Finite Group Databasebeta

← #27151666

Subgroups — #27151666

Order 12416.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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#20261242C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3#20261075C212yesmaximal, complemented
4#20261074C212yesmaximal, complemented
5#20261058C212yesmaximal, complemented
6#20261057C212yesmaximal, complemented
7#20260777C211yesmaximal, characteristic, complemented
8#20260775C211yesmaximal, characteristic, complemented
9#20260736C211yesmaximal, characteristic, complemented
10#20260698C211yesmaximal, characteristic, complemented
11#417×C97C211yesmaximal, characteristic, complemented
12C22×C4×D97C2211yescharacteristic
13C2×#13095495C2211yescharacteristic
14#15690791—21no—
15#15690786C2211yescharacteristic
16#15690786C2211yescharacteristic
17#15690781C2211yescharacteristic
18#15690781C2211yescharacteristic
19#15690778—21no—
20C4×C22⋊D97—21no—
21C4×C22⋊D97—21no—
22#15690769C2211yescharacteristic
23#15690768—21no—
24#15690768—21no—
25#15690768C2212yes—
26#15690768C2212yes—
27C2×C4⋊Dic97C2211yescharacteristic
28C2×#13095488—21no—
29C2×C4×Dic97—21no—
30#15690744C2212yescomplemented
31#15690743C2212yescomplemented
32#15690742C2212yescomplemented
33#15690741C2212yescomplemented
34#15690734C2212yescomplemented
35#15690731C2212yescomplemented
36#15690722C2212yes—
37#15690722C2212yes—
38C2×#41×C97C2211yescharacteristic
39C2×M16×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41#103×C97C2212yescomplemented
42#102×C97C2212yescomplemented
43C23×D97D411yescharacteristic
44C2×C22⋊D97—21no—
45C2×C22⋊D97—21no—
46C22×Dic97—21no—
47C22×Dic97D411yescharacteristic
48C2×C4×D97—22no—
49C2×C4×D97—22no—
50C2×C4×D97D411yescharacteristic
51C2×C4×D97D411yescharacteristic
52C2×C4×D97C2311yescharacteristic
53C2×C4×D97C2311yescharacteristic
54#13095495—22no—
55C2×#11715895C2312yes—
56C4⋊Dic97—21no—
57C4⋊Dic97C2312yes—
58C4⋊Dic97C2311yescharacteristic
59C4⋊Dic97C2311yescharacteristic
60#13095490—21no—
61#13095490—21no—
62C22⋊Dic97—21no—
63C22⋊Dic97—21no—
64C22⋊Dic97—21no—
65#13095488—22no—
66#13095488—22no—
67#13095488—21no—
68#13095488—21no—
69C4×Dic97—21no—
70C4×Dic97—22no—
71C4×Dic97—22no—
72#41×C97—22no—
73#41×C97—22no—
74C2×Q8×C97C2311yescharacteristic
75C2×D4×C97—21no—
76C2×D4×C97C2311yescharacteristic
77C22×C4×C97—21no—
78C22×C4×C97—21no—
79C22×C4×C97C2311yescharacteristic
80M16×C97—22no—
81C2×C8×C97C2312yes—
82C4⋊C4×C97—21no—
83C4⋊C4×C97C2312yes—
84C23×C97—21no—
85C23×C97C2×D411yescharacteristic
86C22×D97—22no—
87C22×D97—41no—
88C22×D97—41no—
89C22×D97C2×D411yescharacteristic
90C22×D97C2×D411yescharacteristic
91Q8×C97—21no—
92Q8×C97—21no—
93D4×C97—21no—
94D4×C97—21no—
95D4×C97—42no—
96C2×C4×C97—22no—
97C2×C4×C97—21no—
98C2×C4×C97—42no—
99C2×C4×C97—42no—
100C2×C4×C97C2×D411yescharacteristic
101C2×C4×C97#4112yes—
102C2×C4×C97#4112yes—
103C2×C4×C97C2411yescharacteristic
104C22⋊D97—42no—
105C22⋊D97—42no—
106C2×Dic97—22no—
107C2×Dic97—22no—
108C2×Dic97—21no—
109C2×Dic97—21no—
110C2×Dic97—42no—
111C2×Dic97C2×D411yescharacteristic
112C2×Dic97C2×D411yescharacteristic
113C4×D97—22no—
114C4×D97—22no—
115C4×D97—21no—
116C4×D97—21no—
117C4×D97—21no—
118C4×D97—21no—
119C8×C97—22no—
120#11715895—22no—
121C22×C97—21no—
122C22×C97—21no—
123C22×C97—41no—
124C22×C97—41no—
125C22×C97—41no—
126C22×C97#12312yes—
127C22×C97C22×D411yescharacteristic
128C2×D97—21no—
129C2×D97—21no—
130C2×D97—21no—
131C2×D97—21no—
132C2×D97—21no—
133C2×D97—21no—
134C2×D97—44no—
135C4×C97—41no—
136C4×C97—42no—
137C4×C97#12312yes—
138C4×C97C2×#4111yescharacteristic
139C4×C97C2×#4111yescharacteristic
140Dic97—21no—
141Dic97—21no—
142Dic97—42no—
143Dic97—41no—
144Dic97—41no—
145C2×C97—22no—
146C2×C97—81no—
147C2×C97C2×#12311yescharacteristic
148C2×C97#57511yescharacteristic
149C2×C97#57511yescharacteristic
150D97—41no—
151D97—41no—
152#3093—971nomaximal, Sylow
153C97#309311yescharacteristic, complemented, Sylow
154#525—971no—
155C2×#117—971no—
156#485—972no—
157#485—972no—
158#482—972no—
159#482—972no—
160#428—971no—
161#428—971no—
162#417—971no—
163#417D9711yescharacteristic, complemented
164#407—971no—
165C2×#41C2×D9711yescharacteristic
166C23×C4—971no—
167C2×M16—971no—
168C2×M16C2×D9711yescharacteristic
169#123—1941no—
170C22⋊Q8—971no—
171C22⋊Q8—971no—
172#121—971no—
173#121—971no—
174#120—1941no—
175C4×D4—1941no—
176C4×D4—1941no—
177#117—971no—
178#117—972no—
179#117—972no—
180#117—1941no—
181#117—1941no—
182C2×C4⋊C4—971no—
183C2×C4⋊C4C2×D9711yescharacteristic
184C2×C22⋊C4—1941no—
185C2×C42—1941no—
186#107—972no—
187#107—972no—
188#106—972no—
189#106—972no—
190#103—972no—
191#103C2×D9712yescomplemented
192#102—972no—
193#102C2×D9712yescomplemented
194C22⋊C8—972no—
195C22⋊C8—972no—
196C24—971no—
197#41—22no—
198#41—22no—
199C2×Q8C22×D9711yescharacteristic
200C2×D4—21no—
201C2×D4—1941no—
202C2×D4—1941no—
203C2×D4C22×D9711yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—971no—
207C22×C4—971no—
208C22×C4—971no—
209C22×C4—971no—
210C22×C4—971no—
211C22×C4—1941no—
212C22×C4—1942no—
213C22×C4—1942no—
214C22×C4C22×D9711yescharacteristic
215M16—22no—
216M16—1942no—
217C2×C8—972no—
218C2×C8C22×D9712yes—
219C4⋊C4—21no—
220C4⋊C4—972no—
221C4⋊C4—971no—
222C4⋊C4—971no—
223C4⋊C4—1941no—
224C4⋊C4—1941no—
225C4⋊C4—1941no—
226C4⋊C4C22×D9712yes—
227C22⋊C4—1942no—
228C22⋊C4—1942no—
229C22⋊C4—1941no—
230C22⋊C4—1941no—
231C22⋊C4—1941no—
232C22⋊C4—1941no—
233C22⋊C4—1941no—
234C42—1942no—
235C42—1942no—
236C42—1941no—
237C23—21no—
238C23—971no—
239C23—971no—
240C23—3881no—
241C23—3881no—
242C23—1942no—
243C23D4×D9711yescharacteristic
244Q8—21no—
245Q8—21no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—3882no—
250D4—3882no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—971no—
256C2×C4—971no—
257C2×C4—3882no—
258C2×C4—1942no—
259C2×C4—1941no—
260C2×C4—1941no—
261C2×C4—1942no—
262C2×C4—1942no—
263C2×C4—1942no—
264C2×C4—1941no—
265C2×C4—1941no—
266C2×C4—1941no—
267C2×C4—1941no—
268C2×C4D4×D9711yescharacteristic
269C2×C4#1309551012yes—
270C2×C4#1309551112yes—
271C2×C4C23×D9711yescharacteristic
272C8—22no—
273C8—1942no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—3884no—
280C22—1941no—
281C22—1941no—
282C22—1941no—
283C22—1941no—
284C22—1941no—
285C22—1941no—
286C22#1569079212yes—
287C22C2×D4×D9711yescharacteristic
288C4—41no—
289C4—42no—
290C4—3881no—
291C4—3882no—
292C4—3881no—
293C4—1941no—
294C4—1941no—
295C4#1569079212yes—
296C4C2×#1309551011yescharacteristic
297C4C2×#1309551111yescharacteristic
298C2—22no—
299C2—81no—
300C2—3881no—
301C2—3881no—
302C2C2×#1569079211yescharacteristic
303C2#2026153111yescharacteristic
304C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#417×C97C2
socleC22×C97C22×D4
prime radical#20261242C2
hypercenter#417D97
hypocenterC97#3093
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#15690768C2semidirect
3. #20261075C2semidirect
4. #20261074C2semidirect
5. #20261058C2semidirect
6. #20261057C2semidirect
7. #20260777C2semidirect
8. #20260775C2semidirect
9. #20260736C2semidirect
10. #20260698C2semidirect
11. #417×C97C2semidirect
30. #15690744C22semidirect
31. #15690743C22semidirect
32. #15690742C22semidirect
33. #15690741C22semidirect
34. #15690734C22semidirect
35. #15690731C22semidirect
41. #103×C97C22semidirect
42. #102×C97C22semidirect
153. C97#3093semidirect
163. #417D97semidirect
191. #103C2×D97semidirect
193. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612422
C2×#156907682
#202610752
#202610742
#202610582
#202610572
#202607772
#202607752
#202607362
#202606982
#417×C972
#309397