Finite Group Databasebeta

← #27151650

Subgroups — #27151650

Order 12416.

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

normal subgroups only

This group has 420 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#20261248C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3#20261074C212yesmaximal, complemented
4#20261068C211yesmaximal, characteristic, complemented
5#20261061C212yesmaximal, complemented
6#20261054C211yesmaximal, characteristic, complemented
7#20261048C211yesmaximal, characteristic, complemented
8#20260770C212yesmaximal, complemented
9#20260730C212yesmaximal, complemented
10C22⋊C8×D97C211yesmaximal, characteristic, complemented
11#483×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14C2×C8×D97C2212yes—
15#15690795—21no—
16#15690786C2211yescharacteristic
17#15690785C2211yescharacteristic
18#15690784C2211yescharacteristic
19#15690780—21no—
20#15690772—21no—
21#15690770—21no—
22#15690768—21no—
23#15690768—21no—
24#15690768C2212yes—
25#15690768C2212yes—
26#15690768C2211yescharacteristic
27C2×C4⋊Dic97C2211yescharacteristic
28C2×#13095488—21no—
29C2×C4×Dic97—21no—
30#15690743C2212yescomplemented
31#15690743C2212yescomplemented
32#15690742C2212yescomplemented
33#15690734C2212yescomplemented
34#15690732C2212yescomplemented
35#15690730C2212yescomplemented
36C22⋊#11715895C2211yescharacteristic
37#15690722C2212yes—
38C22⋊Q8×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic
40#106×C97C2212yescomplemented
41#103×C97C2212yescomplemented
42C22⋊C8×C97C2211yescharacteristic
43C23×D97D411yescharacteristic
44#13095512—24no—
45C2×D388—21no—
46C2×D388—21no—
47C2×D388C2311yescharacteristic
48C2×C22⋊D97—21no—
49C22×Dic97—21no—
50C22×Dic97D411yescharacteristic
51C2×C4×D97—24no—
52C2×C4×D97—21no—
53C2×C4×D97D411yescharacteristic
54C2×C4×D97D411yescharacteristic
55C2×C4×D97C2311yescharacteristic
56C2×C4×D97C2311yescharacteristic
57C2×#11715895C2312yes—
58C8×D97—24no—
59C4⋊Dic97—21no—
60C4⋊Dic97C2312yes—
61C4⋊Dic97C2311yescharacteristic
62#13095490—21no—
63#13095488—22no—
64#13095488—22no—
65#13095488—21no—
66#13095488—21no—
67#13095488—21no—
68#13095488—21no—
69C4×Dic97—22no—
70C4×Dic97—22no—
71C4×Dic97—21no—
72C2×Q8×C97C2311yescharacteristic
73C22×C4×C97—21no—
74C22×C4×C97C2311yescharacteristic
75C2×C8×C97C2312yes—
76C4⋊C4×C97—21no—
77C4⋊C4×C97—21no—
78C4⋊C4×C97C2312yes—
79C4⋊C4×C97C2311yescharacteristic
80C22⋊C4×C97—21no—
81C23×C97C2×D411yescharacteristic
82C22×D97—22no—
83C22×D97—22no—
84C22×D97—21no—
85C22×D97—41no—
86C22×D97C2×D411yescharacteristic
87C22×D97C2×D411yescharacteristic
88Q8×C97—22no—
89C2×C4×C97—22no—
90C2×C4×C97—21no—
91C2×C4×C97—21no—
92C2×C4×C97—22no—
93C2×C4×C97—42no—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—42no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97—21no—
100C2×Dic97—42no—
101C2×Dic97C2×D411yescharacteristic
102C2×Dic97C2×D411yescharacteristic
103D388—22no—
104D388—42no—
105D388—42no—
106C4×D97—22no—
107C4×D97—22no—
108C4×D97—22no—
109C4×D97—42no—
110C4×D97#4114yes—
111C8×C97—22no—
112#11715895—22no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97#13512yes—
116C22×C97C22×D411yescharacteristic
117C2×D97—21no—
118C2×D97—21no—
119C2×D97—21no—
120C2×D97—21no—
121C2×D97—21no—
122C2×D97—44no—
123C2×D97—41no—
124C2×D97—41no—
125C2×D97—41no—
126C2×D97#12312yes—
127C4×C97—21no—
128C4×C97—41no—
129C4×C97—42no—
130C4×C97—41no—
131C4×C97C2×#4112yes—
132Dic97—21no—
133Dic97—41no—
134Dic97—42no—
135Dic97#12312yes—
136C2×C97—22no—
137C2×C97C2×#12311yescharacteristic
138C2×C97C2×#13511yescharacteristic
139C2×C97#57511yescharacteristic
140D97—22no—
141D97—41no—
142D97—81no—
143#3091—971nomaximal, Sylow
144C97#309111yescharacteristic, complemented, Sylow
145#525—971no—
146C2×#117—971no—
147#485—972no—
148#484—971no—
149#483D9711yescharacteristic, complemented
150#482—972no—
151#481—971no—
152#480—971no—
153#427—972no—
154#416—972no—
155C2×C22⋊C8—971no—
156C2×#41—971no—
157C23×C4—971no—
158C22×C8—972no—
159#123—1941no—
160C22⋊Q8—971no—
161C22⋊Q8C2×D9711yescharacteristic
162#121—971no—
163#121—971no—
164#120—1941no—
165C4×D4—1941no—
166C4×D4—1941no—
167#117—972no—
168#117—972no—
169#117—971no—
170#117—1941no—
171#117—1941no—
172C2×C4⋊C4—971no—
173C2×C4⋊C4C2×D9711yescharacteristic
174C2×C22⋊C4—1941no—
175C2×C42—1941no—
176#107—972no—
177#107—972no—
178#106—972no—
179#106C2×D9712yescomplemented
180#103—972no—
181#103C2×D9712yescomplemented
182#102—972no—
183#102—972no—
184C22⋊C8—971no—
185C22⋊C8—972no—
186C22⋊C8C2×D9711yescharacteristic
187C24—971no—
188#41—1944no—
189C2×Q8C22×D9711yescharacteristic
190C2×D4—971no—
191C2×D4—1941no—
192C2×D4—1941no—
193C2×D4—1941no—
194C22×C4—21no—
195C22×C4—971no—
196C22×C4—971no—
197C22×C4—971no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—1941no—
201C22×C4—1944no—
202C22×C4—1941no—
203C22×C4C22×D9711yescharacteristic
204C2×C8—972no—
205C2×C8—1944no—
206C2×C8C22×D9712yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—972no—
210C4⋊C4—971no—
211C4⋊C4—1941no—
212C4⋊C4—1941no—
213C4⋊C4C22×D9712yes—
214C4⋊C4C22×D9711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1941no—
217C22⋊C4—1942no—
218C22⋊C4—1942no—
219C22⋊C4—1941no—
220C22⋊C4—1941no—
221C22⋊C4—1941no—
222C42—1942no—
223C42—1942no—
224C42—1941no—
225C23—971no—
226C23—971no—
227C23—3881no—
228C23—1942no—
229C23—1941no—
230C23—1942no—
231C23D4×D9711yescharacteristic
232Q8—22no—
233D4—3882no—
234D4—3882no—
235D4—3882no—
236D4—1942no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—971no—
243C2×C4—974no—
244C2×C4—971no—
245C2×C4—3882no—
246C2×C4—3882no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4—1942no—
251C2×C4—1941no—
252C2×C4—1942no—
253C2×C4D4×D9711yescharacteristic
254C2×C4C23×D9711yescharacteristic
255C8—22no—
256C8—1942no—
257C22—21no—
258C22—21no—
259C22—972no—
260C22—3884no—
261C22—3881no—
262C22—3881no—
263C22—3881no—
264C22—1941no—
265C22—1941no—
266C22—1941no—
267C22—1941no—
268C22—1941no—
269C22#1569083112yes—
270C22C2×D4×D9711yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—972no—
276C4—3881no—
277C4—3882no—
278C4—1941no—
279C4#41×D9712yes—
280C2—22no—
281C2—7761no—
282C2—3881no—
283C2—1942no—
284C2#123×D9711yescharacteristic
285C2C2×#1569083111yescharacteristic
286C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#483×C97C2
socleC22×C97C22×D4
prime radical#20261248C2
hypercenter#483D97
hypocenterC97#3091
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. C2×#15690768C2semidirect
3. #20261074C2semidirect
4. #20261068C2semidirect
5. #20261061C2semidirect
6. #20261054C2semidirect
7. #20261048C2semidirect
8. #20260770C2semidirect
9. #20260730C2semidirect
10. C22⋊C8×D97C2semidirect
11. #483×C97C2semidirect
30. #15690743C22semidirect
31. #15690743C22semidirect
32. #15690742C22semidirect
33. #15690734C22semidirect
34. #15690732C22semidirect
35. #15690730C22semidirect
40. #106×C97C22semidirect
41. #103×C97C22semidirect
144. C97#3091semidirect
149. #483D97semidirect
179. #106C2×D97semidirect
181. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612482
C2×#156907682
#202610742
#202610682
#202610612
#202610542
#202610482
#202607702
#202607302
C22⋊C8×D972
#483×C972
#309197