Subgroups — C₂³×#812 #11701648

Order 768.

2334 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₈	C₂×D₃
commutator	C₂×C₃	C₂⁴×C₈
Frattini	C₈	C₂⁴×D₃
Fitting	C₂⁴×C₃×C₈	C₂
socle	C₂⁴×C₃	C₂×C₈
prime radical	C₂⁵×C₃	C₈
cosocle	C₃×C₈	C₂⁵
hypercenter	C₂⁴×C₈	D₃
hypocenter	C₃	C₂³×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#812	C₂	direct (N × H)
3. C₂³×C₃⋊C₁₆	C₂	semidirect
10. C₂×#812	C₂²	direct (N × H)
11. C₂²×C₃⋊C₁₆	C₂²	semidirect
25. #812	C₂³	direct (N × H)
26. C₂×C₃⋊C₁₆	C₂³	semidirect
47. C₃⋊C₁₆	C₂⁴	semidirect
61. C₂³×C₃	M₃₂	semidirect
87. C₂²×C₃	C₂×M₃₂	semidirect
94. C₂³	#812	direct (N × H)
106. C₂×C₃	C₂²×M₃₂	semidirect
109. C₂²	C₂×#812	direct (N × H)
115. C₃	C₂³×M₃₂	semidirect
117. C₂	C₂²×#812	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₈	2
C₂²×#812	2
C₂³×C₃⋊C₁₆	2
C₂³×M₃₂	3

Subgroups — C23×#812 #11701648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#812 #11701648