Finite Group Databasebeta

← #35227189

Subgroups — #35227189

Order 20864.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#23782389C211yesmaximal, characteristic, complemented
2#23782372C211yesmaximal, characteristic, complemented
3#23782021C212yesmaximal, complemented
4#23782015C211yesmaximal, characteristic, complemented
5#23781970C212yesmaximal, complemented
6#23781956C211yesmaximal, characteristic, complemented
7#23781882C212yesmaximal
8#23781789C211yesmaximal, characteristic, complemented
9SD16×Dic163C212yesmaximal, complemented
10C4×#14579684C211yesmaximal, characteristic
11#461×C163C211yesmaximal, characteristic, complemented
12C2×#14579684C2211yescharacteristic
13#19104735C2211yescharacteristic
14#19104730—22no—
15#19104726C2212yes—
16#19104724C2212yes—
17#19104715—21no—
18#19104714—21no—
19#19104712—22no—
20Q8×Dic163C2212yes—
21C4×Dic326C2211yescharacteristic
22C4×D652C2211yescharacteristic
23D4×Dic163C2212yes—
24#19104693—21no—
25C2×C4⋊Dic163—21no—
26#19104668C2211yescharacteristic
27#19104667C2212yes—
28Dic163⋊C8C2212yes—
29#19104659C2212yes—
30#19104659C2212yes—
31#19104657C2212yes—
32#19104656C2211yescharacteristic
33#19104655C2211yescharacteristic
34C4×C163⋊C8C2211yescharacteristic
35C8×Dic163C2212yes—
36C2×SD16×C163C2212yes—
37C4⋊Q8×C163C2211yescharacteristic
38C4×D4×C163C2211yescharacteristic
39C4⋊C8×C163C2211yescharacteristic
40#103×C163C2212yes—
41C2×Dic326—21no—
42C2×Dic326D411yescharacteristic
43C2×D652C2311yescharacteristic
44C22×Dic163—22no—
45C2×C4×D163—21no—
46#14579684—22no—
47C2×C163⋊C8C2311yescharacteristic
48C2×C163⋊C8C2311yescharacteristic
49C4⋊Dic163—22no—
50C4⋊Dic163—22no—
51C4⋊Dic163—21no—
52C4⋊Dic163D411yescharacteristic
53C4⋊Dic163C2311yescharacteristic
54C4⋊Dic163C2312yes—
55Dic163⋊C4—22no—
56Dic163⋊C4D412yes—
57C22⋊Dic163—22no—
58(C2×D163)⋊C4—22no—
59(C2×D163)⋊C4—22no—
60(C2×D163)⋊C4—21no—
61C4×Dic163—22no—
62C4×Dic163C2312yes—
63C2×Q8×C163C2312yes—
64C2×D4×C163C2311yescharacteristic
65C22×C4×C163—21no—
66SD16×C163—24no—
67C2×C8×C163C2312yes—
68C4⋊C4×C163—21no—
69C4⋊C4×C163—21no—
70C4⋊C4×C163C2311yescharacteristic
71C22⋊C4×C163—21no—
72C42×C163C2311yescharacteristic
73C22×D163—21no—
74Dic326—22no—
75Dic326—81no—
76D652—21no—
77D652—21no—
78C2×Dic163—21no—
79C2×Dic163—22no—
80C2×Dic163—82no—
81C2×Dic163C2×D412yes—
82C2×Dic163C2×D412yes—
83C4×D163—42no—
84C163⋊C8—21no—
85C163⋊C8—21no—
86C23×C163—21no—
87Q8×C163—22no—
88Q8×C163—22no—
89D4×C163—21no—
90D4×C163#4112yes—
91C2×C4×C163—21no—
92C2×C4×C163—22no—
93C2×C4×C163—42no—
94C2×C4×C163C2×D411yescharacteristic
95C2×C4×C163C2×D411yescharacteristic
96C2×C4×C163C2411yescharacteristic
97C8×C163—22no—
98C22×C163—22no—
99C22×C163—41no—
100C22×C163—41no—
101C22×C163C22×D411yescharacteristic
102C2×D163—41no—
103C2×D163—41no—
104C2×D163—41no—
105C4×C163—21no—
106C4×C163—42no—
107C4×C163—41no—
108C4×C163#13512yes—
109C4×C163C2×#4111yescharacteristic
110C4×C1632−1+411yescharacteristic
111Dic163—22no—
112Dic163—22no—
113Dic163—41no—
114Dic163—42no—
115C2×C163—42no—
116C2×C163#54811yescharacteristic
117C2×C163C2×#13511yescharacteristic
118C2×C163#57711yescharacteristic
119D163—81no—
120C163#336011yescharacteristic, complemented, Sylow
121#3360—1631nomaximal, Sylow
122#556—1631no—
123#554—1631no—
124#488—1632no—
125#487—1631no—
126#479—1632no—
127#476—1631no—
128#463—1632no—
129#461D16311yescharacteristic, complemented
130C8×Q8—1631no—
131C4×SD16—1632no—
132C4×D8—1631no—
133C2×SD16C2×D16312yes—
134C2×D8—1631no—
135C4⋊Q8—1631no—
136C4⋊Q8C2×D16311yescharacteristic
137#126—3262no—
138#125—1632no—
139#124—1632no—
140C22⋊Q8—3262no—
141C22⋊Q8—3261no—
142C22⋊Q8—3261no—
143C4×Q8—1632no—
144C4×Q8—1631no—
145C4×D4—1632no—
146C4×D4—1631no—
147C4×D4C2×D16311yescharacteristic
148#117—3261no—
149C2×C4⋊C4—3261no—
150#107—1631no—
151#106—1632no—
152C4⋊C8—1632no—
153C4⋊C8C2×D16311yescharacteristic
154#103—1632no—
155#103—1632no—
156#103C2×D16312yes—
157#102—1631no—
158#102—1632no—
159#102—1631no—
160C4×C8—1632no—
161C4×C8—1631no—
162C2×Q8—3261no—
163C2×Q8—1631no—
164C2×Q8C22×D16312yes—
165C2×D4—1631no—
166C2×D4C22×D16311yescharacteristic
167C22×C4—21no—
168C22×C4—3262no—
169C22×C4—3261no—
170SD16—24no—
171D8—3262no—
172C2×C8—1631no—
173C2×C8—1631no—
174C2×C8C22×D16312yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—3262no—
178C4⋊C4—3262no—
179C4⋊C4—3262no—
180C4⋊C4—3261no—
181C4⋊C4—1632no—
182C4⋊C4—1631no—
183C4⋊C4—1631no—
184C4⋊C4—1632no—
185C4⋊C4C22×D16311yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—3262no—
188C22⋊C4—3262no—
189C22⋊C4—3262no—
190C22⋊C4—3261no—
191C42—3262no—
192C42—1632no—
193C42C22×D16311yescharacteristic
194C23—21no—
195C23—3261no—
196Q8—22no—
197Q8—22no—
198Q8—13041no—
199Q8—3262no—
200D4—21no—
201D4—3261no—
202D4—3261no—
203D4#1457969812yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—13042no—
208C2×C4—3261no—
209C2×C4—3262no—
210C2×C4—6522no—
211C2×C4—1632no—
212C2×C4—1632no—
213C2×C4D4×D16311yescharacteristic
214C2×C4D4×D16311yescharacteristic
215C2×C4C23×D16311yescharacteristic
216C8—22no—
217C8—3261no—
218C8—3261no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—6521no—
223C22—6521no—
224C22—6521no—
225C22C2×D4×D16311yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—3262no—
230C4—3262no—
231C4—6521no—
232C4—6522no—
233C4#1910475612yes—
234C4C2×#1457969811yescharacteristic
235C4#1910478211yescharacteristic
236C2—42no—
237C2—13041no—
238C2#2378232111yescharacteristic
239C2C2×#1910475611yescharacteristic
240C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#461×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579684C22
hypercenter#461D163
hypocenterC163#3360
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #23782389C2semidirect
2. #23782372C2semidirect
3. #23782021C2semidirect
4. #23782015C2semidirect
5. #23781970C2semidirect
6. #23781956C2semidirect
8. #23781789C2semidirect
9. SD16×Dic163C2semidirect
11. #461×C163C2semidirect
120. C163#3360semidirect
129. #461D163semidirect

Maximal subgroups

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

subgroupindex
#237823892
#237823722
#237820212
#237820152
#237819702
#237819562
#237818822
#237817892
SD16×Dic1632
C4×#145796842
#461×C1632
#3360163