Finite Group Databasebeta

← #35224926

Subgroups — #35224926

Order 20864.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#23782213C211yesmaximal, characteristic
2C4⋊Q8×D163C211yesmaximal, characteristic, complemented
3#23782023C214yesmaximal, complemented
4#23782017C211yesmaximal, characteristic, complemented
5#23782016C211yesmaximal, characteristic, complemented
6#23782011C211yesmaximal, characteristic, complemented
7#23781663C214yesmaximal, complemented
8C4×#14579682C211yesmaximal, characteristic, complemented
9#486×C163C211yesmaximal, characteristic, complemented
10C2×#14579698C2212yes—
11C2×Q8×D163—21no—
12C2×#14579682C2212yes—
13#19104735C2211yescharacteristic
14#19104735C2211yescharacteristic
15#19104734—21no—
16#19104734C2212yes—
17#19104733—21no—
18#19104722C2212yes—
19#19104706—21no—
20#19104706—21no—
21D4×Dic163—21no—
22D4×Dic163—21no—
23C4⋊C4×D163—21no—
24C4⋊C4×D163C2212yes—
25C42×D163C2211yescharacteristic
26#19104660C2214yes—
27#19104658C2214yes—
28#19104656C2214yes—
29#19104645C2212yes—
30C4×C163⋊C8C2211yescharacteristic
31C4⋊Q8×C163C2211yescharacteristic
32#127×C163C2211yescharacteristic
33#102×C163C2214yes—
34C4×C8×C163C2211yescharacteristic
35#14579698—28no—
36C2×Dic326—22no—
37C2×Dic326—21no—
38C2×Dic326C2312yes—
39Q8×D163—44no—
40C2×C22⋊D163—22no—
41C22×Dic163—22no—
42C2×C4×D163—22no—
43C2×C4×D163D411yescharacteristic
44C2×C4×D163D411yescharacteristic
45C2×C4×D163C2311yescharacteristic
46#14579682—24no—
47C2×C163⋊C8C2312yes—
48C4⋊Dic163—21no—
49C4⋊Dic163—21no—
50C4⋊Dic163—21no—
51C4⋊Dic163C2312yes—
52Dic163⋊C4—22no—
53Dic163⋊C4—22no—
54C22⋊Dic163—22no—
55C22⋊Dic163—22no—
56C4×Dic163D411yescharacteristic
57C4×Dic163D411yescharacteristic
58C4×Dic163C2311yescharacteristic
59C2×Q8×C163—21no—
60C2×D4×C163—21no—
61C2×D4×C163—21no—
62C2×D4×C163C2312yes—
63C2×C8×C163C2312yes—
64C4⋊C4×C163—21no—
65C4⋊C4×C163C2312yes—
66C42×C163C2311yescharacteristic
67C22×D163C2×D411yescharacteristic
68Dic326—24no—
69Dic326—44no—
70Dic326—42no—
71C22⋊D163—44no—
72C2×Dic163—22no—
73C2×Dic163—22no—
74C2×Dic163—44no—
75C2×Dic163C2×D411yescharacteristic
76C2×Dic163C2×D411yescharacteristic
77C2×Dic163C2×D411yescharacteristic
78C4×D163—22no—
79C4×D163—22no—
80C4×D163—44no—
81C4×D163#4114yes—
82C163⋊C8—22no—
83C23×C163—22no—
84Q8×C163—42no—
85D4×C163—24no—
86D4×C163—42no—
87D4×C163—42no—
88C2×C4×C163—22no—
89C2×C4×C163C2×D411yescharacteristic
90C2×C4×C163C2×D411yescharacteristic
91C2×C4×C163C2411yescharacteristic
92C8×C163—22no—
93C22×C163—42no—
94C22×C163—42no—
95C22×C163—42no—
96C22×C163C22×D411yescharacteristic
97C2×D163—21no—
98C2×D163—21no—
99C2×D163#12412yes—
100C4×C163—42no—
101C4×C163AGL(1,ℤ/8)12yes—
102C4×C163#13712yes—
103C4×C163C2×#4112yes—
104Dic163—21no—
105Dic163—21no—
106Dic163—42no—
107Dic163—42no—
108Dic163#12412yes—
109C2×C163—82no—
110C2×C163C2×#12411yescharacteristic
111C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C163C2×#13711yescharacteristic
113D163—22no—
114C163#314111yescharacteristic, complemented, Sylow
115#3141—1631nomaximal, Sylow
116#532—1631no—
117C2×C4⋊Q8—1631no—
118#489—1634no—
119#487—1631no—
120#487—1631no—
121#486—1631no—
122#486D16311yescharacteristic, complemented
123#417—1634no—
124C4×M16—1631no—
125C2×#41—1632no—
126C22×Q8—3261no—
127C2×M16—1632no—
128C4⋊Q8—3261no—
129C4⋊Q8—3261no—
130C4⋊Q8—1632no—
131C4⋊Q8—1631no—
132C4⋊Q8—1631no—
133C4⋊Q8C2×D16311yescharacteristic
134#127C2×D16311yescharacteristic
135#124—1632no—
136#121—3261no—
137#121—3261no—
138C4×D4—3261no—
139C4×D4—3261no—
140C2×C4⋊C4—3261no—
141C2×C4⋊C4—1632no—
142C2×C42—1631no—
143#103—1634no—
144#103—1634no—
145#102—1634no—
146#102C2×D16314yes—
147#97—1632no—
148C4×C8—1631no—
149C4×C8C2×D16311yescharacteristic
150#41—3268no—
151C2×Q8—21no—
152C2×Q8—3262no—
153C2×Q8—3261no—
154C2×Q8—6524no—
155C2×Q8—1632no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—3262no—
159C2×D4C22×D16312yes—
160C22×C4—3262no—
161C22×C4—3262no—
162C22×C4—1631no—
163C22×C4—1631no—
164C22×C4—1631no—
165M16—3264no—
166C2×C8—1632no—
167C2×C8C22×D16312yes—
168C4⋊C4—21no—
169C4⋊C4—3262no—
170C4⋊C4—3262no—
171C4⋊C4—3261no—
172C4⋊C4—3261no—
173C4⋊C4—3261no—
174C4⋊C4—1632no—
175C4⋊C4C22×D16312yes—
176C22⋊C4—3262no—
177C22⋊C4—3262no—
178C42—1631no—
179C42—1631no—
180C42—1631no—
181C42C22×D16311yescharacteristic
182C23—22no—
183C23—1631no—
184Q8—42no—
185Q8—3264no—
186Q8—6524no—
187Q8—6522no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—6524no—
192C2×C4—22no—
193C2×C4—3262no—
194C2×C4—3262no—
195C2×C4—3262no—
196C2×C4—3262no—
197C2×C4—6524no—
198C2×C4—6524no—
199C2×C4—1631no—
200C2×C4—1631no—
201C2×C4—1631no—
202C2×C4—1634no—
203C2×C4D4×D16311yescharacteristic
204C2×C4D4×D16311yescharacteristic
205C2×C4C23×D16311yescharacteristic
206C8—22no—
207C8—3262no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—3261no—
212C22—3261no—
213C22—1632no—
214C22C2×D4×D16311yescharacteristic
215C4—42no—
216C4—3261no—
217C4—3261no—
218C4—6522no—
219C4—6522no—
220C4—1632no—
221C4#1910475912yes—
222C4#1910476412yes—
223C4#41×D16312yes—
224C2—82no—
225C2—3262no—
226C2#124×D16311yescharacteristic
227C2C2×#1910475911yescharacteristic
228C2C2×#1910476411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#486×C163C2
socleC22×C163C22×D4
prime radical#23782213C2
hypercenter#486D163
hypocenterC163#3141
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D163C2semidirect
3. #23782023C2semidirect
4. #23782017C2semidirect
5. #23782016C2semidirect
6. #23782011C2semidirect
7. #23781663C2semidirect
8. C4×#14579682C2semidirect
9. #486×C163C2semidirect
114. C163#3141semidirect
122. #486D163semidirect

Maximal subgroups

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

subgroupindex
#237822132
C4⋊Q8×D1632
#237820232
#237820172
#237820162
#237820112
#237816632
C4×#145796822
#486×C1632
#3141163