Finite Group Databasebeta

← #35217873

Subgroups — #35217873

Order 20864.

864 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 169 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×#14579699C211yesmaximal, characteristic
2C2×#19104719C212yesmaximal, complemented
3C22×Dic163⋊C4C211yesmaximal, characteristic, complemented
4#23781586C212yesmaximal, complemented
5#23781583C212yesmaximal, complemented
6#23781542C212yesmaximal, complemented
7#23781539C214yesmaximal, complemented
8#381×C163C211yesmaximal, characteristic, complemented
9C2×#14579699—28no—
10C2×#14579699—24no—
11C22×Dic326C2211yescharacteristic
12C22×D652C2211yescharacteristic
13C22×C22⋊D163C2212yes—
14C23×Dic163C2211yescharacteristic
15C23×Dic163C2211yescharacteristic
16C22×C4×D163C2212yes—
17#19104719—28no—
18C2×C4⋊Dic163C2214yes—
19C2×Dic163⋊C4—24no—
20C2×Dic163⋊C4C2212yes—
21C2×Dic163⋊C4C2212yes—
22C2×(C2×D163)⋊C4C2214yes—
23C2×(C2×D163)⋊C4C2214yes—
24C163⋊#95C2214yes—
25C23×C4×C163C2211yescharacteristic
26C2×C22⋊C4×C163C2212yes—
27#95×C163C2212yes—
28#95×C163C2212yes—
29C23×D163C2312yes—
30#14579699—232no—
31C2×Dic326—24no—
32C2×Dic326—22no—
33C2×D652—24no—
34C2×D652—22no—
35C2×C22⋊D163—24no—
36C2×C22⋊D163—24no—
37C2×C22⋊D163D418yes—
38C22×Dic163—24no—
39C22×Dic163—22no—
40C22×Dic163—22no—
41C22×Dic163—24no—
42C22×Dic163C2312yes—
43C22×Dic163C2312yes—
44C22×Dic163C2312yes—
45C2×C4×D163—24no—
46C2×C4×D163—24no—
47C2×C4×D163—24no—
48C4⋊Dic163—28no—
49Dic163⋊C4—24no—
50Dic163⋊C4—24no—
51(C2×D163)⋊C4—28no—
52(C2×D163)⋊C4—28no—
53C24×C163C2311yescharacteristic
54C22×C4×C163—24no—
55C22×C4×C163D414yes—
56C22×C4×C163C2314yes—
57C22×C4×C163C2311yescharacteristic
58C22×C4×C163C2311yescharacteristic
59C22⋊C4×C163—24no—
60C22×D163—22no—
61C22×D163—22no—
62C22×D163—24no—
63C22×D163—24no—
64C22×D163C2×D414yes—
65Dic326—28no—
66D652—28no—
67C22⋊D163—216no—
68C2×Dic163—28no—
69C2×Dic163—22no—
70C2×Dic163—22no—
71C2×Dic163—22no—
72C2×Dic163—22no—
73C2×Dic163—24no—
74C2×Dic163—22no—
75C2×Dic163—44no—
76C2×Dic163C2×D414yes—
77C2×Dic163#4114yescomplemented
78C4×D163—216no—
79C23×C163—22no—
80C23×C163—22no—
81C23×C163C2×D414yes—
82C23×C163C2×D412yes—
83C23×C163C2411yescharacteristic
84C2×C4×C163—28no—
85C2×C4×C163—22no—
86C2×C4×C163—22no—
87C2×C4×C163—24no—
88C2×C4×C163—24no—
89C2×C4×C163—24no—
90C2×C4×C163C2×D412yes—
91C2×C4×C163C2×D412yes—
92C22×C163—22no—
93C22×C163—22no—
94C22×C163—28no—
95C22×C163#12014yes—
96C22×C163C22×D412yes—
97C22×C163C22×D411yescharacteristic
98C22×C163C2×#4112yes—
99C22×C1632−1+412yes—
100C2×D163—24no—
101C2×D163—28no—
102C2×D163—42no—
103C2×D163—42no—
104C2×D163—44no—
105C4×C163—22no—
106C4×C163—22no—
107C4×C163—44no—
108Dic163—24no—
109Dic163—24no—
110Dic163—42no—
111C2×C163—24no—
112C2×C163C2×#12011yescharacteristic
113C2×C163#52511yescharacteristic
114C2×C163#53611yescharacteristic
115C2×C163#54714yes—
116D163—44no—
117C163#240611yescharacteristic, complemented, Sylow
118#2406—1631nomaximal, Sylow
119C22×#41—1631no—
120C2×#123—1632no—
121C22×C4⋊C4—1631no—
122#395—1632no—
123#394—1632no—
124#386—1632no—
125#385—1634no—
126#381D16311yescharacteristic, complemented
127C2×#41—3268no—
128C2×#41—3264no—
129C22×Q8—1631no—
130C22×D4—1632no—
131C22×D4—1631no—
132C23×C4—1631no—
133C23×C4—1631no—
134C23×C4—1632no—
135C23×C4C2×D16311yescharacteristic
136#123—3268no—
137C2×C4⋊C4—3264no—
138C2×C4⋊C4—1632no—
139C2×C4⋊C4—1632no—
140C2×C4⋊C4—1634no—
141C2×C22⋊C4—1634no—
142C2×C22⋊C4—1634no—
143C2×C22⋊C4C2×D16312yes—
144#95—1634no—
145#95C2×D16312yes—
146#95C2×D16312yes—
147C24—1632no—
148C24C22×D16311yescharacteristic
149#41—32632no—
150C2×Q8—3264no—
151C2×Q8—3262no—
152C2×D4—3264no—
153C2×D4—3264no—
154C2×D4—3264no—
155C2×D4—3262no—
156C2×D4—1638no—
157C22×C4—24no—
158C22×C4—3264no—
159C22×C4—3264no—
160C22×C4—3262no—
161C22×C4—3264no—
162C22×C4—3264no—
163C22×C4—3264no—
164C22×C4—3262no—
165C22×C4—1632no—
166C22×C4—1632no—
167C22×C4—1632no—
168C22×C4D65214yes—
169C22×C4C22×D16314yes—
170C22×C4C22×D16311yescharacteristic
171C22×C4C22×D16311yescharacteristic
172C4⋊C4—3268no—
173C4⋊C4—3264no—
174C4⋊C4—3264no—
175C22⋊C4—24no—
176C22⋊C4—3268no—
177C22⋊C4—3268no—
178C23—22no—
179C23—22no—
180C23—3262no—
181C23—3262no—
182C23—3264no—
183C23—3264no—
184C23—1634no—
185C23D4×D16314yes—
186C23C2×D65212yes—
187C23C23×D16311yescharacteristic
188Q8—3268no—
189D4—3268no—
190D4—32616no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—28no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—32616no—
198C2×C4—3268no—
199C2×C4—3262no—
200C2×C4—3262no—
201C2×C4—3262no—
202C2×C4—3262no—
203C2×C4—3264no—
204C2×C4—3262no—
205C2×C4—6524no—
206C2×C4—1634no—
207C2×C4—1634no—
208C2×C4C2×D65212yes—
209C2×C4C2×D65212yes—
210C22—22no—
211C22—22no—
212C22—28no—
213C22—3264no—
214C22—3268no—
215C22—6522no—
216C22—6522no—
217C22—6524no—
218C22#1910470514yes—
219C22C2×D4×D16312yes—
220C22C22×D65211yescharacteristic
221C22#41×D16312yes—
222C22#1910478212yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—3264no—
227C4—3264no—
228C4—6522no—
229C2—24no—
230C2—6524no—
231C2C2×#1910470511yescharacteristic
232C2#2378217711yescharacteristic
233C2#2378224111yescharacteristic
234C2#2378231314yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#381×C163C2
prime radicalC22×#14579699C2
hypercenter#381D163
hypocenterC163#2406
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#19104719C2semidirect
3. C22×Dic163⋊C4C2semidirect
4. #23781586C2semidirect
5. #23781583C2semidirect
6. #23781542C2semidirect
7. #23781539C2semidirect
8. #381×C163C2semidirect
77. C2×Dic163#41semidirect
117. C163#2406semidirect
126. #381D163semidirect

Maximal subgroups

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

subgroupindex
C22×#145796992
C2×#191047192
C22×Dic163⋊C42
#237815862
#237815832
#237815422
#237815392
#381×C1632
#2406163