Finite Group Databasebeta

← #27151342

Subgroups — #27151342

Order 12416.

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

normal subgroups only

This group has 500 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
1C2×#15690830C211yesmaximal, characteristic, complemented
2C22×#13095495C211yesmaximal, characteristic, complemented
3#20261246C211yesmaximal, characteristic, complemented
4#20260991C212yesmaximal, complemented
5#20260986C212yesmaximal, complemented
6#20260983C212yesmaximal, complemented
7#20260979C212yesmaximal, complemented
8#20260771C211yesmaximal, characteristic, complemented
9#20260731C211yesmaximal, characteristic, complemented
10#20260729C211yesmaximal, characteristic, complemented
11#525×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13#15690830—24no—
14C2×#13095501C2211yescharacteristic, complemented
15C2×#13095499C2211yescharacteristic, complemented
16C2×#13095498C2211yescharacteristic, complemented
17C2×#13095496C2211yescharacteristic, complemented
18C2×#13095495—24no—
19C2×#13095495C2212yes—
20C2×#13095495C2212yes—
21C22×#11715895C2211yescharacteristic
22C22×#11715895C2211yescharacteristic
23#15690793—21no—
24#15690789C2212yes—
25#15690784C2212yes—
26#15690779—21no—
27C4×C22⋊D97—22no—
28#15690769C2211yescharacteristic
29#15690743C2212yescomplemented
30#15690741C2212yescomplemented
31#15690734C2212yescomplemented
32#15690733C2212yescomplemented
33#15690731C2212yescomplemented
34#15690730C2212yescomplemented
35C2×#41×C97C2211yescharacteristic
36C23×C4×C97C2211yescharacteristic
37#123×C97—21no—
38C22⋊Q8×C97C2212yes—
39#121×C97C2212yes—
40#120×C97—21no—
41C4×D4×C97—22no—
42#117×C97C2211yescharacteristic
43#13095511—22no—
44#13095511—22no—
45C2×Dic194C2311yescharacteristic
46C2×D388C2311yescharacteristic
47C2×C22⋊D97—22no—
48C2×C22⋊D97—21no—
49C2×C4×D97—21no—
50#13095501—22no—
51#13095499—22no—
52#13095498—22no—
53#13095496—22no—
54#13095495—24no—
55#13095495—24no—
56C2×#11715895—22no—
57C2×#11715895D414yescomplemented
58C2×#11715895C2312yes—
59C2×#11715895C2311yescharacteristic
60C2×#11715895C2311yescharacteristic
61C4⋊Dic97C2312yes—
62#13095490—22no—
63C22⋊Dic97—21no—
64C22⋊Dic97—22no—
65#13095488—22no—
66C4×Dic97—21no—
67C24×C97D411yescharacteristic
68#41×C97—22no—
69#41×C97—22no—
70C2×Q8×C97C2311yescharacteristic
71C2×D4×C97—22no—
72C2×D4×C97—21no—
73C2×D4×C97C2311yescharacteristic
74C22×C4×C97—22no—
75C22×C4×C97—22no—
76C22×C4×C97—21no—
77C22×C4×C97D412yes—
78C22×C4×C97D411yescharacteristic
79C22×C4×C97C2312yes—
80C22×C4×C97C2311yescharacteristic
81C4⋊C4×C97—22no—
82C4⋊C4×C97C2312yes—
83C22⋊C4×C97—21no—
84C22⋊C4×C97—22no—
85C22⋊C4×C97—22no—
86C42×C97—21no—
87C23×C97—22no—
88C23×C97—21no—
89C23×C97—42no—
90C23×C97C2×D412yes—
91C23×C97C2×D411yescharacteristic
92C22×D97—21no—
93Q8×C97—21no—
94Q8×C97—21no—
95D4×C97—21no—
96D4×C97—21no—
97D4×C97—44no—
98D4×C97—42no—
99C2×C4×C97—22no—
100C2×C4×C97—24no—
101C2×C4×C97—22no—
102C2×C4×C97—22no—
103C2×C4×C97—22no—
104C2×C4×C97—21no—
105C2×C4×C97—42no—
106C2×C4×C97C2×D412yes—
107C2×C4×C97C2×D412yes—
108C2×C4×C97C2×D411yescharacteristic
109C2×C4×C97#4112yes—
110C2×C4×C97C2411yescharacteristic
111C22⋊D97—44no—
112C22⋊D97—42no—
113C2×Dic97—22no—
114C2×Dic97—21no—
115D388—21no—
116D388—21no—
117C4×D97—42no—
118Dic194—21no—
119Dic194—21no—
120#11715895—22no—
121#11715895C2×D412yes—
122#11715895C2×D412yes—
123C22×C97—22no—
124C22×C97—22no—
125C22×C97—22no—
126C22×C97—22no—
127C22×C97—44no—
128C22×C97—41no—
129C22×C97—42no—
130C22×C97#12112yes—
131C22×C97C22×D411yescharacteristic
132C2×D97—41no—
133C2×D97—42no—
134C4×C97—22no—
135C4×C97—41no—
136C4×C97—42no—
137C4×C97#12112yes—
138C4×C97C22×D411yescharacteristic
139C4×C97C2×#4111yescharacteristic
140Dic97—41no—
141Dic97—42no—
142C2×C97—22no—
143C2×C97—42no—
144C2×C97—81no—
145C2×C97C2×#12111yescharacteristic
146C2×C97#57512yes—
147D97—81no—
148#3054—971nomaximal, Sylow
149C97#305411yescharacteristic, complemented, Sylow
150C2×#135—971no—
151C22×M16—971no—
152#525—971no—
153#525D9711yescharacteristic, complemented
154#470—972no—
155#469—972no—
156#468—972no—
157#468—972no—
158#427—971no—
159#416—971no—
160#416—971no—
161C2×#41—971no—
162C2×#41C2×D9711yescharacteristic
163C23×C4C2×D9711yescharacteristic
164#135—1944no—
165C2×Q16—971no—
166C2×SD16—971no—
167C2×SD16—971no—
168C2×D8—971no—
169C2×M16—972no—
170C2×M16—972no—
171C2×M16—1944no—
172C22×C8—971no—
173C22×C8—971no—
174#123—21no—
175#123—1941no—
176C22⋊Q8—972no—
177C22⋊Q8C2×D9712yes—
178#121—972no—
179#121C2×D9712yes—
180#120—21no—
181#120—1941no—
182C4×D4—22no—
183C4×D4—1942no—
184#117—971no—
185#117C2×D9711yescharacteristic
186#107—972no—
187#106—972no—
188#103—972no—
189#103—972no—
190#102—972no—
191#102—972no—
192C24C22⋊D9711yescharacteristic
193#41—22no—
194#41—22no—
195#41—1942no—
196#41—1942no—
197C2×Q8—971no—
198C2×Q8C22×D9711yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—971no—
202C2×D4—1942no—
203C2×D4—1941no—
204C2×D4C22×D9711yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—1941no—
209C22×C4C22⋊D9711yescharacteristic
210C22×C4C22⋊D9712yes—
211C22×C4C22×D9712yes—
212C22×C4C22×D9711yescharacteristic
213Q16—1942no—
214SD16—1942no—
215SD16—1942no—
216D8—1942no—
217M16—1944no—
218M16—1944no—
219C2×C8—974no—
220C2×C8—971no—
221C2×C8—971no—
222C2×C8—972no—
223C2×C8—1942no—
224C4⋊C4—22no—
225C4⋊C4—972no—
226C4⋊C4—1942no—
227C4⋊C4C22×D9712yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—1941no—
232C22⋊C4—1942no—
233C22⋊C4—1942no—
234C42—21no—
235C42—1941no—
236C23—22no—
237C23—21no—
238C23—42no—
239C23—1941no—
240C23C2×C22⋊D9712yes—
241C23C2×C22⋊D9711yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—1941no—
245Q8—1941no—
246D4—21no—
247D4—21no—
248D4—44no—
249D4—42no—
250D4—3884no—
251D4—3882no—
252D4—1941no—
253D4—1941no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—3882no—
262C2×C4—1942no—
263C2×C4—1941no—
264C2×C4D4×D9712yes—
265C2×C4C2×C22⋊D9712yes—
266C2×C4C2×C22⋊D9711yescharacteristic
267C2×C4#1309551012yes—
268C2×C4C23×D9711yescharacteristic
269C8—972no—
270C8—972no—
271C8—1942no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—44no—
277C22—41no—
278C22—42no—
279C22—3881no—
280C22—3882no—
281C22#1569078212yes—
282C22C22×C22⋊D9711yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—3881no—
287C4—3882no—
288C4#1569078212yes—
289C4C2×D4×D9711yescharacteristic
290C4C2×#1309551011yescharacteristic
291C2—22no—
292C2—42no—
293C2—81no—
294C2—7761no—
295C2C2×#1569078211yescharacteristic
296C2#2026153312yes—
subgroupG/H
centerC2×C4C2×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#525×C97C2
socleC22×C97C22×D4
hypercenter#525D97
hypocenterC97#3054
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#15690830C2semidirect
2. C22×#13095495C2semidirect
3. #20261246C2semidirect
4. #20260991C2semidirect
5. #20260986C2semidirect
6. #20260983C2semidirect
7. #20260979C2semidirect
8. #20260771C2semidirect
9. #20260731C2semidirect
10. #20260729C2semidirect
11. #525×C97C2semidirect
14. C2×#13095501C22semidirect
15. C2×#13095499C22semidirect
16. C2×#13095498C22semidirect
17. C2×#13095496C22semidirect
29. #15690743C22semidirect
30. #15690741C22semidirect
31. #15690734C22semidirect
32. #15690733C22semidirect
33. #15690731C22semidirect
34. #15690730C22semidirect
57. C2×#11715895D4semidirect
149. C97#3054semidirect
153. #525D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908302
C22×#130954952
#202612462
#202609912
#202609862
#202609832
#202609792
#202607712
#202607312
#202607292
#525×C972
#305497