Finite Group Databasebeta

← #27150988

Subgroups — #27150988

Order 12416.

806 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 169 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×#15690835C212yesmaximal, complemented
2C2×#15690784C211yesmaximal, characteristic, complemented
3#20260880C214yesmaximal, complemented
4#20260874C212yesmaximal, complemented
5#20260869C212yesmaximal, complemented
6#20260736C212yesmaximal, complemented
7#20260699C211yesmaximal, characteristic, complemented
8C22×#41×C97C211yesmaximal, characteristic, complemented
9C22×D388C2211yescharacteristic, complemented
10C22×C22⋊D97—21no—
11#15690835—28no—
12C2×#13095498C2214yescomplemented
13C2×#13095496C2214yescomplemented
14C2×#13095495C2212yescomplemented
15#15690784—21no—
16#15690784—21no—
17#15690784C2212yes—
18#15690784C2212yescomplemented
19C2×C4⋊Dic97C2211yescharacteristic, complemented
20C2×#13095488—21no—
21#15690734C2214yescomplemented
22#15690731C2214yescomplemented
23C22⋊#11715895C2212yescomplemented
24C22⋊#11715895C2212yescomplemented
25C2×#41×C97—28no—
26C2×#41×C97—21no—
27C2×#41×C97—21no—
28C2×#41×C97C2212yes—
29C2×#41×C97C2212yescomplemented
30C22×Q8×C97C2211yescharacteristic, complemented
31C22×D4×C97—21no—
32C22×D4×C97C2211yescharacteristic, complemented
33C23×C4×C97—21no—
34C23×C4×C97C2211yescharacteristic
35C23×D97—21no—
36C2×D388—24no—
37C2×D388—21no—
38C2×D388C2312yes—
39C2×C22⋊D97—22no—
40C2×C22⋊D97—22no—
41C2×C22⋊D97—44no—
42C22×Dic97—21no—
43#13095498—28no—
44#13095496—28no—
45#13095495—24no—
46C2×#11715895C2314yes—
47C4⋊Dic97—21no—
48C4⋊Dic97C2312yes—
49#13095488—22no—
50#13095488—22no—
51C24×C97—21no—
52C24×C97D411yescharacteristic
53#41×C97—28no—
54#41×C97—28no—
55#41×C97—24no—
56#41×C97—24no—
57#41×C97—24no—
58#41×C97D418yescomplemented
59C2×Q8×C97—24no—
60C2×Q8×C97—21no—
61C2×Q8×C97C2312yes—
62C2×D4×C97—28no—
63C2×D4×C97—22no—
64C2×D4×C97—22no—
65C2×D4×C97—24no—
66C2×D4×C97—21no—
67C2×D4×C97C2312yes—
68C22×C4×C97—28no—
69C22×C4×C97—21no—
70C22×C4×C97—22no—
71C22×C4×C97—22no—
72C22×C4×C97—24no—
73C22×C4×C97D411yescharacteristic
74C22×C4×C97D411yescharacteristic
75C22×C4×C97D411yescharacteristic
76C22×C4×C97C2311yescharacteristic
77C22×C4×C97C2311yescharacteristic
78C22×C4×C97C2311yescharacteristic
79C23×C97—24no—
80C23×C97—22no—
81C23×C97—22no—
82C23×C97—22no—
83C23×C97—41no—
84C23×C97—42no—
85C23×C97—42no—
86C23×C97C2×D411yescharacteristic
87C23×C97C2×D411yescharacteristic
88C23×C97C2×D411yescharacteristic
89C22×D97—22no—
90C22×D97—42no—
91C22×D97—42no—
92C22×D97—42no—
93Q8×C97—24no—
94Q8×C97—22no—
95Q8×C97C2×D414yescomplemented
96D4×C97—28no—
97D4×C97—24no—
98D4×C97—28no—
99D4×C97—22no—
100D4×C97C2×D414yescomplemented
101C2×C4×C97—24no—
102C2×C4×C97—28no—
103C2×C4×C97—22no—
104C2×C4×C97—22no—
105C2×C4×C97—22no—
106C2×C4×C97—28no—
107C2×C4×C97—22no—
108C2×C4×C97—22no—
109C2×C4×C97—22no—
110C2×C4×C97C2×D411yescharacteristic
111C2×C4×C97C2×D411yescharacteristic
112C2×C4×C97C2×D411yescharacteristic
113C2×C4×C97C2×D414yes—
114C2×C4×C97C2411yescharacteristic
115C22⋊D97—44no—
116C22⋊D97—44no—
117C2×Dic97—22no—
118C2×Dic97—42no—
119D388—24no—
120D388—42no—
121#11715895—24no—
122C22×C97—21no—
123C22×C97—21no—
124C22×C97—21no—
125C22×C97—21no—
126C22×C97—21no—
127C22×C97—24no—
128C22×C97—21no—
129C22×C97—21no—
130C22×C97—44no—
131C22×C97—48no—
132C22×C97—42no—
133C22×C97—42no—
134C22×C97#12012yes—
135C22×C97AGL(1,ℤ/8)12yescomplemented
136C22×C97C22×D411yescharacteristic
137C2×D97—42no—
138C2×D97—42no—
139C2×D97—42no—
140C2×D97—84no—
141C4×C97—21no—
142C4×C97—24no—
143C4×C97—21no—
144C4×C97#12012yes—
145C4×C97C22×D412yes—
146Dic97—42no—
147C2×C97—22no—
148C2×C97—22no—
149C2×C97—41no—
150C2×C97—44no—
151C2×C97C2×#12011yescharacteristic
152C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C97#57511yescharacteristic
154D97—82no—
155#3006—971nomaximal, Sylow
156C97#300611yescharacteristic, complemented, Sylow
157C22×#41D9711yescharacteristic, complemented
158C2×AGL(1,ℤ/8)—972no—
159C2×#121—971no—
160#449—974no—
161#448—972no—
162#447—972no—
163#417—972no—
164#407—971no—
165C2×#41—28no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41C2×D9712yes—
169C2×#41C2×D9712yescomplemented
170C22×Q8C2×D9711yescharacteristic, complemented
171C22×D4—21no—
172C22×D4—971no—
173C22×D4—1941no—
174C22×D4C2×D9711yescharacteristic, complemented
175C23×C4—21no—
176C23×C4C2×D9711yescharacteristic
177AGL(1,ℤ/8)—1948no—
178C2×SD16—974no—
179C2×D8—974no—
180C2×M16—972no—
181#121—972no—
182#121—972no—
183#121—1941no—
184#121—1941no—
185C2×C4⋊C4—971no—
186C2×C22⋊C4—1941no—
187#103—974no—
188#102—974no—
189C22⋊C8—972no—
190C22⋊C8—972no—
191C24—21no—
192C24—1941no—
193C24C22⋊D9711yescharacteristic
194#41—28no—
195#41—28no—
196#41—24no—
197#41—24no—
198#41—24no—
199#41C22⋊D9718yescomplemented
200C2×Q8—24no—
201C2×Q8—21no—
202C2×Q8C22×D9712yes—
203C2×D4—28no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—972no—
209C2×D4—3884no—
210C2×D4—1942no—
211C2×D4—1942no—
212C2×D4—1941no—
213C2×D4—1944no—
214C2×D4C22×D9712yes—
215C22×C4—28no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—1941no—
221C22×C4C22⋊D9711yescharacteristic
222C22×C4C22⋊D9711yescharacteristic
223C22×C4C22⋊D9711yescharacteristic
224C22×C4C22×D9711yescharacteristic
225C22×C4C22×D9711yescharacteristic
226C22×C4C22×D9711yescharacteristic
227SD16—1948no—
228D8—1948no—
229M16—1944no—
230C2×C8—974no—
231C4⋊C4—972no—
232C4⋊C4—1941no—
233C22⋊C4—1942no—
234C22⋊C4—1942no—
235C23—24no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—3882no—
243C23—3882no—
244C23—3882no—
245C23—1942no—
246C23C2×C22⋊D9711yescharacteristic
247C23C2×C22⋊D9711yescharacteristic
248C23C2×C22⋊D9711yescharacteristic
249Q8—24no—
250Q8—22no—
251Q8C2×C22⋊D9714yescomplemented
252D4—28no—
253D4—24no—
254D4—28no—
255D4—22no—
256D4—3884no—
257D4—3884no—
258D4—3882no—
259D4—1944no—
260D4C2×C22⋊D9714yescomplemented
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—3882no—
271C2×C4—1942no—
272C2×C4C2×C22⋊D9711yescharacteristic
273C2×C4C2×C22⋊D9711yescharacteristic
274C2×C4C2×C22⋊D9711yescharacteristic
275C2×C4C2×C22⋊D9714yes—
276C2×C4C23×D9711yescharacteristic
277C8—1944no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—48no—
288C22—42no—
289C22—42no—
290C22—7764no—
291C22—3882no—
292C22—3882no—
293C22—3882no—
294C22#1569077912yes—
295C22#1569083512yescomplemented
296C22C22×C22⋊D9711yescharacteristic
297C4—21no—
298C4—24no—
299C4—21no—
300C4—3882no—
301C4#1569077912yes—
302C4C22×C22⋊D9712yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—44no—
307C2—7762no—
308C2C2×#1569077911yescharacteristic
309C2C2×#1569083511yescharacteristic
310C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×#41×C97C2
socleC22×C97C22×D4
hypercenterC22×#41D97
hypocenterC97#3006
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×#15690784C2semidirect
3. #20260880C2semidirect
4. #20260874C2semidirect
5. #20260869C2semidirect
6. #20260736C2semidirect
7. #20260699C2semidirect
8. C22×#41×C97C2semidirect
9. C22×D388C22semidirect
12. C2×#13095498C22semidirect
13. C2×#13095496C22semidirect
14. C2×#13095495C22semidirect
18. #15690784C22semidirect
19. C2×C4⋊Dic97C22semidirect
21. #15690734C22semidirect
22. #15690731C22semidirect
23. C22⋊#11715895C22semidirect
24. C22⋊#11715895C22semidirect
29. C2×#41×C97C22semidirect
30. C22×Q8×C97C22semidirect
32. C22×D4×C97C22semidirect
58. #41×C97D4semidirect
95. Q8×C97C2×D4semidirect
100. D4×C97C2×D4semidirect
135. C22×C97AGL(1,ℤ/8)semidirect
156. C97#3006semidirect
157. C22×#41D97semidirect
169. C2×#41C2×D97semidirect
170. C22×Q8C2×D97semidirect
174. C22×D4C2×D97semidirect
199. #41C22⋊D97semidirect
251. Q8C2×C22⋊D97semidirect
260. D4C2×C22⋊D97semidirect
295. C22#15690835semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×#156907842
#202608802
#202608742
#202608692
#202607362
#202606992
C22×#41×C972
#300697