Finite Group Databasebeta

← #35225141

Subgroups — #35225141

Order 20864.

364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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#23782220C211yesmaximal, characteristic, complemented
2#23782200C211yesmaximal, characteristic
3#23782075C211yesmaximal, characteristic, complemented
4#23782057C211yesmaximal, characteristic, complemented
5#23782025C211yesmaximal, characteristic, complemented
6#23781922C212yesmaximal, complemented
7#23781914C212yesmaximal
8#23781757C212yesmaximal, complemented
9#23781750C212yesmaximal, complemented
10#23781617C211yesmaximal, characteristic, complemented
11#491×C163C211yesmaximal, characteristic, complemented
12C2×#14579689C2212yes—
13C2×#14579686C2212yes—
14C2×#14579682C2212yes—
15#19104735C2211yescharacteristic
16#19104730—21no—
17#19104726—21no—
18#19104723C2211yescharacteristic
19#19104720—21no—
20#19104715—21no—
21#19104715C2211yescharacteristic
22#19104711C2212yes—
23#19104710C2211yescharacteristic
24Q8×Dic163C2212yes—
25#19104700—21no—
26#19104700C2211yescharacteristic
27#19104697—21no—
28#19104697C2211yescharacteristic
29#19104693—21no—
30#19104693—21no—
31C42×D163C2211yescharacteristic
32#19104668C2212yes—
33#19104666C2212yes—
34#19104660C2212yes—
35#19104657C2212yes—
36#19104646C2211yescharacteristic
37C8×Dic163C2212yes—
38C4⋊Q8×C163C2211yescharacteristic
39#125×C163C2211yescharacteristic
40#103×C163C2212yes—
41#103×C163C2212yes—
42#97×C163C2211yescharacteristic
43C2×Dic326—21no—
44C2×Dic326C2311yescharacteristic
45C2×D652—21no—
46C2×D652C2311yescharacteristic
47C2×C4×D163—21no—
48C2×C4×D163D411yescharacteristic
49C2×C4×D163D411yescharacteristic
50C2×C4×D163C2311yescharacteristic
51#14579689—24no—
52#14579686—24no—
53#14579682—24no—
54C2×C163⋊C8C2312yes—
55C4⋊Dic163—21no—
56C4⋊Dic163—22no—
57C4⋊Dic163—21no—
58C4⋊Dic163C2312yes—
59Dic163⋊C4—22no—
60Dic163⋊C4—21no—
61(C2×D163)⋊C4—22no—
62(C2×D163)⋊C4—21no—
63(C2×D163)⋊C4—21no—
64(C2×D163)⋊C4—21no—
65(C2×D163)⋊C4—21no—
66(C2×D163)⋊C4—21no—
67C4×Dic163—21no—
68C4×Dic163—22no—
69C4×Dic163D411yescharacteristic
70C4×Dic163D411yescharacteristic
71C4×Dic163C2311yescharacteristic
72C2×Q8×C163C2312yes—
73C2×C8×C163C2312yes—
74C4⋊C4×C163—21no—
75C4⋊C4×C163—21no—
76C4⋊C4×C163—21no—
77C4⋊C4×C163—21no—
78C4⋊C4×C163C2311yescharacteristic
79C4⋊C4×C163C2311yescharacteristic
80C42×C163C2311yescharacteristic
81C22×D163—21no—
82C22×D163C2×D411yescharacteristic
83Dic326—22no—
84Dic326—42no—
85D652—22no—
86D652—42no—
87C2×Dic163—21no—
88C2×Dic163—22no—
89C2×Dic163C2×D411yescharacteristic
90C2×Dic163C2×D411yescharacteristic
91C2×Dic163C2×D411yescharacteristic
92C4×D163—22no—
93C4×D163—22no—
94C4×D163—41no—
95C4×D163—42no—
96C163⋊C8#4114yes—
97Q8×C163—22no—
98Q8×C163—22no—
99C2×C4×C163—21no—
100C2×C4×C163—21no—
101C2×C4×C163—22no—
102C2×C4×C163C2×D411yescharacteristic
103C2×C4×C163C2×D411yescharacteristic
104C2×C4×C163C2411yescharacteristic
105C8×C163—22no—
106C22×C163C22×D411yescharacteristic
107C2×D163—21no—
108C2×D163—21no—
109C2×D163—21no—
110C2×D163—41no—
111C2×D163—41no—
112C2×D163—41no—
113C4×C163—21no—
114C4×C163—21no—
115C4×C163—42no—
116C4×C163—41no—
117C4×C163—41no—
118C4×C163C2×#4112yes—
119Dic163—21no—
120Dic163—21no—
121Dic163—41no—
122Dic163—42no—
123Dic163#13712yes—
124C2×C163#53211yescharacteristic
125C2×C163C2×#13711yescharacteristic
126C2×C163#57511yescharacteristic
127D163—41no—
128D163—81no—
129C163#317211yescharacteristic, complemented, Sylow
130#3172—1631nomaximal, Sylow
131#533—1631no—
132#529—1631no—
133#501—1631no—
134#497—1631no—
135#491D16311yescharacteristic, complemented
136#489—1631no—
137#470—1632no—
138#468—1632no—
139C4×Q16—1632no—
140C4×SD16—1632no—
141C4×M16—1631no—
142C2×Q16—1632no—
143C2×SD16—1632no—
144C2×M16—1632no—
145C4⋊Q8—1631no—
146C4⋊Q8C2×D16311yescharacteristic
147#126—3261no—
148#125—3261no—
149#125C2×D16311yescharacteristic
150#124—1631no—
151#123—3261no—
152C22⋊Q8—3261no—
153C22⋊Q8—1632no—
154C22⋊Q8—1631no—
155#121—1631no—
156C4×Q8—3261no—
157C4×Q8—1632no—
158C4×Q8—1631no—
159C4×D4—3261no—
160C4×D4—1631no—
161#117—3261no—
162#117—3261no—
163C2×C42—1631no—
164#107—1632no—
165#106—1632no—
166#103—1632no—
167#103C2×D16312yes—
168#103C2×D16312yes—
169#102—1632no—
170#97—1631no—
171#97C2×D16311yescharacteristic
172C4×C8—1632no—
173C2×Q8—3261no—
174C2×Q8—1631no—
175C2×Q8C22×D16312yes—
176C2×D4—3261no—
177C2×D4—1631no—
178C22×C4—3261no—
179C22×C4—1631no—
180C22×C4—1631no—
181C22×C4—1631no—
182Q16—3264no—
183SD16—3264no—
184M16—3264no—
185C2×C8—1632no—
186C2×C8C22×D16312yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—3261no—
192C4⋊C4—3262no—
193C4⋊C4—3262no—
194C4⋊C4—3261no—
195C4⋊C4—3261no—
196C4⋊C4—1632no—
197C4⋊C4C22×D16311yescharacteristic
198C4⋊C4C22×D16311yescharacteristic
199C22⋊C4—3262no—
200C22⋊C4—3261no—
201C22⋊C4—3261no—
202C22⋊C4—3261no—
203C22⋊C4—3261no—
204C22⋊C4—3261no—
205C42—3261no—
206C42—3262no—
207C42—1631no—
208C42—1631no—
209C42—1631no—
210C42C22×D16311yescharacteristic
211C23—3261no—
212C23—1631no—
213Q8—22no—
214Q8—22no—
215Q8—3262no—
216Q8—6522no—
217D4—3262no—
218D4—6522no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—3262no—
223C2×C4—3262no—
224C2×C4—3261no—
225C2×C4—3262no—
226C2×C4—6521no—
227C2×C4—6522no—
228C2×C4—1631no—
229C2×C4—1631no—
230C2×C4—1631no—
231C2×C4D4×D16311yescharacteristic
232C2×C4D4×D16311yescharacteristic
233C2×C4C23×D16311yescharacteristic
234C8—22no—
235C8—1634no—
236C22—3261no—
237C22—3261no—
238C22—3261no—
239C22—6521no—
240C22—6521no—
241C22—6521no—
242C22C2×D4×D16311yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—3261no—
249C4—3261no—
250C4—6522no—
251C4—6521no—
252C4—1632no—
253C4#41×D16312yes—
254C2—13041no—
255C2—6521no—
256C2#2378221411yescharacteristic
257C2#137×D16311yescharacteristic
258C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#491×C163C2
socleC22×C163C22×D4
prime radical#19104710C22
hypercenter#491D163
hypocenterC163#3172
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #23782220C2semidirect
3. #23782075C2semidirect
4. #23782057C2semidirect
5. #23782025C2semidirect
6. #23781922C2semidirect
8. #23781757C2semidirect
9. #23781750C2semidirect
10. #23781617C2semidirect
11. #491×C163C2semidirect
129. C163#3172semidirect
135. #491D163semidirect

Maximal subgroups

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

subgroupindex
#237822202
#237822002
#237820752
#237820572
#237820252
#237819222
#237819142
#237817572
#237817502
#237816172
#491×C1632
#3172163