Subgroups — C₂²×#4461 #10947813

Order 768.

942 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊C₁₆	C₂	1	2	yes	maximal
3	C₂×#4461	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#346	—	3	1	no	maximal, Sylow
5	C₂³×C₃×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄×C₈	C₄	1	6	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	6	yes	—
10	C₂²×C₃⋊C₁₆	C₄	1	16	yes	complemented
11	C₂²×C₃⋊C₁₆	C₂²	1	12	yes	—
12	#4461	C₂²	1	16	yes	direct product factor
13	C₂³×C₁₆	—	3	2	no	—
14	C₂²×C₄×C₈	D₃	1	1	yes	characteristic
15	C₂×#346	—	3	12	no	—
16	C₂³×C₃×C₄	C₈	1	2	yes	—
17	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₈	C₈	1	8	yes	—
19	C₂²×C₃×C₈	C₂×C₄	1	8	yes	—
20	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
21	C₂²×C₃×C₈	C₂×C₄	1	6	yes	—
22	C₂²×C₃×C₈	C₂³	1	3	yes	—
23	C₂×C₃×C₄²	C₈	1	6	yes	—
24	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
25	C₃×C₄×C₈	C₂×C₄	1	12	yes	—
26	C₃×C₄×C₈	C₂³	1	4	yes	—
27	C₂×C₃⋊C₁₆	C₂×C₄	1	48	yes	complemented
28	C₂×C₃⋊C₁₆	C₂³	1	8	yes	—
29	C₂³×C₈	Dic₃	1	1	yes	characteristic
30	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
31	C₂²×C₄²	Dic₃	1	1	yes	characteristic
32	C₂²×C₁₆	—	3	16	no	—
33	C₂²×C₁₆	—	3	12	no	—
34	C₂×C₄×C₈	Dic₃	1	6	yes	—
35	C₂×C₄×C₈	C₂×D₃	1	6	yes	—
36	#346	—	3	16	no	—
37	C₂⁴×C₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₃×C₄	—	2	8	no	—
39	C₂²×C₃×C₄	C₄²	1	4	yes	—
40	C₂²×C₃×C₄	C₂×C₈	1	12	yes	—
41	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
42	C₂²×C₃×C₄	C₂×C₈	1	4	yes	—
43	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
44	C₂×C₃×C₈	C₄²	1	12	yes	—
45	C₂×C₃×C₈	C₂×C₈	1	24	yes	—
46	C₂×C₃×C₈	C₂²×C₄	1	12	yes	—
47	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
48	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
49	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
50	C₃×C₄²	C₂×C₈	1	12	yes	—
51	C₃×C₄²	C₂²×C₄	1	4	yes	—
52	C₃⋊C₁₆	C₂²×C₄	1	16	yes	complemented
53	C₂³×C₄	C₃⋊C₈	1	2	yes	—
54	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
55	C₂²×C₈	C₃⋊C₈	1	8	yes	—
56	C₂²×C₈	C₄×D₃	1	8	yes	—
57	C₂²×C₈	C₂×Dic₃	1	6	yes	—
58	C₂²×C₈	C₂×Dic₃	1	3	yes	—
59	C₂²×C₈	C₂²×D₃	1	3	yes	—
60	C₂×C₄²	C₃⋊C₈	1	6	yes	—
61	C₂×C₄²	C₂×Dic₃	1	6	yes	—
62	C₂×C₁₆	—	3	48	no	—
63	C₂×C₁₆	—	3	8	no	—
64	C₄×C₈	C₂×Dic₃	1	12	yes	—
65	C₄×C₈	C₂²×D₃	1	4	yes	—
66	C₂³×C₃	C₄×C₈	1	4	yes	—
67	C₂³×C₃	M₃₂	1	8	yes	—
68	C₂³×C₃	C₂²×C₈	1	3	yes	—
69	C₂×C₃×C₄	—	2	24	no	—
70	C₂×C₃×C₄	C₄×C₈	1	12	yes	—
71	C₂×C₃×C₄	C₂×C₄²	1	6	yes	—
72	C₂×C₃×C₄	C₂²×C₈	1	8	yes	—
73	C₂×C₃×C₄	C₂²×C₈	1	6	yes	—
74	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
75	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
76	C₃×C₈	C₂×C₄²	1	6	yes	—
77	C₃×C₈	C₂²×C₈	1	8	yes	—
78	C₃×C₈	C₂³×C₄	1	2	yes	—
79	C₂⁴	C₂×C₃⋊C₈	1	1	yes	characteristic
80	C₂²×C₄	—	2	8	no	—
81	C₂²×C₄	C₂×C₃⋊C₈	1	12	yes	—
82	C₂²×C₄	C₂×C₃⋊C₈	1	3	yes	—
83	C₂²×C₄	C₂×C₃⋊C₈	1	4	yes	—
84	C₂²×C₄	C₄×Dic₃	1	4	yes	—
85	C₂²×C₄	C₂²×Dic₃	1	3	yes	—
86	C₂×C₈	C₂×C₃⋊C₈	1	24	yes	—
87	C₂×C₈	C₄×Dic₃	1	12	yes	—
88	C₂×C₈	C₂×C₄×D₃	1	12	yes	—
89	C₂×C₈	C₂²×Dic₃	1	4	yes	—
90	C₂×C₈	C₂²×Dic₃	1	3	yes	—
91	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
92	C₄²	C₂×C₃⋊C₈	1	12	yes	—
93	C₄²	C₂²×Dic₃	1	4	yes	—
94	C₁₆	—	3	16	no	—
95	C₂²×C₃	#346	1	16	yes	complemented
96	C₂²×C₃	C₂×C₄×C₈	1	6	yes	—
97	C₂²×C₃	C₂×M₃₂	1	12	yes	—
98	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
99	C₃×C₄	—	2	8	no	—
100	C₃×C₄	C₂×C₄×C₈	1	6	yes	—
101	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
102	C₃×C₄	C₂³×C₈	1	1	yes	characteristic
103	C₂³	C₄×C₃⋊C₈	1	4	yes	—
104	C₂³	#812	1	8	yes	—
105	C₂³	C₂²×C₃⋊C₈	1	3	yes	—
106	C₂×C₄	—	2	24	no	—
107	C₂×C₄	C₄×C₃⋊C₈	1	12	yes	—
108	C₂×C₄	C₂²×C₃⋊C₈	1	8	yes	—
109	C₂×C₄	C₂²×C₃⋊C₈	1	3	yes	—
110	C₂×C₄	C₂²×C₃⋊C₈	1	6	yes	—
111	C₂×C₄	C₂×C₄×Dic₃	1	6	yes	—
112	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
113	C₈	C₂²×C₃⋊C₈	1	8	yes	—
114	C₈	C₂×C₄×Dic₃	1	6	yes	—
115	C₈	C₂²×C₄×D₃	1	2	yes	—
116	C₂×C₃	C₂×#346	1	12	yes	complemented
117	C₂×C₃	C₂²×C₄×C₈	1	1	yes	characteristic
118	C₂×C₃	C₂²×M₃₂	1	2	yes	—
119	C₂²	#4461	1	16	yes	direct product factor
120	C₂²	C₂×C₄×C₃⋊C₈	1	6	yes	—
121	C₂²	C₂×#812	1	12	yes	—
122	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
123	C₄	—	2	8	no	—
124	C₄	C₂×C₄×C₃⋊C₈	1	6	yes	—
125	C₄	C₂³×C₃⋊C₈	1	1	yes	characteristic
126	C₄	C₂²×C₄×Dic₃	1	1	yes	characteristic
127	C₃	C₂²×#346	1	1	yes	characteristic, complemented, Sylow
128	C₂	C₂×#4461	1	12	yes	direct product factor
129	C₂	C₂²×C₄×C₃⋊C₈	1	1	yes	characteristic
130	C₂	C₂²×#812	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₃
commutator	C₂×C₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂²×C₃×C₄×C₈	C₂
socle, prime radical	C₂⁴×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂²×C₄×C₈	D₃
hypocenter	C₃	C₂²×#346
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#4461	C₂	direct (N × H)
10. C₂²×C₃⋊C₁₆	C₄	semidirect
12. #4461	C₂²	direct (N × H)
27. C₂×C₃⋊C₁₆	C₂×C₄	semidirect
52. C₃⋊C₁₆	C₂²×C₄	semidirect
95. C₂²×C₃	#346	semidirect
116. C₂×C₃	C₂×#346	semidirect
119. C₂²	#4461	direct (N × H)
127. C₃	C₂²×#346	semidirect
128. C₂	C₂×#4461	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×C₄×C₈	2
C₂³×C₃⋊C₁₆	2
C₂×#4461	2
C₂²×#346	3

Subgroups — C22×#4461 #10947813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4461 #10947813