Finite Group Databasebeta

← #27151644

Subgroups — #27151644

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#20261245C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3#20261078C212yesmaximal, complemented
4#20261071C211yesmaximal, characteristic, complemented
5#20261065C211yesmaximal, characteristic, complemented
6#20261057C212yesmaximal, complemented
7#20261051C211yesmaximal, characteristic, complemented
8#20260772C212yesmaximal, complemented
9#20260728C212yesmaximal, complemented
10C22⋊C8×D97C211yesmaximal, characteristic, complemented
11#480×C97C211yesmaximal, characteristic, complemented
12C2×#13095510C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14C2×C8×D97C2212yes—
15#15690792—21no—
16#15690790C2211yescharacteristic
17#15690789C2211yescharacteristic
18#15690781C2211yescharacteristic
19#15690778—21no—
20D4×Dic97—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#15690744C2212yescomplemented
31#15690741C2212yescomplemented
32#15690741C2212yescomplemented
33#15690735C2212yescomplemented
34#15690733C2212yescomplemented
35#15690731C2212yescomplemented
36C22⋊#11715895C2211yescharacteristic
37#15690722C2212yes—
38#121×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic
40#107×C97C2212yescomplemented
41#102×C97C2212yescomplemented
42C22⋊C8×C97C2211yescharacteristic
43C23×D97D411yescharacteristic
44#13095510—24no—
45C2×Dic194C2311yescharacteristic
46C2×C22⋊D97—21no—
47C2×C22⋊D97—21no—
48C22×Dic97—21no—
49C22×Dic97—21no—
50C22×Dic97D411yescharacteristic
51C2×C4×D97—24no—
52C2×C4×D97D411yescharacteristic
53C2×C4×D97D411yescharacteristic
54C2×C4×D97C2311yescharacteristic
55C2×C4×D97C2311yescharacteristic
56C2×#11715895C2312yes—
57C8×D97—24no—
58C4⋊Dic97—21no—
59C4⋊Dic97—21no—
60C4⋊Dic97C2312yes—
61C4⋊Dic97C2311yescharacteristic
62#13095490—21no—
63C22⋊Dic97—21no—
64C22⋊Dic97—21no—
65#13095488—22no—
66#13095488—22no—
67#13095488—21no—
68#13095488—21no—
69C4×Dic97—22no—
70C4×Dic97—22no—
71C4×Dic97—21no—
72C2×D4×C97—21no—
73C2×D4×C97C2311yescharacteristic
74C22×C4×C97—21no—
75C22×C4×C97C2311yescharacteristic
76C2×C8×C97C2312yes—
77C4⋊C4×C97—21no—
78C4⋊C4×C97C2312yes—
79C4⋊C4×C97C2311yescharacteristic
80C22⋊C4×C97—21no—
81C23×C97—21no—
82C23×C97C2×D411yescharacteristic
83C22×D97—22no—
84C22×D97—22no—
85C22×D97—41no—
86C22×D97C2×D411yescharacteristic
87C22×D97C2×D411yescharacteristic
88D4×C97—22no—
89D4×C97—42no—
90C2×C4×C97—22no—
91C2×C4×C97—21no—
92C2×C4×C97—22no—
93C2×C4×C97—42no—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—42no—
97C22⋊D97—42no—
98C2×Dic97—22no—
99C2×Dic97—22no—
100C2×Dic97—21no—
101C2×Dic97—21no—
102C2×Dic97—42no—
103C2×Dic97—42no—
104C2×Dic97C2×D411yescharacteristic
105C2×Dic97C2×D411yescharacteristic
106C4×D97—22no—
107C4×D97—22no—
108C4×D97—22no—
109C4×D97#4114yes—
110Dic194—22no—
111C8×C97—22no—
112#11715895—22no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97—41no—
116C22×C97—41no—
117C22×C97—41no—
118C22×C97#13512yes—
119C22×C97C22×D411yescharacteristic
120C2×D97—21no—
121C2×D97—21no—
122C2×D97—21no—
123C2×D97—21no—
124C2×D97—21no—
125C2×D97—44no—
126C2×D97#12312yes—
127C4×C97—21no—
128C4×C97—42no—
129C4×C97—41no—
130C4×C97C2×#4112yes—
131Dic97—21no—
132Dic97—41no—
133Dic97—42no—
134Dic97—41no—
135Dic97#12312yes—
136C2×C97—22no—
137C2×C97—81no—
138C2×C97C2×#12311yescharacteristic
139C2×C97C2×#13511yescharacteristic
140C2×C97#57511yescharacteristic
141D97—22no—
142D97—41no—
143#3091—971nomaximal, Sylow
144C97#309111yescharacteristic, complemented, Sylow
145#525—971no—
146C2×#117—971no—
147#485—972no—
148#484—971no—
149#483—971no—
150#482—972no—
151#481—971no—
152#480D9711yescharacteristic, complemented
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⋊Q8—971no—
162#121—971no—
163#121C2×D9711yescharacteristic
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#107C2×D9712yescomplemented
178#106—972no—
179#106—972no—
180#103—972no—
181#103—972no—
182#102—972no—
183#102C2×D9712yescomplemented
184C22⋊C8—971no—
185C22⋊C8—972no—
186C22⋊C8C2×D9711yescharacteristic
187C24—971no—
188#41—1944no—
189C2×Q8—971no—
190C2×D4—21no—
191C2×D4—1941no—
192C2×D4—1941no—
193C2×D4C22×D9711yescharacteristic
194C22×C4—21no—
195C22×C4—971no—
196C22×C4—971no—
197C22×C4—971no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—1944no—
201C22×C4—1941no—
202C22×C4—1941no—
203C22×C4C22×D9711yescharacteristic
204C2×C8—972no—
205C2×C8—1944no—
206C2×C8C22×D9712yes—
207C4⋊C4—21no—
208C4⋊C4—972no—
209C4⋊C4—971no—
210C4⋊C4—1941no—
211C4⋊C4—1941no—
212C4⋊C4—1941no—
213C4⋊C4C22×D9712yes—
214C4⋊C4C22×D9711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1942no—
217C22⋊C4—1942no—
218C22⋊C4—1941no—
219C22⋊C4—1941no—
220C22⋊C4—1941no—
221C22⋊C4—1941no—
222C42—1942no—
223C42—1942no—
224C42—1941no—
225C23—21no—
226C23—971no—
227C23—971no—
228C23—3881no—
229C23—1942no—
230C23—1942no—
231C23D4×D9711yescharacteristic
232Q8—1942no—
233D4—22no—
234D4—42no—
235D4—3882no—
236D4—3882no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—971no—
242C2×C4—974no—
243C2×C4—971no—
244C2×C4—3882no—
245C2×C4—3882no—
246C2×C4—1942no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4—1941no—
251C2×C4—1941no—
252C2×C4—1942no—
253C2×C4D4×D9711yescharacteristic
254C2×C4C23×D9711yescharacteristic
255C8—22no—
256C8—1942no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—972no—
263C22—3884no—
264C22—1941no—
265C22—1941no—
266C22—1941no—
267C22—1941no—
268C22—1941no—
269C22#1569082912yes—
270C22C2×D4×D9711yescharacteristic
271C4—21no—
272C4—42no—
273C4—41no—
274C4—972no—
275C4—3881no—
276C4—3882no—
277C4—3881no—
278C4—1941no—
279C4#41×D9712yes—
280C2—22no—
281C2—81no—
282C2—3881no—
283C2—1942no—
284C2#123×D9711yescharacteristic
285C2C2×#1569082911yescharacteristic
286C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#480×C97C2
socleC22×C97C22×D4
prime radical#20261245C2
hypercenter#480D97
hypocenterC97#3091
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. #20261078C2semidirect
4. #20261071C2semidirect
5. #20261065C2semidirect
6. #20261057C2semidirect
7. #20261051C2semidirect
8. #20260772C2semidirect
9. #20260728C2semidirect
10. C22⋊C8×D97C2semidirect
11. #480×C97C2semidirect
30. #15690744C22semidirect
31. #15690741C22semidirect
32. #15690741C22semidirect
33. #15690735C22semidirect
34. #15690733C22semidirect
35. #15690731C22semidirect
40. #107×C97C22semidirect
41. #102×C97C22semidirect
144. C97#3091semidirect
152. #480D97semidirect
177. #107C2×D97semidirect
183. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612452
C2×#156907682
#202610782
#202610712
#202610652
#202610572
#202610512
#202607722
#202607282
C22⋊C8×D972
#480×C972
#309197