Subgroups — C₂²×#5140 #11701651

Order 768.

2118 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#5140	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#812	C₂	1	3	yes	maximal, complemented
4	C₂³×C₃⋊C₁₆	C₂	1	3	yes	maximal, complemented
5	C₂²×#504	—	3	1	no	maximal, Sylow
6	C₂²×C₃×#41	C₄	1	1	yes	characteristic
7	C₂×C₃×#131	C₄	1	12	yes	—
8	C₂×C₃×#131	C₂²	1	12	yes	—
9	C₂²×C₃×M₁₆	C₄	1	3	yes	—
10	C₂³×C₃×C₈	C₂²	1	3	yes	—
11	#5140	C₂²	1	64	yes	direct product factor
12	C₂×#812	C₂²	1	36	yes	complemented
13	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊C₁₆	C₂²	1	3	yes	—
15	C₂²×C₃⋊C₁₆	C₂²	1	36	yes	complemented
16	C₂²×#131	D₃	1	1	yes	characteristic
17	C₂×#504	—	3	24	no	—
18	C₂²×M₃₂	—	3	3	no	—
19	C₂³×C₁₆	—	3	3	no	—
20	C₂×C₃×#41	C₈	1	12	yes	—
21	C₂×C₃×#41	C₂×C₄	1	12	yes	—
22	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
23	C₂²×C₃×D₄	C₈	1	3	yes	—
24	C₂³×C₃×C₄	C₂×C₄	1	3	yes	—
25	C₃×#131	C₂×C₄	1	48	yes	—
26	C₃×#131	C₂³	1	16	yes	—
27	C₂×C₃×M₁₆	C₂×C₄	1	36	yes	—
28	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
29	C₂²×C₃×C₈	C₂×C₄	1	18	yes	—
30	C₂²×C₃×C₈	C₂³	1	18	yes	—
31	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
32	#812	C₂³	1	48	yes	complemented
33	C₂×C₃⋊C₁₆	C₂³	1	6	yes	—
34	C₂×C₃⋊C₁₆	C₂³	1	18	yes	—
35	C₂×C₃⋊C₁₆	C₂³	1	48	yes	complemented
36	C₂²×#41	Dic₃	1	1	yes	characteristic
37	C₂×#131	Dic₃	1	12	yes	—
38	C₂×#131	C₂×D₃	1	12	yes	—
39	C₂²×M₁₆	Dic₃	1	3	yes	—
40	C₂³×C₈	C₂×D₃	1	3	yes	—
41	#504	—	3	64	no	—
42	C₂×M₃₂	—	3	36	no	—
43	C₂²×C₁₆	—	3	36	no	—
44	C₂²×C₁₆	—	3	1	no	—
45	C₂²×C₁₆	—	3	3	no	—
46	C₂⁴×C₃	C₂×C₈	1	3	yes	—
47	C₃×#41	C₂×C₈	1	48	yes	—
48	C₃×#41	C₂²×C₄	1	16	yes	—
49	C₂×C₃×Q₈	C₂×C₈	1	12	yes	—
50	C₂×C₃×D₄	C₂×C₈	1	36	yes	—
51	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
52	C₂²×C₃×C₄	C₂×C₈	1	18	yes	—
53	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂²×C₄	1	18	yes	—
55	C₃×M₁₆	C₂²×C₄	1	48	yes	—
56	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
57	C₂×C₃×C₈	C₂²×C₄	1	18	yes	—
58	C₂×C₃×C₈	C₂²×C₄	1	36	yes	—
59	C₂×C₃×C₈	C₂⁴	1	3	yes	—
60	C₂×C₃×C₈	C₂⁴	1	12	yes	—
61	C₃⋊C₁₆	C₂⁴	1	4	yes	—
62	C₃⋊C₁₆	C₂⁴	1	12	yes	—
63	C₂×#41	C₃⋊C₈	1	12	yes	—
64	C₂×#41	C₂×Dic₃	1	12	yes	—
65	C₂²×Q₈	C₃⋊C₈	1	1	yes	characteristic
66	C₂²×D₄	C₃⋊C₈	1	3	yes	—
67	C₂³×C₄	C₂×Dic₃	1	3	yes	—
68	#131	C₂×Dic₃	1	48	yes	—
69	#131	C₂²×D₃	1	16	yes	—
70	C₂×M₁₆	C₂×Dic₃	1	36	yes	—
71	C₂²×C₈	C₂×Dic₃	1	3	yes	—
72	C₂²×C₈	C₂×Dic₃	1	18	yes	—
73	C₂²×C₈	C₂²×D₃	1	18	yes	—
74	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
75	M₃₂	—	3	48	no	—
76	C₂×C₁₆	—	3	6	no	—
77	C₂×C₁₆	—	3	18	no	—
78	C₂×C₁₆	—	3	48	no	—
79	C₂³×C₃	—	2	12	no	—
80	C₂³×C₃	C₂²×C₈	1	18	yes	—
81	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
82	C₃×Q₈	C₂²×C₈	1	16	yes	—
83	C₃×D₄	C₂²×C₈	1	48	yes	—
84	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
85	C₂×C₃×C₄	C₂²×C₈	1	18	yes	—
86	C₂×C₃×C₄	C₂²×C₈	1	36	yes	—
87	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
88	C₂×C₃×C₄	C₂³×C₄	1	12	yes	—
89	C₃×C₈	C₂³×C₄	1	3	yes	—
90	C₃×C₈	C₂³×C₄	1	12	yes	—
91	C₃×C₈	C₂⁵	1	1	yes	characteristic
92	C₂⁴	C₂×C₃⋊C₈	1	3	yes	—
93	#41	C₂×C₃⋊C₈	1	48	yes	—
94	#41	C₂²×Dic₃	1	16	yes	—
95	C₂×Q₈	C₂×C₃⋊C₈	1	12	yes	—
96	C₂×D₄	C₂×C₃⋊C₈	1	36	yes	—
97	C₂²×C₄	C₂×C₃⋊C₈	1	3	yes	—
98	C₂²×C₄	C₂×C₃⋊C₈	1	18	yes	—
99	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
100	C₂²×C₄	C₂²×Dic₃	1	18	yes	—
101	M₁₆	C₂²×Dic₃	1	48	yes	—
102	C₂×C₈	C₂²×Dic₃	1	3	yes	—
103	C₂×C₈	C₂²×Dic₃	1	18	yes	—
104	C₂×C₈	C₂²×Dic₃	1	36	yes	—
105	C₂×C₈	C₂³×D₃	1	3	yes	—
106	C₂×C₈	C₂³×D₃	1	12	yes	—
107	C₁₆	—	3	4	no	—
108	C₁₆	—	3	12	no	—
109	C₂²×C₃	—	2	36	no	—
110	C₂²×C₃	#504	1	4	yes	complemented
111	C₂²×C₃	C₂³×C₈	1	12	yes	—
112	C₂²×C₃	C₂³×C₈	1	3	yes	—
113	C₃×C₄	C₂³×C₈	1	3	yes	—
114	C₃×C₄	C₂³×C₈	1	12	yes	—
115	C₃×C₄	C₂⁴×C₄	1	1	yes	characteristic
116	C₂³	—	2	12	no	—
117	C₂³	C₂²×C₃⋊C₈	1	18	yes	—
118	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
119	Q₈	C₂²×C₃⋊C₈	1	16	yes	—
120	D₄	C₂²×C₃⋊C₈	1	48	yes	—
121	C₂×C₄	C₂²×C₃⋊C₈	1	3	yes	—
122	C₂×C₄	C₂²×C₃⋊C₈	1	18	yes	—
123	C₂×C₄	C₂²×C₃⋊C₈	1	36	yes	—
124	C₂×C₄	C₂³×Dic₃	1	3	yes	—
125	C₂×C₄	C₂³×Dic₃	1	12	yes	—
126	C₈	C₂³×Dic₃	1	3	yes	—
127	C₈	C₂³×Dic₃	1	12	yes	—
128	C₈	C₂⁴×D₃	1	1	yes	characteristic
129	C₂×C₃	—	2	12	no	—
130	C₂×C₃	C₂×#504	1	6	yes	complemented
131	C₂×C₃	C₂⁴×C₈	1	1	yes	characteristic
132	C₂²	—	2	36	no	—
133	C₂²	#5140	1	4	yes	direct product factor
134	C₂²	C₂³×C₃⋊C₈	1	12	yes	—
135	C₂²	C₂³×C₃⋊C₈	1	3	yes	—
136	C₄	C₂³×C₃⋊C₈	1	3	yes	—
137	C₄	C₂³×C₃⋊C₈	1	12	yes	—
138	C₄	C₂⁴×Dic₃	1	1	yes	characteristic
139	C₃	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
140	C₂	—	2	12	no	—
141	C₂	C₂×#5140	1	6	yes	direct product factor
142	C₂	C₂⁴×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₃
commutator	C₂×C₃	C₂⁴×C₈
Frattini	C₈	C₂⁴×D₃
Fitting	C₂²×C₃×#131	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂²×C₃×#41	C₄
cosocle	C₃×C₈	C₂⁵
hypercenter	C₂²×#131	D₃
hypocenter	C₃	C₂²×#504
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×#131

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#5140	C₂	direct (N × H)
3. C₂²×#812	C₂	semidirect
4. C₂³×C₃⋊C₁₆	C₂	semidirect
11. #5140	C₂²	direct (N × H)
12. C₂×#812	C₂²	semidirect
15. C₂²×C₃⋊C₁₆	C₂²	semidirect
32. #812	C₂³	semidirect
35. C₂×C₃⋊C₁₆	C₂³	semidirect
110. C₂²×C₃	#504	semidirect
130. C₂×C₃	C₂×#504	semidirect
133. C₂²	#5140	direct (N × H)
139. C₃	C₂²×#504	semidirect
141. C₂	C₂×#5140	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×#131	2
C₂×#5140	2
C₂²×#812	2
C₂³×C₃⋊C₁₆	2
C₂²×#504	3

Subgroups — C22×#5140 #11701651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#5140 #11701651