Subgroups — C₂²×C₃×#493 #10679236

Order 768.

4638 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₈	C₂	1	4	yes	maximal, complemented
2	C₂²×C₃×#127	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#493	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#493	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂³×C₃×D₄	—	2	4	no	—
7	C₂³×C₃×D₄	C₂²	1	4	yes	—
8	C₂²×C₃×D₈	—	2	32	no	—
9	C₂²×C₃×D₈	C₂²	1	48	yes	complemented
10	C₂³×C₃×C₈	C₂²	1	2	yes	—
11	C₂×C₃×#127	—	2	12	no	—
12	C₂×C₃×#127	C₂²	1	24	yes	complemented
13	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₃×#493	C₂²	1	64	yes	direct product factor
15	C₂×C₃×C₄×C₈	C₂²	1	12	yes	complemented
16	C₂³×D₈	C₂×C₃	1	4	yes	complemented
17	C₂²×#127	C₂×C₃	1	2	yes	complemented
18	C₂×#493	C₂×C₃	1	24	yes	direct product factor
19	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂⁵×C₃	—	2	4	no	—
21	C₂²×C₃×D₄	—	2	32	no	—
22	C₂²×C₃×D₄	—	2	48	no	—
23	C₂²×C₃×D₄	—	2	12	no	—
24	C₂²×C₃×D₄	—	4	32	no	—
25	C₂²×C₃×D₄	C₂³	1	24	yes	—
26	C₂³×C₃×C₄	D₄	1	2	yes	—
27	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂×C₃×D₈	—	2	192	no	—
29	C₂×C₃×D₈	C₂³	1	64	yes	complemented
30	C₂²×C₃×C₈	D₄	1	16	yes	complemented
31	C₂²×C₃×C₈	C₂³	1	12	yes	—
32	C₃×#127	—	2	48	no	—
33	C₃×#127	C₂³	1	32	yes	complemented
34	C₂×C₃×C₄²	D₄	1	6	yes	—
35	C₂×C₃×C₄²	C₂³	1	6	yes	—
36	C₃×C₄×C₈	C₂³	1	16	yes	complemented
37	C₂³×D₄	—	2	4	no	—
38	C₂³×D₄	C₂²×C₃	1	4	yes	—
39	C₂²×D₈	—	2	32	no	—
40	C₂²×D₈	C₂²×C₃	1	48	yes	complemented
41	C₂³×C₈	C₂²×C₃	1	2	yes	—
42	C₂×#127	—	2	12	no	—
43	C₂×#127	C₂²×C₃	1	24	yes	complemented
44	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
45	#493	C₂²×C₃	1	64	yes	direct product factor
46	C₂×C₄×C₈	C₂²×C₃	1	12	yes	complemented
47	C₂⁴×C₃	—	2	24	no	—
48	C₂⁴×C₃	—	4	16	no	—
49	C₂⁴×C₃	—	4	32	no	—
50	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×D₄	—	2	96	no	—
52	C₂×C₃×D₄	—	2	24	no	—
53	C₂×C₃×D₄	—	2	64	no	—
54	C₂×C₃×D₄	—	4	48	no	—
55	C₂×C₃×D₄	—	4	192	no	—
56	C₂×C₃×D₄	C₂⁴	1	16	yes	—
57	C₂²×C₃×C₄	D₈	1	16	yes	complemented
58	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
59	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
60	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
61	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
62	C₃×D₈	—	2	128	no	—
63	C₂×C₃×C₈	C₂×D₄	1	48	yes	complemented
64	C₂×C₃×C₈	C₂⁴	1	8	yes	—
65	C₃×C₄²	C₂×D₄	1	12	yes	—
66	C₃×C₄²	C₂⁴	1	4	yes	—
67	C₂⁵	—	2	4	no	—
68	C₂²×D₄	—	2	32	no	—
69	C₂²×D₄	—	2	12	no	—
70	C₂²×D₄	—	2	48	no	—
71	C₂²×D₄	—	4	32	no	—
72	C₂²×D₄	C₂³×C₃	1	24	yes	—
73	C₂³×C₄	C₃×D₄	1	2	yes	—
74	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
75	C₂×D₈	—	2	192	no	—
76	C₂×D₈	C₂³×C₃	1	64	yes	complemented
77	C₂²×C₈	C₃×D₄	1	16	yes	complemented
78	C₂²×C₈	C₂³×C₃	1	12	yes	—
79	#127	—	2	48	no	—
80	#127	C₂³×C₃	1	32	yes	complemented
81	C₂×C₄²	C₃×D₄	1	6	yes	—
82	C₂×C₄²	C₂³×C₃	1	6	yes	—
83	C₄×C₈	C₂³×C₃	1	16	yes	complemented
84	C₂³×C₃	—	2	16	no	—
85	C₂³×C₃	—	4	48	no	—
86	C₂³×C₃	—	4	96	no	—
87	C₂³×C₃	—	8	64	no	—
88	C₂³×C₃	#127	1	4	yes	—
89	C₂³×C₃	C₂×D₈	1	8	yes	—
90	C₂³×C₃	C₂²×D₄	1	3	yes	—
91	C₃×D₄	—	2	32	no	—
92	C₃×D₄	—	4	48	no	—
93	C₃×D₄	—	4	128	no	—
94	C₂×C₃×C₄	#127	1	12	yes	—
95	C₂×C₃×C₄	C₂×D₈	1	48	yes	complemented
96	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
97	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
98	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
99	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
100	C₃×C₈	C₂²×D₄	1	16	yes	complemented
101	C₂⁴	—	2	24	no	—
102	C₂⁴	—	4	16	no	—
103	C₂⁴	—	4	32	no	—
104	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
105	C₂×D₄	—	2	96	no	—
106	C₂×D₄	—	2	24	no	—
107	C₂×D₄	—	2	64	no	—
108	C₂×D₄	—	4	48	no	—
109	C₂×D₄	—	4	192	no	—
110	C₂×D₄	C₂⁴×C₃	1	16	yes	—
111	C₂²×C₄	C₃×D₈	1	16	yes	complemented
112	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
113	C₂²×C₄	C₂×C₃×D₄	1	8	yes	—
114	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
115	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
116	D₈	—	2	128	no	—
117	C₂×C₈	C₂×C₃×D₄	1	48	yes	complemented
118	C₂×C₈	C₂⁴×C₃	1	8	yes	—
119	C₄²	C₂×C₃×D₄	1	12	yes	—
120	C₄²	C₂⁴×C₃	1	4	yes	—
121	C₂²×C₃	—	4	16	no	—
122	C₂²×C₃	—	4	32	no	—
123	C₂²×C₃	—	8	96	no	—
124	C₂²×C₃	#493	1	16	yes	direct product factor
125	C₂²×C₃	C₂×#127	1	6	yes	—
126	C₂²×C₃	C₂²×D₈	1	12	yes	—
127	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
128	C₃×C₄	C₂×#127	1	6	yes	—
129	C₃×C₄	C₂²×D₈	1	16	yes	complemented
130	C₃×C₄	C₂³×D₄	1	2	yes	—
131	C₂³	—	2	16	no	—
132	C₂³	—	4	96	no	—
133	C₂³	—	4	48	no	—
134	C₂³	—	8	64	no	—
135	C₂³	C₃×#127	1	4	yes	—
136	C₂³	C₂×C₃×D₈	1	8	yes	—
137	C₂³	C₂²×C₃×D₄	1	3	yes	—
138	D₄	—	2	32	no	—
139	D₄	—	4	128	no	—
140	D₄	—	4	48	no	—
141	C₂×C₄	C₃×#127	1	12	yes	—
142	C₂×C₄	C₂×C₃×D₈	1	48	yes	complemented
143	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
144	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
145	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
146	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
147	C₈	C₂²×C₃×D₄	1	16	yes	complemented
148	C₂×C₃	—	8	16	no	—
149	C₂×C₃	C₂×#493	1	12	yes	direct product factor
150	C₂×C₃	C₂²×#127	1	1	yes	characteristic
151	C₂×C₃	C₂³×D₈	1	2	yes	—
152	C₂²	—	4	16	no	—
153	C₂²	—	4	32	no	—
154	C₂²	—	8	96	no	—
155	C₂²	C₃×#493	1	16	yes	direct product factor
156	C₂²	C₂×C₃×#127	1	6	yes	—
157	C₂²	C₂²×C₃×D₈	1	12	yes	—
158	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
159	C₄	C₂×C₃×#127	1	6	yes	—
160	C₄	C₂²×C₃×D₈	1	16	yes	complemented
161	C₄	C₂³×C₃×D₄	1	2	yes	—
162	C₃	C₂²×#493	1	1	yes	characteristic, direct product factor, Sylow
163	C₂	—	8	16	no	—
164	C₂	C₂×C₃×#493	1	12	yes	direct product factor
165	C₂	C₂²×C₃×#127	1	1	yes	characteristic
166	C₂	C₂³×C₃×D₈	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×D₈	C₂	semidirect
2. C₂²×C₃×#127	C₂	semidirect
3. C₂×C₃×#493	C₂	direct (N × H)
4. C₂²×C₃×C₄×C₈	C₂	semidirect
5. C₂²×#493	C₃	direct (N × H)
9. C₂²×C₃×D₈	C₂²	semidirect
12. C₂×C₃×#127	C₂²	semidirect
14. C₃×#493	C₂²	direct (N × H)
15. C₂×C₃×C₄×C₈	C₂²	semidirect
16. C₂³×D₈	C₂×C₃	semidirect
17. C₂²×#127	C₂×C₃	semidirect
18. C₂×#493	C₂×C₃	direct (N × H)
19. C₂²×C₄×C₈	C₂×C₃	semidirect
29. C₂×C₃×D₈	C₂³	semidirect
30. C₂²×C₃×C₈	D₄	semidirect
33. C₃×#127	C₂³	semidirect
36. C₃×C₄×C₈	C₂³	semidirect
40. C₂²×D₈	C₂²×C₃	semidirect
43. C₂×#127	C₂²×C₃	semidirect
45. #493	C₂²×C₃	direct (N × H)
46. C₂×C₄×C₈	C₂²×C₃	semidirect
57. C₂²×C₃×C₄	D₈	semidirect
63. C₂×C₃×C₈	C₂×D₄	semidirect
76. C₂×D₈	C₂³×C₃	semidirect
77. C₂²×C₈	C₃×D₄	semidirect
80. #127	C₂³×C₃	semidirect
83. C₄×C₈	C₂³×C₃	semidirect
95. C₂×C₃×C₄	C₂×D₈	semidirect
100. C₃×C₈	C₂²×D₄	semidirect
111. C₂²×C₄	C₃×D₈	semidirect
117. C₂×C₈	C₂×C₃×D₄	semidirect
124. C₂²×C₃	#493	direct (N × H)
129. C₃×C₄	C₂²×D₈	semidirect
142. C₂×C₄	C₂×C₃×D₈	semidirect
147. C₈	C₂²×C₃×D₄	semidirect
149. C₂×C₃	C₂×#493	direct (N × H)
155. C₂²	C₃×#493	direct (N × H)
160. C₄	C₂²×C₃×D₈	semidirect
162. C₃	C₂²×#493	direct (N × H)
164. C₂	C₂×C₃×#493	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×D₈	2
C₂²×C₃×#127	2
C₂×C₃×#493	2
C₂²×C₃×C₄×C₈	2
C₂²×#493	3

Subgroups — C22×C3×#493 #10679236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#493 #10679236