Subgroups — C₂×Q₈×C₃⋊C₁₆ #10948669

Order 768.

582 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	C₂×C₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊C₁₆	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	3	yes	maximal
4	C₂×C₄×C₃⋊C₁₆	C₂	1	3	yes	maximal
5	C₂×Q₈×C₁₆	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₃×C₈×Q₈	C₄	1	4	yes	—
8	C₃×C₈×Q₈	C₂²	1	4	yes	—
9	C₂×C₃×C₄⋊C₈	C₄	1	3	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
11	C₂²×C₃⋊C₁₆	C₂²	1	3	yes	—
12	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊C₁₆)	C₂²	1	12	yes	—
14	C₄×C₃⋊C₁₆	C₂²	1	12	yes	—
15	C₂×C₈×Q₈	D₃	1	1	yes	characteristic
16	Q₈×C₁₆	—	3	8	no	—
17	C₂×C₄⋊C₁₆	—	3	3	no	—
18	C₂×C₄×C₁₆	—	3	3	no	—
19	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
20	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
21	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
22	C₃×C₄×Q₈	C₈	1	4	yes	—
23	C₃×C₄×Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₄⋊C₄	C₈	1	3	yes	—
25	C₂×C₃×C₄²	C₂×C₄	1	3	yes	—
26	C₃×C₄⋊C₈	C₂×C₄	1	12	yes	—
27	C₃×C₄×C₈	C₂×C₄	1	6	yes	—
28	C₃×C₄×C₈	C₂³	1	6	yes	—
29	C₂×C₃⋊C₁₆	—	2	6	no	—
30	C₂×C₃⋊C₁₆	Q₈	1	4	yes	direct product factor
31	C₂×C₃⋊C₁₆	C₂³	1	6	yes	—
32	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
33	C₂×C₄×Q₈	Dic₃	1	1	yes	characteristic
34	C₂²×C₁₆	—	3	1	no	—
35	C₂²×C₁₆	—	3	3	no	—
36	C₈×Q₈	Dic₃	1	4	yes	—
37	C₈×Q₈	C₂×D₃	1	4	yes	—
38	C₂×C₄⋊C₈	Dic₃	1	3	yes	—
39	C₂×C₄×C₈	C₂×D₃	1	3	yes	—
40	C₄⋊C₁₆	—	3	12	no	—
41	C₄×C₁₆	—	3	12	no	—
42	C₂×C₃×Q₈	C₁₆	1	8	yes	complemented
43	C₂×C₃×Q₈	C₂×C₈	1	4	yes	—
44	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
45	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
46	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃×C₈	—	2	6	no	—
48	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
49	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
51	C₂×C₃×C₈	#41	1	2	yes	—
52	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
53	C₃×C₄⋊C₄	C₂×C₈	1	12	yes	—
54	C₃×C₄²	C₂×C₈	1	6	yes	—
55	C₃×C₄²	C₂²×C₄	1	6	yes	—
56	C₃⋊C₁₆	—	2	6	no	—
57	C₃⋊C₁₆	C₂×Q₈	1	4	yes	direct product factor
58	C₂²×Q₈	C₃⋊C₈	1	1	yes	characteristic
59	C₂²×C₈	C₂×Dic₃	1	3	yes	—
60	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
61	C₄×Q₈	C₃⋊C₈	1	4	yes	—
62	C₄×Q₈	C₂×Dic₃	1	4	yes	—
63	C₂×C₄⋊C₄	C₃⋊C₈	1	3	yes	—
64	C₂×C₄²	C₂×Dic₃	1	3	yes	—
65	C₂×C₁₆	—	3	4	no	—
66	C₂×C₁₆	—	3	2	no	—
67	C₂×C₁₆	—	3	6	no	—
68	C₂×C₁₆	—	6	6	no	—
69	C₄⋊C₈	C₂×Dic₃	1	12	yes	—
70	C₄×C₈	C₂×Dic₃	1	6	yes	—
71	C₄×C₈	C₂²×D₃	1	6	yes	—
72	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
73	C₃×Q₈	C₂×C₁₆	1	16	yes	complemented
74	C₂×C₃×C₄	—	2	6	no	—
75	C₂×C₃×C₄	C₂×C₁₆	1	12	yes	—
76	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
77	C₂×C₃×C₄	C₂²×C₈	1	6	yes	—
78	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂²×C₈	1	6	yes	—
80	C₂×C₃×C₄	#131	1	2	yes	—
81	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
82	C₃×C₈	—	2	6	no	—
83	C₃×C₈	C₄×Q₈	1	2	yes	—
84	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
85	C₃×C₈	C₂×#41	1	1	yes	characteristic
86	C₂×Q₈	C₃⋊C₁₆	1	8	yes	direct product factor
87	C₂×Q₈	C₂×C₃⋊C₈	1	4	yes	—
88	C₂²×C₄	C₂×C₃⋊C₈	1	3	yes	—
89	C₂²×C₄	C₂×C₃⋊C₈	1	3	yes	—
90	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
91	C₂×C₈	—	2	6	no	—
92	C₂×C₈	D₃×Q₈	1	2	yes	—
93	C₂×C₈	#239	1	2	yes	—
94	C₂×C₈	C₂²×Dic₃	1	6	yes	—
95	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
96	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
97	C₄⋊C₄	C₂×C₃⋊C₈	1	12	yes	—
98	C₄²	C₂×C₃⋊C₈	1	6	yes	—
99	C₄²	C₂²×Dic₃	1	6	yes	—
100	C₁₆	—	3	4	no	—
101	C₁₆	—	6	6	no	—
102	C₂²×C₃	C₈×Q₈	1	2	yes	—
103	C₂²×C₃	C₂²×C₁₆	1	2	yes	—
104	C₂²×C₃	#504	1	2	yes	—
105	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
106	C₃×C₄	—	2	6	no	—
107	C₃×C₄	C₈×Q₈	1	2	yes	—
108	C₃×C₄	C₂²×C₁₆	1	12	yes	—
109	C₃×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
110	C₃×C₄	C₂×#131	1	1	yes	characteristic
111	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
112	Q₈	C₂×C₃⋊C₁₆	1	16	yes	direct product factor
113	C₂×C₄	—	2	6	no	—
114	C₂×C₄	C₂×C₃⋊C₁₆	1	12	yes	—
115	C₂×C₄	C₂²×C₃⋊C₈	1	6	yes	—
116	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
117	C₂×C₄	C₂²×C₃⋊C₈	1	6	yes	—
118	C₂×C₄	Q₈×Dic₃	1	2	yes	—
119	C₂×C₄	#948	1	2	yes	—
120	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
121	C₈	—	2	6	no	—
122	C₈	Q₈×Dic₃	1	2	yes	—
123	C₈	C₂×D₃×Q₈	1	1	yes	characteristic
124	C₈	C₂×#239	1	1	yes	characteristic
125	C₂×C₃	Q₈×C₁₆	1	4	yes	complemented
126	C₂×C₃	C₂×C₈×Q₈	1	1	yes	characteristic
127	C₂×C₃	C₂³×C₁₆	1	1	yes	characteristic
128	C₂×C₃	C₂×#504	1	1	yes	characteristic
129	C₂²	Q₈×C₃⋊C₈	1	2	yes	—
130	C₂²	C₂²×C₃⋊C₁₆	1	2	yes	—
131	C₂²	#5140	1	2	yes	—
132	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
133	C₄	—	2	6	no	—
134	C₄	Q₈×C₃⋊C₈	1	2	yes	—
135	C₄	C₂²×C₃⋊C₁₆	1	12	yes	—
136	C₄	C₂×Q₈×Dic₃	1	1	yes	characteristic
137	C₄	C₂×#948	1	1	yes	characteristic
138	C₃	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
139	C₂	Q₈×C₃⋊C₁₆	1	4	yes	direct product factor
140	C₂	C₂×Q₈×C₃⋊C₈	1	1	yes	characteristic
141	C₂	C₂³×C₃⋊C₁₆	1	1	yes	characteristic
142	C₂	C₂×#5140	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₃
commutator	C₂×C₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂×C₃×C₈×Q₈	C₂
socle, prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×C₈×Q₈	D₃
hypocenter	C₃	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃⋊C₁₆	C₂	direct (N × H)
30. C₂×C₃⋊C₁₆	Q₈	direct (N × H)
42. C₂×C₃×Q₈	C₁₆	semidirect
57. C₃⋊C₁₆	C₂×Q₈	direct (N × H)
73. C₃×Q₈	C₂×C₁₆	semidirect
86. C₂×Q₈	C₃⋊C₁₆	direct (N × H)
112. Q₈	C₂×C₃⋊C₁₆	direct (N × H)
125. C₂×C₃	Q₈×C₁₆	semidirect
138. C₃	C₂×Q₈×C₁₆	semidirect
139. C₂	Q₈×C₃⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Q₈	2
Q₈×C₃⋊C₁₆	2
C₂×C₄⋊(C₃⋊C₁₆)	2
C₂×C₄×C₃⋊C₁₆	2
C₂×Q₈×C₁₆	3

Subgroups — C2×Q8×C3⋊C16 #10948669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₃⋊C₁₆ #10948669