Subgroups — C₂²×C₃×#490 #10679214

Order 768.

2830 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#490	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#102	C₂	1	4	yes	maximal, complemented
5	C₂²×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#490	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂³×C₃×D₄	—	2	2	no	—
8	C₂³×C₃×D₄	C₂²	1	2	yes	—
9	C₂³×C₃×C₈	C₂²	1	2	yes	—
10	C₂×C₃×#127	—	2	6	no	—
11	C₂×C₃×#127	C₂²	1	12	yes	—
12	C₂×C₃×#125	—	2	6	no	—
13	C₂×C₃×#125	C₂²	1	12	yes	complemented
14	C₂²×C₃×C₄⋊C₄	—	2	2	no	—
15	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₃×#490	C₂²	1	64	yes	direct product factor
18	C₂×C₃×#102	C₂²	1	48	yes	complemented
19	C₂×C₃×#97	C₂²	1	12	yes	complemented
20	C₂²×#127	C₂×C₃	1	1	yes	characteristic
21	C₂²×#125	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×#490	C₂×C₃	1	24	yes	direct product factor
23	C₂²×#102	C₂×C₃	1	4	yes	complemented
24	C₂²×#97	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂⁵×C₃	—	2	2	no	—
26	C₂²×C₃×D₄	—	2	24	no	—
27	C₂²×C₃×D₄	—	2	6	no	—
28	C₂²×C₃×D₄	—	2	16	no	—
29	C₂²×C₃×D₄	—	4	16	no	—
30	C₂²×C₃×D₄	C₂³	1	12	yes	—
31	C₂³×C₃×C₄	—	2	2	no	—
32	C₂³×C₃×C₄	D₄	1	2	yes	—
33	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₈	—	2	8	no	—
35	C₂²×C₃×C₈	C₂³	1	12	yes	—
36	C₃×#127	—	2	24	no	—
37	C₃×#127	C₂³	1	16	yes	—
38	C₃×#125	—	2	24	no	—
39	C₃×#125	C₂³	1	16	yes	complemented
40	C₂×C₃×C₄⋊C₄	—	2	24	no	—
41	C₂×C₃×C₄⋊C₄	—	2	6	no	—
42	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
43	C₂×C₃×C₄²	D₄	1	6	yes	—
44	C₂×C₃×C₄²	C₂³	1	6	yes	—
45	C₃×#102	C₂³	1	64	yes	complemented
46	C₃×#97	C₂³	1	16	yes	complemented
47	C₂³×D₄	—	2	2	no	—
48	C₂³×D₄	C₂²×C₃	1	2	yes	—
49	C₂³×C₈	C₂²×C₃	1	2	yes	—
50	C₂×#127	—	2	6	no	—
51	C₂×#127	C₂²×C₃	1	12	yes	—
52	C₂×#125	—	2	6	no	—
53	C₂×#125	C₂²×C₃	1	12	yes	complemented
54	C₂²×C₄⋊C₄	—	2	2	no	—
55	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
57	#490	C₂²×C₃	1	64	yes	direct product factor
58	C₂×#102	C₂²×C₃	1	48	yes	complemented
59	C₂×#97	C₂²×C₃	1	12	yes	complemented
60	C₂⁴×C₃	—	2	12	no	—
61	C₂⁴×C₃	—	4	8	no	—
62	C₂⁴×C₃	—	4	16	no	—
63	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×D₄	—	2	12	no	—
65	C₂×C₃×D₄	—	2	32	no	—
66	C₂×C₃×D₄	—	2	48	no	—
67	C₂×C₃×D₄	—	4	96	no	—
68	C₂×C₃×D₄	—	4	24	no	—
69	C₂×C₃×D₄	C₂⁴	1	8	yes	—
70	C₂²×C₃×C₄	—	2	8	no	—
71	C₂²×C₃×C₄	—	2	12	no	—
72	C₂²×C₃×C₄	—	4	8	no	—
73	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
74	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
75	C₂²×C₃×C₄	#41	1	8	yes	—
76	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
77	C₂×C₃×C₈	—	2	24	no	—
78	C₂×C₃×C₈	C₂⁴	1	8	yes	—
79	C₃×C₄⋊C₄	—	2	32	no	—
80	C₃×C₄⋊C₄	—	2	12	no	—
81	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
82	C₃×C₄²	C₂×D₄	1	12	yes	—
83	C₃×C₄²	C₂⁴	1	4	yes	—
84	C₂⁵	—	2	2	no	—
85	C₂²×D₄	—	2	6	no	—
86	C₂²×D₄	—	2	24	no	—
87	C₂²×D₄	—	2	16	no	—
88	C₂²×D₄	—	4	16	no	—
89	C₂²×D₄	C₂³×C₃	1	12	yes	—
90	C₂³×C₄	—	2	2	no	—
91	C₂³×C₄	C₃×D₄	1	2	yes	—
92	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
93	C₂²×C₈	—	2	8	no	—
94	C₂²×C₈	C₂³×C₃	1	12	yes	—
95	#127	—	2	24	no	—
96	#127	C₂³×C₃	1	16	yes	—
97	#125	—	2	24	no	—
98	#125	C₂³×C₃	1	16	yes	complemented
99	C₂×C₄⋊C₄	—	2	24	no	—
100	C₂×C₄⋊C₄	—	2	6	no	—
101	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
102	C₂×C₄²	C₃×D₄	1	6	yes	—
103	C₂×C₄²	C₂³×C₃	1	6	yes	—
104	#102	C₂³×C₃	1	64	yes	complemented
105	#97	C₂³×C₃	1	16	yes	complemented
106	C₂³×C₃	—	2	8	no	—
107	C₂³×C₃	—	4	24	no	—
108	C₂³×C₃	—	4	48	no	—
109	C₂³×C₃	—	8	32	no	—
110	C₂³×C₃	#124	1	4	yes	—
111	C₂³×C₃	AGL(1,ℤ/8)	1	8	yes	—
112	C₂³×C₃	C₂²×D₄	1	3	yes	—
113	C₃×D₄	—	2	16	no	—
114	C₃×D₄	—	4	64	no	—
115	C₃×D₄	—	4	24	no	—
116	C₂×C₃×C₄	—	2	24	no	—
117	C₂×C₃×C₄	—	2	8	no	—
118	C₂×C₃×C₄	—	4	24	no	—
119	C₂×C₃×C₄	#124	1	12	yes	—
120	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
121	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
122	C₂×C₃×C₄	C₂×#41	1	12	yes	—
123	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
124	C₃×C₈	—	2	8	no	—
125	C₂⁴	—	2	12	no	—
126	C₂⁴	—	4	8	no	—
127	C₂⁴	—	4	16	no	—
128	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
129	C₂×D₄	—	2	12	no	—
130	C₂×D₄	—	2	32	no	—
131	C₂×D₄	—	2	48	no	—
132	C₂×D₄	—	4	96	no	—
133	C₂×D₄	—	4	24	no	—
134	C₂×D₄	C₂⁴×C₃	1	8	yes	—
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	12	no	—
137	C₂²×C₄	—	4	8	no	—
138	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
139	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
140	C₂²×C₄	C₃×#41	1	8	yes	—
141	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
142	C₂×C₈	—	2	24	no	—
143	C₂×C₈	C₂⁴×C₃	1	8	yes	—
144	C₄⋊C₄	—	2	32	no	—
145	C₄⋊C₄	—	2	12	no	—
146	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
147	C₄²	C₂×C₃×D₄	1	12	yes	—
148	C₄²	C₂⁴×C₃	1	4	yes	—
149	C₂²×C₃	—	4	8	no	—
150	C₂²×C₃	—	4	16	no	—
151	C₂²×C₃	—	8	48	no	—
152	C₂²×C₃	#490	1	16	yes	direct product factor
153	C₂²×C₃	C₂×#124	1	6	yes	—
154	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
155	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
156	C₃×C₄	—	2	8	no	—
157	C₃×C₄	—	4	8	no	—
158	C₃×C₄	C₂×#124	1	6	yes	—
159	C₃×C₄	C₂²×#41	1	2	yes	—
160	C₂³	—	2	8	no	—
161	C₂³	—	4	48	no	—
162	C₂³	—	4	24	no	—
163	C₂³	—	8	32	no	—
164	C₂³	C₃×#124	1	4	yes	—
165	C₂³	C₃×AGL(1,ℤ/8)	1	8	yes	—
166	C₂³	C₂²×C₃×D₄	1	3	yes	—
167	D₄	—	2	16	no	—
168	D₄	—	4	64	no	—
169	D₄	—	4	24	no	—
170	C₂×C₄	—	2	24	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	4	24	no	—
173	C₂×C₄	C₃×#124	1	12	yes	—
174	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
175	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
176	C₂×C₄	C₂×C₃×#41	1	12	yes	—
177	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
178	C₈	—	2	8	no	—
179	C₂×C₃	—	8	8	no	—
180	C₂×C₃	C₂×#490	1	12	yes	direct product factor
181	C₂×C₃	C₂²×#124	1	1	yes	characteristic
182	C₂×C₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
183	C₂²	—	4	8	no	—
184	C₂²	—	4	16	no	—
185	C₂²	—	8	48	no	—
186	C₂²	C₃×#490	1	16	yes	direct product factor
187	C₂²	C₂×C₃×#124	1	6	yes	—
188	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	12	yes	—
189	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	—	4	8	no	—
192	C₄	C₂×C₃×#124	1	6	yes	—
193	C₄	C₂²×C₃×#41	1	2	yes	—
194	C₃	C₂²×#490	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	8	8	no	—
196	C₂	C₂×C₃×#490	1	12	yes	direct product factor
197	C₂	C₂²×C₃×#124	1	1	yes	characteristic
198	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
prime radical	C₂²×C₃×#127	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
2. C₂²×C₃×#125	C₂	semidirect
3. C₂×C₃×#490	C₂	direct (N × H)
4. C₂²×C₃×#102	C₂	semidirect
5. C₂²×C₃×#97	C₂	semidirect
6. C₂²×#490	C₃	direct (N × H)
13. C₂×C₃×#125	C₂²	semidirect
17. C₃×#490	C₂²	direct (N × H)
18. C₂×C₃×#102	C₂²	semidirect
19. C₂×C₃×#97	C₂²	semidirect
21. C₂²×#125	C₂×C₃	semidirect
22. C₂×#490	C₂×C₃	direct (N × H)
23. C₂²×#102	C₂×C₃	semidirect
24. C₂²×#97	C₂×C₃	semidirect
39. C₃×#125	C₂³	semidirect
45. C₃×#102	C₂³	semidirect
46. C₃×#97	C₂³	semidirect
53. C₂×#125	C₂²×C₃	semidirect
57. #490	C₂²×C₃	direct (N × H)
58. C₂×#102	C₂²×C₃	semidirect
59. C₂×#97	C₂²×C₃	semidirect
98. #125	C₂³×C₃	semidirect
104. #102	C₂³×C₃	semidirect
105. #97	C₂³×C₃	semidirect
152. C₂²×C₃	#490	direct (N × H)
180. C₂×C₃	C₂×#490	direct (N × H)
186. C₂²	C₃×#490	direct (N × H)
194. C₃	C₂²×#490	direct (N × H)
196. C₂	C₂×C₃×#490	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×#127	2
C₂²×C₃×#125	2
C₂×C₃×#490	2
C₂²×C₃×#102	2
C₂²×C₃×#97	2
C₂²×#490	3

Subgroups — C22×C3×#490 #10679214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#490 #10679214