Finite Group Databasebeta

← #27150986

Subgroups — #27150986

Order 12416.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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×#15690836C212yesmaximal, complemented
3#20260878C214yesmaximal, complemented
4#20260873C212yesmaximal, complemented
5#20260868C212yesmaximal, complemented
6#20260734C212yesmaximal, complemented
7#20260699C211yesmaximal, characteristic, complemented
8C2×#121×C97C211yesmaximal, characteristic, complemented
9C2×#13095511—28no—
10C2×#13095511—21no—
11C2×#13095511—21no—
12C2×#13095511C2212yes—
13C2×#13095511C2212yescomplemented
14C22×Dic194C2211yescharacteristic, complemented
15C22×D388C2211yescharacteristic, complemented
16C22×C22⋊D97—21no—
17C22×C4×D97—21no—
18#15690836—28no—
19C2×#13095499C2214yescomplemented
20C2×#13095496C2214yescomplemented
21C2×#13095495C2212yescomplemented
22#15690733C2214yescomplemented
23#15690730C2214yescomplemented
24C22⋊#11715895C2212yescomplemented
25C22⋊#11715895C2212yescomplemented
26C22×D4×C97—21no—
27C22×D4×C97C2211yescharacteristic, complemented
28C23×C4×C97C2211yescharacteristic
29#121×C97—21no—
30#121×C97—21no—
31#121×C97C2212yes—
32#121×C97C2212yescomplemented
33C2×C4⋊C4×C97C2211yescharacteristic, complemented
34C2×C22⋊C4×C97—21no—
35C23×D97—21no—
36#13095511—28no—
37#13095511—28no—
38#13095511—24no—
39#13095511—24no—
40#13095511—24no—
41#13095511D418yescomplemented
42C2×Dic194—21no—
43C2×Dic194—24no—
44C2×Dic194C2312yes—
45C2×D388—21no—
46C2×D388—24no—
47C2×D388C2312yes—
48C2×C22⋊D97—28no—
49C2×C22⋊D97—22no—
50C2×C22⋊D97—22no—
51C22×Dic97—21no—
52C2×C4×D97—28no—
53C2×C4×D97—22no—
54C2×C4×D97—22no—
55#13095499—28no—
56#13095496—28no—
57#13095495—24no—
58C2×#11715895C2314yes—
59C24×C97—21no—
60C24×C97D411yescharacteristic
61C2×D4×C97—22no—
62C2×D4×C97—22no—
63C2×D4×C97—24no—
64C2×D4×C97—21no—
65C2×D4×C97—44no—
66C2×D4×C97C2312yes—
67C22×C4×C97—24no—
68C22×C4×C97—21no—
69C22×C4×C97D411yescharacteristic
70C22×C4×C97D411yescharacteristic
71C22×C4×C97D411yescharacteristic
72C22×C4×C97C2311yescharacteristic
73C22×C4×C97C2311yescharacteristic
74C22×C4×C97C2311yescharacteristic
75C4⋊C4×C97—21no—
76C4⋊C4×C97C2312yes—
77C22⋊C4×C97—22no—
78C22⋊C4×C97—22no—
79C23×C97—22no—
80C23×C97—22no—
81C23×C97—22no—
82C23×C97—41no—
83C23×C97—42no—
84C23×C97—42no—
85C23×C97—42no—
86C23×C97C2×D411yescharacteristic
87C23×C97C2×D411yescharacteristic
88C23×C97C2×D411yescharacteristic
89C22×D97—24no—
90C22×D97—22no—
91C22×D97—42no—
92C22×D97—42no—
93D4×C97—24no—
94D4×C97—44no—
95D4×C97—44no—
96D4×C97—42no—
97C2×C4×C97—22no—
98C2×C4×C97—22no—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97—22no—
102C2×C4×C97—22no—
103C2×C4×C97—42no—
104C2×C4×C97C2×D414yes—
105C2×C4×C97C2×D411yescharacteristic
106C2×C4×C97C2×D411yescharacteristic
107C2×C4×C97C2×D411yescharacteristic
108C2×C4×C97C2411yescharacteristic
109C22⋊D97—28no—
110C22⋊D97—28no—
111C2×Dic97—24no—
112C2×Dic97—22no—
113D388—24no—
114D388—22no—
115D388C2×D414yescomplemented
116C4×D97—28no—
117C4×D97—28no—
118Dic194—24no—
119Dic194—22no—
120Dic194C2×D414yescomplemented
121#11715895—24no—
122C22×C97—21no—
123C22×C97—21no—
124C22×C97—21no—
125C22×C97—21no—
126C22×C97—21no—
127C22×C97—21no—
128C22×C97—21no—
129C22×C97—44no—
130C22×C97—42no—
131C22×C97—42no—
132C22×C97—42no—
133C22×C97—84no—
134C22×C97#12012yes—
135C22×C97AGL(1,ℤ/8)12yescomplemented
136C22×C97C22×D411yescharacteristic
137C2×D97—24no—
138C2×D97—48no—
139C2×D97—42no—
140C2×D97—42no—
141C4×C97—21no—
142C4×C97—21no—
143C4×C97—42no—
144C4×C97#12012yes—
145C4×C97C22×D412yes—
146Dic97—24no—
147C2×C97—22no—
148C2×C97—22no—
149C2×C97—41no—
150C2×C97—82no—
151C2×C97C2×#12011yescharacteristic
152C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C97#57511yescharacteristic
154D97—44no—
155#3006—971nomaximal, Sylow
156C97#300611yescharacteristic, complemented, Sylow
157C22×#41—971no—
158C2×AGL(1,ℤ/8)—972no—
159C2×#121D9711yescharacteristic, complemented
160#449—974no—
161#448—972no—
162#447—972no—
163#417—972no—
164#407—971no—
165C2×#41—972no—
166C2×#41—972no—
167C2×#41—1948no—
168C2×#41—1941no—
169C2×#41—1941no—
170C22×Q8—971no—
171C22×D4—21no—
172C22×D4—971no—
173C22×D4—1941no—
174C22×D4C2×D9711yescharacteristic, complemented
175C23×C4—1941no—
176C23×C4C2×D9711yescharacteristic
177AGL(1,ℤ/8)—1948no—
178C2×SD16—974no—
179C2×D8—974no—
180C2×M16—972no—
181#121—21no—
182#121—21no—
183#121C2×D9712yes—
184#121C2×D9712yescomplemented
185C2×C4⋊C4C2×D9711yescharacteristic, complemented
186C2×C22⋊C4—21no—
187#103—974no—
188#102—974no—
189C22⋊C8—972no—
190C22⋊C8—972no—
191C24—21no—
192C24—1941no—
193C24C22⋊D9711yescharacteristic
194#41—978no—
195#41—1944no—
196#41—1944no—
197#41—1948no—
198#41—1948no—
199#41—1944no—
200C2×Q8—972no—
201C2×Q8—1944no—
202C2×Q8—1941no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—44no—
208C2×D4—972no—
209C2×D4—1944no—
210C2×D4—1948no—
211C2×D4—1942no—
212C2×D4—1942no—
213C2×D4—1941no—
214C2×D4C22×D9712yes—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—1948no—
218C22×C4—1941no—
219C22×C4—1942no—
220C22×C4—1942no—
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—21no—
232C4⋊C4C22×D9712yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—3882no—
243C23—3882no—
244C23—1944no—
245C23—1942no—
246C23C2×C22⋊D9711yescharacteristic
247C23C2×C22⋊D9711yescharacteristic
248C23C2×C22⋊D9711yescharacteristic
249Q8—974no—
250Q8—1944no—
251Q8—1942no—
252D4—24no—
253D4—44no—
254D4—44no—
255D4—42no—
256D4—974no—
257D4—1944no—
258D4—1948no—
259D4—1948no—
260D4—1942no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—1948no—
269C2×C4—1944no—
270C2×C4—1942no—
271C2×C4—1948no—
272C2×C4D4×D9714yes—
273C2×C4C2×C22⋊D9711yescharacteristic
274C2×C4C2×C22⋊D9711yescharacteristic
275C2×C4C2×C22⋊D9711yescharacteristic
276C2×C4C23×D9711yescharacteristic
277C8—1944no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—84no—
290C22—3888no—
291C22—3882no—
292C22—3882no—
293C22—1944no—
294C22#1569077812yes—
295C22#1569083612yescomplemented
296C22C22×C22⋊D9711yescharacteristic
297C4—21no—
298C4—21no—
299C4—42no—
300C4—1944no—
301C4#1569077812yes—
302C4C2×D4×D9712yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—82no—
307C2—3884no—
308C2C2×#1569077811yescharacteristic
309C2C2×#1569083611yescharacteristic
310C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#121×C97C2
socleC22×C97C22×D4
hypercenterC2×#121D97
hypocenterC97#3006
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13095511C2semidirect
2. C2×#15690836C2semidirect
3. #20260878C2semidirect
4. #20260873C2semidirect
5. #20260868C2semidirect
6. #20260734C2semidirect
7. #20260699C2semidirect
8. C2×#121×C97C2semidirect
13. C2×#13095511C22semidirect
14. C22×Dic194C22semidirect
15. C22×D388C22semidirect
19. C2×#13095499C22semidirect
20. C2×#13095496C22semidirect
21. C2×#13095495C22semidirect
22. #15690733C22semidirect
23. #15690730C22semidirect
24. C22⋊#11715895C22semidirect
25. C22⋊#11715895C22semidirect
27. C22×D4×C97C22semidirect
32. #121×C97C22semidirect
33. C2×C4⋊C4×C97C22semidirect
41. #13095511D4semidirect
115. D388C2×D4semidirect
120. Dic194C2×D4semidirect
135. C22×C97AGL(1,ℤ/8)semidirect
156. C97#3006semidirect
159. C2×#121D97semidirect
174. C22×D4C2×D97semidirect
184. #121C2×D97semidirect
185. C2×C4⋊C4C2×D97semidirect
295. C22#15690836semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×#156908362
#202608782
#202608732
#202608682
#202607342
#202606992
C2×#121×C972
#300697