Finite Group Databasebeta

← #27155142

Subgroups — #27155142

Order 12416.

1160 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261361C211yesmaximal, characteristic, complemented
2#20261330C212yesmaximal, complemented
3#20261329C214yesmaximal, complemented
4#20261322C212yesmaximal, complemented
5#20261320C214yesmaximal, complemented
6#20261294C212yesmaximal, complemented
7#20261266C212yesmaximal, complemented
8#20261262C214yesmaximal, complemented
9#20261245C212yesmaximal, complemented
10#20261241C214yesmaximal, complemented
11C2×D4×Dic97C212yesmaximal, complemented
12C2×#15690770C211yesmaximal, characteristic, complemented
13#534×C97C211yesmaximal, characteristic, complemented
14C2×#13095510C2212yescomplemented
15C22×C22⋊D97C2211yescharacteristic
16C23×Dic97C2211yescharacteristic
17C23×Dic97C2212yescomplemented
18C22×C4×D97C2211yescharacteristic, complemented
19#15690805C2214yescomplemented
20#15690803C2214yescomplemented
21#15690803C2214yescomplemented
22#15690801C2212yescomplemented
23#15690800C2214yescomplemented
24#15690797C2212yescomplemented
25#15690796C2214yescomplemented
26#15690794C2214yescomplemented
27#15690793C2214yescomplemented
28#15690792C2214yescomplemented
29#15690791C2214yescomplemented
30#15690791C2214yescomplemented
31#15690790C2212yescomplemented
32#15690789C2212yescomplemented
33#15690789C2212yescomplemented
34#15690787C2214yescomplemented
35#15690787C2214yescomplemented
36#15690787C2214yescomplemented
37#15690782C2214yes—
38#15690782C2214yes—
39#15690782C2214yescomplemented
40#15690781C2212yescomplemented
41#15690779C2214yescomplemented
42#15690778C2214yescomplemented
43#15690775C2212yescomplemented
44D4×Dic97C2214yescomplemented
45D4×Dic97C2214yescomplemented
46D4×Dic97C2214yescomplemented
47D4×Dic97C2214yescomplemented
48D4×Dic97C2212yescomplemented
49#15690770C2214yescomplemented
50#15690770C2214yescomplemented
51#15690770C2214yescomplemented
52#15690768C2212yescomplemented
53#15690766C2214yescomplemented
54C2×C4⋊Dic97C2212yescomplemented
55C2×#13095490C2211yescharacteristic, complemented
56C2×C22⋊Dic97C2212yes—
57C2×C22⋊Dic97C2212yescomplemented
58C2×#13095488C2211yescharacteristic, complemented
59C2×C4×Dic97C2211yescharacteristic, complemented
60C2×C4×Dic97C2212yescomplemented
61C22×D4×C97C2212yes—
62#123×C97C2214yescomplemented
63#121×C97C2212yes—
64#121×C97C2212yes—
65#120×C97C2214yescomplemented
66C2×C22⋊C4×C97C2211yescharacteristic
67C23×D97C2311yescharacteristic
68#13095510—44no—
69C2×Dic194C2312yes—
70C2×Dic194C2312yes—
71C2×C22⋊D97—24no—
72C2×C22⋊D97C2314yes—
73C2×C22⋊D97C2314yes—
74C2×C22⋊D97C2314yes—
75C22×Dic97—42no—
76C22×Dic97—44no—
77C22×Dic97C2311yescharacteristic
78C22×Dic97C2312yes—
79C22×Dic97C2311yescharacteristic
80C22×Dic97C2314yes—
81C22×Dic97C2312yes—
82C22×Dic97C2311yescharacteristic
83C22×Dic97C2312yes—
84C22×Dic97C2312yes—
85C22×Dic97C2314yes—
86C2×C4×D97—42no—
87C2×C4×D97C2312yes—
88C2×C4×D97C2312yes—
89C4⋊Dic97C2314yes—
90C4⋊Dic97C2314yes—
91C4⋊Dic97C2312yes—
92C4⋊Dic97C2312yes—
93#13095490C2314yes—
94#13095490C2314yes—
95#13095490C2314yes—
96#13095490C2314yes—
97#13095490C2314yes—
98C22⋊Dic97C2314yes—
99C22⋊Dic97C2314yes—
100C22⋊Dic97C2314yes—
101C22⋊Dic97C2314yes—
102C22⋊Dic97C2314yes—
103C22⋊Dic97C2314yes—
104#13095488C2314yes—
105#13095488C2314yes—
106#13095488C2314yes—
107C4×Dic97C2312yes—
108C4×Dic97C2314yes—
109C4×Dic97C2312yes—
110C4×Dic97C2314yes—
111C24×C97C2312yes—
112C24×C97C2311yescharacteristic
113C2×D4×C97—28no—
114C2×D4×C97C2314yes—
115C2×D4×C97C2314yes—
116C2×D4×C97C2312yes—
117C2×D4×C97C2312yes—
118C22×C4×C97C2312yes—
119C22×C4×C97C2311yescharacteristic
120C22×C4×C97C2311yescharacteristic
121C4⋊C4×C97C2312yes—
122C4⋊C4×C97C2312yes—
123C22⋊C4×C97C2314yes—
124C22⋊C4×C97C2314yes—
125C22⋊C4×C97C2314yes—
126C23×C97—24no—
127C23×C97—41no—
128C23×C97—42no—
129C23×C97—41no—
130C23×C97—42no—
131C23×C97#4114yescomplemented
132C23×C97C2414yes—
133C23×C97C2412yes—
134C23×C97C2411yescharacteristic
135C22×D97—22no—
136C22×D97—41no—
137C22×D97—41no—
138C22×D97C2412yes—
139D4×C97—28no—
140D4×C97—28no—
141D4×C97—42no—
142D4×C97—42no—
143C2×C4×C97—24no—
144C2×C4×C97—41no—
145C2×C4×C97—41no—
146C2×C4×C97C2412yes—
147C2×C4×C97C2412yes—
148C2×C4×C97C2414yes—
149C22⋊D97—28no—
150C22⋊D97—44no—
151C22⋊D97—44no—
152C2×Dic97—22no—
153C2×Dic97—22no—
154C2×Dic97—24no—
155C2×Dic97—28no—
156C2×Dic97—42no—
157C2×Dic97—44no—
158C2×Dic97—42no—
159C2×Dic97—41no—
160C2×Dic97—42no—
161C2×Dic97C2412yes—
162C2×Dic97C2414yes—
163C2×Dic97C2412yes—
164C2×Dic97C2411yescharacteristic
165C2×Dic97C2414yes—
166C2×Dic97C2411yescharacteristic
167C4×D97—24no—
168C4×D97—42no—
169Dic194—42no—
170Dic194—42no—
171C22×C97—22no—
172C22×C97—21no—
173C22×C97—22no—
174C22×C97—21no—
175C22×C97—24no—
176C22×C97—24no—
177C22×C97—24no—
178C22×C97—44no—
179C22×C97—48no—
180C22×C97C2×#4112yes—
181C22×C97C2×#4114yes—
182C22×C97C2511yescharacteristic
183C2×D97—22no—
184C2×D97—22no—
185C2×D97—22no—
186C2×D97—44no—
187C4×C97—24no—
188C4×C97—22no—
189C4×C97—22no—
190Dic97—22no—
191Dic97—24no—
192Dic97—21no—
193Dic97—22no—
194Dic97—24no—
195Dic97H(5,𝔽2)12yes—
196C2×C97—22no—
197C2×C97—24no—
198C2×C97—44no—
199C2×C97C22×#4111yescharacteristic
200C2×C97C2×H(5,𝔽2)11yescharacteristic
201C2×C97#58511yescharacteristic
202D97—42no—
203#3459—971nomaximal, Sylow
204C97#345911yescharacteristic, complemented, Sylow
205#543—971no—
206#539—974no—
207#539—972no—
208#538—972no—
209#538—974no—
210#534—972no—
211#534D9711yescharacteristic, complemented
212#529—974no—
213#529—972no—
214#525—972no—
215#525—974no—
216C2×C4×D4—972no—
217C2×C4×D4—971no—
218C2×#41—972no—
219C22×D4—971no—
220C22×D4C2×D9712yes—
221C23×C4—971no—
222C23×C4—972no—
223C23×C4—971no—
224#126—974no—
225#126—974no—
226#126—974no—
227#125—974no—
228#125—972no—
229#124—972no—
230#124—974no—
231#123—974no—
232#123—974no—
233#123—974no—
234#123—974no—
235#123—974no—
236#123C2×D9714yescomplemented
237C22⋊Q8—972no—
238C22⋊Q8—972no—
239C22⋊Q8—972no—
240C22⋊Q8—974no—
241C22⋊Q8—974no—
242C22⋊Q8—974no—
243#121—974no—
244#121—972no—
245#121—974no—
246#121—974no—
247#121C2×D9712yes—
248#121C2×D9712yes—
249#120—974no—
250#120—974no—
251#120C2×D9714yescomplemented
252C4×Q8—972no—
253C4×D4—974no—
254C4×D4—974no—
255C4×D4—974no—
256C4×D4—974no—
257C4×D4—972no—
258C4×D4—974no—
259C4×D4—974no—
260C4×D4—974no—
261#117—972no—
262#117—974no—
263C2×C4⋊C4—971no—
264C2×C4⋊C4—972no—
265C2×C22⋊C4—972no—
266C2×C22⋊C4—972no—
267C2×C22⋊C4—971no—
268C2×C22⋊C4C2×D9711yescharacteristic
269C2×C42—972no—
270C2×C42—971no—
271C24—971no—
272C24C22×D9712yes—
273C24C22×D9711yescharacteristic
274#41—3884no—
275C2×Q8—972no—
276C2×Q8—972no—
277C2×D4—28no—
278C2×D4—974no—
279C2×D4—974no—
280C2×D4—974no—
281C2×D4—1944no—
282C2×D4C22×D9714yes—
283C2×D4C22×D9714yes—
284C2×D4C22×D9712yes—
285C2×D4C22×D9712yes—
286C22×C4—972no—
287C22×C4—971no—
288C22×C4—972no—
289C22×C4—972no—
290C22×C4—974no—
291C22×C4—971no—
292C22×C4—971no—
293C22×C4—972no—
294C22×C4—974no—
295C22×C4—972no—
296C22×C4—972no—
297C22×C4—3882no—
298C22×C4—3882no—
299C22×C4—3884no—
300C22×C4C22×D9712yes—
301C22×C4C22×D9711yescharacteristic
302C22×C4C22×D9711yescharacteristic
303C4⋊C4—974no—
304C4⋊C4—974no—
305C4⋊C4—974no—
306C4⋊C4—974no—
307C4⋊C4—974no—
308C4⋊C4—974no—
309C4⋊C4—974no—
310C4⋊C4—972no—
311C4⋊C4—972no—
312C4⋊C4C22×D9712yes—
313C4⋊C4C22×D9712yes—
314C22⋊C4—974no—
315C22⋊C4—974no—
316C22⋊C4—974no—
317C22⋊C4—974no—
318C22⋊C4—974no—
319C22⋊C4—974no—
320C22⋊C4—974no—
321C22⋊C4—974no—
322C22⋊C4—974no—
323C22⋊C4C22×D9714yes—
324C22⋊C4C22×D9714yes—
325C22⋊C4C22×D9714yes—
326C42—974no—
327C42—972no—
328C42—974no—
329C42—972no—
330C23—24no—
331C23—41no—
332C23—42no—
333C23—41no—
334C23—42no—
335C23—972no—
336C23—3881no—
337C23—3881no—
338C23—1942no—
339C23#1309551014yescomplemented
340C23C23×D9714yes—
341C23C23×D9711yescharacteristic
342C23C23×D9712yes—
343Q8—3882no—
344Q8—3882no—
345D4—28no—
346D4—28no—
347D4—42no—
348D4—42no—
349D4—3884no—
350D4—3884no—
351D4—1948no—
352C2×C4—24no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—971no—
356C2×C4—974no—
357C2×C4—972no—
358C2×C4—971no—
359C2×C4—972no—
360C2×C4—974no—
361C2×C4—3882no—
362C2×C4—3881no—
363C2×C4—3882no—
364C2×C4—3884no—
365C2×C4—3882no—
366C2×C4—3882no—
367C2×C4—1944no—
368C2×C4—1942no—
369C2×C4—1944no—
370C2×C4—1948no—
371C2×C4—1942no—
372C2×C4C23×D9712yes—
373C2×C4C23×D9714yes—
374C2×C4C23×D9712yes—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—24no—
380C22—24no—
381C22—24no—
382C22—44no—
383C22—48no—
384C22—3884no—
385C22—1942no—
386C22—1942no—
387C22—1942no—
388C22C2×#1309551014yes—
389C22C2×#1309551012yes—
390C22C24×D9711yescharacteristic
391C4—24no—
392C4—22no—
393C4—22no—
394C4—972no—
395C4—1942no—
396C4—1941no—
397C4—1942no—
398C4—1944no—
399C4—1944no—
400C2—24no—
401C2—22no—
402C2—44no—
403C2—3882no—
404C2C22×#1309551011yescharacteristic
405C2H(5,𝔽2)×D9711yescharacteristic
406C2#2026157011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#534×C97C2
hypercenter#534D97
hypocenterC97#3459
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #20261361C2semidirect
2. #20261330C2semidirect
3. #20261329C2semidirect
4. #20261322C2semidirect
5. #20261320C2semidirect
6. #20261294C2semidirect
7. #20261266C2semidirect
8. #20261262C2semidirect
9. #20261245C2semidirect
10. #20261241C2semidirect
11. C2×D4×Dic97C2semidirect
12. C2×#15690770C2semidirect
13. #534×C97C2semidirect
14. C2×#13095510C22semidirect
17. C23×Dic97C22semidirect
18. C22×C4×D97C22semidirect
19. #15690805C22semidirect
20. #15690803C22semidirect
21. #15690803C22semidirect
22. #15690801C22semidirect
23. #15690800C22semidirect
24. #15690797C22semidirect
25. #15690796C22semidirect
26. #15690794C22semidirect
27. #15690793C22semidirect
28. #15690792C22semidirect
29. #15690791C22semidirect
30. #15690791C22semidirect
31. #15690790C22semidirect
32. #15690789C22semidirect
33. #15690789C22semidirect
34. #15690787C22semidirect
35. #15690787C22semidirect
36. #15690787C22semidirect
39. #15690782C22semidirect
40. #15690781C22semidirect
41. #15690779C22semidirect
42. #15690778C22semidirect
43. #15690775C22semidirect
44. D4×Dic97C22semidirect
45. D4×Dic97C22semidirect
46. D4×Dic97C22semidirect
47. D4×Dic97C22semidirect
48. D4×Dic97C22semidirect
49. #15690770C22semidirect
50. #15690770C22semidirect
51. #15690770C22semidirect
52. #15690768C22semidirect
53. #15690766C22semidirect
54. C2×C4⋊Dic97C22semidirect
55. C2×#13095490C22semidirect
57. C2×C22⋊Dic97C22semidirect
58. C2×#13095488C22semidirect
59. C2×C4×Dic97C22semidirect
60. C2×C4×Dic97C22semidirect
62. #123×C97C22semidirect
65. #120×C97C22semidirect
131. C23×C97#41semidirect
204. C97#3459semidirect
211. #534D97semidirect
236. #123C2×D97semidirect
251. #120C2×D97semidirect
339. C23#13095510semidirect

Maximal subgroups

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

subgroupindex
#202613612
#202613302
#202613292
#202613222
#202613202
#202612942
#202612662
#202612622
#202612452
#202612412
C2×D4×Dic972
C2×#156907702
#534×C972
#345997