Finite Group Databasebeta

← #27151671

Subgroups — #27151671

Order 12416.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D97C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3#20261062C212yesmaximal, complemented
4#20261058C212yesmaximal, complemented
5#20261054C211yesmaximal, characteristic, complemented
6#20261051C211yesmaximal, characteristic, complemented
7#20261048C211yesmaximal, characteristic, complemented
8#20260776C212yesmaximal, complemented
9#20260738C212yesmaximal, complemented
10#20260697C211yesmaximal, characteristic, complemented
11#480×C97C211yesmaximal, characteristic, complemented
12C2×D4×D97—21no—
13C2×D4×D97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15C2×#13095494C2212yes—
16#15690785C2211yescharacteristic
17#15690784C2211yescharacteristic
18#15690783—21no—
19#15690782—21no—
20#15690781C2211yescharacteristic
21#15690768—21no—
22#15690768—21no—
23#15690768C2212yes—
24#15690768C2212yescomplemented
25C2×C4⋊Dic97C2211yescharacteristic, complemented
26C4⋊C4×D97C2211yescharacteristic
27C2×#13095488—21no—
28C22⋊C4×D97—21no—
29C2×C4×Dic97—21no—
30#15690743C2212yescomplemented
31#15690742C2212yescomplemented
32#15690741C2212yescomplemented
33#15690732C2212yescomplemented
34#15690731C2212yescomplemented
35#15690730C2212yescomplemented
36C22⋊#11715895C2211yescharacteristic, complemented
37#15690722C2212yescomplemented
38#121×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic, complemented
40#107×C97C2212yescomplemented
41#102×C97C2212yescomplemented
42C22⋊C8×C97C2211yescharacteristic, complemented
43C23×D97—21no—
44C23×D97D411yescharacteristic
45C2×D388—21no—
46C2×D388C2311yescharacteristic
47C2×C22⋊D97—21no—
48C2×C22⋊D97—21no—
49C2×C22⋊D97—21no—
50D4×D97—24no—
51D4×D97—44no—
52C22×Dic97—21no—
53C22×Dic97D411yescharacteristic
54C2×C4×D97—24no—
55C2×C4×D97—21no—
56C2×C4×D97D411yescharacteristic
57C2×C4×D97D411yescharacteristic
58C2×C4×D97C2311yescharacteristic
59C2×C4×D97C2311yescharacteristic
60#13095494—24no—
61C2×#11715895C2312yes—
62C4⋊Dic97—21no—
63C4⋊Dic97C2311yescharacteristic
64C4⋊Dic97C2312yes—
65#13095490—21no—
66C22⋊Dic97—21no—
67#13095488—22no—
68#13095488—22no—
69#13095488—21no—
70#13095488—21no—
71C4×Dic97—22no—
72C4×Dic97—22no—
73C2×D4×C97—21no—
74C2×D4×C97C2311yescharacteristic
75C22×C4×C97—21no—
76C22×C4×C97C2311yescharacteristic
77C2×C8×C97C2312yes—
78C4⋊C4×C97—21no—
79C4⋊C4×C97C2311yescharacteristic
80C4⋊C4×C97C2312yes—
81C22⋊C4×C97—21no—
82C23×C97—21no—
83C23×C97C2×D411yescharacteristic
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—21no—
87C22×D97—41no—
88C22×D97—42no—
89C22×D97—42no—
90C22×D97—42no—
91C22×D97C2×D411yescharacteristic
92C22×D97C2×D411yescharacteristic
93D4×C97—22no—
94D4×C97—42no—
95C2×C4×C97—22no—
96C2×C4×C97—21no—
97C2×C4×C97—22no—
98C2×C4×C97—42no—
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2411yescharacteristic
101C22⋊D97—42no—
102C22⋊D97—42no—
103C22⋊D97—42no—
104C2×Dic97—22no—
105C2×Dic97—22no—
106C2×Dic97—21no—
107C2×Dic97—42no—
108C2×Dic97C2×D411yescharacteristic
109C2×Dic97C2×D411yescharacteristic
110D388—22no—
111D388—42no—
112C4×D97—22no—
113C4×D97—22no—
114C4×D97—22no—
115C4×D97—42no—
116C4×D97#4114yes—
117C8×C97—22no—
118#11715895—22no—
119C22×C97—21no—
120C22×C97—21no—
121C22×C97—41no—
122C22×C97—41no—
123C22×C97—41no—
124C22×C97AGL(1,ℤ/8)12yes—
125C22×C97C22×D411yescharacteristic
126C2×D97—21no—
127C2×D97—21no—
128C2×D97—21no—
129C2×D97—21no—
130C2×D97—21no—
131C2×D97—44no—
132C2×D97—41no—
133C2×D97—41no—
134C2×D97—41no—
135C2×D97—84no—
136C2×D97#12312yes—
137C4×C97—21no—
138C4×C97—41no—
139C4×C97—42no—
140C4×C97C2×#4112yes—
141Dic97—21no—
142Dic97—41no—
143Dic97—42no—
144Dic97#12312yes—
145C2×C97—22no—
146C2×C97—81no—
147C2×C97C2×#12311yescharacteristic
148C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C97#57511yescharacteristic
150D97—22no—
151D97—41no—
152D97—81no—
153#3094—971nomaximal, Sylow
154C97#309411yescharacteristic, complemented, Sylow
155C2×#121—971no—
156C2×#117—971no—
157#482—972no—
158#482—972no—
159#481—971no—
160#481—971no—
161#480—971no—
162#480D9711yescharacteristic, complemented
163#428—972no—
164#418—972no—
165#407—971no—
166C22×D4—971no—
167C22×D4—1941no—
168C23×C4—971no—
169C2×M16—972no—
170#121—971no—
171#121—971no—
172#121—971no—
173#121—1941no—
174#121—1941no—
175#121C2×D9711yescharacteristic
176#117—972no—
177#117—972no—
178#117—1941no—
179#117—1941no—
180C2×C4⋊C4—971no—
181C2×C4⋊C4—971no—
182C2×C4⋊C4C2×D9711yescharacteristic, complemented
183C2×C22⋊C4—1941no—
184C2×C22⋊C4—1941no—
185C2×C42—1941no—
186#107—972no—
187#107C2×D9712yescomplemented
188#106—972no—
189#106—972no—
190#102—972no—
191#102—972no—
192#102—972no—
193#102C2×D9712yescomplemented
194C22⋊C8—971no—
195C22⋊C8—972no—
196C22⋊C8C2×D9711yescharacteristic, complemented
197C24—971no—
198C24—1941no—
199C2×D4—21no—
200C2×D4—971no—
201C2×D4—3884no—
202C2×D4—1941no—
203C2×D4—1944no—
204C2×D4—1941no—
205C2×D4—1941no—
206C2×D4—1941no—
207C2×D4C22×D9711yescharacteristic
208C22×C4—21no—
209C22×C4—971no—
210C22×C4—971no—
211C22×C4—971no—
212C22×C4—971no—
213C22×C4—971no—
214C22×C4—1944no—
215C22×C4—1941no—
216C22×C4—1941no—
217C22×C4C22×D9711yescharacteristic
218M16—1944no—
219C2×C8—972no—
220C2×C8C22×D9712yes—
221C4⋊C4—21no—
222C4⋊C4—972no—
223C4⋊C4—971no—
224C4⋊C4—1941no—
225C4⋊C4—1941no—
226C4⋊C4C22×D9712yes—
227C4⋊C4C22×D9711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—1941no—
230C22⋊C4—1942no—
231C22⋊C4—1942no—
232C22⋊C4—1941no—
233C22⋊C4—1941no—
234C42—1942no—
235C42—1942no—
236C23—21no—
237C23—971no—
238C23—971no—
239C23—3881no—
240C23—3882no—
241C23—3882no—
242C23—3882no—
243C23—1942no—
244C23—1941no—
245C23—1942no—
246C23D4×D9711yescharacteristic
247D4—22no—
248D4—42no—
249D4—3882no—
250D4—3882no—
251D4—3882no—
252D4—3882no—
253D4—1942no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—971no—
259C2×C4—974no—
260C2×C4—971no—
261C2×C4—3882no—
262C2×C4—3882no—
263C2×C4—1942no—
264C2×C4—1942no—
265C2×C4—1942no—
266C2×C4—1942no—
267C2×C4—1941no—
268C2×C4—1942no—
269C2×C4D4×D9711yescharacteristic
270C2×C4C23×D9711yescharacteristic
271C8—22no—
272C8—1942no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—7764no—
279C22—972no—
280C22—3884no—
281C22—3881no—
282C22—3881no—
283C22—3881no—
284C22—1941no—
285C22—1941no—
286C22—1941no—
287C22—1941no—
288C22—1941no—
289C22#1569083412yes—
290C22C2×D4×D9711yescharacteristic
291C4—21no—
292C4—42no—
293C4—41no—
294C4—972no—
295C4—3881no—
296C4—3882no—
297C4—1941no—
298C4#41×D9712yes—
299C2—22no—
300C2—81no—
301C2—7761no—
302C2—3881no—
303C2—1942no—
304C2#123×D9711yescharacteristic
305C2C2×#1569083411yescharacteristic
306C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#480×C97C2
socleC22×C97C22×D4
prime radical#121×D97C2
hypercenter#480D97
hypocenterC97#3094
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#15690768C2semidirect
3. #20261062C2semidirect
4. #20261058C2semidirect
5. #20261054C2semidirect
6. #20261051C2semidirect
7. #20261048C2semidirect
8. #20260776C2semidirect
9. #20260738C2semidirect
10. #20260697C2semidirect
11. #480×C97C2semidirect
24. #15690768C22semidirect
25. C2×C4⋊Dic97C22semidirect
30. #15690743C22semidirect
31. #15690742C22semidirect
32. #15690741C22semidirect
33. #15690732C22semidirect
34. #15690731C22semidirect
35. #15690730C22semidirect
36. C22⋊#11715895C22semidirect
37. #15690722C22semidirect
39. C2×C4⋊C4×C97C22semidirect
40. #107×C97C22semidirect
41. #102×C97C22semidirect
42. C22⋊C8×C97C22semidirect
154. C97#3094semidirect
162. #480D97semidirect
182. C2×C4⋊C4C2×D97semidirect
187. #107C2×D97semidirect
193. #102C2×D97semidirect
196. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#121×D972
C2×#156907682
#202610622
#202610582
#202610542
#202610512
#202610482
#202607762
#202607382
#202606972
#480×C972
#309497