Finite Group Databasebeta

← #27153565

Subgroups — #27153565

Order 12416.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261381C211yesmaximal, characteristic
2#20261092C211yesmaximal, characteristic, complemented
3#20261065C211yesmaximal, characteristic, complemented
4#20261059C211yesmaximal, characteristic, complemented
5#20261032C211yesmaximal, characteristic, complemented
6#20260974C211yesmaximal, characteristic, complemented
7#20260966C211yesmaximal, characteristic, complemented
8#20260957C211yesmaximal, characteristic, complemented
9#20260895C211yesmaximal, characteristic, complemented
10#20260885C211yesmaximal, characteristic, complemented
11C4×#13095498C211yesmaximal, characteristic, complemented
12D4×#11715895C211yesmaximal, characteristic, complemented
13#20260729C211yesmaximal, characteristic, complemented
14C2×#15690730C211yesmaximal, characteristic, complemented
15#552×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C2×#13095501C2211yescharacteristic, complemented
19C2×#13095498C2211yescharacteristic
20C2×#13095498C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22C22×#11715895C2211yescharacteristic, complemented
23#15690798C2211yescharacteristic
24#15690795—21no—
25#15690789C2211yescharacteristic
26#15690784C2211yescharacteristic
27#15690783—21no—
28#15690778—21no—
29C4×D388C2211yescharacteristic
30C22⋊C4×D97—21no—
31#15690743C2211yescharacteristic, complemented
32#15690741C2211yescharacteristic, complemented
33#15690741C2211yescharacteristic, complemented
34C4⋊#11715895C2211yescharacteristic, complemented
35#15690734C2211yescharacteristic, complemented
36#15690733C2211yescharacteristic, complemented
37#15690733C2211yescharacteristic, complemented
38#15690733C2211yescharacteristic, complemented
39#15690730C2211yescharacteristic
40#15690730C2211yescharacteristic
41#15690730C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic, complemented
47C4×#11715895C2211yescharacteristic, complemented
48#126×C97—21no—
49#125×C97—21no—
50#125×C97C2211yescharacteristic, complemented
51#123×C97—21no—
52C22⋊Q8×C97C2211yescharacteristic
53C22⋊Q8×C97C2211yescharacteristic, complemented
54C4×Q8×C97C2211yescharacteristic, complemented
55C4×D4×C97C2211yescharacteristic
56#117×C97—21no—
57#117×C97C2211yescharacteristic, complemented
58C2×C4⋊C4×C97C2211yescharacteristic
59C23×D97—21no—
60#13095511—22no—
61#13095511—41no—
62C2×Dic194C2311yescharacteristic
63C2×D388—22no—
64C2×D388—21no—
65C2×D388—41no—
66C2×D388C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72#13095501—41no—
73#13095498—22no—
74#13095498—41no—
75C2×#11715895—41no—
76C2×#11715895—41no—
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C2×#11715895C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82#13095490—21no—
83C22⋊Dic97—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87#13095488—21no—
88C2×Q8×C97C2311yescharacteristic
89C2×D4×C97D411yescharacteristic
90C22×C4×C97—21no—
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93C4⋊C4×C97—21no—
94C4⋊C4×C97—21no—
95C4⋊C4×C97—21no—
96C4⋊C4×C97—21no—
97C4⋊C4×C97—21no—
98C4⋊C4×C97—21no—
99C4⋊C4×C97D411yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97D411yescharacteristic
107C22⋊C4×C97D411yescharacteristic
108C42×C97—21no—
109C42×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97C2×D411yescharacteristic
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114C22×D97—21no—
115C22×D97—42no—
116C22×D97—42no—
117C22×D97—81no—
118Q8×C97—21no—
119Q8×C97—21no—
120D4×C97—22no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—42no—
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2×D411yescharacteristic
132C2×C4×C97C2×D411yescharacteristic
133C2×C4×C97C2×D411yescharacteristic
134C2×C4×C97C2411yescharacteristic
135C22⋊D97—42no—
136C22⋊D97—81no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139D388—21no—
140D388—21no—
141D388—21no—
142D388—42no—
143D388#4112yes—
144C4×D97—42no—
145C4×D97—81no—
146Dic194—21no—
147Dic194—21no—
148#11715895—21no—
149#11715895—21no—
150#11715895—21no—
151#11715895—21no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97#13512yes—
158C22×C97C22×D411yescharacteristic
159C2×D97—22no—
160C2×D97—41no—
161C2×D97—41no—
162C2×D97—41no—
163C2×D97—41no—
164C2×D97—41no—
165C2×D97—84no—
166C4×C97—21no—
167C4×C97—21no—
168C4×C97—21no—
169C4×C97—21no—
170C4×C97—41no—
171C4×C97—41no—
172C4×C97—41no—
173C4×C97—41no—
174C4×C97C2×#4111yescharacteristic
175C4×C97H(5,𝔽2)11yescharacteristic
176Dic97—41no—
177Dic97—41no—
178C2×C97—22no—
179C2×C97—41no—
180C2×C97#54611yescharacteristic
181C2×C97C2×#13511yescharacteristic
182C2×C97#57711yescharacteristic
183D97—42no—
184D97—81no—
185#3296—971nomaximal, Sylow
186C97#329611yescharacteristic, complemented, Sylow
187#552D9711yescharacteristic, complemented
188#546—971no—
189#488—971no—
190#483—971no—
191#482—971no—
192#478—971no—
193#467—971no—
194#465—971no—
195#464—971no—
196#452—971no—
197#450—971no—
198C4×SD16—971no—
199C8×D4—971no—
200#416—971no—
201C2×#102—971no—
202C2×#41—971no—
203C22×D4—971no—
204C2×Q16—971no—
205C2×SD16—971no—
206C2×SD16—971no—
207C22×C8—971no—
208C22×C8—971no—
209#126—21no—
210#125—21no—
211#125C2×D9711yescharacteristic, complemented
212#124—971no—
213#123—21no—
214#123—1941no—
215C22⋊Q8—971no—
216C22⋊Q8C2×D9711yescharacteristic
217C22⋊Q8C2×D9711yescharacteristic, complemented
218#121—971no—
219#121—1941no—
220#120—1941no—
221C4×Q8C2×D9711yescharacteristic, complemented
222C4×D4—971no—
223C4×D4C2×D9711yescharacteristic
224#117—21no—
225#117C2×D9711yescharacteristic, complemented
226C2×C4⋊C4C2×D9711yescharacteristic
227C2×C22⋊C4—1941no—
228#107—971no—
229#106—971no—
230#106—971no—
231C4⋊C8—971no—
232#103—971no—
233#103—971no—
234#103—971no—
235#103—971no—
236#102—971no—
237#102—971no—
238#102—971no—
239#102—971no—
240#102—971no—
241#102—971no—
242C22⋊C8—971no—
243C22⋊C8—971no—
244C4×C8—971no—
245C24—1941no—
246#41—3881no—
247#41—1942no—
248C2×Q8—971no—
249C2×Q8C22×D9711yescharacteristic
250C2×D4—971no—
251C2×D4—971no—
252C2×D4—3881no—
253C2×D4—1942no—
254C2×D4—1941no—
255C2×D4—1941no—
256C2×D4—1941no—
257C2×D4C22⋊D9711yescharacteristic
258C22×C4—21no—
259C22×C4—1941no—
260C22×C4—1941no—
261C22×C4C22×D9711yescharacteristic
262C22×C4C22×D9711yescharacteristic
263Q16—3881no—
264SD16—3881no—
265SD16—1942no—
266C2×C8—971no—
267C2×C8—971no—
268C2×C8—971no—
269C2×C8—971no—
270C2×C8—3881no—
271C2×C8—3881no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—971no—
279C4⋊C4—1941no—
280C4⋊C4C22⋊D9711yescharacteristic
281C4⋊C4C22×D9711yescharacteristic
282C4⋊C4C22×D9711yescharacteristic
283C4⋊C4C22×D9711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4C22⋊D9711yescharacteristic
293C22⋊C4C22⋊D9711yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D9711yescharacteristic
297C23—7761no—
298C23—3882no—
299C23—3882no—
300C23—1941no—
301C23—1941no—
302C23C2×C22⋊D9711yescharacteristic
303C23C2×C22⋊D9711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1941no—
307Q8—1941no—
308D4—22no—
309D4—7761no—
310D4—972no—
311D4—3882no—
312D4—3882no—
313D4—1941no—
314D4—1941no—
315D4—1941no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—7761no—
326C2×C4—3882no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4C2×C22⋊D9711yescharacteristic
330C2×C4C2×C22⋊D9711yescharacteristic
331C2×C4C2×C22⋊D9711yescharacteristic
332C2×C4C2×C22⋊D9711yescharacteristic
333C2×C4C23×D9711yescharacteristic
334C8—1941no—
335C8—1941no—
336C8—1941no—
337C8—1941no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—7764no—
344C22—3881no—
345C22—3881no—
346C22—3881no—
347C22—3881no—
348C22—3881no—
349C22—1942no—
350C22#1569083012yes—
351C22C22×C22⋊D9711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3881no—
361C4—3881no—
362C4#41×D9711yescharacteristic
363C4#1569085611yescharacteristic
364C2—22no—
365C2—41no—
366C2—7761no—
367C2—3882no—
368C2#2026137311yescharacteristic
369C2C2×#1569083011yescharacteristic
370C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#552×C97C2
socleC22×C97C22×D4
prime radical#20261381C2
hypercenter#552D97
hypocenterC97#3296
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20261092C2semidirect
3. #20261065C2semidirect
4. #20261059C2semidirect
5. #20261032C2semidirect
6. #20260974C2semidirect
7. #20260966C2semidirect
8. #20260957C2semidirect
9. #20260895C2semidirect
10. #20260885C2semidirect
11. C4×#13095498C2semidirect
12. D4×#11715895C2semidirect
13. #20260729C2semidirect
14. C2×#15690730C2semidirect
15. #552×C97C2semidirect
18. C2×#13095501C22semidirect
22. C22×#11715895C22semidirect
31. #15690743C22semidirect
32. #15690741C22semidirect
33. #15690741C22semidirect
34. C4⋊#11715895C22semidirect
35. #15690734C22semidirect
36. #15690733C22semidirect
37. #15690733C22semidirect
38. #15690733C22semidirect
46. C22⋊#11715895C22semidirect
47. C4×#11715895C22semidirect
50. #125×C97C22semidirect
53. C22⋊Q8×C97C22semidirect
54. C4×Q8×C97C22semidirect
57. #117×C97C22semidirect
186. C97#3296semidirect
187. #552D97semidirect
211. #125C2×D97semidirect
217. C22⋊Q8C2×D97semidirect
221. C4×Q8C2×D97semidirect
225. #117C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613812
#202610922
#202610652
#202610592
#202610322
#202609742
#202609662
#202609572
#202608952
#202608852
C4×#130954982
D4×#117158952
#202607292
C2×#156907302
#552×C972
#329697