Finite Group Databasebeta

← #35214568

Subgroups — #35214568

Order 20864.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14579682C211yesmaximal, characteristic
2C2×#19104668C211yesmaximal, characteristic, complemented
3C2×#19104666C211yesmaximal, characteristic, complemented
4#23781522C211yesmaximal, characteristic
5#23781522C211yesmaximal, characteristic
6#23781428C211yesmaximal, characteristic, complemented
7#340×C163C211yesmaximal, characteristic, complemented
8C22×C4×D163C2211yescharacteristic
9C2×#14579682—22no—
10C2×#14579682—22no—
11C2×#14579682C414yescomplemented
12C22×C163⋊C8C2211yescharacteristic
13C2×C4⋊Dic163C2211yescharacteristic
14C2×C4⋊Dic163C2211yescharacteristic
15C2×(C2×D163)⋊C4—21no—
16C2×(C2×D163)⋊C4—21no—
17#19104668—22no—
18#19104666—22no—
19C163⋊#95—21no—
20C163⋊#95—21no—
21C22×C8×C163C2211yescharacteristic
22C2×C4⋊C4×C163C2211yescharacteristic
23C2×C4⋊C4×C163C2211yescharacteristic
24C23×D163D411yescharacteristic
25C22×Dic163—21no—
26C22×Dic163—21no—
27C22×Dic163D411yescharacteristic
28C2×C4×D163—21no—
29C2×C4×D163—21no—
30C2×C4×D163—21no—
31C2×C4×D163—21no—
32C2×C4×D163C2×C412yes—
33C2×C4×D163Q812yes—
34#14579682—28no—
35C2×C163⋊C8C2×C412yes—
36C2×C163⋊C8D412yes—
37C2×C163⋊C8D412yes—
38C4⋊Dic163—21no—
39C4⋊Dic163—21no—
40C4⋊Dic163—21no—
41C4⋊Dic163—21no—
42(C2×D163)⋊C4—42no—
43(C2×D163)⋊C4—42no—
44C22×C4×C163—21no—
45C22×C4×C163—21no—
46C22×C4×C163C2311yescharacteristic
47C2×C8×C163—21no—
48C2×C8×C163—21no—
49C2×C8×C163C2×C412yes—
50C4⋊C4×C163—21no—
51C4⋊C4×C163—21no—
52C4⋊C4×C163—21no—
53C4⋊C4×C163—21no—
54C22×D163—21no—
55C22×D163—21no—
56C22×D163—21no—
57C22×D163—21no—
58C22×D163—21no—
59C22×D163—21no—
60C22×D163C4⋊C412yes—
61C2×Dic163—21no—
62C2×Dic163—21no—
63C2×Dic163—41no—
64C2×Dic163—41no—
65C2×Dic163—41no—
66C2×Dic163—41no—
67C2×Dic163—41no—
68C2×Dic163—41no—
69C2×Dic163C4⋊C412yes—
70C4×D163—22no—
71C4×D163—22no—
72C4×D163—22no—
73C4×D163—22no—
74C163⋊C8C22⋊C414yescomplemented
75C23×C163C2×D411yescharacteristic
76C2×C4×C163—41no—
77C2×C4×C163—41no—
78C2×C4×C163—41no—
79C2×C4×C163—41no—
80C2×C4×C163—41no—
81C2×C4×C163—41no—
82C2×C4×C163C22×C411yescharacteristic
83C2×C4×C163C2×D411yescharacteristic
84C2×C4×C163C2×D411yescharacteristic
85C2×C4×C163C2×Q811yescharacteristic
86C2×C4×C163#4111yescharacteristic
87C2×C4×C163#4111yescharacteristic
88C8×C163—22no—
89C22×C163C2×C4⋊C411yescharacteristic
90C22×C163#12111yescharacteristic
91C22×C163#12111yescharacteristic
92C22×C163AGL(1,ℤ/8)11yescharacteristic
93C22×C163AGL(1,ℤ/8)11yescharacteristic
94C22×C163#13711yescharacteristic
95C22×C163#13711yescharacteristic
96C2×D163—22no—
97C2×D163—22no—
98C2×D163—22no—
99C2×D163—41no—
100C2×D163—41no—
101C2×D163—41no—
102C2×D163—41no—
103C4×C163—81no—
104C4×C163—81no—
105C4×C163C2×C22⋊C411yescharacteristic
106C4×C163#11711yescharacteristic
107C4×C163C22⋊Q811yescharacteristic
108C4×C163C22⋊Q811yescharacteristic
109Dic163—22no—
110Dic163—81no—
111Dic163—81no—
112C2×C163#38011yescharacteristic
113C2×C163#42811yescharacteristic
114C2×C163#42811yescharacteristic
115C2×C163#46811yescharacteristic
116C2×C163#46811yescharacteristic
117C2×C163#46911yescharacteristic
118C2×C163#47011yescharacteristic
119D163—42no—
120C163#180911yescharacteristic, complemented, Sylow
121#1809—1631nomaximal, Sylow
122C22×M16—1631no—
123C2×#107—1631no—
124C2×#106—1631no—
125#380—1631no—
126#380—1631no—
127#340—1631no—
128#340D16311yescharacteristic, complemented
129C23×C4—1631no—
130C2×M16—3262no—
131C2×M16—3262no—
132C2×M16—1634no—
133C22×C8—1631no—
134C22×C8C2×D16311yescharacteristic
135C2×C4⋊C4—1631no—
136C2×C4⋊C4—1631no—
137C2×C4⋊C4C2×D16311yescharacteristic
138C2×C4⋊C4C2×D16311yescharacteristic
139C2×C22⋊C4—3261no—
140C2×C22⋊C4—3261no—
141#107—3262no—
142#106—3262no—
143#95—3261no—
144#95—3261no—
145C24—1631no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3261no—
149C22×C4—3261no—
150C22×C4—3261no—
151C22×C4—3261no—
152C22×C4—3261no—
153C22×C4—3261no—
154C22×C4—1631no—
155C22×C4—1632no—
156C22×C4—1632no—
157C22×C4C22×D16311yescharacteristic
158M16—3268no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—1632no—
162C2×C8—1632no—
163C2×C8—1632no—
164C2×C8C4×D16312yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—3261no—
170C4⋊C4—3261no—
171C4⋊C4—3261no—
172C4⋊C4—3261no—
173C22⋊C4—6522no—
174C22⋊C4—6522no—
175C23—3261no—
176C23—3261no—
177C23—3261no—
178C23—3261no—
179C23—3261no—
180C23—3261no—
181C23—1632no—
182C23D4×D16311yescharacteristic
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3261no—
190C2×C4—3261no—
191C2×C4—3262no—
192C2×C4—3262no—
193C2×C4—3262no—
194C2×C4—3262no—
195C2×C4—6521no—
196C2×C4—6521no—
197C2×C4—6521no—
198C2×C4—6521no—
199C2×C4—6521no—
200C2×C4—6521no—
201C2×C4—1632no—
202C2×C4C2×C4×D16311yescharacteristic
203C2×C4D4×D16311yescharacteristic
204C2×C4D4×D16311yescharacteristic
205C2×C4Q8×D16311yescharacteristic
206C2×C4#1457969911yescharacteristic
207C2×C4#1457969911yescharacteristic
208C8—22no—
209C8—1634no—
210C22—3262no—
211C22—3262no—
212C22—3262no—
213C22—6521no—
214C22—6521no—
215C22—6521no—
216C22—6521no—
217C22C4⋊C4×D16311yescharacteristic
218C22#1910470811yescharacteristic
219C22#1910470811yescharacteristic
220C22#1910475911yescharacteristic
221C22#1910476211yescharacteristic
222C22#1910476411yescharacteristic
223C22#1910476711yescharacteristic
224C4—81no—
225C4—81no—
226C4—13041no—
227C4—13041no—
228C4—3262no—
229C4C22⋊C4×D16311yescharacteristic
230C4#1910469211yescharacteristic
231C4#1910471311yescharacteristic
232C4#1910471311yescharacteristic
233C2—6522no—
234C2#2378152411yescharacteristic
235C2#2378170411yescharacteristic
236C2#2378170611yescharacteristic
237C2#2378191011yescharacteristic
238C2#2378191411yescharacteristic
239C2#2378191711yescharacteristic
240C2#2378192211yescharacteristic
subgroupG/H
centerC23D4×D163
commutatorC2×C4×C163C22×C4
FrattiniC22×C4C22×D163
Fitting#340×C163C2
socleC23×C163C2×D4
prime radicalC23×D163D4
cosocleC22×C4×C163C23
hypercenter#340D163
hypocenterC163#1809
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#19104668C2semidirect
3. C2×#19104666C2semidirect
6. #23781428C2semidirect
7. #340×C163C2semidirect
11. C2×#14579682C4semidirect
74. C163⋊C8C22⋊C4semidirect
120. C163#1809semidirect
128. #340D163semidirect

Maximal subgroups

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

subgroupindex
C22×#145796822
C2×#191046682
C2×#191046662
#237815222
#237814282
#340×C1632
#1809163