Subgroups — C₂²×C₃×#496 #10679240

Order 768.

4038 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×SD₁₆	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃×D₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#496	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#496	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂³×C₃×D₄	—	2	2	no	—
10	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×D₄	C₂²	1	2	yes	—
12	C₂²×C₃×SD₁₆	—	2	16	no	—
13	C₂²×C₃×SD₁₆	C₂²	1	24	yes	complemented
14	C₂²×C₃×D₈	—	2	16	no	—
15	C₂²×C₃×D₈	C₂²	1	24	yes	complemented
16	C₂³×C₃×C₈	C₂²	1	2	yes	—
17	C₂×C₃×#127	—	2	6	no	—
18	C₂×C₃×#127	C₂²	1	12	yes	complemented
19	C₂×C₃×#124	—	2	6	no	—
20	C₂×C₃×#124	C₂²	1	12	yes	complemented
21	C₂²×C₃×C₂²⋊C₄	—	2	2	no	—
22	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#496	C₂²	1	64	yes	direct product factor
24	C₂×C₃×#97	C₂²	1	12	yes	complemented
25	C₂³×SD₁₆	C₂×C₃	1	2	yes	complemented
26	C₂³×D₈	C₂×C₃	1	2	yes	complemented
27	C₂²×#127	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×#124	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#496	C₂×C₃	1	24	yes	direct product factor
30	C₂²×#97	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁵×C₃	—	2	2	no	—
32	C₂⁵×C₃	—	2	1	no	—
33	C₂²×C₃×Q₈	—	2	3	no	—
34	C₂²×C₃×Q₈	—	2	8	no	—
35	C₂²×C₃×Q₈	C₂³	1	6	yes	—
36	C₂²×C₃×D₄	—	2	8	no	—
37	C₂²×C₃×D₄	—	2	8	no	—
38	C₂²×C₃×D₄	—	2	8	no	—
39	C₂²×C₃×D₄	—	2	24	no	—
40	C₂²×C₃×D₄	—	2	3	no	—
41	C₂²×C₃×D₄	—	2	6	no	—
42	C₂²×C₃×D₄	—	4	16	no	—
43	C₂²×C₃×D₄	C₂³	1	6	yes	—
44	C₂²×C₃×D₄	C₂³	1	12	yes	—
45	C₂³×C₃×C₄	—	2	1	no	—
46	C₂³×C₃×C₄	D₄	1	2	yes	—
47	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃×SD₁₆	—	2	96	no	—
49	C₂×C₃×SD₁₆	C₂³	1	32	yes	complemented
50	C₂×C₃×D₈	—	2	96	no	—
51	C₂×C₃×D₈	C₂³	1	32	yes	complemented
52	C₂²×C₃×C₈	D₄	1	16	yes	complemented
53	C₂²×C₃×C₈	C₂³	1	12	yes	—
54	C₃×#127	—	2	24	no	—
55	C₃×#127	C₂³	1	16	yes	complemented
56	C₃×#124	—	2	24	no	—
57	C₃×#124	C₂³	1	16	yes	complemented
58	C₂×C₃×C₂²⋊C₄	—	2	24	no	—
59	C₂×C₃×C₄²	D₄	1	6	yes	—
60	C₂×C₃×C₄²	C₂³	1	6	yes	—
61	C₃×#97	C₂³	1	16	yes	complemented
62	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
63	C₂³×D₄	—	2	2	no	—
64	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
65	C₂³×D₄	C₂²×C₃	1	2	yes	—
66	C₂²×SD₁₆	—	2	16	no	—
67	C₂²×SD₁₆	C₂²×C₃	1	24	yes	complemented
68	C₂²×D₈	—	2	16	no	—
69	C₂²×D₈	C₂²×C₃	1	24	yes	complemented
70	C₂³×C₈	C₂²×C₃	1	2	yes	—
71	C₂×#127	—	2	6	no	—
72	C₂×#127	C₂²×C₃	1	12	yes	complemented
73	C₂×#124	—	2	6	no	—
74	C₂×#124	C₂²×C₃	1	12	yes	complemented
75	C₂²×C₂²⋊C₄	—	2	2	no	—
76	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
77	#496	C₂²×C₃	1	64	yes	direct product factor
78	C₂×#97	C₂²×C₃	1	12	yes	complemented
79	C₂⁴×C₃	—	2	6	no	—
80	C₂⁴×C₃	—	2	12	no	—
81	C₂⁴×C₃	—	4	8	no	—
82	C₂⁴×C₃	—	4	4	no	—
83	C₂⁴×C₃	—	4	8	no	—
84	C₂⁴×C₃	—	4	16	no	—
85	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×Q₈	—	2	6	no	—
87	C₂×C₃×Q₈	—	2	24	no	—
88	C₂×C₃×Q₈	—	4	12	no	—
89	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
90	C₂×C₃×D₄	—	2	24	no	—
91	C₂×C₃×D₄	—	2	24	no	—
92	C₂×C₃×D₄	—	2	24	no	—
93	C₂×C₃×D₄	—	2	6	no	—
94	C₂×C₃×D₄	—	2	12	no	—
95	C₂×C₃×D₄	—	2	32	no	—
96	C₂×C₃×D₄	—	4	96	no	—
97	C₂×C₃×D₄	—	4	12	no	—
98	C₂×C₃×D₄	—	4	24	no	—
99	C₂×C₃×D₄	C₂⁴	1	4	yes	—
100	C₂×C₃×D₄	C₂⁴	1	8	yes	—
101	C₂²×C₃×C₄	—	2	8	no	—
102	C₂²×C₃×C₄	—	2	6	no	—
103	C₂²×C₃×C₄	—	4	4	no	—
104	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
105	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
106	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
107	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
108	C₃×SD₁₆	—	2	64	no	—
109	C₃×D₈	—	2	64	no	—
110	C₂×C₃×C₈	C₂×D₄	1	48	yes	complemented
111	C₂×C₃×C₈	C₂⁴	1	8	yes	—
112	C₃×C₂²⋊C₄	—	2	32	no	—
113	C₃×C₄²	C₂×D₄	1	12	yes	—
114	C₃×C₄²	C₂⁴	1	4	yes	—
115	C₂⁵	—	2	1	no	—
116	C₂⁵	—	2	2	no	—
117	C₂²×Q₈	—	2	3	no	—
118	C₂²×Q₈	—	2	8	no	—
119	C₂²×Q₈	C₂³×C₃	1	6	yes	—
120	C₂²×D₄	—	2	8	no	—
121	C₂²×D₄	—	2	8	no	—
122	C₂²×D₄	—	2	8	no	—
123	C₂²×D₄	—	2	3	no	—
124	C₂²×D₄	—	2	6	no	—
125	C₂²×D₄	—	2	24	no	—
126	C₂²×D₄	—	4	16	no	—
127	C₂²×D₄	C₂³×C₃	1	6	yes	—
128	C₂²×D₄	C₂³×C₃	1	12	yes	—
129	C₂³×C₄	—	2	1	no	—
130	C₂³×C₄	C₃×D₄	1	2	yes	—
131	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂×SD₁₆	—	2	96	no	—
133	C₂×SD₁₆	C₂³×C₃	1	32	yes	complemented
134	C₂×D₈	—	2	96	no	—
135	C₂×D₈	C₂³×C₃	1	32	yes	complemented
136	C₂²×C₈	C₃×D₄	1	16	yes	complemented
137	C₂²×C₈	C₂³×C₃	1	12	yes	—
138	#127	—	2	24	no	—
139	#127	C₂³×C₃	1	16	yes	complemented
140	#124	—	2	24	no	—
141	#124	C₂³×C₃	1	16	yes	complemented
142	C₂×C₂²⋊C₄	—	2	24	no	—
143	C₂×C₄²	C₃×D₄	1	6	yes	—
144	C₂×C₄²	C₂³×C₃	1	6	yes	—
145	#97	C₂³×C₃	1	16	yes	complemented
146	C₂³×C₃	—	2	4	no	—
147	C₂³×C₃	—	2	8	no	—
148	C₂³×C₃	—	4	12	no	—
149	C₂³×C₃	—	4	24	no	—
150	C₂³×C₃	—	4	48	no	—
151	C₂³×C₃	—	4	24	no	—
152	C₂³×C₃	—	8	16	no	—
153	C₂³×C₃	—	8	32	no	—
154	C₂³×C₃	#127	1	4	yes	—
155	C₂³×C₃	AGL(1,ℤ/8)	1	8	yes	—
156	C₂³×C₃	C₂²×D₄	1	3	yes	—
157	C₃×Q₈	—	2	8	no	—
158	C₃×Q₈	—	4	12	no	—
159	C₃×D₄	—	2	8	no	—
160	C₃×D₄	—	2	8	no	—
161	C₃×D₄	—	2	8	no	—
162	C₃×D₄	—	4	64	no	—
163	C₃×D₄	—	4	12	no	—
164	C₃×D₄	—	4	24	no	—
165	C₂×C₃×C₄	—	2	24	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	4	12	no	—
168	C₂×C₃×C₄	#127	1	12	yes	—
169	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
170	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
171	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
172	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
173	C₃×C₈	C₂²×D₄	1	16	yes	complemented
174	C₂⁴	—	2	6	no	—
175	C₂⁴	—	2	12	no	—
176	C₂⁴	—	4	4	no	—
177	C₂⁴	—	4	8	no	—
178	C₂⁴	—	4	16	no	—
179	C₂⁴	—	4	8	no	—
180	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×Q₈	—	2	6	no	—
182	C₂×Q₈	—	2	24	no	—
183	C₂×Q₈	—	4	12	no	—
184	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
185	C₂×D₄	—	2	24	no	—
186	C₂×D₄	—	2	24	no	—
187	C₂×D₄	—	2	24	no	—
188	C₂×D₄	—	2	6	no	—
189	C₂×D₄	—	2	12	no	—
190	C₂×D₄	—	2	32	no	—
191	C₂×D₄	—	4	96	no	—
192	C₂×D₄	—	4	12	no	—
193	C₂×D₄	—	4	24	no	—
194	C₂×D₄	C₂⁴×C₃	1	4	yes	—
195	C₂×D₄	C₂⁴×C₃	1	8	yes	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	4	4	no	—
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	8	yes	—
202	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
203	SD₁₆	—	2	64	no	—
204	D₈	—	2	64	no	—
205	C₂×C₈	C₂×C₃×D₄	1	48	yes	complemented
206	C₂×C₈	C₂⁴×C₃	1	8	yes	—
207	C₂²⋊C₄	—	2	32	no	—
208	C₄²	C₂×C₃×D₄	1	12	yes	—
209	C₄²	C₂⁴×C₃	1	4	yes	—
210	C₂²×C₃	—	4	4	no	—
211	C₂²×C₃	—	4	8	no	—
212	C₂²×C₃	—	4	16	no	—
213	C₂²×C₃	—	4	8	no	—
214	C₂²×C₃	—	8	48	no	—
215	C₂²×C₃	—	8	24	no	—
216	C₂²×C₃	#496	1	16	yes	direct product factor
217	C₂²×C₃	C₂×#127	1	6	yes	—
218	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
219	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
220	C₃×C₄	—	2	8	no	—
221	C₃×C₄	—	4	4	no	—
222	C₃×C₄	C₂×#127	1	6	yes	—
223	C₃×C₄	C₂³×D₄	1	2	yes	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	4	48	no	—
227	C₂³	—	4	24	no	—
228	C₂³	—	4	12	no	—
229	C₂³	—	4	24	no	—
230	C₂³	—	8	32	no	—
231	C₂³	—	8	16	no	—
232	C₂³	C₃×#127	1	4	yes	—
233	C₂³	C₃×AGL(1,ℤ/8)	1	8	yes	—
234	C₂³	C₂²×C₃×D₄	1	3	yes	—
235	Q₈	—	2	8	no	—
236	Q₈	—	4	12	no	—
237	D₄	—	2	8	no	—
238	D₄	—	2	8	no	—
239	D₄	—	2	8	no	—
240	D₄	—	4	64	no	—
241	D₄	—	4	12	no	—
242	D₄	—	4	24	no	—
243	C₂×C₄	—	2	24	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	12	no	—
246	C₂×C₄	C₃×#127	1	12	yes	—
247	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
248	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
249	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
250	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
251	C₈	C₂²×C₃×D₄	1	16	yes	complemented
252	C₂×C₃	—	8	8	no	—
253	C₂×C₃	—	8	4	no	—
254	C₂×C₃	C₂×#496	1	12	yes	direct product factor
255	C₂×C₃	C₂²×#127	1	1	yes	characteristic
256	C₂×C₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
257	C₂²	—	4	4	no	—
258	C₂²	—	4	8	no	—
259	C₂²	—	4	16	no	—
260	C₂²	—	4	8	no	—
261	C₂²	—	8	48	no	—
262	C₂²	—	8	24	no	—
263	C₂²	C₃×#496	1	16	yes	direct product factor
264	C₂²	C₂×C₃×#127	1	6	yes	—
265	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	12	yes	—
266	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
267	C₄	—	2	8	no	—
268	C₄	—	4	4	no	—
269	C₄	C₂×C₃×#127	1	6	yes	—
270	C₄	C₂³×C₃×D₄	1	2	yes	—
271	C₃	C₂²×#496	1	1	yes	characteristic, direct product factor, Sylow
272	C₂	—	8	8	no	—
273	C₂	—	8	4	no	—
274	C₂	C₂×C₃×#496	1	12	yes	direct product factor
275	C₂	C₂²×C₃×#127	1	1	yes	characteristic
276	C₂	C₂²×C₃×AGL(1,ℤ/8)	1	2	yes	—

	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 ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×SD₁₆	C₂	semidirect
2. C₂³×C₃×D₈	C₂	semidirect
3. C₂²×C₃×#127	C₂	semidirect
4. C₂²×C₃×#124	C₂	semidirect
5. C₂×C₃×#496	C₂	direct (N × H)
6. C₂²×C₃×#97	C₂	semidirect
7. C₂²×#496	C₃	direct (N × H)
13. C₂²×C₃×SD₁₆	C₂²	semidirect
15. C₂²×C₃×D₈	C₂²	semidirect
18. C₂×C₃×#127	C₂²	semidirect
20. C₂×C₃×#124	C₂²	semidirect
23. C₃×#496	C₂²	direct (N × H)
24. C₂×C₃×#97	C₂²	semidirect
25. C₂³×SD₁₆	C₂×C₃	semidirect
26. C₂³×D₈	C₂×C₃	semidirect
27. C₂²×#127	C₂×C₃	semidirect
28. C₂²×#124	C₂×C₃	semidirect
29. C₂×#496	C₂×C₃	direct (N × H)
30. C₂²×#97	C₂×C₃	semidirect
49. C₂×C₃×SD₁₆	C₂³	semidirect
51. C₂×C₃×D₈	C₂³	semidirect
52. C₂²×C₃×C₈	D₄	semidirect
55. C₃×#127	C₂³	semidirect
57. C₃×#124	C₂³	semidirect
61. C₃×#97	C₂³	semidirect
67. C₂²×SD₁₆	C₂²×C₃	semidirect
69. C₂²×D₈	C₂²×C₃	semidirect
72. C₂×#127	C₂²×C₃	semidirect
74. C₂×#124	C₂²×C₃	semidirect
77. #496	C₂²×C₃	direct (N × H)
78. C₂×#97	C₂²×C₃	semidirect
110. C₂×C₃×C₈	C₂×D₄	semidirect
133. C₂×SD₁₆	C₂³×C₃	semidirect
135. C₂×D₈	C₂³×C₃	semidirect
136. C₂²×C₈	C₃×D₄	semidirect
139. #127	C₂³×C₃	semidirect
141. #124	C₂³×C₃	semidirect
145. #97	C₂³×C₃	semidirect
173. C₃×C₈	C₂²×D₄	semidirect
205. C₂×C₈	C₂×C₃×D₄	semidirect
216. C₂²×C₃	#496	direct (N × H)
251. C₈	C₂²×C₃×D₄	semidirect
254. C₂×C₃	C₂×#496	direct (N × H)
263. C₂²	C₃×#496	direct (N × H)
271. C₃	C₂²×#496	direct (N × H)
274. C₂	C₂×C₃×#496	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×SD₁₆	2
C₂³×C₃×D₈	2
C₂²×C₃×#127	2
C₂²×C₃×#124	2
C₂×C₃×#496	2
C₂²×C₃×#97	2
C₂²×#496	3

Subgroups — C22×C3×#496 #10679240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#496 #10679240