Finite Group Databasebeta

← #27150976

Subgroups — #27150976

Order 12416.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13095511C211yesmaximal, characteristic, complemented
2C2×#15690840C211yesmaximal, characteristic, complemented
3C2×#15690835C211yesmaximal, characteristic, complemented
4#20260895C214yesmaximal, complemented
5#20260878C214yesmaximal, complemented
6#20260743C211yesmaximal, characteristic, complemented
7#20260739C211yesmaximal, characteristic, complemented
8#20260699C211yesmaximal, characteristic, complemented
9#525×C97C211yesmaximal, characteristic, complemented
10C2×#13095511—24no—
11C2×#13095511—24no—
12C2×#13095511—22no—
13C2×#13095511C2214yes—
14C22×Dic194C2211yescharacteristic
15C22×D388C2211yescharacteristic
16C22×C22⋊D97—21no—
17C22×C4×D97—21no—
18#15690840—24no—
19#15690835—24no—
20C2×#13095501C2212yescomplemented
21C2×#13095499C2212yescomplemented
22C2×#13095498C2212yescomplemented
23C2×#13095496C2212yescomplemented
24C2×#13095495C2211yescharacteristic
25C2×#13095495C2211yescharacteristic
26#15690733C2214yescomplemented
27#15690730C2214yescomplemented
28C22⋊#11715895C2214yescomplemented
29C2×#41×C97C2211yescharacteristic
30C23×C4×C97C2211yescharacteristic
31#123×C97—21no—
32C22⋊Q8×C97C2212yescomplemented
33#121×C97C2212yescomplemented
34#120×C97—21no—
35C4×D4×C97—22no—
36#117×C97C2211yescharacteristic
37C23×D97—21no—
38#13095511—216no—
39#13095511—28no—
40#13095511—28no—
41C2×Dic194—22no—
42C2×Dic194—21no—
43C2×Dic194D414yescomplemented
44C2×Dic194C2312yes—
45C2×D388—22no—
46C2×D388—21no—
47C2×D388D414yescomplemented
48C2×D388C2312yes—
49C2×C22⋊D97—28no—
50C2×C22⋊D97—24no—
51C22×Dic97—21no—
52C2×C4×D97—28no—
53C2×C4×D97—24no—
54#13095501—24no—
55#13095499—24no—
56#13095498—24no—
57#13095496—24no—
58#13095495—22no—
59#13095495—22no—
60C2×#11715895C2312yes—
61C2×#11715895C2312yes—
62C24×C97D411yescharacteristic
63#41×C97—22no—
64#41×C97—22no—
65C2×Q8×C97C2311yescharacteristic
66C2×D4×C97—22no—
67C2×D4×C97—21no—
68C2×D4×C97C2311yescharacteristic
69C22×C4×C97—22no—
70C22×C4×C97—22no—
71C22×C4×C97—21no—
72C22×C4×C97D411yescharacteristic
73C22×C4×C97D412yes—
74C22×C4×C97C2312yes—
75C22×C4×C97C2311yescharacteristic
76C4⋊C4×C97—22no—
77C4⋊C4×C97C2312yes—
78C22⋊C4×C97—21no—
79C22⋊C4×C97—22no—
80C22⋊C4×C97—22no—
81C42×C97—21no—
82C23×C97—22no—
83C23×C97—21no—
84C23×C97—42no—
85C23×C97C2×D412yes—
86C23×C97C2×D411yescharacteristic
87C22×D97—24no—
88C22×D97—22no—
89C22×D97—44no—
90Q8×C97—21no—
91Q8×C97—21no—
92D4×C97—21no—
93D4×C97—21no—
94D4×C97—44no—
95D4×C97—42no—
96C2×C4×C97—22no—
97C2×C4×C97—24no—
98C2×C4×C97—22no—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97—21no—
102C2×C4×C97—42no—
103C2×C4×C97C2×D412yes—
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2×D412yes—
106C2×C4×C97C2×D411yescharacteristic
107C2×C4×C97C2411yescharacteristic
108C22⋊D97—216no—
109C2×Dic97—24no—
110C2×Dic97—22no—
111D388—24no—
112D388—22no—
113D388C2×D414yes—
114C4×D97—216no—
115Dic194—24no—
116Dic194—22no—
117Dic194C2×D414yes—
118#11715895—22no—
119#11715895—22no—
120C22×C97—22no—
121C22×C97—22no—
122C22×C97—22no—
123C22×C97—22no—
124C22×C97—44no—
125C22×C97—41no—
126C22×C97—42no—
127C22×C97#12012yes—
128C22×C97C22×D411yescharacteristic
129C2×D97—24no—
130C2×D97—48no—
131C2×D97—44no—
132C4×C97—22no—
133C4×C97—41no—
134C4×C97—42no—
135C4×C97#12012yes—
136C4×C97C22×D411yescharacteristic
137C4×C97C22×D411yescharacteristic
138Dic97—24no—
139C2×C97—22no—
140C2×C97—42no—
141C2×C97—81no—
142C2×C97C2×#12011yescharacteristic
143C2×C97#57512yes—
144D97—44no—
145#3005—971nomaximal, Sylow
146C97#300511yescharacteristic, complemented, Sylow
147C22×#41—971no—
148C2×#137—971no—
149C2×AGL(1,ℤ/8)—971no—
150#525D9711yescharacteristic, complemented
151#452—974no—
152#449—974no—
153#419—971no—
154#418—971no—
155#407—971no—
156C2×#41—974no—
157C2×#41—1944no—
158C2×#41—1944no—
159C2×#41—1942no—
160C2×#41C2×D9711yescharacteristic
161C22×Q8—971no—
162C22×D4—971no—
163C22×D4—1941no—
164C23×C4—1941no—
165C23×C4C2×D9711yescharacteristic
166#137—1944no—
167AGL(1,ℤ/8)—1944no—
168C2×Q16—972no—
169C2×SD16—972no—
170C2×SD16—972no—
171C2×D8—972no—
172C2×M16—971no—
173C2×M16—971no—
174#123—21no—
175C22⋊Q8C2×D9712yescomplemented
176#121C2×D9712yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D9711yescharacteristic
180#103—974no—
181#102—974no—
182C22⋊C8—974no—
183C24—1941no—
184C24C22⋊D9711yescharacteristic
185#41—22no—
186#41—22no—
187#41—1948no—
188#41—1948no—
189#41—19416no—
190C2×Q8—974no—
191C2×Q8—972no—
192C2×Q8—1942no—
193C2×Q8—1941no—
194C2×Q8C22×D9711yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—974no—
198C2×D4—972no—
199C2×D4—1948no—
200C2×D4—1944no—
201C2×D4—1942no—
202C2×D4—1941no—
203C2×D4C22×D9711yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—1948no—
208C22×C4—1941no—
209C22×C4—1944no—
210C22×C4C22⋊D9712yes—
211C22×C4C22⋊D9711yescharacteristic
212C22×C4C22×D9712yes—
213C22×C4C22×D9711yescharacteristic
214Q16—1944no—
215SD16—1944no—
216SD16—1944no—
217D8—1944no—
218M16—1942no—
219M16—1942no—
220C2×C8—972no—
221C2×C8—972no—
222C4⋊C4—22no—
223C4⋊C4C22×D9712yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—3884no—
232C23—1944no—
233C23—1942no—
234C23C2×C22⋊D9712yes—
235C23C2×C22⋊D9711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—974no—
239Q8—1944no—
240Q8—1942no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—974no—
246D4—19416no—
247D4—1944no—
248D4—1942no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—19416no—
257C2×C4—1944no—
258C2×C4—1942no—
259C2×C4D4×D9712yes—
260C2×C4D4×D9712yes—
261C2×C4C2×C22⋊D9712yes—
262C2×C4C2×C22⋊D9711yescharacteristic
263C2×C4C23×D9711yescharacteristic
264C8—1942no—
265C8—1942no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—3888no—
274C22—3884no—
275C22—1944no—
276C22#1569077812yes—
277C22C22×C22⋊D9711yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—1944no—
282C4#1569077812yes—
283C4C2×D4×D9711yescharacteristic
284C4C2×D4×D9711yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—3884no—
289C2C2×#1569077811yescharacteristic
290C2#2026153312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#525×C97C2
socleC22×C97C22×D4
hypercenter#525D97
hypocenterC97#3005
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#13095511C2semidirect
2. C2×#15690840C2semidirect
3. C2×#15690835C2semidirect
4. #20260895C2semidirect
5. #20260878C2semidirect
6. #20260743C2semidirect
7. #20260739C2semidirect
8. #20260699C2semidirect
9. #525×C97C2semidirect
20. C2×#13095501C22semidirect
21. C2×#13095499C22semidirect
22. C2×#13095498C22semidirect
23. C2×#13095496C22semidirect
26. #15690733C22semidirect
27. #15690730C22semidirect
28. C22⋊#11715895C22semidirect
32. C22⋊Q8×C97C22semidirect
33. #121×C97C22semidirect
43. C2×Dic194D4semidirect
47. C2×D388D4semidirect
146. C97#3005semidirect
150. #525D97semidirect
175. C22⋊Q8C2×D97semidirect
176. #121C2×D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×#156908402
C2×#156908352
#202608952
#202608782
#202607432
#202607392
#202606992
#525×C972
#300597