Subgroups — C₂²×C₃×#497 #10679241

Order 768.

3350 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	2	yes	maximal, complemented
2	C₂³×C₃×SD₁₆	C₂	1	2	yes	maximal
3	C₂²×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×#124	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×#497	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#497	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂³×C₃×Q₈	C₂²	1	2	yes	—
10	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Q₁₆	—	2	16	no	—
12	C₂²×C₃×Q₁₆	C₂²	1	24	yes	complemented
13	C₂²×C₃×SD₁₆	—	2	16	no	—
14	C₂²×C₃×SD₁₆	C₂²	1	24	yes	—
15	C₂³×C₃×C₈	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
17	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
18	C₂×C₃×#124	—	2	6	no	—
19	C₂×C₃×#124	C₂²	1	12	yes	—
20	C₂²×C₃×C₄⋊C₄	—	2	2	no	—
21	C₂²×C₃×C₂²⋊C₄	—	2	2	no	—
22	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#497	C₂²	1	64	yes	direct product factor
24	C₂×C₃×#97	C₂²	1	12	yes	complemented
25	C₂³×Q₁₆	C₂×C₃	1	2	yes	complemented
26	C₂³×SD₁₆	C₂×C₃	1	2	yes	—
27	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×#124	C₂×C₃	1	1	yes	characteristic
29	C₂×#497	C₂×C₃	1	24	yes	direct product factor
30	C₂²×#97	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁵×C₃	—	2	1	no	—
32	C₂²×C₃×Q₈	—	2	3	no	—
33	C₂²×C₃×Q₈	—	2	6	no	—
34	C₂²×C₃×Q₈	—	2	8	no	—
35	C₂²×C₃×Q₈	—	2	8	no	—
36	C₂²×C₃×Q₈	—	2	8	no	—
37	C₂²×C₃×Q₈	C₂³	1	12	yes	—
38	C₂²×C₃×Q₈	C₂³	1	6	yes	—
39	C₂²×C₃×D₄	—	2	8	no	—
40	C₂²×C₃×D₄	—	2	3	no	—
41	C₂²×C₃×D₄	C₂³	1	6	yes	—
42	C₂³×C₃×C₄	—	2	1	no	—
43	C₂³×C₃×C₄	—	2	2	no	—
44	C₂³×C₃×C₄	D₄	1	2	yes	—
45	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂×C₃×Q₁₆	—	2	96	no	—
47	C₂×C₃×Q₁₆	C₂³	1	32	yes	complemented
48	C₂×C₃×SD₁₆	—	2	96	no	—
49	C₂×C₃×SD₁₆	C₂³	1	32	yes	—
50	C₂²×C₃×C₈	D₄	1	16	yes	—
51	C₂²×C₃×C₈	C₂³	1	12	yes	—
52	C₃×C₄⋊Q₈	—	2	24	no	—
53	C₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
54	C₃×#124	—	2	24	no	—
55	C₃×#124	C₂³	1	16	yes	—
56	C₂×C₃×C₄⋊C₄	—	2	24	no	—
57	C₂×C₃×C₂²⋊C₄	—	2	24	no	—
58	C₂×C₃×C₄²	D₄	1	6	yes	—
59	C₂×C₃×C₄²	C₂³	1	6	yes	—
60	C₃×#97	C₂³	1	16	yes	complemented
61	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
62	C₂³×Q₈	C₂²×C₃	1	2	yes	—
63	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
64	C₂²×Q₁₆	—	2	16	no	—
65	C₂²×Q₁₆	C₂²×C₃	1	24	yes	complemented
66	C₂²×SD₁₆	—	2	16	no	—
67	C₂²×SD₁₆	C₂²×C₃	1	24	yes	—
68	C₂³×C₈	C₂²×C₃	1	2	yes	—
69	C₂×C₄⋊Q₈	—	2	6	no	—
70	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	complemented
71	C₂×#124	—	2	6	no	—
72	C₂×#124	C₂²×C₃	1	12	yes	—
73	C₂²×C₄⋊C₄	—	2	2	no	—
74	C₂²×C₂²⋊C₄	—	2	2	no	—
75	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
76	#497	C₂²×C₃	1	64	yes	direct product factor
77	C₂×#97	C₂²×C₃	1	12	yes	complemented
78	C₂⁴×C₃	—	2	6	no	—
79	C₂⁴×C₃	—	4	4	no	—
80	C₂⁴×C₃	—	4	8	no	—
81	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×Q₈	—	2	6	no	—
83	C₂×C₃×Q₈	—	2	12	no	—
84	C₂×C₃×Q₈	—	2	24	no	—
85	C₂×C₃×Q₈	—	2	24	no	—
86	C₂×C₃×Q₈	—	2	24	no	—
87	C₂×C₃×Q₈	—	4	12	no	—
88	C₂×C₃×Q₈	—	4	24	no	—
89	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
90	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
91	C₂×C₃×D₄	—	2	24	no	—
92	C₂×C₃×D₄	—	2	6	no	—
93	C₂×C₃×D₄	—	4	12	no	—
94	C₂×C₃×D₄	C₂⁴	1	4	yes	—
95	C₂²×C₃×C₄	—	2	8	no	—
96	C₂²×C₃×C₄	—	2	6	no	—
97	C₂²×C₃×C₄	—	2	12	no	—
98	C₂²×C₃×C₄	—	4	4	no	—
99	C₂²×C₃×C₄	—	4	8	no	—
100	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
101	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
102	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
103	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
104	C₃×Q₁₆	—	2	64	no	—
105	C₃×SD₁₆	—	2	64	no	—
106	C₂×C₃×C₈	C₂×D₄	1	48	yes	—
107	C₂×C₃×C₈	C₂⁴	1	8	yes	—
108	C₃×C₄⋊C₄	—	2	32	no	—
109	C₃×C₂²⋊C₄	—	2	32	no	—
110	C₃×C₄²	C₂×D₄	1	12	yes	—
111	C₃×C₄²	C₂⁴	1	4	yes	—
112	C₂⁵	—	2	1	no	—
113	C₂²×Q₈	—	2	3	no	—
114	C₂²×Q₈	—	2	6	no	—
115	C₂²×Q₈	—	2	8	no	—
116	C₂²×Q₈	—	2	8	no	—
117	C₂²×Q₈	—	2	8	no	—
118	C₂²×Q₈	C₂³×C₃	1	6	yes	—
119	C₂²×Q₈	C₂³×C₃	1	12	yes	—
120	C₂²×D₄	—	2	8	no	—
121	C₂²×D₄	—	2	3	no	—
122	C₂²×D₄	C₂³×C₃	1	6	yes	—
123	C₂³×C₄	—	2	1	no	—
124	C₂³×C₄	—	2	2	no	—
125	C₂³×C₄	C₃×D₄	1	2	yes	—
126	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
127	C₂×Q₁₆	—	2	96	no	—
128	C₂×Q₁₆	C₂³×C₃	1	32	yes	complemented
129	C₂×SD₁₆	—	2	96	no	—
130	C₂×SD₁₆	C₂³×C₃	1	32	yes	—
131	C₂²×C₈	C₃×D₄	1	16	yes	—
132	C₂²×C₈	C₂³×C₃	1	12	yes	—
133	C₄⋊Q₈	—	2	24	no	—
134	C₄⋊Q₈	C₂³×C₃	1	16	yes	complemented
135	#124	—	2	24	no	—
136	#124	C₂³×C₃	1	16	yes	—
137	C₂×C₄⋊C₄	—	2	24	no	—
138	C₂×C₂²⋊C₄	—	2	24	no	—
139	C₂×C₄²	C₃×D₄	1	6	yes	—
140	C₂×C₄²	C₂³×C₃	1	6	yes	—
141	#97	C₂³×C₃	1	16	yes	complemented
142	C₂³×C₃	—	2	4	no	—
143	C₂³×C₃	—	4	12	no	—
144	C₂³×C₃	—	4	24	no	—
145	C₂³×C₃	—	8	16	no	—
146	C₂³×C₃	#127	1	4	yes	—
147	C₂³×C₃	#137	1	8	yes	—
148	C₂³×C₃	C₂²×D₄	1	3	yes	—
149	C₃×Q₈	—	2	8	no	—
150	C₃×Q₈	—	2	8	no	—
151	C₃×Q₈	—	2	8	no	—
152	C₃×Q₈	—	4	12	no	—
153	C₃×Q₈	—	4	24	no	—
154	C₃×D₄	—	2	8	no	—
155	C₃×D₄	—	4	12	no	—
156	C₂×C₃×C₄	—	2	24	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	8	no	—
159	C₂×C₃×C₄	—	4	12	no	—
160	C₂×C₃×C₄	—	4	24	no	—
161	C₂×C₃×C₄	#127	1	12	yes	—
162	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
163	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
164	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
165	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
166	C₃×C₈	C₂²×D₄	1	16	yes	—
167	C₂⁴	—	2	6	no	—
168	C₂⁴	—	4	4	no	—
169	C₂⁴	—	4	8	no	—
170	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×Q₈	—	2	6	no	—
172	C₂×Q₈	—	2	12	no	—
173	C₂×Q₈	—	2	24	no	—
174	C₂×Q₈	—	2	24	no	—
175	C₂×Q₈	—	2	24	no	—
176	C₂×Q₈	—	4	12	no	—
177	C₂×Q₈	—	4	24	no	—
178	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
179	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
180	C₂×D₄	—	2	24	no	—
181	C₂×D₄	—	2	6	no	—
182	C₂×D₄	—	4	12	no	—
183	C₂×D₄	C₂⁴×C₃	1	4	yes	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	6	no	—
186	C₂²×C₄	—	2	12	no	—
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
190	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
191	C₂²×C₄	C₂×C₃×D₄	1	8	yes	—
192	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
193	Q₁₆	—	2	64	no	—
194	SD₁₆	—	2	64	no	—
195	C₂×C₈	C₂×C₃×D₄	1	48	yes	—
196	C₂×C₈	C₂⁴×C₃	1	8	yes	—
197	C₄⋊C₄	—	2	32	no	—
198	C₂²⋊C₄	—	2	32	no	—
199	C₄²	C₂×C₃×D₄	1	12	yes	—
200	C₄²	C₂⁴×C₃	1	4	yes	—
201	C₂²×C₃	—	4	4	no	—
202	C₂²×C₃	—	4	8	no	—
203	C₂²×C₃	—	8	24	no	—
204	C₂²×C₃	#497	1	16	yes	direct product factor
205	C₂²×C₃	C₂×#127	1	6	yes	—
206	C₂²×C₃	C₂×#137	1	12	yes	—
207	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
208	C₃×C₄	—	2	8	no	—
209	C₃×C₄	—	4	4	no	—
210	C₃×C₄	—	4	8	no	—
211	C₃×C₄	C₂×#127	1	6	yes	—
212	C₃×C₄	C₂³×D₄	1	2	yes	—
213	C₂³	—	2	4	no	—
214	C₂³	—	4	24	no	—
215	C₂³	—	4	12	no	—
216	C₂³	—	8	16	no	—
217	C₂³	C₃×#127	1	4	yes	—
218	C₂³	C₃×#137	1	8	yes	—
219	C₂³	C₂²×C₃×D₄	1	3	yes	—
220	Q₈	—	2	8	no	—
221	Q₈	—	2	8	no	—
222	Q₈	—	2	8	no	—
223	Q₈	—	4	12	no	—
224	Q₈	—	4	24	no	—
225	D₄	—	2	8	no	—
226	D₄	—	4	12	no	—
227	C₂×C₄	—	2	24	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	4	12	no	—
231	C₂×C₄	—	4	24	no	—
232	C₂×C₄	C₃×#127	1	12	yes	—
233	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
234	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
235	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
236	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
237	C₈	C₂²×C₃×D₄	1	16	yes	—
238	C₂×C₃	—	8	4	no	—
239	C₂×C₃	C₂×#497	1	12	yes	direct product factor
240	C₂×C₃	C₂²×#127	1	1	yes	characteristic
241	C₂×C₃	C₂²×#137	1	2	yes	—
242	C₂²	—	4	4	no	—
243	C₂²	—	4	8	no	—
244	C₂²	—	8	24	no	—
245	C₂²	C₃×#497	1	16	yes	direct product factor
246	C₂²	C₂×C₃×#127	1	6	yes	—
247	C₂²	C₂×C₃×#137	1	12	yes	—
248	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
249	C₄	—	2	8	no	—
250	C₄	—	4	4	no	—
251	C₄	—	4	8	no	—
252	C₄	C₂×C₃×#127	1	6	yes	—
253	C₄	C₂³×C₃×D₄	1	2	yes	—
254	C₃	C₂²×#497	1	1	yes	characteristic, direct product factor, Sylow
255	C₂	—	8	4	no	—
256	C₂	C₂×C₃×#497	1	12	yes	direct product factor
257	C₂	C₂²×C₃×#127	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	C₂³×C₃×D₄	C₂²
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×Q₁₆	C₂	semidirect
3. C₂²×C₃×C₄⋊Q₈	C₂	semidirect
5. C₂×C₃×#497	C₂	direct (N × H)
6. C₂²×C₃×#97	C₂	semidirect
7. C₂²×#497	C₃	direct (N × H)
12. C₂²×C₃×Q₁₆	C₂²	semidirect
17. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
23. C₃×#497	C₂²	direct (N × H)
24. C₂×C₃×#97	C₂²	semidirect
25. C₂³×Q₁₆	C₂×C₃	semidirect
27. C₂²×C₄⋊Q₈	C₂×C₃	semidirect
29. C₂×#497	C₂×C₃	direct (N × H)
30. C₂²×#97	C₂×C₃	semidirect
47. C₂×C₃×Q₁₆	C₂³	semidirect
53. C₃×C₄⋊Q₈	C₂³	semidirect
60. C₃×#97	C₂³	semidirect
65. C₂²×Q₁₆	C₂²×C₃	semidirect
70. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
76. #497	C₂²×C₃	direct (N × H)
77. C₂×#97	C₂²×C₃	semidirect
128. C₂×Q₁₆	C₂³×C₃	semidirect
134. C₄⋊Q₈	C₂³×C₃	semidirect
141. #97	C₂³×C₃	semidirect
204. C₂²×C₃	#497	direct (N × H)
239. C₂×C₃	C₂×#497	direct (N × H)
245. C₂²	C₃×#497	direct (N × H)
254. C₃	C₂²×#497	direct (N × H)
256. C₂	C₂×C₃×#497	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×SD₁₆	2
C₂²×C₃×C₄⋊Q₈	2
C₂²×C₃×#124	2
C₂×C₃×#497	2
C₂²×C₃×#97	2
C₂²×#497	3

Subgroups — C22×C3×#497 #10679241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#497 #10679241