Finite Group Databasebeta

← #35215788

Subgroups — #35215788

Order 20864.

382 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 67 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#23782227C211yesmaximal, characteristic
2C2×#19104649C212yesmaximal, complemented
3#23781542C212yesmaximal, complemented
4#23781537C211yesmaximal, characteristic, complemented
5#342×C163C211yesmaximal, characteristic, complemented
6C22×C22⋊D163C2212yes—
7C23×Dic163C2211yescharacteristic
8#19104719—22no—
9#19104709—21no—
10#19104709—21no—
11#19104705—22no—
12C2×Dic163⋊C4—21no—
13C2×C22⋊Dic163C2211yescharacteristic
14C2×(C2×D163)⋊C4—22no—
15#19104649—24no—
16C163⋊#95—22no—
17C163⋊#95—21no—
18C2×C22⋊C4×C163C2212yes—
19C2×C22⋊C4×C163C2211yescharacteristic
20C23×D163D412yes—
21C2×D652—21no—
22C2×D652—21no—
23C2×C22⋊D163—22no—
24C2×C22⋊D163—22no—
25C2×C22⋊D163—22no—
26C2×C22⋊D163—22no—
27C2×C22⋊D163—22no—
28C2×C22⋊D163—22no—
29C22×Dic163—22no—
30C22×Dic163—21no—
31C22×Dic163—21no—
32C22×Dic163—42no—
33C22×Dic163D412yes—
34C4⋊Dic163—21no—
35C4⋊Dic163—21no—
36Dic163⋊C4—42no—
37C22⋊Dic163—21no—
38C22⋊Dic163D412yescomplemented
39(C2×D163)⋊C4—22no—
40(C2×D163)⋊C4—22no—
41(C2×D163)⋊C4—44no—
42C24×C163C2311yescharacteristic
43C22×C4×C163—22no—
44C22×C4×C163D411yescharacteristic
45C22×C4×C163D411yescharacteristic
46C22⋊C4×C163—22no—
47C22⋊C4×C163—22no—
48C22⋊C4×C163—22no—
49C22×D163—22no—
50C22×D163—22no—
51C22×D163—22no—
52C22×D163—42no—
53C22×D163—42no—
54D652—81no—
55D652—81no—
56C22⋊D163—42no—
57C22⋊D163—42no—
58C22⋊D163—42no—
59C22⋊D163—42no—
60C2×Dic163—24no—
61C2×Dic163—22no—
62C2×Dic163—22no—
63C2×Dic163—22no—
64C2×Dic163—22no—
65C2×Dic163—22no—
66C2×Dic163—41no—
67C2×Dic163—41no—
68C2×Dic163—41no—
69C2×Dic163—42no—
70C2×Dic163—41no—
71C23×C163—41no—
72C23×C163—41no—
73C23×C163C2×D412yes—
74C23×C163C2×D411yescharacteristic
75C23×C163C2×D411yescharacteristic
76C23×C163#4112yes—
77C23×C163#4111yescharacteristic
78C2×C4×C163—21no—
79C2×C4×C163—21no—
80C2×C4×C163—41no—
81C2×C4×C163—41no—
82C2×C4×C163—42no—
83C2×C4×C163—42no—
84C2×C4×C163—42no—
85C22×C163—21no—
86C22×C163—22no—
87C22×C163—21no—
88C22×C163—22no—
89C22×C163—41no—
90C22×C163—42no—
91C22×C163—41no—
92C22×C163#12011yescharacteristic
93C22×C163#12112yes—
94C22×C163#12111yescharacteristic
95C22×C163#12312yes—
96C22×C163#12411yescharacteristic
97C2×D163—44no—
98C2×D163—42no—
99C2×D163—42no—
100C2×D163—42no—
101C2×D163—82no—
102C4×C163—41no—
103C4×C163—41no—
104C4×C163—82no—
105Dic163—42no—
106Dic163—42no—
107Dic163—42no—
108C2×C163—21no—
109C2×C163—22no—
110C2×C163—22no—
111C2×C163—21no—
112C2×C163#39411yescharacteristic
113C2×C163B(4,𝔽2)11yescharacteristic
114C2×C163#45811yescharacteristic
115D163—82no—
116C163#204411yescharacteristic, complemented, Sylow
117#2044—1631nomaximal, Sylow
118#534—1631no—
119C2×#99—1632no—
120#386—1632no—
121#385—1631no—
122#342D16311yescharacteristic, complemented
123C22×D4—1632no—
124C23×C4—1631no—
125#123—3262no—
126#121—3261no—
127#121—3261no—
128#120—3262no—
129C2×C4⋊C4—3261no—
130C2×C22⋊C4—3262no—
131C2×C22⋊C4—1631no—
132C2×C22⋊C4C2×D16312yes—
133C2×C22⋊C4C2×D16311yescharacteristic
134#99—3264no—
135#95—3262no—
136#95—3261no—
137C24—1632no—
138C24C22×D16311yescharacteristic
139C2×D4—3261no—
140C2×D4—3261no—
141C2×D4—3262no—
142C2×D4—3262no—
143C2×D4—3262no—
144C2×D4—3262no—
145C2×D4—3262no—
146C2×D4—3262no—
147C22×C4—22no—
148C22×C4—3262no—
149C22×C4—3261no—
150C22×C4—3261no—
151C22×C4—6522no—
152C22×C4—1632no—
153C22×C4D65211yescharacteristic
154C22×C4D65211yescharacteristic
155C4⋊C4—3261no—
156C4⋊C4—3261no—
157C4⋊C4—6522no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—3262no—
162C22⋊C4—3262no—
163C22⋊C4—3261no—
164C22⋊C4—6524no—
165C22⋊C4—1632no—
166C23—41no—
167C23—41no—
168C23—3262no—
169C23—3262no—
170C23—3262no—
171C23—6522no—
172C23—6522no—
173C23D4×D16312yes—
174C23D4×D16311yescharacteristic
175C23C2×D65211yescharacteristic
176C23#1457969912yes—
177C23#1457970011yescharacteristic
178D4—13041no—
179D4—13041no—
180D4—6522no—
181D4—6522no—
182D4—6522no—
183D4—6522no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—3264no—
192C2×C4—3262no—
193C2×C4—3262no—
194C2×C4—3262no—
195C2×C4—3262no—
196C2×C4—3262no—
197C2×C4—6521no—
198C2×C4—6521no—
199C2×C4—6522no—
200C2×C4—6521no—
201C2×C4—6521no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—42no—
208C22—41no—
209C22—13042no—
210C22—6524no—
211C22—6522no—
212C22—6522no—
213C22—6522no—
214C22#1910470511yescharacteristic
215C22#1910470812yes—
216C22#1910471011yescharacteristic
217C22#1910472012yes—
218C22#1910472311yescharacteristic
219C4—41no—
220C4—41no—
221C4—82no—
222C4—6522no—
223C4—6522no—
224C4—6522no—
225C2—22no—
226C2—21no—
227C2—22no—
228C2—21no—
229C2—13042no—
230C2#2378158311yescharacteristic
231C2#2378185311yescharacteristic
232C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC24×C163C23
FrattiniC24C22×D163
Fitting#342×C163C2
socleC22×C163#120
prime radical#23782227C2
hypercenter#342D163
hypocenterC163#2044
LCSG ▷ C24×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#19104649C2semidirect
3. #23781542C2semidirect
4. #23781537C2semidirect
5. #342×C163C2semidirect
38. C22⋊Dic163D4semidirect
116. C163#2044semidirect
122. #342D163semidirect

Maximal subgroups

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

subgroupindex
#237822272
C2×#191046492
#237815422
#237815372
#342×C1632
#2044163