Finite Group Databasebeta

← #328×D181 #37532200

Subgroups — #328×D181 #37532200

Order 23168.

336 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 77 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⋊Q8×D181C211yesmaximal, characteristic
2C22⋊C8×D181C211yesmaximal, characteristic
3#95×D181C211yesmaximal, characteristic
4#26041017C211yesmaximal, characteristic, complemented
5#26041016C211yesmaximal, characteristic, complemented
6#26041015C211yesmaximal, characteristic, complemented
7#328×C181C211yesmaximal, characteristic, complemented
8C2×Q8×D181C411yescharacteristic, complemented
9C22×C4×D181—21no—
10C22×C4×D181C2211yescharacteristic
11C2×C8×D181—21no—
12#19153796C411yescharacteristic, complemented
13#19153795C2211yescharacteristic
14#19153794—21no—
15#19153793—21no—
16#19153792C411yescharacteristic, complemented
17C4⋊C4×D181—21no—
18C4⋊C4×D181C411yescharacteristic, complemented
19C22⋊C4×D181—21no—
20C22⋊#13089678C2211yescharacteristic
21#19153728—21no—
22#19153723—21no—
23#19153723C2211yescharacteristic
24C22⋊Q8×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic
26#95×C181C2211yescharacteristic
27C23×D181D411yescharacteristic
28C2×Dic362—21no—
29C2×Dic362C2×C411yescharacteristic, complemented
30Q8×D181—42no—
31C22×Dic181—21no—
32C22×Dic181D411yescharacteristic
33C2×C4×D181—24no—
34C2×C4×D181—21no—
35C2×C4×D181—21no—
36C2×C4×D181—42no—
37C2×C4×D181—42no—
38C2×C4×D181—42no—
39C2×C4×D181C2×C411yescharacteristic
40C2×C4×D181C2×C411yescharacteristic
41C2×C4×D181D411yescharacteristic
42C2×C4×D181D411yescharacteristic
43C2×#13089678—21no—
44C8×D181—42no—
45C4⋊Dic181—21no—
46C4⋊Dic181C2×C411yescharacteristic, complemented
47#15668428—21no—
48#15668428—21no—
49#15668428—21no—
50C22⋊Dic181—21no—
51#15668426—21no—
52#15668426—21no—
53C2×Q8×C181C2×C411yescharacteristic, complemented
54C22×C4×C181—21no—
55C22×C4×C181C2311yescharacteristic
56C2×C8×C181—21no—
57C4⋊C4×C181—21no—
58C4⋊C4×C181C2×C411yescharacteristic, complemented
59C22⋊C4×C181—21no—
60C22×D181—22no—
61C22×D181—22no—
62C22×D181C22⋊C411yescharacteristic
63C22×D181C22⋊C411yescharacteristic
64C22×D181SD1612yes—
65C22×D181Q1612yes—
66Dic362—42no—
67Dic362—41no—
68C2×Dic181—22no—
69C2×Dic181—21no—
70C2×Dic181—21no—
71C2×Dic181—41no—
72C2×Dic181—41no—
73C2×Dic181—41no—
74C2×Dic181C22⋊C411yescharacteristic
75C2×Dic181C22⋊C411yescharacteristic
76C4×D181—22no—
77C4×D181—22no—
78C4×D181—41no—
79C4×D181—41no—
80C4×D181—48no—
81C4×D181—42no—
82C4×D181—42no—
83#13089678—41no—
84C23×C181C2×D411yescharacteristic
85Q8×C181—41no—
86C2×C4×C181—22no—
87C2×C4×C181—21no—
88C2×C4×C181—21no—
89C2×C4×C181—41no—
90C2×C4×C181—41no—
91C2×C4×C181—41no—
92C2×C4×C181C22×C411yescharacteristic
93C2×C4×C181C2×D411yescharacteristic
94C8×C181—41no—
95C22×C181—21no—
96C22×C181—21no—
97C22×C181C2×C22⋊C411yescharacteristic
98C22×C181C2×SD1611yescharacteristic
99C22×C181C2×Q1611yescharacteristic
100C2×D181—21no—
101C2×D181—21no—
102C2×D181—24no—
103C2×D181—24no—
104C2×D181#9912yes—
105C2×D181#10312yes—
106C2×D181#10312yes—
107C2×D181#10412yes—
108C4×C181—21no—
109C4×C181—21no—
110C4×C181—42no—
111C4×C181—41no—
112C4×C181—41no—
113Dic181—21no—
114Dic181—21no—
115Dic181—42no—
116Dic181—41no—
117Dic181—41no—
118C2×C181—21no—
119C2×C181—21no—
120C2×C181C2×#9911yescharacteristic
121C2×C181C2×#10311yescharacteristic
122C2×C181C2×#10411yescharacteristic
123D181—22no—
124D181#32814yesdirect product factor
125C181C2×#32811yescharacteristic, complemented, Sylow
126C2×#328—1811nomaximal, Sylow
127C2×C22⋊Q8—1811no—
128C2×C22⋊C8—1811no—
129C2×#95—1811no—
130#328—1811no—
131#328—1811no—
132#328—1811no—
133#328D18111yescharacteristic, direct product factor
134C22×Q8—1811no—
135C23×C4—3621no—
136C23×C4—1811no—
137C22×C8—3621no—
138C22⋊Q8—3621no—
139C22⋊Q8—3621no—
140C22⋊Q8—1811no—
141C22⋊Q8—1811no—
142C22⋊Q8—1811no—
143C22⋊Q8C2×D18111yescharacteristic
144C2×C4⋊C4—3621no—
145C2×C4⋊C4—1811no—
146C2×C22⋊C4—3621no—
147C22⋊C8—3621no—
148C22⋊C8—1811no—
149C22⋊C8C2×D18111yescharacteristic
150#95—3621no—
151#95—1811no—
152#95C2×D18111yescharacteristic
153C24—1811no—
154C2×Q8—3621no—
155C2×Q8—1811no—
156C2×Q8—7242no—
157C2×Q8C4×D18111yescharacteristic, complemented
158C22×C4—21no—
159C22×C4—3624no—
160C22×C4—3621no—
161C22×C4—3621no—
162C22×C4—3621no—
163C22×C4—1811no—
164C22×C4—1811no—
165C22×C4—1811no—
166C22×C4—1811no—
167C22×C4—1811no—
168C22×C4—7242no—
169C22×C4—7242no—
170C22×C4—7242no—
171C22×C4C22×D18111yescharacteristic
172C2×C8—21no—
173C2×C8—3621no—
174C2×C8—7242no—
175C4⋊C4—21no—
176C4⋊C4—3621no—
177C4⋊C4—3621no—
178C4⋊C4—3621no—
179C4⋊C4—3621no—
180C4⋊C4—1811no—
181C4⋊C4C4×D18111yescharacteristic, complemented
182C22⋊C4—21no—
183C22⋊C4—3621no—
184C22⋊C4—3621no—
185C22⋊C4—3621no—
186C23—3622no—
187C23—3622no—
188C23—1811no—
189C23—1812no—
190C23—1812no—
191C23—1811no—
192C23D4×D18111yescharacteristic
193Q8—41no—
194Q8—7241no—
195Q8—7242no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3622no—
203C2×C4—3622no—
204C2×C4—3622no—
205C2×C4—3621no—
206C2×C4—3621no—
207C2×C4—1811no—
208C2×C4—1811no—
209C2×C4—7241no—
210C2×C4—7241no—
211C2×C4—7248no—
212C2×C4—7242no—
213C2×C4—7242no—
214C2×C4—7241no—
215C2×C4—7241no—
216C2×C4—7241no—
217C2×C4C2×C4×D18111yescharacteristic
218C2×C4D4×D18111yescharacteristic
219C8—41no—
220C8—7241no—
221C22—21no—
222C22—21no—
223C22—3624no—
224C22—3624no—
225C22—3621no—
226C22—3621no—
227C22—1812no—
228C22—1812no—
229C22—1812no—
230C22—1812no—
231C22C22⋊C4×D18111yescharacteristic
232C22SD16×D18111yescharacteristic
233C22Q16×D18111yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—3621no—
240C4—3621no—
241C4—7242no—
242C4—7241no—
243C4—7241no—
244C2—21no—
245C2—21no—
246C2—3622no—
247C2—1814no—
248C2#99×D18111yescharacteristic
249C2#103×D18111yescharacteristic
250C2#104×D18111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D181
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#328×C181C2
socleC22×C181C2×C22⋊C4
prime radicalC23×D181D4
cosocleC22×C4×C181C23
hypercenter#328D181
hypocenterC181C2×#328
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #26041017C2semidirect
5. #26041016C2semidirect
6. #26041015C2semidirect
7. #328×C181C2semidirect
8. C2×Q8×D181C4semidirect
12. #19153796C4semidirect
16. #19153792C4semidirect
18. C4⋊C4×D181C4semidirect
29. C2×Dic362C2×C4semidirect
46. C4⋊Dic181C2×C4semidirect
53. C2×Q8×C181C2×C4semidirect
58. C4⋊C4×C181C2×C4semidirect
124. D181#328direct (N × H)
125. C181C2×#328semidirect
133. #328D181direct (N × H)
157. C2×Q8C4×D181semidirect
181. C4⋊C4C4×D181semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1812
C22⋊C8×D1812
#95×D1812
#260410172
#260410162
#260410152
#328×C1812
C2×#328181