C₃×#107	C₂	semidirect
6. C₂²×C₃×#103	C₂	semidirect
7. C₂²×#467	C₃	direct (N × H)
11. C₂²×C₃×Q₁₆	C₂²	semidirect
21. C₃×#467	C₂²	direct (N × H)
22. C₂×C₃×#107	C₂²	semidirect
23. C₂×C₃×#103	C₂²	semidirect
24. C₂³×Q₁₆	C₂×C₃	semidirect
27. C₂×#467	C₂×C₃	direct (N × H)
28. C₂²×#107	C₂×C₃	semidirect
29. C₂²×#103	C₂×C₃	semidirect
44. C₂×C₃×Q₁₆	C₂³	semidirect
57. C₃×#107	C₂³	semidirect
58. C₃×#103	C₂³	semidirect
62. C₂²×Q₁₆	C₂²×C₃	semidirect
72. #467	C₂²×C₃	direct (N × H)
73. C₂×#107	C₂²×C₃	semidirect
74. C₂×#103	C₂²×C₃	semidirect
77. C₂⁴×C₃	Q₁₆	semidirect
124. C₂×Q₁₆	C₂³×C₃	semidirect
137. #107	C₂³×C₃	semidirect
138. #103	C₂³×C₃	semidirect
144. C₂³×C₃	C₂×Q₁₆	semidirect
169. C₂⁴	C₃×Q₁₆	semidirect
205. C₂²×C₃	#467	direct (N × H)
208. C₂²×C₃	C₂²×Q₁₆	semidirect
222. C₂³	C₂×C₃×Q₁₆	semidirect
246. C₂×C₃	C₂×#467	direct (N × H)
253. C₂²	C₃×#467	direct (N × H)
256. C₂²	C₂²×C₃×Q₁₆	semidirect
265. C₃	C₂²×#467	direct (N × H)
267. C₂	C₂×C₃×#467	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
C₂⁴×C₃×C₈	2
C₂²×C₃×C₂²⋊Q₈	2
C₂×C₃×#467	2
C₂²×C₃×#107	2
C₂²×C₃×#103	2
C₂²×#467	3

Subgroups — C22×C3×#467 #10679043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#467 #10679043