Finite Group Databasebeta

← #27153571

Subgroups — #27153571

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#20261425C211yesmaximal, characteristic, complemented
2#20261089C211yesmaximal, characteristic, complemented
3#20261063C211yesmaximal, characteristic, complemented
4#20261056C211yesmaximal, characteristic, complemented
5#20261035C211yesmaximal, characteristic, complemented
6#20260974C211yesmaximal, characteristic, complemented
7#20260962C211yesmaximal, characteristic, complemented
8#20260954C211yesmaximal, characteristic, complemented
9#20260897C211yesmaximal, characteristic, complemented
10#20260886C211yesmaximal, characteristic, complemented
11C4×#13095499C211yesmaximal, characteristic, complemented
12D4×#11715895C211yesmaximal, characteristic, complemented
13#20260731C211yesmaximal, characteristic, complemented
14C2×#15690731C211yesmaximal, characteristic, complemented
15#546×C97C211yesmaximal, characteristic
16C2×#13095501C2211yescharacteristic, complemented
17C2×#13095499C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C22×#11715895C2211yescharacteristic
20C22×#11715895C2211yescharacteristic, complemented
21#15690803—21no—
22#15690802C2211yescharacteristic, complemented
23#15690800—21no—
24#15690791—21no—
25#15690789C2211yescharacteristic
26#15690789C2211yescharacteristic, complemented
27C4×Dic194C2211yescharacteristic, complemented
28#15690769C2211yescharacteristic, complemented
29#15690766—21no—
30C2×C4⋊Dic97C2211yescharacteristic
31#15690743C2211yescharacteristic, complemented
32#15690741C2211yescharacteristic, complemented
33#15690741C2211yescharacteristic, complemented
34C4⋊#11715895C2211yescharacteristic, complemented
35#15690734C2211yescharacteristic, complemented
36#15690734C2211yescharacteristic, complemented
37#15690734C2211yescharacteristic, complemented
38#15690733C2211yescharacteristic, complemented
39#15690731C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690731C2211yescharacteristic
44#15690731C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic, complemented
47C4×#11715895C2211yescharacteristic, complemented
48C2×#41×C97C2211yescharacteristic
49C22×D4×C97C2211yescharacteristic
50#124×C97C2211yescharacteristic
51#123×C97—21no—
52C22⋊Q8×C97C2211yescharacteristic
53#121×C97—21no—
54#121×C97C2211yescharacteristic
55#120×C97—21no—
56C4×D4×C97C2211yescharacteristic
57C4×D4×C97C2211yescharacteristic
58C2×C22⋊C4×C97—21no—
59C2×Dic194C2311yescharacteristic
60C22×Dic97—21no—
61#13095501—41no—
62#13095499—22no—
63#13095499—41no—
64C2×#11715895—41no—
65C2×#11715895—41no—
66C2×#11715895C2311yescharacteristic
67C2×#11715895C2311yescharacteristic
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C4⋊Dic97—21no—
71C4⋊Dic97C2311yescharacteristic
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74#13095490—21no—
75#13095490—21no—
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79C22⋊Dic97—21no—
80C22⋊Dic97—21no—
81C22⋊Dic97—21no—
82C4×Dic97—21no—
83C4×Dic97—21no—
84C24×C97—21no—
85#41×C97—22no—
86#41×C97—41no—
87C2×Q8×C97C2311yescharacteristic
88C2×D4×C97—22no—
89C2×D4×C97—21no—
90C2×D4×C97—21no—
91C2×D4×C97—21no—
92C2×D4×C97—41no—
93C2×D4×C97D411yescharacteristic
94C2×D4×C97C2311yescharacteristic
95C2×D4×C97C2311yescharacteristic
96C22×C4×C97—21no—
97C22×C4×C97—21no—
98C22×C4×C97C2311yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C4⋊C4×C97—21no—
101C4⋊C4×C97D411yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C22⋊C4×C97—21no—
108C22⋊C4×C97D411yescharacteristic
109C22⋊C4×C97D411yescharacteristic
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—42no—
114C23×C97—42no—
115C23×C97—81no—
116C23×C97C2×D411yescharacteristic
117C23×C97C2×D411yescharacteristic
118Q8×C97—21no—
119Q8×C97—21no—
120D4×C97—22no—
121D4×C97—21no—
122D4×C97—21no—
123D4×C97—21no—
124D4×C97—42no—
125D4×C97—42no—
126D4×C97—81no—
127D4×C97#4112yes—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—42no—
135C2×C4×C97—81no—
136C2×C4×C97C2×D411yescharacteristic
137C2×C4×C97C2×D411yescharacteristic
138C2×C4×C97C2×D411yescharacteristic
139C2×C4×C97C2×D411yescharacteristic
140C2×C4×C97C2411yescharacteristic
141C2×Dic97—21no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—42no—
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—22no—
156C22×C97—21no—
157C22×C97—21no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97—84no—
164C22×C97#13512yes—
165C22×C97C22×D411yescharacteristic
166C4×C97—21no—
167C4×C97—21no—
168C4×C97—21no—
169C4×C97—21no—
170C4×C97—41no—
171C4×C97—41no—
172C4×C97C2×#4111yescharacteristic
173C4×C97H(5,𝔽2)11yescharacteristic
174Dic97—41no—
175Dic97—41no—
176Dic97—41no—
177Dic97—41no—
178C2×C97—22no—
179C2×C97—41no—
180C2×C97—42no—
181C2×C97—81no—
182C2×C97#54611yescharacteristic
183C2×C97C2×#13511yescharacteristic
184C2×C97#57711yescharacteristic
185#3296—971nomaximal, Sylow
186C97#329611yescharacteristic, complemented, Sylow
187#552—971no—
188#546D9711yescharacteristic
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×#41C2×D9711yescharacteristic
203C22×D4C2×D9711yescharacteristic
204C2×Q16—971no—
205C2×SD16—971no—
206C2×SD16—971no—
207C22×C8—971no—
208C22×C8—971no—
209#126—1941no—
210#125—971no—
211#125—1941no—
212#124C2×D9711yescharacteristic
213#123—21no—
214#123—1941no—
215C22⋊Q8—971no—
216C22⋊Q8—971no—
217C22⋊Q8C2×D9711yescharacteristic
218#121—21no—
219#121C2×D9711yescharacteristic
220#120—21no—
221C4×Q8—971no—
222C4×D4C2×D9711yescharacteristic
223C4×D4C2×D9711yescharacteristic
224#117—971no—
225#117—1941no—
226C2×C4⋊C4—971no—
227C2×C22⋊C4—21no—
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—21no—
246#41—22no—
247#41—41no—
248C2×Q8—971no—
249C2×Q8C22×D9711yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D9711yescharacteristic
256C2×D4C22×D9711yescharacteristic
257C2×D4C22×D9711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
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—971no—
274C4⋊C4—971no—
275C4⋊C4—971no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4—1941no—
282C4⋊C4C22⋊D9711yescharacteristic
283C4⋊C4C22×D9711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4C22⋊D9711yescharacteristic
293C22⋊C4C22⋊D9711yescharacteristic
294C42—1941no—
295C42—1941no—
296C42C22×D9711yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D9711yescharacteristic
303C23C2×C22⋊D9711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1941no—
307Q8—1941no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1309551012yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—3882no—
325C2×C4—1941no—
326C2×C4—1941no—
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—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#1569083012yes—
351C22C22×C22⋊D9711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—3881no—
359C4—3881no—
360C4—3881no—
361C4—3881no—
362C4C2×#1309551011yescharacteristic
363C4#1569085511yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#2026137911yescharacteristic
369C2C2×#1569083011yescharacteristic
370C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radical#546×C97C2
socleC22×C97C22×D4
hypercenter#546D97
hypocenterC97#3296
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #20261425C2semidirect
2. #20261089C2semidirect
3. #20261063C2semidirect
4. #20261056C2semidirect
5. #20261035C2semidirect
6. #20260974C2semidirect
7. #20260962C2semidirect
8. #20260954C2semidirect
9. #20260897C2semidirect
10. #20260886C2semidirect
11. C4×#13095499C2semidirect
12. D4×#11715895C2semidirect
13. #20260731C2semidirect
14. C2×#15690731C2semidirect
16. C2×#13095501C22semidirect
20. C22×#11715895C22semidirect
22. #15690802C22semidirect
26. #15690789C22semidirect
27. C4×Dic194C22semidirect
28. #15690769C22semidirect
31. #15690743C22semidirect
32. #15690741C22semidirect
33. #15690741C22semidirect
34. C4⋊#11715895C22semidirect
35. #15690734C22semidirect
36. #15690734C22semidirect
37. #15690734C22semidirect
38. #15690733C22semidirect
46. C22⋊#11715895C22semidirect
47. C4×#11715895C22semidirect
186. C97#3296semidirect

Maximal subgroups

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

subgroupindex
#202614252
#202610892
#202610632
#202610562
#202610352
#202609742
#202609622
#202609542
#202608972
#202608862
C4×#130954992
D4×#117158952
#202607312
C2×#156907312
#546×C972
#329697