Finite Group Databasebeta

← #27153500

Subgroups — #27153500

Order 12416.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261546C211yesmaximal, characteristic, complemented
2#20261542C211yesmaximal, characteristic, complemented
3#20261538C211yesmaximal, characteristic, complemented
4#20261123C211yesmaximal, characteristic, complemented
5#20261003C211yesmaximal, characteristic, complemented
6#20261002C211yesmaximal, characteristic, complemented
7#20260998C211yesmaximal, characteristic, complemented
8#20260996C211yesmaximal, characteristic, complemented
9#20260913C211yesmaximal, characteristic, complemented
10#20260906C211yesmaximal, characteristic, complemented
11#20260903C211yesmaximal, characteristic, complemented
12#20260901C211yesmaximal, characteristic, complemented
13#20260857C211yesmaximal, characteristic, complemented
14#20260855C211yesmaximal, characteristic, complemented
15#576×C97C211yesmaximal, characteristic, complemented
16#15690857C2211yescharacteristic, complemented
17#15690856C2211yescharacteristic, complemented
18#15690842—21no—
19#15690840—21no—
20#15690840C2211yescharacteristic, complemented
21#15690840C2211yescharacteristic, complemented
22#15690836—21no—
23#15690836—21no—
24#15690836C2211yescharacteristic, complemented
25#15690836C2211yescharacteristic, complemented
26#15690835—21no—
27#15690835C2211yescharacteristic
28#15690835C2211yescharacteristic, complemented
29#15690834—21no—
30#15690831—21no—
31#15690830—21no—
32#15690830—21no—
33#15690830C2211yescharacteristic
34C2×#13095500C2211yescharacteristic, complemented
35C2×#13095499—21no—
36C2×#13095499C2211yescharacteristic, complemented
37SD16×D97—21no—
38C2×#13095496—21no—
39C2×#13095496C2211yescharacteristic, complemented
40#15690817C2211yescharacteristic, complemented
41#15690817C2211yescharacteristic, complemented
42#15690816C2211yescharacteristic, complemented
43#15690806C2211yescharacteristic, complemented
44#15690797C2211yescharacteristic, complemented
45#15690746C2211yescharacteristic, complemented
46#15690745C2211yescharacteristic, complemented
47#15690737C2211yescharacteristic, complemented
48#15690737C2211yescharacteristic, complemented
49#15690737C2211yescharacteristic, complemented
50#15690736C2211yescharacteristic, complemented
51#15690728C2211yescharacteristic, complemented
52#15690727C2211yescharacteristic, complemented
53#15690727C2211yescharacteristic, complemented
54#15690726C2211yescharacteristic, complemented
55#15690720C2211yescharacteristic, complemented
56H(5,𝔽2)×C97C2211yescharacteristic
57H(5,𝔽2)×C97C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C97—21no—
59AGL(1,ℤ/8)×C97—21no—
60AGL(1,ℤ/8)×C97C2211yescharacteristic
61AGL(1,ℤ/8)×C97C2211yescharacteristic, complemented
62#135×C97—21no—
63#135×C97C2211yescharacteristic
64C2×D8×C97—21no—
65C2×D8×C97C2211yescharacteristic, complemented
66#131×C97C2211yescharacteristic, complemented
67#13095512—21no—
68#13095511—21no—
69#13095511—21no—
70#13095511C2311yescharacteristic
71#13095510—21no—
72#13095510—41no—
73C2×Dic194—21no—
74C2×Dic194C2311yescharacteristic
75Q8×D97—21no—
76C2×D388—21no—
77C2×D388C2311yescharacteristic
78C2×C22⋊D97—41no—
79D4×D97—21no—
80D4×D97—41no—
81#13095501—21no—
82#13095501—21no—
83#13095501D411yescharacteristic, complemented
84#13095500—21no—
85#13095500—21no—
86#13095499—21no—
87#13095499—21no—
88#13095499—21no—
89#13095499—21no—
90#13095499—21no—
91#13095499—21no—
92#13095499D411yescharacteristic, complemented
93#13095498—21no—
94#13095498—21no—
95#13095498D411yescharacteristic, complemented
96#13095496—21no—
97#13095496—21no—
98#13095496—21no—
99#13095496—21no—
100#13095496—21no—
101#13095496—21no—
102#13095496D411yescharacteristic, complemented
103#13095495—21no—
104#13095495—21no—
105#13095495C2311yescharacteristic
106#13095495C2311yescharacteristic
107#13095495C2311yescharacteristic
108#13095494—21no—
109C2×#11715895—21no—
110C2×#11715895—21no—
111C2×#11715895C2311yescharacteristic
112C8×D97—21no—
113C22⋊Dic97—41no—
114C4×Dic97C2311yescharacteristic
115#41×C97—21no—
116#41×C97—21no—
117#41×C97—21no—
118#41×C97—21no—
119#41×C97C2311yescharacteristic
120#41×C97C2311yescharacteristic
121#41×C97C2311yescharacteristic
122C2×D4×C97—21no—
123C2×D4×C97—21no—
124C2×D4×C97—21no—
125C2×D4×C97—21no—
126C2×D4×C97—41no—
127C2×D4×C97—41no—
128C2×D4×C97C2311yescharacteristic
129C2×D4×C97C2311yescharacteristic
130Q16×C97D411yescharacteristic, complemented
131SD16×C97—21no—
132SD16×C97—21no—
133SD16×C97D411yescharacteristic, complemented
134SD16×C97D411yescharacteristic, complemented
135D8×C97—21no—
136D8×C97—21no—
137D8×C97—21no—
138D8×C97—21no—
139D8×C97D411yescharacteristic, complemented
140M16×C97—21no—
141M16×C97C2311yescharacteristic
142C2×C8×C97—21no—
143C2×C8×C97C2311yescharacteristic
144C23×C97—21no—
145C23×C97—21no—
146C23×C97—41no—
147C23×C97—41no—
148C22×D97—21no—
149C22×D97—41no—
150Q8×C97C2×D411yescharacteristic
151Q8×C97C2×D411yescharacteristic
152Q8×C97C2×D411yescharacteristic
153D4×C97—21no—
154D4×C97—21no—
155D4×C97—21no—
156D4×C97—21no—
157D4×C97—21no—
158D4×C97—21no—
159D4×C97—21no—
160D4×C97—41no—
161D4×C97—41no—
162D4×C97—41no—
163D4×C97—41no—
164D4×C97C2×D411yescharacteristic
165D4×C97C2×D411yescharacteristic
166D4×C97C2×D411yescharacteristic
167C2×C4×C97—21no—
168C2×C4×C97—21no—
169C2×C4×C97—21no—
170C2×C4×C97—41no—
171C2×C4×C97—41no—
172C2×C4×C97C2411yescharacteristic
173C22⋊D97—21no—
174C22⋊D97—41no—
175C22⋊D97—41no—
176C22⋊D97—82no—
177C2×Dic97—21no—
178C2×Dic97—21no—
179C2×Dic97—41no—
180D388—21no—
181D388—21no—
182D388—41no—
183D388C2×D411yescharacteristic
184C4×D97—21no—
185C4×D97—41no—
186C4×D97—41no—
187Dic194—21no—
188Dic194—21no—
189Dic194—41no—
190Dic194C2×D411yescharacteristic
191C8×C97—21no—
192C8×C97C2×D411yescharacteristic
193C8×C97C2×D411yescharacteristic
194#11715895—21no—
195#11715895—21no—
196#11715895—21no—
197#11715895C2×D411yescharacteristic
198#11715895C2×D411yescharacteristic
199C22×C97—21no—
200C22×C97—21no—
201C22×C97—21no—
202C22×C97—41no—
203C22×C97—41no—
204C22×C97—81no—
205C22×C97—81no—
206C22×C97—161no—
207C22×C97—161no—
208C22×C97H(5,𝔽2)11yescharacteristic
209C2×D97—21no—
210C2×D97—41no—
211C2×D97—81no—
212C2×D97—161no—
213C4×C97—21no—
214C4×C97—21no—
215C4×C97—21no—
216C4×C97C22×D411yescharacteristic
217C4×C97C22×D411yescharacteristic
218Dic97—21no—
219Dic97—41no—
220Dic97—41no—
221C2×C97—21no—
222C2×C97—41no—
223C2×C97—41no—
224C2×C97—41no—
225C2×C97—81no—
226C2×C97—81no—
227C2×C97D4211yescharacteristic
228D97—41no—
229D97—81no—
230#3288—971nomaximal, Sylow
231C97#328811yescharacteristic, complemented, Sylow
232#577—971no—
233#577—971no—
234#576—971no—
235#576D9711yescharacteristic, complemented
236#496—971no—
237#472—971no—
238#472—971no—
239#471—971no—
240#471—971no—
241#455—971no—
242#454—971no—
243#453—971no—
244#453—971no—
245#444—971no—
246#444—971no—
2472−1+4—971no—
248H(5,𝔽2)—971no—
249H(5,𝔽2)C2×D9711yescharacteristic
250H(5,𝔽2)C2×D9711yescharacteristic, complemented
251#137—971no—
252#137—971no—
253#137—1941no—
254#137—1941no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—971no—
258AGL(1,ℤ/8)—971no—
259AGL(1,ℤ/8)—971no—
260AGL(1,ℤ/8)—971no—
261AGL(1,ℤ/8)—1941no—
262AGL(1,ℤ/8)—1941no—
263AGL(1,ℤ/8)—1941no—
264AGL(1,ℤ/8)—1941no—
265AGL(1,ℤ/8)C2×D9711yescharacteristic
266AGL(1,ℤ/8)C2×D9711yescharacteristic, complemented
267#135—21no—
268#135—971no—
269#135—1941no—
270#135—1941no—
271#135—1941no—
272#135C2×D9711yescharacteristic
273C2×SD16—971no—
274C2×SD16—971no—
275C2×SD16—1941no—
276C2×SD16—1941no—
277C2×D8—21no—
278C2×D8—971no—
279C2×D8—1941no—
280C2×D8C2×D9711yescharacteristic, complemented
281#131—971no—
282#131—971no—
283#131—971no—
284#131C2×D9711yescharacteristic, complemented
285#127—971no—
286#124—971no—
287#108—971no—
288#108—971no—
289#104—971no—
290#104—971no—
291#104—971no—
292#104—971no—
293#101—971no—
294#100—971no—
295#100—971no—
296#100—971no—
297#97—971no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—971no—
303#41—3881no—
304#41—1941no—
305#41—1941no—
306#41—1941no—
307#41—1941no—
308#41C22×D9711yescharacteristic
309#41C22×D9711yescharacteristic
310#41C22×D9711yescharacteristic
311C2×Q8—971no—
312C2×Q8—1941no—
313C2×Q8—1941no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—971no—
321C2×D4—3881no—
322C2×D4—3881no—
323C2×D4—1941no—
324C2×D4—1941no—
325C2×D4C22×D9711yescharacteristic
326C2×D4C22×D9711yescharacteristic
327Q16—971no—
328Q16—1941no—
329Q16—1941no—
330Q16C22⋊D9711yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—971no—
334SD16—971no—
335SD16—1941no—
336SD16—1941no—
337SD16—1941no—
338SD16—1941no—
339SD16—1941no—
340SD16—1941no—
341SD16—1941no—
342SD16—1941no—
343SD16—1941no—
344SD16—1941no—
345SD16C22⋊D9711yescharacteristic, complemented
346SD16C22⋊D9711yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—971no—
352D8—1941no—
353D8—1941no—
354D8—1941no—
355D8—1941no—
356D8—1941no—
357D8—1941no—
358D8C22⋊D9711yescharacteristic, complemented
359M16—21no—
360M16—971no—
361M16—971no—
362M16—971no—
363M16—1941no—
364M16—1941no—
365M16—1941no—
366M16C22×D9711yescharacteristic
367C2×C8—21no—
368C2×C8—971no—
369C2×C8—1941no—
370C2×C8—1941no—
371C2×C8—1941no—
372C2×C8C22×D9711yescharacteristic
373C22⋊C4—3881no—
374C42—971no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—3881no—
380C23—1941no—
381Q8—971no—
382Q8—3881no—
383Q8—1941no—
384Q8—1941no—
385Q8D4×D9711yescharacteristic
386Q8C2×C22⋊D9711yescharacteristic
387Q8C2×C22⋊D9711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—7762no—
400D4—971no—
401D4—3881no—
402D4—3881no—
403D4—3881no—
404D4—1941no—
405D4—1941no—
406D4—1941no—
407D4D4×D9711yescharacteristic
408D4C2×C22⋊D9711yescharacteristic
409D4C2×C22⋊D9711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—3881no—
416C2×C4—3881no—
417C2×C4—3881no—
418C2×C4—1941no—
419C2×C4—1941no—
420C2×C4—1941no—
421C2×C4C23×D9711yescharacteristic
422C8—21no—
423C8—971no—
424C8—971no—
425C8—1941no—
426C8—1941no—
427C8—1941no—
428C8C2×C22⋊D9711yescharacteristic
429C8C2×C22⋊D9711yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—7761no—
438C22—161no—
439C22—161no—
440C22—15521no—
441C22—3881no—
442C22—1941no—
443C22#1569085511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—3881no—
448C4—3881no—
449C4—1941no—
450C4C2×D4×D9711yescharacteristic
451C4C22×C22⋊D9711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—7761no—
459C2—3881no—
460C2D4×C22⋊D9711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#576×C97C2
socleC2×C97D42
hypercenter#576D97
hypocenterC97#3288
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20261546C2semidirect
2. #20261542C2semidirect
3. #20261538C2semidirect
4. #20261123C2semidirect
5. #20261003C2semidirect
6. #20261002C2semidirect
7. #20260998C2semidirect
8. #20260996C2semidirect
9. #20260913C2semidirect
10. #20260906C2semidirect
11. #20260903C2semidirect
12. #20260901C2semidirect
13. #20260857C2semidirect
14. #20260855C2semidirect
15. #576×C97C2semidirect
16. #15690857C22semidirect
17. #15690856C22semidirect
20. #15690840C22semidirect
21. #15690840C22semidirect
24. #15690836C22semidirect
25. #15690836C22semidirect
28. #15690835C22semidirect
34. C2×#13095500C22semidirect
36. C2×#13095499C22semidirect
39. C2×#13095496C22semidirect
40. #15690817C22semidirect
41. #15690817C22semidirect
42. #15690816C22semidirect
43. #15690806C22semidirect
44. #15690797C22semidirect
45. #15690746C22semidirect
46. #15690745C22semidirect
47. #15690737C22semidirect
48. #15690737C22semidirect
49. #15690737C22semidirect
50. #15690736C22semidirect
51. #15690728C22semidirect
52. #15690727C22semidirect
53. #15690727C22semidirect
54. #15690726C22semidirect
55. #15690720C22semidirect
57. H(5,𝔽2)×C97C22semidirect
61. AGL(1,ℤ/8)×C97C22semidirect
65. C2×D8×C97C22semidirect
66. #131×C97C22semidirect
83. #13095501D4semidirect
92. #13095499D4semidirect
95. #13095498D4semidirect
102. #13095496D4semidirect
130. Q16×C97D4semidirect
133. SD16×C97D4semidirect
134. SD16×C97D4semidirect
139. D8×C97D4semidirect
231. C97#3288semidirect
235. #576D97semidirect
250. H(5,𝔽2)C2×D97semidirect
266. AGL(1,ℤ/8)C2×D97semidirect
280. C2×D8C2×D97semidirect
284. #131C2×D97semidirect
330. Q16C22⋊D97semidirect
345. SD16C22⋊D97semidirect
346. SD16C22⋊D97semidirect
358. D8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202615462
#202615422
#202615382
#202611232
#202610032
#202610022
#202609982
#202609962
#202609132
#202609062
#202609032
#202609012
#202608572
#202608552
#576×C972
#328897