Finite Group Databasebeta

← #483×D181 #37543595

Subgroups — #483×D181 #37543595

Order 23168.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
2C2×C4⋊C4×D181C211yesmaximal, characteristic
3#26041611C212yesmaximal, complemented
4#26041610C211yesmaximal, characteristic, complemented
5#26041609C211yesmaximal, characteristic, complemented
6#26041608C212yesmaximal, complemented
7#26041607C211yesmaximal, characteristic, complemented
8#106×D181C212yesmaximal, complemented
9#103×D181C212yesmaximal, complemented
10C22⋊C8×D181C211yesmaximal, characteristic
11#483×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181C2211yescharacteristic
13C22×C4×D181—21no—
14C22×C4×D181C2211yescharacteristic
15C2×C8×D181C2212yes—
16#19153796C2211yescharacteristic
17#19153795C2211yescharacteristic
18#19153794—21no—
19#19153793—21no—
20#19153792C2211yescharacteristic
21C2×C4⋊Dic181C2211yescharacteristic
22C2×#15668428—21no—
23C4⋊C4×D181—21no—
24C4⋊C4×D181—21no—
25C4⋊C4×D181—21no—
26C4⋊C4×D181C2212yes—
27C4⋊C4×D181C2212yes—
28C4⋊C4×D181C2211yescharacteristic
29C22⋊C4×D181—21no—
30#19153748C2212yescomplemented
31#19153747C2212yescomplemented
32#19153747C2212yescomplemented
33#19153741C2212yescomplemented
34#19153740C2212yescomplemented
35#19153739C2212yescomplemented
36C22⋊#13089678C2211yescharacteristic
37#19153728C2212yes—
38C22⋊Q8×C181C2211yescharacteristic
39C2×C4⋊C4×C181C2211yescharacteristic
40#106×C181C2212yescomplemented
41#103×C181C2212yescomplemented
42C22⋊C8×C181C2211yescharacteristic
43C23×D181D411yescharacteristic
44C2×Dic362—21no—
45C2×Dic362C2311yescharacteristic
46Q8×D181—24no—
47C22×Dic181—21no—
48C22×Dic181D411yescharacteristic
49C2×C4×D181—24no—
50C2×C4×D181—21no—
51C2×C4×D181—21no—
52C2×C4×D181—22no—
53C2×C4×D181—22no—
54C2×C4×D181—44no—
55C2×C4×D181D411yescharacteristic
56C2×C4×D181D411yescharacteristic
57C2×C4×D181C2311yescharacteristic
58C2×C4×D181C2311yescharacteristic
59C2×#13089678C2312yes—
60C8×D181—24no—
61C4⋊Dic181—21no—
62C4⋊Dic181—21no—
63C4⋊Dic181C2312yes—
64C4⋊Dic181C2311yescharacteristic
65#15668428—22no—
66#15668428—22no—
67#15668428—21no—
68#15668428—21no—
69#15668428—21no—
70C22⋊Dic181—21no—
71#15668426—21no—
72#15668426—21no—
73C2×Q8×C181C2311yescharacteristic
74C22×C4×C181—21no—
75C22×C4×C181C2311yescharacteristic
76C2×C8×C181C2312yes—
77C4⋊C4×C181—21no—
78C4⋊C4×C181—21no—
79C4⋊C4×C181C2312yes—
80C4⋊C4×C181C2311yescharacteristic
81C22⋊C4×C181—21no—
82C22×D181—22no—
83C22×D181—22no—
84C22×D181SD1614yes—
85C22×D181C2×D411yescharacteristic
86C22×D181C2×D411yescharacteristic
87Dic362—22no—
88Dic362—42no—
89C2×Dic181—22no—
90C2×Dic181—21no—
91C2×Dic181—21no—
92C2×Dic181—22no—
93C2×Dic181—42no—
94C2×Dic181C2×D411yescharacteristic
95C2×Dic181C2×D411yescharacteristic
96C4×D181—22no—
97C4×D181—22no—
98C4×D181—22no—
99C4×D181—42no—
100C4×D181—44no—
101C4×D181—44no—
102C4×D181—42no—
103C4×D181#4114yes—
104#13089678—22no—
105C23×C181C2×D411yescharacteristic
106Q8×C181—22no—
107C2×C4×C181—22no—
108C2×C4×C181—21no—
109C2×C4×C181—21no—
110C2×C4×C181—22no—
111C2×C4×C181—42no—
112C2×C4×C181C2×D411yescharacteristic
113C2×C4×C181C2411yescharacteristic
114C8×C181—22no—
115C22×C181—21no—
116C22×C181—21no—
117C22×C181C2×SD1612yes—
118C22×C181C22×D411yescharacteristic
119C2×D181—21no—
120C2×D181—21no—
121C2×D181—28no—
122C2×D181#12312yes—
123C2×D181C2×SD1612yes—
124C2×D181C2×SD1612yes—
125C2×D181#13712yes—
126C4×C181—21no—
127C4×C181—41no—
128C4×C181—42no—
129C4×C181—41no—
130C4×C181C2×#4112yes—
131Dic181—21no—
132Dic181—41no—
133Dic181—42no—
134Dic181—41no—
135Dic181#12312yes—
136C2×C181—22no—
137C2×C181C2×#12311yescharacteristic
138C2×C181C22×SD1611yescharacteristic
139C2×C181C2×#13711yescharacteristic
140D181—22no—
141D181#48314yesdirect product factor
142C181C2×#48311yescharacteristic, complemented, Sylow
143C2×#483—1811nomaximal, Sylow
144C2×C22⋊Q8—1811no—
145C22×C4⋊C4—1811no—
146#483—1811no—
147#483—1811no—
148#483—1812no—
149#483—1812no—
150#483—1811no—
151#483D18111yescharacteristic, direct product factor
152C2×#106—1812no—
153C2×#103—1812no—
154C2×C22⋊C8—1811no—
155C22×Q8—1811no—
156C23×C4—3621no—
157C23×C4—1811no—
158C22×C8—1812no—
159C22⋊Q8—3621no—
160C22⋊Q8—3621no—
161C22⋊Q8—1811no—
162C22⋊Q8—1811no—
163C22⋊Q8—1811no—
164C22⋊Q8C2×D18111yescharacteristic
165C2×C4⋊C4—3621no—
166C2×C4⋊C4—3621no—
167C2×C4⋊C4—3621no—
168C2×C4⋊C4—3621no—
169C2×C4⋊C4—1811no—
170C2×C4⋊C4—1812no—
171C2×C4⋊C4—1812no—
172C2×C4⋊C4—1811no—
173C2×C4⋊C4C2×D18111yescharacteristic
174C2×C22⋊C4—3621no—
175#106—1812no—
176#106—1812no—
177#106—1812no—
178#106C2×D18112yescomplemented
179#103—1812no—
180#103—1812no—
181#103—1812no—
182#103C2×D18112yescomplemented
183C22⋊C8—1811no—
184C22⋊C8—1812no—
185C22⋊C8C2×D18111yescharacteristic
186C24—1811no—
187C2×Q8—3621no—
188C2×Q8—3624no—
189C2×Q8—1811no—
190C2×Q8C22×D18111yescharacteristic
191C22×C4—21no—
192C22×C4—3624no—
193C22×C4—3621no—
194C22×C4—3621no—
195C22×C4—3622no—
196C22×C4—3622no—
197C22×C4—3621no—
198C22×C4—1811no—
199C22×C4—1811no—
200C22×C4—1811no—
201C22×C4—1811no—
202C22×C4—1811no—
203C22×C4—7244no—
204C22×C4C22×D18111yescharacteristic
205C2×C8—3624no—
206C2×C8—1812no—
207C2×C8C22×D18112yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—3622no—
211C4⋊C4—3622no—
212C4⋊C4—3621no—
213C4⋊C4—3621no—
214C4⋊C4—3621no—
215C4⋊C4—3621no—
216C4⋊C4—3621no—
217C4⋊C4—1812no—
218C4⋊C4—1811no—
219C4⋊C4C22×D18112yes—
220C4⋊C4C22×D18111yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—3621no—
223C22⋊C4—3621no—
224C22⋊C4—3621no—
225C23—3622no—
226C23—3622no—
227C23—1814no—
228C23—1811no—
229C23—1811no—
230C23D4×D18111yescharacteristic
231Q8—22no—
232Q8—3622no—
233Q8—7242no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3622no—
240C2×C4—3622no—
241C2×C4—3622no—
242C2×C4—3622no—
243C2×C4—3621no—
244C2×C4—3621no—
245C2×C4—3622no—
246C2×C4—1811no—
247C2×C4—1811no—
248C2×C4—1814no—
249C2×C4—7244no—
250C2×C4—7244no—
251C2×C4—7242no—
252C2×C4—7242no—
253C2×C4—7242no—
254C2×C4D4×D18111yescharacteristic
255C2×C4C23×D18111yescharacteristic
256C8—22no—
257C8—3622no—
258C22—21no—
259C22—21no—
260C22—3628no—
261C22—3621no—
262C22—3621no—
263C22—1812no—
264C22—1812no—
265C22—1812no—
266C22—1812no—
267C22SD16×D18112yes—
268C22C2×D4×D18111yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—3621no—
274C4—1812no—
275C4—7241no—
276C4—7242no—
277C4—7241no—
278C4#41×D18112yes—
279C2—22no—
280C2—3622no—
281C2—1814no—
282C2#123×D18111yescharacteristic
283C2C2×SD16×D18111yescharacteristic
284C2#137×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#483×C181C2
socleC22×C181C22×D4
prime radicalC23×D181D4
hypercenter#483D181
hypocenterC181C2×#483
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #26041611C2semidirect
4. #26041610C2semidirect
5. #26041609C2semidirect
6. #26041608C2semidirect
7. #26041607C2semidirect
8. #106×D181C2semidirect
9. #103×D181C2semidirect
11. #483×C181C2semidirect
30. #19153748C22semidirect
31. #19153747C22semidirect
32. #19153747C22semidirect
33. #19153741C22semidirect
34. #19153740C22semidirect
35. #19153739C22semidirect
40. #106×C181C22semidirect
41. #103×C181C22semidirect
141. D181#483direct (N × H)
142. C181C2×#483semidirect
151. #483D181direct (N × H)
178. #106C2×D181semidirect
182. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1812
C2×C4⋊C4×D1812
#260416112
#260416102
#260416092
#260416082
#260416072
#106×D1812
#103×D1812
C22⋊C8×D1812
#483×C1812
C2×#483181