Finite Group Databasebeta

← #35214731

Subgroups — #35214731

Order 20864.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊Dic163C211yesmaximal, characteristic
2C2×#19104667C212yesmaximal, complemented
3#23781525C211yesmaximal, characteristic
4#23781428C212yesmaximal, complemented
5C2×C22⋊C8×C163C211yesmaximal, characteristic
6C23×Dic163—21no—
7C2×C4⋊Dic163—24no—
8C2×C4⋊Dic163—21no—
9C2×C4⋊Dic163C2211yescharacteristic
10C2×C4⋊Dic163C2211yescharacteristic
11C2×C4⋊Dic163C2212yes—
12C2×C22⋊Dic163—21no—
13#19104667—24no—
14C163⋊#95—21no—
15C23×C4×C163C2211yescharacteristic
16C22×C8×C163C2212yes—
17C22⋊C8×C163C414yescomplemented
18C22×Dic163—24no—
19C22×Dic163—22no—
20C22×Dic163—21no—
21C22×Dic163—21no—
22C22×Dic163—42no—
23C22×Dic163—42no—
24C4⋊Dic163—24no—
25C4⋊Dic163—22no—
26C4⋊Dic163—22no—
27C4⋊Dic163—22no—
28C4⋊Dic163D414yes—
29C22⋊Dic163—42no—
30C24×C163D411yescharacteristic
31C22×C4×C163—22no—
32C22×C4×C163—22no—
33C22×C4×C163C2×C412yes—
34C22×C4×C163D411yescharacteristic
35C22×C4×C163Q812yes—
36C22×C4×C163C2311yescharacteristic
37C2×C8×C163—22no—
38C2×C8×C163—22no—
39C2×C8×C163C2×C414yescomplemented
40C2×Dic163—24no—
41C2×Dic163—48no—
42C2×Dic163—42no—
43C2×Dic163—42no—
44C2×Dic163—41no—
45C2×Dic163—41no—
46C2×Dic163—41no—
47C2×Dic163—41no—
48C2×Dic163—41no—
49C2×Dic163—41no—
50C23×C163—21no—
51C23×C163—21no—
52C23×C163—21no—
53C23×C163—21no—
54C23×C163C4⋊C412yes—
55C23×C163SD1612yes—
56C23×C163SD1612yes—
57C23×C163C2×D411yescharacteristic
58C2×C4×C163—21no—
59C2×C4×C163—21no—
60C2×C4×C163—24no—
61C2×C4×C163—24no—
62C2×C4×C163C4⋊C412yes—
63C2×C4×C163C22×C411yescharacteristic
64C2×C4×C163C2×D412yes—
65C2×C4×C163C2×Q811yescharacteristic
66C2×C4×C163#4112yes—
67C8×C163—24no—
68C22×C163—22no—
69C22×C163—22no—
70C22×C163—22no—
71C22×C163—22no—
72C22×C163—22no—
73C22×C163—22no—
74C22×C163#10614yescomplemented
75C22×C163C2×C4⋊C411yescharacteristic
76C22×C163#12011yescharacteristic
77C22×C163#12311yescharacteristic
78C22×C163C2×SD1611yescharacteristic
79C22×C163C2×SD1611yescharacteristic
80C22×C163AGL(1,ℤ/8)11yescharacteristic
81C22×C163#13711yescharacteristic
82C4×C163—22no—
83C4×C163C4×D412yes—
84C4×C163C22⋊Q812yes—
85Dic163—44no—
86Dic163—81no—
87Dic163—81no—
88C2×C163—24no—
89C2×C163#38511yescharacteristic
90C2×C163C2×#10611yescharacteristic
91C2×C163#42811yescharacteristic
92C2×C163#44811yescharacteristic
93C2×C163#45011yescharacteristic
94C2×C163#48111yescharacteristic
95C2×C163#48311yescharacteristic
96C163#185311yescharacteristic, complemented, Sylow
97#1853—1631nomaximal, Sylow
98C22×C4⋊C4—1631no—
99C2×#106—1632no—
100C2×C22⋊C8D16311yescharacteristic
101#380—1631no—
102#340—1632no—
103C23×C4—3261no—
104C23×C4C2×D16311yescharacteristic
105C22×C8C2×D16312yes—
106C2×C4⋊C4—3261no—
107C2×C4⋊C4—3264no—
108C2×C4⋊C4—1631no—
109C2×C4⋊C4—1631no—
110C2×C4⋊C4—1632no—
111C2×C22⋊C4—3261no—
112#106—3264no—
113C22⋊C8Dic16314yescomplemented
114#95—3261no—
115C24D65211yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—3264no—
119C22×C4—3262no—
120C22×C4—3261no—
121C22×C4—3261no—
122C22×C4—6522no—
123C22×C4—6522no—
124C22×C4C2×Dic16312yes—
125C22×C4D65211yescharacteristic
126C22×C4Dic32612yes—
127C22×C4C22×D16311yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×Dic16314yescomplemented
131C4⋊C4—3264no—
132C4⋊C4—3262no—
133C4⋊C4—3262no—
134C4⋊C4—3262no—
135C4⋊C4—1634no—
136C22⋊C4—6522no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊Dic16312yes—
142C23#1457968812yes—
143C23#1457968812yes—
144C23C2×D65211yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—3264no—
150C2×C4—6521no—
151C2×C4—6521no—
152C2×C4—6521no—
153C2×C4—6521no—
154C2×C4—6521no—
155C2×C4—6521no—
156C2×C4—6528no—
157C2×C4—6522no—
158C2×C4—6522no—
159C2×C4C4⋊Dic16312yes—
160C2×C4C22×Dic16311yescharacteristic
161C2×C4D4×D16312yes—
162C2×C4C2×Dic32611yescharacteristic
163C2×C4#1457969812yes—
164C8—24no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#1910466714yescomplemented
172C22C2×C4⋊Dic16311yescharacteristic
173C22#1910470511yescharacteristic
174C22#1910471911yescharacteristic
175C22C2×#1457968811yescharacteristic
176C22C2×#1457968811yescharacteristic
177C22#1910476311yescharacteristic
178C22#1910476811yescharacteristic
179C4—22no—
180C4—13041no—
181C4—13041no—
182C4—6524no—
183C4D4×Dic16312yes—
184C4#1910471212yes—
185C2—24no—
186C2#2378154011yescharacteristic
187C2C2×#1910466711yescharacteristic
188C2#2378170711yescharacteristic
189C2#2378180611yescharacteristic
190C2#2378181811yescharacteristic
191C2#2378198411yescharacteristic
192C2#2378199611yescharacteristic
subgroupG/H
centerC23C2×D652
commutatorC2×C4×C163C22×C4
FrattiniC22×C4C22×D163
FittingC2×C22⋊C8×C163C2
socleC23×C163C2×D4
prime radicalC24×C163D4
cosocleC22×C4×C163C23
hypercenterC2×C22⋊C8D163
hypocenterC163#1853
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#19104667C2semidirect
4. #23781428C2semidirect
17. C22⋊C8×C163C4semidirect
39. C2×C8×C163C2×C4semidirect
74. C22×C163#106semidirect
96. C163#1853semidirect
113. C22⋊C8Dic163semidirect
130. C2×C8C2×Dic163semidirect
171. C22#19104667semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1632
C2×#191046672
#237815252
#237814282
C2×C22⋊C8×C1632
#1853163