Finite Group Databasebeta

← #26032267

Subgroups — #26032267

Order 11392.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D356C211yesmaximal, characteristic
2#19151964C211yesmaximal, characteristic, complemented
3#19151922C211yesmaximal, characteristic, complemented
4#19151911C211yesmaximal, characteristic, complemented
5#19151895C211yesmaximal, characteristic, complemented
6#19151846C211yesmaximal, characteristic, complemented
7#19151829C211yesmaximal, characteristic, complemented
8#19151798C211yesmaximal, characteristic, complemented
9#19151748C211yesmaximal, characteristic, complemented
10#19151748C211yesmaximal, characteristic, complemented
11#19151696C211yesmaximal, characteristic, complemented
12#19151670C211yesmaximal, characteristic, complemented
13#19151602C211yesmaximal, characteristic, complemented
14C2×#15667932C211yesmaximal, characteristic, complemented
15#552×C89C211yesmaximal, characteristic, complemented
16C22×D356C2211yescharacteristic
17C22×D356C2211yescharacteristic
18C2×D4×D89—21no—
19C2×#13083561C2211yescharacteristic
20C2×#13083561C2211yescharacteristic, complemented
21C2×#13083561C2211yescharacteristic, complemented
22C2×#13083558C2211yescharacteristic, complemented
23C22×#10623494C2211yescharacteristic, complemented
24#15668009C2211yescharacteristic
25#15667987—21no—
26#15667986C2211yescharacteristic
27#15667986C2211yescharacteristic
28#15667982—21no—
29#15667982—21no—
30C4×D356C2211yescharacteristic
31#15667945C2211yescharacteristic, complemented
32#15667945C2211yescharacteristic, complemented
33#15667943C2211yescharacteristic, complemented
34C4⋊#10623494C2211yescharacteristic, complemented
35#15667936C2211yescharacteristic, complemented
36#15667932C2211yescharacteristic
37#15667932C2211yescharacteristic
38#15667932C2211yescharacteristic
39#15667932C2211yescharacteristic, complemented
40#15667932C2211yescharacteristic, complemented
41#15667932C2211yescharacteristic, complemented
42#15667932C2211yescharacteristic, complemented
43#15667932C2211yescharacteristic, complemented
44#15667932C2211yescharacteristic, complemented
45C22⋊#10623494C2211yescharacteristic, complemented
46C22⋊#10623494C2211yescharacteristic, complemented
47#15667923C2211yescharacteristic, complemented
48#126×C89—21no—
49#125×C89—21no—
50#125×C89C2211yescharacteristic, complemented
51#123×C89—21no—
52C22⋊Q8×C89C2211yescharacteristic, complemented
53C22⋊Q8×C89C2211yescharacteristic, complemented
54C4×Q8×C89C2211yescharacteristic, complemented
55C4×D4×C89C2211yescharacteristic
56#117×C89—21no—
57#117×C89C2211yescharacteristic, complemented
58C2×C4⋊C4×C89C2211yescharacteristic, complemented
59C23×D89—21no—
60C23×D89—21no—
61C2×D356—22no—
62C2×D356—22no—
63C2×D356—21no—
64C2×D356—21no—
65C2×D356—21no—
66C2×D356—21no—
67C2×D356—41no—
68C2×D356—41no—
69C2×D356C2311yescharacteristic
70C2×D356C2311yescharacteristic
71C2×D356C2311yescharacteristic
72C2×C22⋊D89—21no—
73C2×C22⋊D89—21no—
74D4×D89—44no—
75C2×C4×D89—21no—
76#13083561—22no—
77#13083561—41no—
78#13083561—41no—
79#13083558—41no—
80C2×#10623494—41no—
81C2×#10623494C2311yescharacteristic
82C2×#10623494C2311yescharacteristic
83C2×#10623494C2311yescharacteristic
84C2×#10623494C2311yescharacteristic
85C4⋊Dic89C2311yescharacteristic
86#13083551—21no—
87#13083551—21no—
88#13083551—21no—
89C2×Q8×C89C2311yescharacteristic
90C2×D4×C89D411yescharacteristic
91C22×C4×C89—21no—
92C22×C4×C89C2311yescharacteristic
93C22×C4×C89C2311yescharacteristic
94C4⋊C4×C89—21no—
95C4⋊C4×C89—21no—
96C4⋊C4×C89—21no—
97C4⋊C4×C89—21no—
98C4⋊C4×C89—21no—
99C4⋊C4×C89—21no—
100C4⋊C4×C89D411yescharacteristic
101C4⋊C4×C89C2311yescharacteristic
102C4⋊C4×C89C2311yescharacteristic
103C4⋊C4×C89C2311yescharacteristic
104C22⋊C4×C89—21no—
105C22⋊C4×C89—21no—
106C22⋊C4×C89—21no—
107C22⋊C4×C89D411yescharacteristic
108C22⋊C4×C89D411yescharacteristic
109C42×C89—21no—
110C42×C89—21no—
111C42×C89C2311yescharacteristic
112C23×C89C2×D411yescharacteristic
113C23×C89C2×D411yescharacteristic
114C22×D89—21no—
115C22×D89—21no—
116C22×D89—21no—
117C22×D89—42no—
118C22×D89—42no—
119C22×D89—42no—
120C22×D89—42no—
121C22×D89—81no—
122C22×D89—81no—
123Q8×C89—21no—
124Q8×C89—21no—
125D4×C89—22no—
126C2×C4×C89—21no—
127C2×C4×C89—21no—
128C2×C4×C89—21no—
129C2×C4×C89—21no—
130C2×C4×C89—21no—
131C2×C4×C89—21no—
132C2×C4×C89—21no—
133C2×C4×C89—21no—
134C2×C4×C89—42no—
135C2×C4×C89C2×D411yescharacteristic
136C2×C4×C89C2×D411yescharacteristic
137C2×C4×C89C2×D411yescharacteristic
138C2×C4×C89C2×D411yescharacteristic
139C2×C4×C89C2411yescharacteristic
140C22⋊D89—42no—
141C22⋊D89—42no—
142C2×Dic89—21no—
143D356—21no—
144D356—21no—
145D356—21no—
146D356—21no—
147D356—21no—
148D356—42no—
149D356—42no—
150D356—42no—
151D356—81no—
152D356#4112yes—
153C4×D89—42no—
154#10623494—21no—
155#10623494—21no—
156#10623494—21no—
157#10623494—21no—
158C22×C89—21no—
159C22×C89—21no—
160C22×C89—21no—
161C22×C89—21no—
162C22×C89—21no—
163C22×C89AGL(1,ℤ/8)12yes—
164C22×C89C22×D411yescharacteristic
165C2×D89—22no—
166C2×D89—41no—
167C2×D89—41no—
168C2×D89—41no—
169C2×D89—41no—
170C2×D89—41no—
171C2×D89—41no—
172C2×D89—41no—
173C2×D89—41no—
174C2×D89—84no—
175C2×D89—84no—
176C4×C89—21no—
177C4×C89—21no—
178C4×C89—21no—
179C4×C89—21no—
180C4×C89—41no—
181C4×C89—41no—
182C4×C89—41no—
183C4×C89—41no—
184C4×C89C2×#4111yescharacteristic
185C4×C89H(5,𝔽2)11yescharacteristic
186Dic89—41no—
187C2×C89—22no—
188C2×C89—41no—
189C2×C89#54611yescharacteristic
190C2×C89C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C89#57711yescharacteristic
192D89—42no—
193D89—81no—
194D89—81no—
195#3310—891nomaximal, Sylow
196C89#331011yescharacteristic, complemented, Sylow
197#552D8911yescharacteristic, complemented
198D42—891no—
199#490—891no—
200#482—891no—
201#480—891no—
202#478—891no—
203#468—891no—
204#465—891no—
205#460—891no—
206#450—891no—
207#450—891no—
208#440—891no—
209#435—891no—
210#418—891no—
211C2×#102—891no—
212C22×D4—891no—
213C22×D4—891no—
214C22×D4—1781no—
215C2×SD16—891no—
216C2×SD16—891no—
217C2×SD16—891no—
218C2×M16—891no—
219C22×C8—891no—
220#127—891no—
221#126—21no—
222#125—21no—
223#125C2×D8911yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D8911yescharacteristic, complemented
226C22⋊Q8C2×D8911yescharacteristic, complemented
227#121—891no—
228#121—891no—
229#121—1781no—
230#120—1781no—
231#120—1781no—
232C4×Q8C2×D8911yescharacteristic, complemented
233C4×D4—891no—
234C4×D4C2×D8911yescharacteristic
235#117—21no—
236#117C2×D8911yescharacteristic, complemented
237C2×C4⋊C4C2×D8911yescharacteristic, complemented
238#107—891no—
239#107—891no—
240#106—891no—
241C4⋊C8—891no—
242#103—891no—
243#102—891no—
244#102—891no—
245#102—891no—
246#102—891no—
247#102—891no—
248#102—891no—
249#102—891no—
250#102—891no—
251#102—891no—
252C22⋊C8—891no—
253C22⋊C8—891no—
254#97—891no—
255C24—1781no—
256C24—1781no—
257C2×Q8C22×D8911yescharacteristic
258C2×D4—891no—
259C2×D4—891no—
260C2×D4—891no—
261C2×D4—3561no—
262C2×D4—3561no—
263C2×D4—3564no—
264C2×D4—1782no—
265C2×D4—1782no—
266C2×D4—1781no—
267C2×D4—1781no—
268C2×D4—1781no—
269C2×D4—1781no—
270C2×D4—1781no—
271C2×D4—1781no—
272C2×D4C22⋊D8911yescharacteristic
273C22×C4—21no—
274C22×C4—1781no—
275C22×C4C22×D8911yescharacteristic
276C22×C4C22×D8911yescharacteristic
277SD16—3561no—
278SD16—3561no—
279SD16—1782no—
280M16—3561no—
281C2×C8—891no—
282C2×C8—891no—
283C2×C8—891no—
284C2×C8—891no—
285C2×C8—3561no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—891no—
293C4⋊C4C22⋊D8911yescharacteristic
294C4⋊C4C22×D8911yescharacteristic
295C4⋊C4C22×D8911yescharacteristic
296C4⋊C4C22×D8911yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—1781no—
301C22⋊C4—1781no—
302C22⋊C4—1781no—
303C22⋊C4C22⋊D8911yescharacteristic
304C22⋊C4C22⋊D8911yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D8911yescharacteristic
308C23—3562no—
309C23—3562no—
310C23—3562no—
311C23—3562no—
312C23—1781no—
313C23—1781no—
314C23—1781no—
315C23—7121no—
316C23—7121no—
317C23C2×C22⋊D8911yescharacteristic
318C23C2×C22⋊D8911yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—892no—
323D4—3562no—
324D4—3562no—
325D4—3562no—
326D4—3562no—
327D4—3562no—
328D4—1781no—
329D4—1781no—
330D4—1781no—
331D4—1781no—
332D4—1781no—
333D4—7121no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3562no—
344C2×C4—1781no—
345C2×C4C2×C22⋊D8911yescharacteristic
346C2×C4C2×C22⋊D8911yescharacteristic
347C2×C4C2×C22⋊D8911yescharacteristic
348C2×C4C2×C22⋊D8911yescharacteristic
349C2×C4C23×D8911yescharacteristic
350C8—1781no—
351C8—1781no—
352C8—1781no—
353C8—1781no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—3561no—
360C22—3561no—
361C22—3561no—
362C22—3561no—
363C22—3561no—
364C22—3561no—
365C22—3561no—
366C22—3561no—
367C22—1782no—
368C22—7124no—
369C22—7124no—
370C22#1566803712yes—
371C22C22×C22⋊D8911yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3561no—
381C4#41×D8911yescharacteristic
382C4#1566805811yescharacteristic
383C2—22no—
384C2—41no—
385C2—3562no—
386C2—7121no—
387C2—7121no—
388C2#1915223611yescharacteristic
389C2C2×#1566803711yescharacteristic
390C2#1915240611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D89
commutator, cosocleC2×C4×C89C24
FrattiniC2×C4C23×D89
Fitting#552×C89C2
socleC22×C89C22×D4
prime radicalD4×D356C2
hypercenter#552D89
hypocenterC89#3310
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #19151964C2semidirect
3. #19151922C2semidirect
4. #19151911C2semidirect
5. #19151895C2semidirect
6. #19151846C2semidirect
7. #19151829C2semidirect
8. #19151798C2semidirect
9. #19151748C2semidirect
10. #19151748C2semidirect
11. #19151696C2semidirect
12. #19151670C2semidirect
13. #19151602C2semidirect
14. C2×#15667932C2semidirect
15. #552×C89C2semidirect
20. C2×#13083561C22semidirect
21. C2×#13083561C22semidirect
22. C2×#13083558C22semidirect
23. C22×#10623494C22semidirect
31. #15667945C22semidirect
32. #15667945C22semidirect
33. #15667943C22semidirect
34. C4⋊#10623494C22semidirect
35. #15667936C22semidirect
39. #15667932C22semidirect
40. #15667932C22semidirect
41. #15667932C22semidirect
42. #15667932C22semidirect
43. #15667932C22semidirect
44. #15667932C22semidirect
45. C22⋊#10623494C22semidirect
46. C22⋊#10623494C22semidirect
47. #15667923C22semidirect
50. #125×C89C22semidirect
52. C22⋊Q8×C89C22semidirect
53. C22⋊Q8×C89C22semidirect
54. C4×Q8×C89C22semidirect
57. #117×C89C22semidirect
58. C2×C4⋊C4×C89C22semidirect
196. C89#3310semidirect
197. #552D89semidirect
223. #125C2×D89semidirect
225. C22⋊Q8C2×D89semidirect
226. C22⋊Q8C2×D89semidirect
232. C4×Q8C2×D89semidirect
236. #117C2×D89semidirect
237. C2×C4⋊C4C2×D89semidirect

Maximal subgroups

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

subgroupindex
D4×D3562
#191519642
#191519222
#191519112
#191518952
#191518462
#191518292
#191517982
#191517482
#191516962
#191516702
#191516022
C2×#156679322
#552×C892
#331089