Subgroups — C₂²×C₃×#494 #10679237

Order 768.

3262 conjugacy classes of subgroups.

The subgroup lattice has 150 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	4	yes	maximal
2	C₂²×C₃×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₂×C₃×#494	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×#494	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂³×C₃×Q₈	C₂²	1	4	yes	—
7	C₂²×C₃×Q₁₆	—	2	32	no	—
8	C₂²×C₃×Q₁₆	C₂²	1	48	yes	—
9	C₂³×C₃×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Q₈	—	2	12	no	—
11	C₂×C₃×C₄⋊Q₈	C₂²	1	24	yes	—
12	C₂²×C₃×C₄⋊C₄	—	2	4	no	—
13	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₃×#494	C₂²	1	64	yes	direct product factor
15	C₂×C₃×C₄×C₈	C₂²	1	12	yes	—
16	C₂³×Q₁₆	C₂×C₃	1	4	yes	—
17	C₂²×C₄⋊Q₈	C₂×C₃	1	2	yes	—
18	C₂×#494	C₂×C₃	1	24	yes	direct product factor
19	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₃×Q₈	—	2	12	no	—
21	C₂²×C₃×Q₈	—	2	32	no	—
22	C₂²×C₃×Q₈	C₂³	1	24	yes	—
23	C₂³×C₃×C₄	—	2	4	no	—
24	C₂³×C₃×C₄	D₄	1	2	yes	—
25	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×C₃×Q₁₆	—	2	192	no	—
27	C₂×C₃×Q₁₆	C₂³	1	64	yes	—
28	C₂²×C₃×C₈	D₄	1	16	yes	—
29	C₂²×C₃×C₈	C₂³	1	12	yes	—
30	C₃×C₄⋊Q₈	—	2	48	no	—
31	C₃×C₄⋊Q₈	C₂³	1	32	yes	—
32	C₂×C₃×C₄⋊C₄	—	2	48	no	—
33	C₂×C₃×C₄²	D₄	1	6	yes	—
34	C₂×C₃×C₄²	C₂³	1	6	yes	—
35	C₃×C₄×C₈	C₂³	1	16	yes	—
36	C₂³×Q₈	C₂²×C₃	1	4	yes	—
37	C₂²×Q₁₆	—	2	32	no	—
38	C₂²×Q₁₆	C₂²×C₃	1	48	yes	—
39	C₂³×C₈	C₂²×C₃	1	2	yes	—
40	C₂×C₄⋊Q₈	—	2	12	no	—
41	C₂×C₄⋊Q₈	C₂²×C₃	1	24	yes	—
42	C₂²×C₄⋊C₄	—	2	4	no	—
43	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
44	#494	C₂²×C₃	1	64	yes	direct product factor
45	C₂×C₄×C₈	C₂²×C₃	1	12	yes	—
46	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×Q₈	—	2	24	no	—
48	C₂×C₃×Q₈	—	2	96	no	—
49	C₂×C₃×Q₈	—	4	48	no	—
50	C₂×C₃×Q₈	C₂⁴	1	16	yes	—
51	C₂²×C₃×C₄	—	2	24	no	—
52	C₂²×C₃×C₄	—	4	16	no	—
53	C₂²×C₃×C₄	Q₁₆	1	16	yes	complemented
54	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
55	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
56	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
57	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
58	C₃×Q₁₆	—	2	128	no	—
59	C₂×C₃×C₈	C₂×D₄	1	48	yes	—
60	C₂×C₃×C₈	C₂⁴	1	8	yes	—
61	C₃×C₄⋊C₄	—	2	64	no	—
62	C₃×C₄²	C₂×D₄	1	12	yes	—
63	C₃×C₄²	C₂⁴	1	4	yes	—
64	C₂²×Q₈	—	2	12	no	—
65	C₂²×Q₈	—	2	32	no	—
66	C₂²×Q₈	C₂³×C₃	1	24	yes	—
67	C₂³×C₄	—	2	4	no	—
68	C₂³×C₄	C₃×D₄	1	2	yes	—
69	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
70	C₂×Q₁₆	—	2	192	no	—
71	C₂×Q₁₆	C₂³×C₃	1	64	yes	—
72	C₂²×C₈	C₃×D₄	1	16	yes	—
73	C₂²×C₈	C₂³×C₃	1	12	yes	—
74	C₄⋊Q₈	—	2	48	no	—
75	C₄⋊Q₈	C₂³×C₃	1	32	yes	—
76	C₂×C₄⋊C₄	—	2	48	no	—
77	C₂×C₄²	C₃×D₄	1	6	yes	—
78	C₂×C₄²	C₂³×C₃	1	6	yes	—
79	C₄×C₈	C₂³×C₃	1	16	yes	—
80	C₂³×C₃	#127	1	4	yes	—
81	C₂³×C₃	C₂×Q₁₆	1	8	yes	—
82	C₂³×C₃	C₂²×D₄	1	3	yes	—
83	C₃×Q₈	—	2	32	no	—
84	C₃×Q₈	—	4	48	no	—
85	C₂×C₃×C₄	—	2	16	no	—
86	C₂×C₃×C₄	—	4	48	no	—
87	C₂×C₃×C₄	#127	1	12	yes	—
88	C₂×C₃×C₄	C₂×Q₁₆	1	48	yes	complemented
89	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
90	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
91	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
92	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
93	C₃×C₈	C₂²×D₄	1	16	yes	—
94	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
95	C₂×Q₈	—	2	24	no	—
96	C₂×Q₈	—	2	96	no	—
97	C₂×Q₈	—	4	48	no	—
98	C₂×Q₈	C₂⁴×C₃	1	16	yes	—
99	C₂²×C₄	—	2	24	no	—
100	C₂²×C₄	—	4	16	no	—
101	C₂²×C₄	C₃×Q₁₆	1	16	yes	complemented
102	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
103	C₂²×C₄	C₂×C₃×D₄	1	8	yes	—
104	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
105	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
106	Q₁₆	—	2	128	no	—
107	C₂×C₈	C₂×C₃×D₄	1	48	yes	—
108	C₂×C₈	C₂⁴×C₃	1	8	yes	—
109	C₄⋊C₄	—	2	64	no	—
110	C₄²	C₂×C₃×D₄	1	12	yes	—
111	C₄²	C₂⁴×C₃	1	4	yes	—
112	C₂²×C₃	#494	1	16	yes	direct product factor
113	C₂²×C₃	C₂×#127	1	6	yes	—
114	C₂²×C₃	C₂²×Q₁₆	1	12	yes	—
115	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
116	C₃×C₄	—	4	16	no	—
117	C₃×C₄	C₂×#127	1	6	yes	—
118	C₃×C₄	C₂²×Q₁₆	1	16	yes	complemented
119	C₃×C₄	C₂³×D₄	1	2	yes	—
120	C₂³	C₃×#127	1	4	yes	—
121	C₂³	C₂×C₃×Q₁₆	1	8	yes	—
122	C₂³	C₂²×C₃×D₄	1	3	yes	—
123	Q₈	—	2	32	no	—
124	Q₈	—	4	48	no	—
125	C₂×C₄	—	2	16	no	—
126	C₂×C₄	—	4	48	no	—
127	C₂×C₄	C₃×#127	1	12	yes	—
128	C₂×C₄	C₂×C₃×Q₁₆	1	48	yes	complemented
129	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
130	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
131	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
132	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
133	C₈	C₂²×C₃×D₄	1	16	yes	—
134	C₂×C₃	C₂×#494	1	12	yes	direct product factor
135	C₂×C₃	C₂²×#127	1	1	yes	characteristic
136	C₂×C₃	C₂³×Q₁₆	1	2	yes	—
137	C₂²	C₃×#494	1	16	yes	direct product factor
138	C₂²	C₂×C₃×#127	1	6	yes	—
139	C₂²	C₂²×C₃×Q₁₆	1	12	yes	—
140	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
141	C₄	—	4	16	no	—
142	C₄	C₂×C₃×#127	1	6	yes	—
143	C₄	C₂²×C₃×Q₁₆	1	16	yes	complemented
144	C₄	C₂³×C₃×D₄	1	2	yes	—
145	C₃	C₂²×#494	1	1	yes	characteristic, direct product factor, Sylow
146	C₂	C₂×C₃×#494	1	12	yes	direct product factor
147	C₂	C₂²×C₃×#127	1	1	yes	characteristic
148	C₂	C₂³×C₃×Q₁₆	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	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
3. C₂×C₃×#494	C₂	direct (N × H)
5. C₂²×#494	C₃	direct (N × H)
14. C₃×#494	C₂²	direct (N × H)
18. C₂×#494	C₂×C₃	direct (N × H)
44. #494	C₂²×C₃	direct (N × H)
53. C₂²×C₃×C₄	Q₁₆	semidirect
88. C₂×C₃×C₄	C₂×Q₁₆	semidirect
101. C₂²×C₄	C₃×Q₁₆	semidirect
112. C₂²×C₃	#494	direct (N × H)
118. C₃×C₄	C₂²×Q₁₆	semidirect
128. C₂×C₄	C₂×C₃×Q₁₆	semidirect
134. C₂×C₃	C₂×#494	direct (N × H)
137. C₂²	C₃×#494	direct (N × H)
143. C₄	C₂²×C₃×Q₁₆	semidirect
145. C₃	C₂²×#494	direct (N × H)
146. C₂	C₂×C₃×#494	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
C₂²×C₃×C₄⋊Q₈	2
C₂×C₃×#494	2
C₂²×C₃×C₄×C₈	2
C₂²×#494	3

Subgroups — C22×C3×#494 #10679237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#494 #10679237