Finite Group Databasebeta

← #37549866

Subgroups — #37549866

Order 23168.

348 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C2×#19153884C211yesmaximal, characteristic, complemented
2C2×#19153879C211yesmaximal, characteristic, complemented
3C2×#19153878C211yesmaximal, characteristic
4C2×C8×#10624382C211yesmaximal, characteristic, complemented
5(C2×#10624382)⋊Q8C211yesmaximal, characteristic, complemented
6#26042231C211yesmaximal, characteristic
7C2×SD16×D181C211yesmaximal, characteristic
8C2×D181⋊C8C2211yescharacteristic
9C2×C4⋊#10624382C2211yescharacteristic
10C2×C4⋊#10624382C2211yescharacteristic
11C2×C22⋊#10624382—22no—
12C2×C4×#10624382C2211yescharacteristic
13#19153884—22no—
14#19153879—22no—
15#19153878—22no—
16C8×#10624382—22no—
17(C2×#10624382)⋊C4—22no—
18C2×Q8×D181C2211yescharacteristic
19C2×D4×D181C2211yescharacteristic
20C2×#15668438C411yescharacteristic, complemented
21C2×#15668437C411yescharacteristic, complemented
22C2×#15668436C411yescharacteristic, complemented
23SD16×D181—22no—
24SD16×D181—22no—
25C2×C8×D181C2211yescharacteristic
26C2×SD16×C181C411yescharacteristic, complemented
27C22×#10624382—21no—
28C22×#10624382—21no—
29C22×#10624382D412yes—
30D181⋊C8—21no—
31D181⋊C8D412yes—
32C2×#13089685—21no—
33C4⋊#10624382—21no—
34C4⋊#10624382—21no—
35C4⋊#10624382—21no—
36C4⋊#10624382—21no—
37C22⋊#10624382—44no—
38C4×#10624382—21no—
39C4×#10624382—21no—
40C23×D181—21no—
41C2×Dic362—21no—
42C2×Dic362C2×C411yescharacteristic
43Q8×D181—21no—
44Q8×D181—21no—
45Q8×D181—21no—
46Q8×D181—21no—
47C2×D724C2×C411yescharacteristic
48C2×C22⋊D181—21no—
49D4×D181—21no—
50D4×D181—21no—
51D4×D181—21no—
52D4×D181—21no—
53C2×C4×D181—21no—
54C2×C4×D181C2311yescharacteristic
55#15668438—22no—
56#15668437—22no—
57#15668436—22no—
58C2×#13089678C2×C411yescharacteristic
59C8×D181D412yes—
60C8×D181D412yes—
61C2×Q8×C181C2×C411yescharacteristic
62C2×D4×C181C2×C411yescharacteristic
63SD16×C181—22no—
64C2×C8×C181C2×C411yescharacteristic
65C2×#10624382—22no—
66C2×#10624382—22no—
67C2×#10624382—41no—
68C2×#10624382—41no—
69C2×#10624382—42no—
70C2×#10624382—42no—
71C2×#10624382SD1614yescomplemented
72#13089685—22no—
73C22×D181—21no—
74C22×D181—41no—
75C22×D181—41no—
76C22×D181—42no—
77C22×D181—42no—
78C22×D181C2×D411yescharacteristic
79Dic362—21no—
80Dic362—21no—
81Dic362—41no—
82Dic362—41no—
83D724—21no—
84D724—21no—
85C22⋊D181—41no—
86C22⋊D181—41no—
87C2×Dic181—21no—
88C2×Dic181#4111yescharacteristic
89C4×D181—41no—
90C4×D181—41no—
91C4×D181C2×D411yescharacteristic
92C4×D181C2×D411yescharacteristic
93C4×D181C2×D411yescharacteristic
94C4×D181#4111yescharacteristic
95#13089678C22⋊C412yescomplemented
96C23×C181—21no—
97Q8×C181—21no—
98Q8×C181—21no—
99D4×C181—21no—
100D4×C181—21no—
101C2×C4×C181—21no—
102C2×C4×C181C22×C411yescharacteristic
103C8×C181C22⋊C412yescomplemented
104C22×C181—41no—
105C22×C181—42no—
106C22×C181C4×D411yescharacteristic
107C2×D181—41no—
108C2×D181—42no—
109C2×D181—82no—
110C2×D181—81no—
111C2×D181—81no—
112C2×D181#12111yescharacteristic
113C2×D181#12711yescharacteristic
114C2×D181C2×SD1612yes—
115C2×D181#13512yes—
116#10624382—24no—
117#10624382—81no—
118#10624382—81no—
119C4×C181—41no—
120C4×C181C2×C22⋊C411yescharacteristic
121C4×C181C4×D411yescharacteristic
122Dic181—41no—
123Dic181#12111yescharacteristic
124Dic181#12411yescharacteristic
125C2×C181—81no—
126C2×C181#39011yescharacteristic
127C2×C181C4×SD1612yes—
128D181—81no—
129D181#46512yes—
130D181#49211yescharacteristic
131D181#49511yescharacteristic
132C181#196811yescharacteristic, complemented, Sylow
133#1968—1811nomaximal, Sylow
134C22×SD16—1811no—
135C2×#106—1811no—
136C2×#103—1811no—
137C2×#102—1811no—
138C2×C4×C8—1811no—
139#391—1811no—
140#390—1811no—
141C22×Q8—1811no—
142C22×D4—1811no—
143C2×SD16—3622no—
144C2×SD16—3622no—
145C2×SD16—1811no—
146C2×SD16—1811no—
147C2×SD16—1811no—
148C2×SD16#1062438211yescharacteristic, complemented
149C22×C8—1811no—
150C22×C8—1811no—
151C2×C4⋊C4—1811no—
152C2×C4⋊C4—1811no—
153C2×C22⋊C4—3622no—
154C2×C42—1811no—
155#106—3622no—
156#103—3622no—
157#102—3622no—
158C4×C8—3622no—
159#95—3622no—
160C24—3621no—
161C2×Q8—3621no—
162C2×Q8—3621no—
163C2×Q8—3621no—
164C2×Q8—3621no—
165C2×Q8—3621no—
166C2×Q8—1811no—
167C2×Q8C2×#1062438211yescharacteristic
168C2×D4—3621no—
169C2×D4—3621no—
170C2×D4—3621no—
171C2×D4—3621no—
172C2×D4—3621no—
173C2×D4—1811no—
174C2×D4C2×#1062438211yescharacteristic
175C22×C4—3621no—
176C22×C4—3621no—
177C22×C4—3621no—
178C22×C4—1812no—
179C22×C4—1811no—
180SD16—22no—
181SD16—3622no—
182SD16—3622no—
183SD16—3622no—
184C2×C8—3621no—
185C2×C8—3621no—
186C2×C8—1812no—
187C2×C8—1811no—
188C2×C8—1812no—
189C2×C8—1812no—
190C2×C8C2×#1062438211yescharacteristic
191C4⋊C4—3621no—
192C4⋊C4—3621no—
193C4⋊C4—3621no—
194C4⋊C4—3621no—
195C22⋊C4—7244no—
196C42—3621no—
197C42—3621no—
198C23—21no—
199C23—3621no—
200C23—1811no—
201C23—7242no—
202C23—7241no—
203C23—7241no—
204C23—7242no—
205Q8—21no—
206Q8—21no—
207Q8—3621no—
208Q8—3621no—
209Q8—7241no—
210Q8—7241no—
211D4—21no—
212D4—21no—
213D4—3621no—
214D4—3621no—
215D4—7241no—
216D4—7241no—
217C2×C4—21no—
218C2×C4—3622no—
219C2×C4—3622no—
220C2×C4—3621no—
221C2×C4—1814no—
222C2×C4—1811no—
223C2×C4—1811no—
224C2×C4—1811no—
225C2×C4—1811no—
226C2×C4—1811no—
227C2×C4—7241no—
228C2×C4—7241no—
229C2×C4—7242no—
230C2×C4—7241no—
231C2×C4—7241no—
232C2×C4—7242no—
233C2×C4C22×#1062438211yescharacteristic
234C8—3622no—
235C8—1812no—
236C8C22⋊#1062438212yescomplemented
237C22—41no—
238C22—42no—
239C22—14482no—
240C22—14481no—
241C22—14481no—
242C22—1811no—
243C22—1811no—
244C22—1812no—
245C22—1812no—
246C22—7241no—
247C22—7242no—
248C22D4×#1062438211yescharacteristic
249C4—41no—
250C4—3624no—
251C4—14481no—
252C4—14481no—
253C4—1811no—
254C4—1811no—
255C4—7241no—
256C4C2×C22⋊#1062438211yescharacteristic
257C4D4×#1062438211yescharacteristic
258C2—81no—
259C2—14481no—
260C2—1812no—
261C2—1811no—
262C2—1811no—
263C2#2604223111yescharacteristic
264C2SD16×#1062438212yes—
subgroupG/H
centerC22D4×#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC2×SD16×C181C4
socleC22×C181C4×D4
prime radicalC2×D4×D181C22
cosocleC2×C4×D181C23
hypercenterC2×SD16#10624382
hypocenterC181#1968
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19153884C2semidirect
2. C2×#19153879C2semidirect
4. C2×C8×#10624382C2semidirect
5. (C2×#10624382)⋊Q8C2semidirect
20. C2×#15668438C4semidirect
21. C2×#15668437C4semidirect
22. C2×#15668436C4semidirect
26. C2×SD16×C181C4semidirect
71. C2×#10624382SD16semidirect
95. #13089678C22⋊C4semidirect
103. C8×C181C22⋊C4semidirect
132. C181#1968semidirect
148. C2×SD16#10624382semidirect
236. C8C22⋊#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538842
C2×#191538792
C2×#191538782
C2×C8×#106243822
(C2×#10624382)⋊Q82
#260422312
C2×SD16×D1812
#1968181