Finite Group Databasebeta

← #35213461

Subgroups — #35213461

Order 20864.

316 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#23782291C211yesmaximal, characteristic, complemented
2#23781818C212yesmaximal, complemented
3#23781574C211yesmaximal, characteristic
4#23781403C212yesmaximal, complemented
5#323×C163C211yesmaximal, characteristic, complemented
6C22×D652C2211yescharacteristic
7C22×C22⋊D163—21no—
8C2×#14579688—22no—
9#19104714—21no—
10#19104714C2212yes—
11#19104712—22no—
12#19104712—22no—
13C2×C22⋊Dic163—21no—
14C2×(C2×D163)⋊C4—21no—
15#19104657—22no—
16C163⋊#95C2211yescharacteristic
17C2×C22⋊C4×C163C2211yescharacteristic
18C22⋊C8×C163C2212yes—
19C23×D163—21no—
20C2×Dic326—21no—
21C2×Dic326D412yescomplemented
22C2×D652—22no—
23C2×D652—42no—
24C2×C22⋊D163—42no—
25C2×C22⋊D163—42no—
26C2×C22⋊D163—42no—
27C22×Dic163—21no—
28C22×Dic163—21no—
29C22×Dic163—21no—
30#14579688—44no—
31C4⋊Dic163—21no—
32C4⋊Dic163D412yescomplemented
33Dic163⋊C4—22no—
34Dic163⋊C4—22no—
35C22⋊Dic163—22no—
36C22⋊Dic163—22no—
37(C2×D163)⋊C4—42no—
38C24×C163D411yescharacteristic
39C22×C4×C163D411yescharacteristic
40C22×C4×C163C2311yescharacteristic
41C2×C8×C163—22no—
42C22⋊C4×C163—22no—
43C22×D163—22no—
44C22×D163—41no—
45C22×D163—41no—
46C22×D163—41no—
47C22×D163—41no—
48C22×D163—41no—
49C22×D163—41no—
50Dic326—42no—
51Dic326—81no—
52D652—24no—
53D652—42no—
54C22⋊D163—48no—
55C2×Dic163—22no—
56C2×Dic163—22no—
57C2×Dic163—22no—
58C2×Dic163—41no—
59C2×Dic163—41no—
60C2×Dic163—81no—
61C2×Dic163—81no—
62C23×C163—21no—
63C23×C163—21no—
64C23×C163—41no—
65C23×C163—41no—
66C23×C163SD1612yescomplemented
67C23×C163C2×D411yescharacteristic
68C2×C4×C163—21no—
69C2×C4×C163—41no—
70C2×C4×C163—41no—
71C2×C4×C163C2×D412yes—
72C8×C163—42no—
73C22×C163—21no—
74C22×C163—21no—
75C22×C163—22no—
76C22×C163—42no—
77C22×C163—41no—
78C22×C163—41no—
79C22×C163—81no—
80C22×C163#12011yescharacteristic
81C22×C163C2×SD1611yescharacteristic
82C22×C163#13711yescharacteristic
83C2×D163—42no—
84C2×D163—42no—
85C2×D163—42no—
86C2×D163—81no—
87C2×D163—81no—
88C2×D163—81no—
89C2×D163—81no—
90C4×C163—22no—
91C4×C163—41no—
92Dic163—42no—
93Dic163—42no—
94Dic163—42no—
95C2×C163—21no—
96C2×C163—21no—
97C2×C163—42no—
98C2×C163#44811yescharacteristic
99C2×C163#45311yescharacteristic
100C2×C163B(4,𝔽2)11yescharacteristic
101D163—82no—
102C163#159611yescharacteristic, complemented, Sylow
103#1596—1631nomaximal, Sylow
104#543—1631no—
105#450—1632no—
106#392—1631no—
107#328—1632no—
108#323D16311yescharacteristic, complemented
109C22×D4—3261no—
110C22×D4—1631no—
111C2×SD16—3262no—
112C22⋊Q8—3262no—
113C22⋊Q8—3262no—
114C22⋊Q8—3261no—
115C22⋊Q8—1632no—
116C2×C22⋊C4—3261no—
117C2×C22⋊C4—3261no—
118C2×C22⋊C4C2×D16311yescharacteristic
119#102—3262no—
120C22⋊C8C2×D16312yes—
121#95—1631no—
122C24—3261no—
123C24D65211yescharacteristic
124C2×Q8—3261no—
125C2×Q8—1632no—
126C2×D4—3262no—
127C2×D4—6522no—
128C2×D4—6522no—
129C2×D4—6522no—
130C2×D4—6522no—
131C22×C4—3261no—
132C22×C4—3261no—
133C22×C4—3261no—
134C22×C4D65211yescharacteristic
135C22×C4C22×D16311yescharacteristic
136SD16—6524no—
137C2×C8—22no—
138C4⋊C4—3262no—
139C4⋊C4—3261no—
140C4⋊C4—3262no—
141C4⋊C4—1632no—
142C22⋊C4—22no—
143C22⋊C4—3262no—
144C22⋊C4—3262no—
145C22⋊C4—6522no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—3262no—
151C23—6521no—
152C23—6521no—
153C23—6521no—
154C23—6521no—
155C23—6521no—
156C23—6521no—
157C23#1457968812yescomplemented
158C23C2×D65211yescharacteristic
159Q8—13041no—
160Q8—6522no—
161D4—3264no—
162D4—6522no—
163D4—6528no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—13041no—
168C2×C4—13041no—
169C2×C4—3262no—
170C2×C4—3262no—
171C2×C4—3262no—
172C2×C4—6521no—
173C2×C4—6521no—
174C2×C4D4×D16312yes—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—13041no—
184C22—13041no—
185C22—13041no—
186C22—13041no—
187C22—6522no—
188C22—6522no—
189C22—6522no—
190C22#1910470511yescharacteristic
191C22C2×#1457968811yescharacteristic
192C22#1910476811yescharacteristic
193C4—22no—
194C4—41no—
195C4—6522no—
196C4—6522no—
197C4—6522no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—13042no—
202C2#2378180611yescharacteristic
203C2#2378183511yescharacteristic
204C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#323×C163C2
socleC22×C163#120
prime radical#23781574C2
hypercenter#323D163
hypocenterC163#1596
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #23782291C2semidirect
2. #23781818C2semidirect
4. #23781403C2semidirect
5. #323×C163C2semidirect
21. C2×Dic326D4semidirect
32. C4⋊Dic163D4semidirect
66. C23×C163SD16semidirect
102. C163#1596semidirect
108. #323D163semidirect
157. C23#14579688semidirect

Maximal subgroups

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

subgroupindex
#237822912
#237818182
#237815742
#237814032
#323×C1632
#1596163