Subgroups — C₂³×C₄⋊(C₃⋊C₈) #11658795

Order 768.

3870 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₃⋊C₈	C₂	1	2	yes	maximal
3	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄⋊C₈	—	3	1	no	maximal, Sylow
5	C₂⁴×C₃×C₄	C₄	1	1	yes	characteristic
6	C₂⁴×C₃×C₄	C₄	1	1	yes	characteristic
7	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄²	C₄	1	14	yes	—
9	C₂²×C₃×C₄²	C₂²	1	14	yes	—
10	C₂³×C₃⋊C₈	—	2	16	no	—
11	C₂³×C₃⋊C₈	C₂²	1	28	yes	—
12	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	112	yes	direct product factor
13	C₂⁴×C₈	—	3	2	no	—
14	C₂³×C₄²	D₃	1	1	yes	characteristic
15	C₂²×C₄⋊C₈	—	3	28	no	—
16	C₂⁵×C₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₃×C₄	—	2	8	no	—
18	C₂³×C₃×C₄	C₈	1	16	yes	complemented
19	C₂³×C₃×C₄	C₂×C₄	1	14	yes	—
20	C₂³×C₃×C₄	C₂×C₄	1	7	yes	—
21	C₂³×C₃×C₄	C₂×C₄	1	14	yes	—
22	C₂³×C₃×C₄	D₄	1	8	yes	—
23	C₂³×C₃×C₄	Q₈	1	8	yes	—
24	C₂³×C₃×C₄	C₂³	1	7	yes	—
25	C₂×C₃×C₄²	C₂×C₄	1	84	yes	—
26	C₂×C₃×C₄²	C₂³	1	28	yes	—
27	C₂²×C₃⋊C₈	—	2	112	no	—
28	C₂²×C₃⋊C₈	C₂³	1	56	yes	—
29	C₄⋊(C₃⋊C₈)	C₂³	1	64	yes	direct product factor
30	C₂⁴×C₄	Dic₃	1	1	yes	characteristic
31	C₂⁴×C₄	Dic₃	1	1	yes	characteristic
32	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
33	C₂³×C₈	—	3	28	no	—
34	C₂³×C₈	—	6	16	no	—
35	C₂²×C₄²	Dic₃	1	14	yes	—
36	C₂²×C₄²	C₂×D₃	1	14	yes	—
37	C₂×C₄⋊C₈	—	3	112	no	—
38	C₂⁴×C₃	C₄⋊C₄	1	8	yes	—
39	C₂⁴×C₃	C₂×C₈	1	8	yes	—
40	C₂⁴×C₃	M₁₆	1	8	yes	—
41	C₂⁴×C₃	C₂²×C₄	1	7	yes	—
42	C₂²×C₃×C₄	—	2	56	no	—
43	C₂²×C₃×C₄	C₄⋊C₄	1	56	yes	—
44	C₂²×C₃×C₄	C₂×C₈	1	112	yes	complemented
45	C₂²×C₃×C₄	C₂²×C₄	1	28	yes	—
46	C₂²×C₃×C₄	C₂²×C₄	1	21	yes	—
47	C₂²×C₃×C₄	C₂²×C₄	1	28	yes	—
48	C₂²×C₃×C₄	C₂×D₄	1	28	yes	—
49	C₂²×C₃×C₄	C₂×Q₈	1	28	yes	—
50	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
51	C₃×C₄²	C₂²×C₄	1	56	yes	—
52	C₃×C₄²	C₂⁴	1	8	yes	—
53	C₂×C₃⋊C₈	—	2	112	no	—
54	C₂×C₃⋊C₈	C₂⁴	1	16	yes	—
55	C₂⁵	C₂×Dic₃	1	1	yes	characteristic
56	C₂³×C₄	—	2	8	no	—
57	C₂³×C₄	C₃⋊C₈	1	16	yes	complemented
58	C₂³×C₄	Dic₆	1	8	yes	—
59	C₂³×C₄	D₁₂	1	8	yes	—
60	C₂³×C₄	C₂×Dic₃	1	14	yes	—
61	C₂³×C₄	C₂×Dic₃	1	14	yes	—
62	C₂³×C₄	C₂×Dic₃	1	7	yes	—
63	C₂³×C₄	C₂²×D₃	1	7	yes	—
64	C₂²×C₈	—	3	56	no	—
65	C₂²×C₈	—	6	112	no	—
66	C₂×C₄²	C₂×Dic₃	1	84	yes	—
67	C₂×C₄²	C₂²×D₃	1	28	yes	—
68	C₄⋊C₈	—	3	64	no	—
69	C₂³×C₃	C₄⋊C₈	1	64	yes	complemented
70	C₂³×C₃	C₂×C₄⋊C₄	1	28	yes	—
71	C₂³×C₃	C₂²×C₈	1	28	yes	—
72	C₂³×C₃	C₂×M₁₆	1	28	yes	—
73	C₂³×C₃	C₂³×C₄	1	7	yes	—
74	C₂×C₃×C₄	—	2	56	no	—
75	C₂×C₃×C₄	C₂×C₄⋊C₄	1	84	yes	—
76	C₂×C₃×C₄	C₂²×C₈	1	112	yes	complemented
77	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
78	C₂×C₃×C₄	C₂³×C₄	1	7	yes	—
79	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
80	C₂×C₃×C₄	C₂²×D₄	1	14	yes	—
81	C₂×C₃×C₄	C₂²×Q₈	1	14	yes	—
82	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
83	C₃⋊C₈	—	2	16	no	—
84	C₂⁴	C₂×C₃⋊C₈	1	8	yes	—
85	C₂⁴	#208	1	8	yes	—
86	C₂⁴	C₄⋊Dic₃	1	8	yes	—
87	C₂⁴	C₂²×Dic₃	1	7	yes	—
88	C₂²×C₄	—	2	56	no	—
89	C₂²×C₄	C₂×C₃⋊C₈	1	112	yes	complemented
90	C₂²×C₄	C₄⋊Dic₃	1	56	yes	—
91	C₂²×C₄	C₂×Dic₆	1	28	yes	—
92	C₂²×C₄	C₂×D₁₂	1	28	yes	—
93	C₂²×C₄	C₂²×Dic₃	1	28	yes	—
94	C₂²×C₄	C₂²×Dic₃	1	28	yes	—
95	C₂²×C₄	C₂²×Dic₃	1	21	yes	—
96	C₂²×C₄	C₂³×D₃	1	7	yes	—
97	C₂×C₈	—	3	16	no	—
98	C₂×C₈	—	6	112	no	—
99	C₄²	C₂²×Dic₃	1	56	yes	—
100	C₄²	C₂³×D₃	1	8	yes	—
101	C₂²×C₃	C₂×C₄⋊C₈	1	112	yes	complemented
102	C₂²×C₃	C₂²×C₄⋊C₄	1	14	yes	—
103	C₂²×C₃	C₂³×C₈	1	14	yes	—
104	C₂²×C₃	C₂²×M₁₆	1	14	yes	—
105	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
106	C₃×C₄	—	2	8	no	—
107	C₃×C₄	C₂²×C₄⋊C₄	1	14	yes	—
108	C₃×C₄	C₂³×C₈	1	16	yes	complemented
109	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
110	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
111	C₂³	C₄⋊(C₃⋊C₈)	1	64	yes	direct product factor
112	C₂³	C₂²×C₃⋊C₈	1	28	yes	—
113	C₂³	C₂×#208	1	28	yes	—
114	C₂³	C₂×C₄⋊Dic₃	1	28	yes	—
115	C₂³	C₂³×Dic₃	1	7	yes	—
116	C₂×C₄	—	2	56	no	—
117	C₂×C₄	C₂²×C₃⋊C₈	1	112	yes	complemented
118	C₂×C₄	C₂×C₄⋊Dic₃	1	84	yes	—
119	C₂×C₄	C₂²×Dic₆	1	14	yes	—
120	C₂×C₄	C₂²×D₁₂	1	14	yes	—
121	C₂×C₄	C₂³×Dic₃	1	8	yes	—
122	C₂×C₄	C₂³×Dic₃	1	8	yes	—
123	C₂×C₄	C₂³×Dic₃	1	7	yes	—
124	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
125	C₈	—	6	16	no	—
126	C₂×C₃	C₂²×C₄⋊C₈	1	28	yes	complemented
127	C₂×C₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₃	C₂⁴×C₈	1	1	yes	characteristic
129	C₂×C₃	C₂³×M₁₆	1	1	yes	characteristic
130	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	112	yes	direct product factor
131	C₂²	C₂³×C₃⋊C₈	1	14	yes	—
132	C₂²	C₂²×#208	1	14	yes	—
133	C₂²	C₂²×C₄⋊Dic₃	1	14	yes	—
134	C₂²	C₂⁴×Dic₃	1	1	yes	characteristic
135	C₄	—	2	8	no	—
136	C₄	C₂³×C₃⋊C₈	1	16	yes	complemented
137	C₄	C₂²×C₄⋊Dic₃	1	14	yes	—
138	C₄	C₂³×Dic₆	1	1	yes	characteristic
139	C₄	C₂³×D₁₂	1	1	yes	characteristic
140	C₃	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
141	C₂	C₂²×C₄⋊(C₃⋊C₈)	1	28	yes	direct product factor
142	C₂	C₂⁴×C₃⋊C₈	1	1	yes	characteristic
143	C₂	C₂³×#208	1	1	yes	characteristic
144	C₂	C₂³×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₃
commutator	C₂×C₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂³×C₃×C₄²	C₂
socle, prime radical	C₂⁵×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₂³×C₄²	D₃
hypocenter	C₃	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊(C₃⋊C₈)	C₂	direct (N × H)
12. C₂×C₄⋊(C₃⋊C₈)	C₂²	direct (N × H)
18. C₂³×C₃×C₄	C₈	semidirect
29. C₄⋊(C₃⋊C₈)	C₂³	direct (N × H)
44. C₂²×C₃×C₄	C₂×C₈	semidirect
57. C₂³×C₄	C₃⋊C₈	semidirect
69. C₂³×C₃	C₄⋊C₈	semidirect
76. C₂×C₃×C₄	C₂²×C₈	semidirect
89. C₂²×C₄	C₂×C₃⋊C₈	semidirect
101. C₂²×C₃	C₂×C₄⋊C₈	semidirect
108. C₃×C₄	C₂³×C₈	semidirect
111. C₂³	C₄⋊(C₃⋊C₈)	direct (N × H)
117. C₂×C₄	C₂²×C₃⋊C₈	semidirect
126. C₂×C₃	C₂²×C₄⋊C₈	semidirect
130. C₂²	C₂×C₄⋊(C₃⋊C₈)	direct (N × H)
136. C₄	C₂³×C₃⋊C₈	semidirect
140. C₃	C₂³×C₄⋊C₈	semidirect
141. C₂	C₂²×C₄⋊(C₃⋊C₈)	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×C₄²	2
C₂⁴×C₃⋊C₈	2
C₂²×C₄⋊(C₃⋊C₈)	2
C₂³×C₄⋊C₈	3

Subgroups — C23×C4⋊(C3⋊C8) #11658795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊(C₃⋊C₈) #11658795