Subgroups — C₂²×C₃×#495 #10679238

Order 768.

3518 conjugacy classes of subgroups.

The subgroup lattice has 248 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	C₂³×C₃×SD₁₆	C₂	1	2	yes	maximal, complemented
3	C₂³×C₃×D₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃×#124	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#495	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#495	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	C₂²	1	2	yes	—
9	C₂³×C₃×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×Q₁₆	—	2	8	no	—
11	C₂²×C₃×Q₁₆	C₂²	1	12	yes	complemented
12	C₂²×C₃×SD₁₆	—	2	16	no	—
13	C₂²×C₃×SD₁₆	C₂²	1	24	yes	complemented
14	C₂²×C₃×D₈	—	2	8	no	—
15	C₂²×C₃×D₈	C₂²	1	12	yes	—
16	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×#124	—	2	12	no	—
19	C₂×C₃×#124	C₂²	1	24	yes	complemented
20	C₂²×C₃×C₂²⋊C₄	—	2	4	no	—
21	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
22	C₃×#495	C₂²	1	64	yes	direct product factor
23	C₂×C₃×C₄×C₈	C₂²	1	12	yes	complemented
24	C₂³×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂³×SD₁₆	C₂×C₃	1	2	yes	complemented
26	C₂³×D₈	C₂×C₃	1	1	yes	characteristic
27	C₂²×#124	C₂×C₃	1	2	yes	complemented
28	C₂×#495	C₂×C₃	1	24	yes	direct product factor
29	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂⁵×C₃	—	2	2	no	—
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	C₂²×C₃×D₄	—	2	8	no	—
36	C₂²×C₃×D₄	—	2	8	no	—
37	C₂²×C₃×D₄	—	2	6	no	—
38	C₂²×C₃×D₄	C₂³	1	12	yes	—
39	C₂³×C₃×C₄	—	2	2	no	—
40	C₂³×C₃×C₄	D₄	1	2	yes	—
41	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃×Q₁₆	—	2	48	no	—
43	C₂×C₃×Q₁₆	C₂³	1	16	yes	complemented
44	C₂×C₃×SD₁₆	—	2	96	no	—
45	C₂×C₃×SD₁₆	C₂³	1	32	yes	complemented
46	C₂×C₃×D₈	—	2	48	no	—
47	C₂×C₃×D₈	C₂³	1	16	yes	—
48	C₂²×C₃×C₈	D₄	1	8	yes	—
49	C₂²×C₃×C₈	D₄	1	8	yes	—
50	C₂²×C₃×C₈	C₂³	1	6	yes	—
51	C₂²×C₃×C₈	C₂³	1	6	yes	—
52	C₃×#124	—	2	48	no	—
53	C₃×#124	C₂³	1	32	yes	complemented
54	C₂×C₃×C₂²⋊C₄	—	2	48	no	—
55	C₂×C₃×C₄²	D₄	1	6	yes	—
56	C₂×C₃×C₄²	C₂³	1	6	yes	—
57	C₃×C₄×C₈	C₂³	1	16	yes	complemented
58	C₂³×Q₈	C₂²×C₃	1	2	yes	—
59	C₂³×D₄	C₂²×C₃	1	2	yes	—
60	C₂²×Q₁₆	—	2	8	no	—
61	C₂²×Q₁₆	C₂²×C₃	1	12	yes	complemented
62	C₂²×SD₁₆	—	2	16	no	—
63	C₂²×SD₁₆	C₂²×C₃	1	24	yes	complemented
64	C₂²×D₈	—	2	8	no	—
65	C₂²×D₈	C₂²×C₃	1	12	yes	—
66	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
67	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×#124	—	2	12	no	—
69	C₂×#124	C₂²×C₃	1	24	yes	complemented
70	C₂²×C₂²⋊C₄	—	2	4	no	—
71	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
72	#495	C₂²×C₃	1	64	yes	direct product factor
73	C₂×C₄×C₈	C₂²×C₃	1	12	yes	complemented
74	C₂⁴×C₃	—	2	12	no	—
75	C₂⁴×C₃	—	4	8	no	—
76	C₂⁴×C₃	—	4	16	no	—
77	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×Q₈	—	2	12	no	—
79	C₂×C₃×Q₈	—	2	24	no	—
80	C₂×C₃×Q₈	—	2	24	no	—
81	C₂×C₃×Q₈	—	4	24	no	—
82	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
83	C₂×C₃×D₄	—	2	24	no	—
84	C₂×C₃×D₄	—	2	24	no	—
85	C₂×C₃×D₄	—	2	12	no	—
86	C₂×C₃×D₄	—	4	24	no	—
87	C₂×C₃×D₄	C₂⁴	1	8	yes	—
88	C₂²×C₃×C₄	—	2	8	no	—
89	C₂²×C₃×C₄	—	2	12	no	—
90	C₂²×C₃×C₄	—	4	8	no	—
91	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
92	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
93	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
94	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
95	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
96	C₃×Q₁₆	—	2	32	no	—
97	C₃×SD₁₆	—	2	64	no	—
98	C₃×D₈	—	2	32	no	—
99	C₂×C₃×C₈	C₂×D₄	1	24	yes	—
100	C₂×C₃×C₈	C₂×D₄	1	24	yes	—
101	C₂×C₃×C₈	C₂⁴	1	4	yes	—
102	C₂×C₃×C₈	C₂⁴	1	4	yes	—
103	C₃×C₂²⋊C₄	—	2	64	no	—
104	C₃×C₄²	C₂×D₄	1	12	yes	—
105	C₃×C₄²	C₂⁴	1	4	yes	—
106	C₂⁵	—	2	2	no	—
107	C₂²×Q₈	—	2	6	no	—
108	C₂²×Q₈	—	2	8	no	—
109	C₂²×Q₈	—	2	8	no	—
110	C₂²×Q₈	C₂³×C₃	1	12	yes	—
111	C₂²×D₄	—	2	8	no	—
112	C₂²×D₄	—	2	8	no	—
113	C₂²×D₄	—	2	6	no	—
114	C₂²×D₄	C₂³×C₃	1	12	yes	—
115	C₂³×C₄	—	2	2	no	—
116	C₂³×C₄	C₃×D₄	1	2	yes	—
117	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂×Q₁₆	—	2	48	no	—
119	C₂×Q₁₆	C₂³×C₃	1	16	yes	complemented
120	C₂×SD₁₆	—	2	96	no	—
121	C₂×SD₁₆	C₂³×C₃	1	32	yes	complemented
122	C₂×D₈	—	2	48	no	—
123	C₂×D₈	C₂³×C₃	1	16	yes	—
124	C₂²×C₈	C₃×D₄	1	8	yes	—
125	C₂²×C₈	C₃×D₄	1	8	yes	—
126	C₂²×C₈	C₂³×C₃	1	6	yes	—
127	C₂²×C₈	C₂³×C₃	1	6	yes	—
128	#124	—	2	48	no	—
129	#124	C₂³×C₃	1	32	yes	complemented
130	C₂×C₂²⋊C₄	—	2	48	no	—
131	C₂×C₄²	C₃×D₄	1	6	yes	—
132	C₂×C₄²	C₂³×C₃	1	6	yes	—
133	C₄×C₈	C₂³×C₃	1	16	yes	complemented
134	C₂³×C₃	—	2	8	no	—
135	C₂³×C₃	—	4	24	no	—
136	C₂³×C₃	—	4	48	no	—
137	C₂³×C₃	—	8	32	no	—
138	C₂³×C₃	#127	1	4	yes	—
139	C₂³×C₃	#135	1	8	yes	—
140	C₂³×C₃	C₂²×D₄	1	3	yes	—
141	C₃×Q₈	—	2	8	no	—
142	C₃×Q₈	—	2	8	no	—
143	C₃×Q₈	—	4	24	no	—
144	C₃×D₄	—	2	8	no	—
145	C₃×D₄	—	2	8	no	—
146	C₃×D₄	—	4	24	no	—
147	C₂×C₃×C₄	—	2	24	no	—
148	C₂×C₃×C₄	—	2	8	no	—
149	C₂×C₃×C₄	—	4	24	no	—
150	C₂×C₃×C₄	#127	1	12	yes	—
151	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
152	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
153	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
154	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
155	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
156	C₃×C₈	C₂²×D₄	1	8	yes	—
157	C₃×C₈	C₂²×D₄	1	8	yes	—
158	C₂⁴	—	2	12	no	—
159	C₂⁴	—	4	8	no	—
160	C₂⁴	—	4	16	no	—
161	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂×Q₈	—	2	12	no	—
163	C₂×Q₈	—	2	24	no	—
164	C₂×Q₈	—	2	24	no	—
165	C₂×Q₈	—	4	24	no	—
166	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
167	C₂×D₄	—	2	24	no	—
168	C₂×D₄	—	2	24	no	—
169	C₂×D₄	—	2	12	no	—
170	C₂×D₄	—	4	24	no	—
171	C₂×D₄	C₂⁴×C₃	1	8	yes	—
172	C₂²×C₄	—	2	8	no	—
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	—	4	8	no	—
175	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
176	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
177	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
178	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
179	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
180	Q₁₆	—	2	32	no	—
181	SD₁₆	—	2	64	no	—
182	D₈	—	2	32	no	—
183	C₂×C₈	C₂×C₃×D₄	1	24	yes	—
184	C₂×C₈	C₂×C₃×D₄	1	24	yes	—
185	C₂×C₈	C₂⁴×C₃	1	4	yes	—
186	C₂×C₈	C₂⁴×C₃	1	4	yes	—
187	C₂²⋊C₄	—	2	64	no	—
188	C₄²	C₂×C₃×D₄	1	12	yes	—
189	C₄²	C₂⁴×C₃	1	4	yes	—
190	C₂²×C₃	—	4	8	no	—
191	C₂²×C₃	—	4	16	no	—
192	C₂²×C₃	—	8	48	no	—
193	C₂²×C₃	#495	1	16	yes	direct product factor
194	C₂²×C₃	C₂×#127	1	6	yes	—
195	C₂²×C₃	C₂×#135	1	12	yes	—
196	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
197	C₃×C₄	—	2	8	no	—
198	C₃×C₄	—	4	8	no	—
199	C₃×C₄	C₂×#127	1	6	yes	—
200	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
201	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
202	C₂³	—	2	8	no	—
203	C₂³	—	4	48	no	—
204	C₂³	—	4	24	no	—
205	C₂³	—	8	32	no	—
206	C₂³	C₃×#127	1	4	yes	—
207	C₂³	C₃×#135	1	8	yes	—
208	C₂³	C₂²×C₃×D₄	1	3	yes	—
209	Q₈	—	2	8	no	—
210	Q₈	—	2	8	no	—
211	Q₈	—	4	24	no	—
212	D₄	—	2	8	no	—
213	D₄	—	2	8	no	—
214	D₄	—	4	24	no	—
215	C₂×C₄	—	2	24	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	4	24	no	—
218	C₂×C₄	C₃×#127	1	12	yes	—
219	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
220	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
221	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
222	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
223	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
224	C₈	C₂²×C₃×D₄	1	8	yes	—
225	C₈	C₂²×C₃×D₄	1	8	yes	—
226	C₂×C₃	—	8	8	no	—
227	C₂×C₃	C₂×#495	1	12	yes	direct product factor
228	C₂×C₃	C₂²×#127	1	1	yes	characteristic
229	C₂×C₃	C₂²×#135	1	2	yes	—
230	C₂²	—	4	8	no	—
231	C₂²	—	4	16	no	—
232	C₂²	—	8	48	no	—
233	C₂²	C₃×#495	1	16	yes	direct product factor
234	C₂²	C₂×C₃×#127	1	6	yes	—
235	C₂²	C₂×C₃×#135	1	12	yes	—
236	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
237	C₄	—	2	8	no	—
238	C₄	—	4	8	no	—
239	C₄	C₂×C₃×#127	1	6	yes	—
240	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
241	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
242	C₃	C₂²×#495	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	8	8	no	—
244	C₂	C₂×C₃×#495	1	12	yes	direct product factor
245	C₂	C₂²×C₃×#127	1	1	yes	characteristic
246	C₂	C₂²×C₃×#135	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂³×C₃×D₈	C₂
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
2. C₂³×C₃×SD₁₆	C₂	semidirect
4. C₂²×C₃×#124	C₂	semidirect
5. C₂×C₃×#495	C₂	direct (N × H)
6. C₂²×C₃×C₄×C₈	C₂	semidirect
7. C₂²×#495	C₃	direct (N × H)
11. C₂²×C₃×Q₁₆	C₂²	semidirect
13. C₂²×C₃×SD₁₆	C₂²	semidirect
19. C₂×C₃×#124	C₂²	semidirect
22. C₃×#495	C₂²	direct (N × H)
23. C₂×C₃×C₄×C₈	C₂²	semidirect
24. C₂³×Q₁₆	C₂×C₃	semidirect
25. C₂³×SD₁₆	C₂×C₃	semidirect
27. C₂²×#124	C₂×C₃	semidirect
28. C₂×#495	C₂×C₃	direct (N × H)
29. C₂²×C₄×C₈	C₂×C₃	semidirect
43. C₂×C₃×Q₁₆	C₂³	semidirect
45. C₂×C₃×SD₁₆	C₂³	semidirect
53. C₃×#124	C₂³	semidirect
57. C₃×C₄×C₈	C₂³	semidirect
61. C₂²×Q₁₆	C₂²×C₃	semidirect
63. C₂²×SD₁₆	C₂²×C₃	semidirect
69. C₂×#124	C₂²×C₃	semidirect
72. #495	C₂²×C₃	direct (N × H)
73. C₂×C₄×C₈	C₂²×C₃	semidirect
119. C₂×Q₁₆	C₂³×C₃	semidirect
121. C₂×SD₁₆	C₂³×C₃	semidirect
129. #124	C₂³×C₃	semidirect
133. C₄×C₈	C₂³×C₃	semidirect
193. C₂²×C₃	#495	direct (N × H)
227. C₂×C₃	C₂×#495	direct (N × H)
233. C₂²	C₃×#495	direct (N × H)
242. C₃	C₂²×#495	direct (N × H)
244. C₂	C₂×C₃×#495	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
C₂³×C₃×D₈	2
C₂²×C₃×#124	2
C₂×C₃×#495	2
C₂²×C₃×C₄×C₈	2
C₂²×#495	3

Subgroups — C22×C3×#495 #10679238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#495 #10679238