Finite Group Databasebeta

← #37545545

Subgroups — #37545545

Order 23168.

462 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 141 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#26041986C211yesmaximal, characteristic, complemented
2#26041625C211yesmaximal, characteristic, complemented
3#26041595C212yesmaximal, complemented
4#26041569C211yesmaximal, characteristic, complemented
5#26041506C212yesmaximal, complemented
6#26041476C211yesmaximal, characteristic, complemented
7#26041430C212yesmaximal, complemented
8C4×#15668437C211yesmaximal, characteristic, complemented
9D4×#13089678C211yesmaximal, characteristic, complemented
10C2×#19153737C212yesmaximal, complemented
11D42×C181C211yesmaximal, characteristic
12C2×#15668437C2211yescharacteristic
13C2×#15668437C2212yescomplemented
14C22×#13089678C2212yescomplemented
15#19153816C2211yescharacteristic, complemented
16#19153807—21no—
17#19153795C2212yescomplemented
18#19153793—22no—
19C4×Dic362C2211yescharacteristic, complemented
20D4×Dic181—21no—
21C2×C4⋊Dic181C2212yescomplemented
22#19153747C2211yescharacteristic, complemented
23#19153747C2212yescomplemented
24C4⋊#13089678C2211yescharacteristic, complemented
25#19153739C2211yescharacteristic, complemented
26#19153737C2211yescharacteristic
27#19153737C2212yes—
28#19153737C2212yescomplemented
29#19153737C2212yescomplemented
30#19153737C2212yescomplemented
31C22⋊#13089678C2212yescomplemented
32C4×#13089678C2211yescharacteristic, complemented
33C22×D4×C181—21no—
34C22×D4×C181C2212yes—
35#127×C181C2211yescharacteristic
36#121×C181—21no—
37#121×C181C2212yes—
38#120×C181—22no—
39C4×D4×C181C2211yescharacteristic
40C4×D4×C181C2211yescharacteristic
41C2×Dic362—21no—
42C2×Dic362C2311yescharacteristic
43C22×Dic181—22no—
44#15668437—22no—
45#15668437—42no—
46C2×#13089678—42no—
47C2×#13089678C2311yescharacteristic
48C2×#13089678C2311yescharacteristic
49C2×#13089678C2312yes—
50C4⋊Dic181—22no—
51C4⋊Dic181—21no—
52C4⋊Dic181C2311yescharacteristic
53C4⋊Dic181C2312yes—
54#15668428—21no—
55#15668428—22no—
56C22⋊Dic181—22no—
57C4×Dic181—21no—
58C24×C181—22no—
59C2×D4×C181—24no—
60C2×D4×C181—22no—
61C2×D4×C181—22no—
62C2×D4×C181—21no—
63C2×D4×C181—21no—
64C2×D4×C181—42no—
65C2×D4×C181—44no—
66C2×D4×C181D411yescharacteristic
67C2×D4×C181C2312yes—
68C2×D4×C181C2311yescharacteristic
69C22×C4×C181—21no—
70C22×C4×C181C2312yes—
71C4⋊C4×C181D411yescharacteristic
72C4⋊C4×C181C2311yescharacteristic
73C22⋊C4×C181—21no—
74C22⋊C4×C181—22no—
75C22⋊C4×C181D412yes—
76C42×C181C2311yescharacteristic
77Dic362—21no—
78Dic362—21no—
79Dic362—81no—
80C2×Dic181—21no—
81C2×Dic181—21no—
82C2×Dic181—22no—
83C2×Dic181—82no—
84#13089678—21no—
85#13089678—21no—
86#13089678—22no—
87C23×C181—21no—
88C23×C181—22no—
89C23×C181—44no—
90C23×C181—44no—
91C23×C181—82no—
92C23×C181C2×D412yes—
93D4×C181—22no—
94D4×C181—21no—
95D4×C181—22no—
96D4×C181—44no—
97D4×C181—44no—
98D4×C181—42no—
99D4×C181—81no—
100D4×C181SD1614yescomplemented
101D4×C181#4112yes—
102C2×C4×C181—22no—
103C2×C4×C181—22no—
104C2×C4×C181—21no—
105C2×C4×C181—42no—
106C2×C4×C181C2×D411yescharacteristic
107C2×C4×C181C2×D412yes—
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2411yescharacteristic
110C22×C181—22no—
111C22×C181—22no—
112C22×C181—22no—
113C22×C181—41no—
114C22×C181—41no—
115C22×C181—42no—
116C22×C181—42no—
117C22×C181—42no—
118C22×C181—88no—
119C22×C181C2×SD1614yes—
120C22×C181C22×D411yescharacteristic
121C4×C181—22no—
122C4×C181—21no—
123C4×C181—41no—
124C4×C181C2×SD1612yes—
125C4×C181C2×#4111yescharacteristic
126C4×C181H(5,𝔽2)11yescharacteristic
127Dic181—41no—
128Dic181—41no—
129Dic181—42no—
130C2×C181—24no—
131C2×C181—42no—
132C2×C181—82no—
133C2×C181#54611yescharacteristic
134C2×C181C22×SD1611yescharacteristic
135C2×C181#57611yescharacteristic
136C181#329511yescharacteristic, complemented, Sylow
137#3295—1811nomaximal, Sylow
138#554—1811no—
139D42D18111yescharacteristic
140#486—1811no—
141#481—1812no—
142#476—1811no—
143#465—1812no—
144#460—1811no—
145#450—1812no—
146C4×SD16—1811no—
147C8×D4—1811no—
148C2×#102—1812no—
149C22×D4—21no—
150C22×D4C2×D18112yes—
151C2×SD16—1811no—
152C2×SD16—1812no—
153C22×C8—1812no—
154C4⋊Q8—1811no—
155#127C2×D18111yescharacteristic
156#125—3621no—
157C22⋊Q8—3622no—
158C22⋊Q8—1812no—
159#121—21no—
160#121C2×D18112yes—
161#120—22no—
162C4×Q8—1811no—
163C4×D4—3621no—
164C4×D4C2×D18111yescharacteristic
165C4×D4C2×D18111yescharacteristic
166C2×C4⋊C4—1812no—
167#106—1811no—
168#106—1812no—
169C4⋊C8—1811no—
170#103—1811no—
171#102—1811no—
172#102—1812no—
173#102—1812no—
174#102—1812no—
175#102—1812no—
176C22⋊C8—1812no—
177C4×C8—1811no—
178C24—22no—
179C2×Q8—3621no—
180C2×Q8—1811no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4C22⋊D18111yescharacteristic
189C2×D4C22×D18112yes—
190C2×D4C22×D18111yescharacteristic
191C22×C4—21no—
192C22×C4—3622no—
193C22×C4C22×D18112yes—
194SD16—3622no—
195SD16—7242no—
196C2×C8—1811no—
197C2×C8—1811no—
198C2×C8—1812no—
199C2×C8—7242no—
200C4⋊C4—3622no—
201C4⋊C4—3621no—
202C4⋊C4—3621no—
203C4⋊C4—3622no—
204C4⋊C4—1811no—
205C4⋊C4—1812no—
206C4⋊C4C22⋊D18111yescharacteristic
207C4⋊C4C22×D18111yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—3622no—
211C22⋊C4C22⋊D18112yes—
212C42—3621no—
213C42C22×D18111yescharacteristic
214C23—21no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—82no—
219C23C2×C22⋊D18112yes—
220Q8—3621no—
221Q8—3621no—
222Q8—14481no—
223D4—22no—
224D4—21no—
225D4—22no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—81no—
230D4#1566843714yescomplemented
231D4#1566844812yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—3621no—
237C2×C4—3621no—
238C2×C4—3622no—
239C2×C4—14482no—
240C2×C4C2×C22⋊D18111yescharacteristic
241C2×C4C2×C22⋊D18112yes—
242C2×C4C2×C22⋊D18111yescharacteristic
243C2×C4C23×D18111yescharacteristic
244C8—3621no—
245C8—3621no—
246C8—3622no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—88no—
256C22C2×#1566843714yes—
257C22C22×C22⋊D18111yescharacteristic
258C4—22no—
259C4—21no—
260C4—41no—
261C4—7241no—
262C4—7241no—
263C4—7242no—
264C4C2×#1566843712yes—
265C4C2×#1566844811yescharacteristic
266C4#1915386111yescharacteristic
267C2—24no—
268C2—42no—
269C2—82no—
270C2#2604192311yescharacteristic
271C2C22×#1566843711yescharacteristic
272C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radicalD42×C181C2
socleC22×C181C22×D4
hypercenterD42D181
hypocenterC181#3295
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #26041986C2semidirect
2. #26041625C2semidirect
3. #26041595C2semidirect
4. #26041569C2semidirect
5. #26041506C2semidirect
6. #26041476C2semidirect
7. #26041430C2semidirect
8. C4×#15668437C2semidirect
9. D4×#13089678C2semidirect
10. C2×#19153737C2semidirect
13. C2×#15668437C22semidirect
14. C22×#13089678C22semidirect
15. #19153816C22semidirect
17. #19153795C22semidirect
19. C4×Dic362C22semidirect
21. C2×C4⋊Dic181C22semidirect
22. #19153747C22semidirect
23. #19153747C22semidirect
24. C4⋊#13089678C22semidirect
25. #19153739C22semidirect
28. #19153737C22semidirect
29. #19153737C22semidirect
30. #19153737C22semidirect
31. C22⋊#13089678C22semidirect
32. C4×#13089678C22semidirect
100. D4×C181SD16semidirect
136. C181#3295semidirect
230. D4#15668437semidirect

Maximal subgroups

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

subgroupindex
#260419862
#260416252
#260415952
#260415692
#260415062
#260414762
#260414302
C4×#156684372
D4×#130896782
C2×#191537372
D42×C1812
#3295181