Finite Group Databasebeta

← #27153697

Subgroups — #27153697

Order 12416.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261389C211yesmaximal, characteristic, complemented
2#20261094C211yesmaximal, characteristic, complemented
3#20261063C211yesmaximal, characteristic, complemented
4#20261051C211yesmaximal, characteristic, complemented
5#20261025C211yesmaximal, characteristic, complemented
6#20260986C211yesmaximal, characteristic, complemented
7#20260966C211yesmaximal, characteristic, complemented
8#20260947C211yesmaximal, characteristic, complemented
9#20260880C211yesmaximal, characteristic, complemented
10#20260869C211yesmaximal, characteristic, complemented
11#20260845C211yesmaximal, characteristic, complemented
12#20260807C211yesmaximal, characteristic, complemented
13#20260736C211yesmaximal, characteristic, complemented
14C2×#15690731C211yesmaximal, characteristic, complemented
15#546×C97C211yesmaximal, characteristic, complemented
16C2×#13095510—21no—
17C2×#13095498C2211yescharacteristic, complemented
18C2×#13095496C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×#13095495C2211yescharacteristic, complemented
21C22×#11715895C2211yescharacteristic
22#15690810C2211yescharacteristic, complemented
23#15690794—21no—
24#15690794—21no—
25#15690790—21no—
26#15690784C2211yescharacteristic
27#15690784C2211yescharacteristic, complemented
28C4×D388C2211yescharacteristic, complemented
29C2×C4⋊Dic97C2211yescharacteristic
30C2×C4⋊Dic97C2211yescharacteristic, complemented
31#15690741C2211yescharacteristic, complemented
32#15690741C2211yescharacteristic, complemented
33#15690741C2211yescharacteristic, complemented
34C4⋊#11715895C2211yescharacteristic, complemented
35#15690734C2211yescharacteristic, complemented
36#15690734C2211yescharacteristic, complemented
37#15690734C2211yescharacteristic, complemented
38#15690731C2211yescharacteristic
39#15690731C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690731C2211yescharacteristic
44#15690730C2211yescharacteristic, complemented
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic, complemented
47#15690721C2211yescharacteristic, 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—
59#13095510—44no—
60C2×Dic194—21no—
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C22⋊D97—21no—
64C22×Dic97—21no—
65C22×Dic97—21no—
66C2×C4×D97—21no—
67#13095498—41no—
68#13095496—22no—
69#13095496—41no—
70#13095495—41no—
71C2×#11715895—41no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C4⋊Dic97—21no—
77C4⋊Dic97—21no—
78C4⋊Dic97—21no—
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82#13095488—21no—
83#13095488—21no—
84#13095488—21no—
85C24×C97—21no—
86#41×C97—22no—
87#41×C97—41no—
88C2×Q8×C97C2311yescharacteristic
89C2×D4×C97—22no—
90C2×D4×C97—21no—
91C2×D4×C97—21no—
92C2×D4×C97—21no—
93C2×D4×C97—41no—
94C2×D4×C97D411yescharacteristic
95C2×D4×C97C2311yescharacteristic
96C2×D4×C97C2311yescharacteristic
97C22×C4×C97—21no—
98C22×C4×C97—21no—
99C22×C4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C4⋊C4×C97—21no—
102C4⋊C4×C97D411yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C22⋊C4×C97—21no—
108C22⋊C4×C97—21no—
109C22⋊C4×C97D411yescharacteristic
110C22⋊C4×C97D411yescharacteristic
111C42×C97C2311yescharacteristic
112C23×C97—21no—
113C23×C97—21no—
114C23×C97—42no—
115C23×C97—42no—
116C23×C97—81no—
117C23×C97C2×D411yescharacteristic
118C23×C97C2×D411yescharacteristic
119C22×D97—21no—
120Q8×C97—21no—
121Q8×C97—21no—
122D4×C97—22no—
123D4×C97—21no—
124D4×C97—21no—
125D4×C97—21no—
126D4×C97—42no—
127D4×C97—42no—
128D4×C97—81no—
129D4×C97#4112yescomplemented
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—42no—
137C2×C4×C97—81no—
138C2×C4×C97C2×D411yescharacteristic
139C2×C4×C97C2×D411yescharacteristic
140C2×C4×C97C2×D411yescharacteristic
141C2×C4×C97C2×D411yescharacteristic
142C2×C4×C97C2411yescharacteristic
143C22⋊D97—42no—
144C22⋊D97—42no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—21no—
148C2×Dic97—42no—
149C2×Dic97—42no—
150D388—21no—
151D388—21no—
152C4×D97—42no—
153Dic194—42no—
154#11715895—21no—
155#11715895—21no—
156#11715895—21no—
157#11715895—21no—
158C22×C97—21no—
159C22×C97—21no—
160C22×C97—21no—
161C22×C97—22no—
162C22×C97—21no—
163C22×C97—21no—
164C22×C97—41no—
165C22×C97—41no—
166C22×C97—41no—
167C22×C97—41no—
168C22×C97—41no—
169C22×C97—84no—
170C22×C97AGL(1,ℤ/8)12yes—
171C22×C97C22×D411yescharacteristic
172C2×D97—41no—
173C2×D97—41no—
174C2×D97—41no—
175C4×C97—21no—
176C4×C97—21no—
177C4×C97—21no—
178C4×C97—21no—
179C4×C97—41no—
180C4×C97—41no—
181C4×C97C2×#4111yescharacteristic
182C4×C97H(5,𝔽2)11yescharacteristic
183Dic97—41no—
184Dic97—41no—
185Dic97—41no—
186C2×C97—22no—
187C2×C97—41no—
188C2×C97—42no—
189C2×C97—81no—
190C2×C97#54611yescharacteristic
191C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C97#57711yescharacteristic
193D97—81no—
194#3306—971nomaximal, Sylow
195C97#330611yescharacteristic, complemented, Sylow
196#547—971no—
197#546D9711yescharacteristic, complemented
198#489—971no—
199#483—971no—
200#481—971no—
201#476—971no—
202#469—971no—
203#465—971no—
204#463—971no—
205#449—971no—
206#447—971no—
207#442—971no—
208#435—971no—
209#417—971no—
210C2×#102—971no—
211C2×#41—1941no—
212C2×#41C2×D9711yescharacteristic
213C22×D4C2×D9711yescharacteristic
214C2×SD16—971no—
215C2×D8—971no—
216C2×D8—971no—
217C2×M16—971no—
218C22×C8—971no—
219C4⋊Q8—971no—
220#124C2×D9711yescharacteristic
221#123—21no—
222#123—1941no—
223#123—1941no—
224C22⋊Q8—1941no—
225C22⋊Q8C2×D9711yescharacteristic
226#121—21no—
227#121—971no—
228#121—971no—
229#121C2×D9711yescharacteristic
230#120—21no—
231C4×D4—971no—
232C4×D4C2×D9711yescharacteristic
233C4×D4C2×D9711yescharacteristic
234C2×C4⋊C4—971no—
235C2×C4⋊C4—971no—
236C2×C22⋊C4—21no—
237#106—971no—
238#106—971no—
239#106—971no—
240C4⋊C8—971no—
241#103—971no—
242#103—971no—
243#103—971no—
244#102—971no—
245#102—971no—
246#102—971no—
247#102—971no—
248#102—971no—
249#102—971no—
250#102—971no—
251C22⋊C8—971no—
252C22⋊C8—971no—
253#97—971no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—3884no—
258C2×Q8—1941no—
259C2×Q8C22×D9711yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—971no—
266C2×D4—1941no—
267C2×D4—1941no—
268C2×D4C22⋊D9711yescharacteristic
269C2×D4C22×D9711yescharacteristic
270C2×D4C22×D9711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—1941no—
274C22×C4—1941no—
275C22×C4—1941no—
276C22×C4C22×D9711yescharacteristic
277C22×C4C22×D9711yescharacteristic
278SD16—3881no—
279D8—3881no—
280D8—1942no—
281M16—3881no—
282C2×C8—971no—
283C2×C8—971no—
284C2×C8—971no—
285C2×C8—971no—
286C2×C8—3881no—
287C4⋊C4—21no—
288C4⋊C4—971no—
289C4⋊C4—971no—
290C4⋊C4—971no—
291C4⋊C4—1941no—
292C4⋊C4—1941no—
293C4⋊C4—1941no—
294C4⋊C4C22⋊D9711yescharacteristic
295C4⋊C4C22×D9711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—1941no—
302C22⋊C4—1941no—
303C22⋊C4—1941no—
304C22⋊C4C22⋊D9711yescharacteristic
305C22⋊C4C22⋊D9711yescharacteristic
306C42C22×D9711yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—1941no—
313C23C2×C22⋊D9711yescharacteristic
314C23C2×C22⋊D9711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—3882no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—3882no—
326D4—3882no—
327D4—1941no—
328D4—1941no—
329D4#1309551012yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—3882no—
339C2×C4—3882no—
340C2×C4—3882no—
341C2×C4—1941no—
342C2×C4—1941no—
343C2×C4—1941no—
344C2×C4C2×C22⋊D9711yescharacteristic
345C2×C4C2×C22⋊D9711yescharacteristic
346C2×C4C2×C22⋊D9711yescharacteristic
347C2×C4C2×C22⋊D9711yescharacteristic
348C2×C4C23×D9711yescharacteristic
349C8—1941no—
350C8—1941no—
351C8—1941no—
352C8—1941no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—3881no—
366C22—3881no—
367C22—3881no—
368C22#1569083512yes—
369C22C22×C22⋊D9711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—3881no—
377C4—3881no—
378C4—3881no—
379C4C2×#1309551011yescharacteristic
380C4#1569085511yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—7761no—
386C2#2026137911yescharacteristic
387C2C2×#1569083511yescharacteristic
388C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#546×C97C2
socleC22×C97C22×D4
hypercenter#546D97
hypocenterC97#3306
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. #20261389C2semidirect
2. #20261094C2semidirect
3. #20261063C2semidirect
4. #20261051C2semidirect
5. #20261025C2semidirect
6. #20260986C2semidirect
7. #20260966C2semidirect
8. #20260947C2semidirect
9. #20260880C2semidirect
10. #20260869C2semidirect
11. #20260845C2semidirect
12. #20260807C2semidirect
13. #20260736C2semidirect
14. C2×#15690731C2semidirect
15. #546×C97C2semidirect
17. C2×#13095498C22semidirect
20. C2×#13095495C22semidirect
22. #15690810C22semidirect
27. #15690784C22semidirect
28. C4×D388C22semidirect
30. C2×C4⋊Dic97C22semidirect
31. #15690741C22semidirect
32. #15690741C22semidirect
33. #15690741C22semidirect
34. C4⋊#11715895C22semidirect
35. #15690734C22semidirect
36. #15690734C22semidirect
37. #15690734C22semidirect
44. #15690730C22semidirect
46. C22⋊#11715895C22semidirect
47. #15690721C22semidirect
129. D4×C97#41semidirect
195. C97#3306semidirect
197. #546D97semidirect
329. D4#13095510semidirect

Maximal subgroups

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

subgroupindex
#202613892
#202610942
#202610632
#202610512
#202610252
#202609862
#202609662
#202609472
#202608802
#202608692
#202608452
#202608072
#202607362
C2×#156907312
#546×C972
#330697