Finite Group Databasebeta

← #28374796

Subgroups — #28374796

Order 13952.

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#20295837C211yesmaximal, characteristic, complemented
2#20295833C211yesmaximal, characteristic, complemented
3#20295829C211yesmaximal, characteristic, complemented
4#20295414C211yesmaximal, characteristic, complemented
5#20295294C211yesmaximal, characteristic, complemented
6#20295293C211yesmaximal, characteristic, complemented
7#20295289C211yesmaximal, characteristic, complemented
8#20295287C211yesmaximal, characteristic, complemented
9#20295204C211yesmaximal, characteristic, complemented
10#20295197C211yesmaximal, characteristic, complemented
11#20295194C211yesmaximal, characteristic, complemented
12#20295192C211yesmaximal, characteristic, complemented
13#20295148C211yesmaximal, characteristic, complemented
14#20295146C211yesmaximal, characteristic, complemented
15#576×C109C211yesmaximal, characteristic, complemented
16#16812746C2211yescharacteristic, complemented
17#16812745C2211yescharacteristic, complemented
18#16812731—21no—
19#16812729—21no—
20#16812729C2211yescharacteristic, complemented
21#16812729C2211yescharacteristic, complemented
22#16812725—21no—
23#16812725—21no—
24#16812725C2211yescharacteristic, complemented
25#16812725C2211yescharacteristic, complemented
26#16812724—21no—
27#16812724C2211yescharacteristic
28#16812724C2211yescharacteristic, complemented
29#16812723—21no—
30#16812720—21no—
31#16812719—21no—
32#16812719—21no—
33#16812719C2211yescharacteristic
34C2×#13182018C2211yescharacteristic, complemented
35C2×#13182017—21no—
36C2×#13182017C2211yescharacteristic, complemented
37SD16×D109—21no—
38C2×#13182014—21no—
39C2×#13182014C2211yescharacteristic, complemented
40#16812706C2211yescharacteristic, complemented
41#16812706C2211yescharacteristic, complemented
42#16812705C2211yescharacteristic, complemented
43#16812695C2211yescharacteristic, complemented
44#16812686C2211yescharacteristic, complemented
45#16812635C2211yescharacteristic, complemented
46#16812634C2211yescharacteristic, complemented
47#16812626C2211yescharacteristic, complemented
48#16812626C2211yescharacteristic, complemented
49#16812626C2211yescharacteristic, complemented
50#16812625C2211yescharacteristic, complemented
51#16812617C2211yescharacteristic, complemented
52#16812616C2211yescharacteristic, complemented
53#16812616C2211yescharacteristic, complemented
54#16812615C2211yescharacteristic, complemented
55#16812609C2211yescharacteristic, complemented
56H(5,𝔽2)×C109C2211yescharacteristic
57H(5,𝔽2)×C109C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C109—21no—
59AGL(1,ℤ/8)×C109—21no—
60AGL(1,ℤ/8)×C109C2211yescharacteristic
61AGL(1,ℤ/8)×C109C2211yescharacteristic, complemented
62#135×C109—21no—
63#135×C109C2211yescharacteristic
64C2×D8×C109—21no—
65C2×D8×C109C2211yescharacteristic, complemented
66#131×C109C2211yescharacteristic, complemented
67#13182030—21no—
68#13182029—21no—
69#13182029—21no—
70#13182029C2311yescharacteristic
71#13182028—21no—
72#13182028—41no—
73C2×Dic218—21no—
74C2×Dic218C2311yescharacteristic
75Q8×D109—21no—
76C2×D436—21no—
77C2×D436C2311yescharacteristic
78C2×C22⋊D109—41no—
79D4×D109—21no—
80D4×D109—41no—
81#13182019—21no—
82#13182019—21no—
83#13182019D411yescharacteristic, complemented
84#13182018—21no—
85#13182018—21no—
86#13182017—21no—
87#13182017—21no—
88#13182017—21no—
89#13182017—21no—
90#13182017—21no—
91#13182017—21no—
92#13182017D411yescharacteristic, complemented
93#13182016—21no—
94#13182016—21no—
95#13182016D411yescharacteristic, complemented
96#13182014—21no—
97#13182014—21no—
98#13182014—21no—
99#13182014—21no—
100#13182014—21no—
101#13182014—21no—
102#13182014D411yescharacteristic, complemented
103#13182013—21no—
104#13182013—21no—
105#13182013C2311yescharacteristic
106#13182013C2311yescharacteristic
107#13182013C2311yescharacteristic
108#13182012—21no—
109C2×#11724784—21no—
110C2×#11724784—21no—
111C2×#11724784C2311yescharacteristic
112C8×D109—21no—
113C22⋊Dic109—41no—
114C4×Dic109C2311yescharacteristic
115#41×C109—21no—
116#41×C109—21no—
117#41×C109—21no—
118#41×C109—21no—
119#41×C109C2311yescharacteristic
120#41×C109C2311yescharacteristic
121#41×C109C2311yescharacteristic
122C2×D4×C109—21no—
123C2×D4×C109—21no—
124C2×D4×C109—21no—
125C2×D4×C109—21no—
126C2×D4×C109—41no—
127C2×D4×C109—41no—
128C2×D4×C109C2311yescharacteristic
129C2×D4×C109C2311yescharacteristic
130Q16×C109D411yescharacteristic, complemented
131SD16×C109—21no—
132SD16×C109—21no—
133SD16×C109D411yescharacteristic, complemented
134SD16×C109D411yescharacteristic, complemented
135D8×C109—21no—
136D8×C109—21no—
137D8×C109—21no—
138D8×C109—21no—
139D8×C109D411yescharacteristic, complemented
140M16×C109—21no—
141M16×C109C2311yescharacteristic
142C2×C8×C109—21no—
143C2×C8×C109C2311yescharacteristic
144C23×C109—21no—
145C23×C109—21no—
146C23×C109—41no—
147C23×C109—41no—
148C22×D109—21no—
149C22×D109—41no—
150Q8×C109C2×D411yescharacteristic
151Q8×C109C2×D411yescharacteristic
152Q8×C109C2×D411yescharacteristic
153D4×C109—21no—
154D4×C109—21no—
155D4×C109—21no—
156D4×C109—21no—
157D4×C109—21no—
158D4×C109—21no—
159D4×C109—21no—
160D4×C109—41no—
161D4×C109—41no—
162D4×C109—41no—
163D4×C109—41no—
164D4×C109C2×D411yescharacteristic
165D4×C109C2×D411yescharacteristic
166D4×C109C2×D411yescharacteristic
167C2×C4×C109—21no—
168C2×C4×C109—21no—
169C2×C4×C109—21no—
170C2×C4×C109—41no—
171C2×C4×C109—41no—
172C2×C4×C109C2411yescharacteristic
173C22⋊D109—21no—
174C22⋊D109—41no—
175C22⋊D109—41no—
176C22⋊D109—82no—
177C2×Dic109—21no—
178C2×Dic109—21no—
179C2×Dic109—41no—
180D436—21no—
181D436—21no—
182D436—41no—
183D436C2×D411yescharacteristic
184C4×D109—21no—
185C4×D109—41no—
186C4×D109—41no—
187Dic218—21no—
188Dic218—21no—
189Dic218—41no—
190Dic218C2×D411yescharacteristic
191C8×C109—21no—
192C8×C109C2×D411yescharacteristic
193C8×C109C2×D411yescharacteristic
194#11724784—21no—
195#11724784—21no—
196#11724784—21no—
197#11724784C2×D411yescharacteristic
198#11724784C2×D411yescharacteristic
199C22×C109—21no—
200C22×C109—21no—
201C22×C109—21no—
202C22×C109—41no—
203C22×C109—41no—
204C22×C109—81no—
205C22×C109—81no—
206C22×C109—161no—
207C22×C109—161no—
208C22×C109H(5,𝔽2)11yescharacteristic
209C2×D109—21no—
210C2×D109—41no—
211C2×D109—81no—
212C2×D109—161no—
213C4×C109—21no—
214C4×C109—21no—
215C4×C109—21no—
216C4×C109C22×D411yescharacteristic
217C4×C109C22×D411yescharacteristic
218Dic109—21no—
219Dic109—41no—
220Dic109—41no—
221C2×C109—21no—
222C2×C109—41no—
223C2×C109—41no—
224C2×C109—41no—
225C2×C109—81no—
226C2×C109—81no—
227C2×C109D4211yescharacteristic
228D109—41no—
229D109—81no—
230#3288—1091nomaximal, Sylow
231C109#328811yescharacteristic, complemented, Sylow
232#577—1091no—
233#577—1091no—
234#576—1091no—
235#576D10911yescharacteristic, complemented
236#496—1091no—
237#472—1091no—
238#472—1091no—
239#471—1091no—
240#471—1091no—
241#455—1091no—
242#454—1091no—
243#453—1091no—
244#453—1091no—
245#444—1091no—
246#444—1091no—
2472−1+4—1091no—
248H(5,𝔽2)—1091no—
249H(5,𝔽2)C2×D10911yescharacteristic
250H(5,𝔽2)C2×D10911yescharacteristic, complemented
251#137—1091no—
252#137—1091no—
253#137—2181no—
254#137—2181no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—1091no—
258AGL(1,ℤ/8)—1091no—
259AGL(1,ℤ/8)—1091no—
260AGL(1,ℤ/8)—1091no—
261AGL(1,ℤ/8)—2181no—
262AGL(1,ℤ/8)—2181no—
263AGL(1,ℤ/8)—2181no—
264AGL(1,ℤ/8)—2181no—
265AGL(1,ℤ/8)C2×D10911yescharacteristic
266AGL(1,ℤ/8)C2×D10911yescharacteristic, complemented
267#135—21no—
268#135—1091no—
269#135—2181no—
270#135—2181no—
271#135—2181no—
272#135C2×D10911yescharacteristic
273C2×SD16—1091no—
274C2×SD16—1091no—
275C2×SD16—2181no—
276C2×SD16—2181no—
277C2×D8—21no—
278C2×D8—1091no—
279C2×D8—2181no—
280C2×D8C2×D10911yescharacteristic, complemented
281#131—1091no—
282#131—1091no—
283#131—1091no—
284#131C2×D10911yescharacteristic, complemented
285#127—1091no—
286#124—1091no—
287#108—1091no—
288#108—1091no—
289#104—1091no—
290#104—1091no—
291#104—1091no—
292#104—1091no—
293#101—1091no—
294#100—1091no—
295#100—1091no—
296#100—1091no—
297#97—1091no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—1091no—
303#41—4361no—
304#41—2181no—
305#41—2181no—
306#41—2181no—
307#41—2181no—
308#41C22×D10911yescharacteristic
309#41C22×D10911yescharacteristic
310#41C22×D10911yescharacteristic
311C2×Q8—1091no—
312C2×Q8—2181no—
313C2×Q8—2181no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—1091no—
321C2×D4—4361no—
322C2×D4—4361no—
323C2×D4—2181no—
324C2×D4—2181no—
325C2×D4C22×D10911yescharacteristic
326C2×D4C22×D10911yescharacteristic
327Q16—1091no—
328Q16—2181no—
329Q16—2181no—
330Q16C22⋊D10911yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—1091no—
334SD16—1091no—
335SD16—2181no—
336SD16—2181no—
337SD16—2181no—
338SD16—2181no—
339SD16—2181no—
340SD16—2181no—
341SD16—2181no—
342SD16—2181no—
343SD16—2181no—
344SD16—2181no—
345SD16C22⋊D10911yescharacteristic, complemented
346SD16C22⋊D10911yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—1091no—
352D8—2181no—
353D8—2181no—
354D8—2181no—
355D8—2181no—
356D8—2181no—
357D8—2181no—
358D8C22⋊D10911yescharacteristic, complemented
359M16—21no—
360M16—1091no—
361M16—1091no—
362M16—1091no—
363M16—2181no—
364M16—2181no—
365M16—2181no—
366M16C22×D10911yescharacteristic
367C2×C8—21no—
368C2×C8—1091no—
369C2×C8—2181no—
370C2×C8—2181no—
371C2×C8—2181no—
372C2×C8C22×D10911yescharacteristic
373C22⋊C4—4361no—
374C42—1091no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—4361no—
380C23—2181no—
381Q8—1091no—
382Q8—4361no—
383Q8—2181no—
384Q8—2181no—
385Q8D4×D10911yescharacteristic
386Q8C2×C22⋊D10911yescharacteristic
387Q8C2×C22⋊D10911yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—8722no—
400D4—1091no—
401D4—4361no—
402D4—4361no—
403D4—4361no—
404D4—2181no—
405D4—2181no—
406D4—2181no—
407D4D4×D10911yescharacteristic
408D4C2×C22⋊D10911yescharacteristic
409D4C2×C22⋊D10911yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—4361no—
416C2×C4—4361no—
417C2×C4—4361no—
418C2×C4—2181no—
419C2×C4—2181no—
420C2×C4—2181no—
421C2×C4C23×D10911yescharacteristic
422C8—21no—
423C8—1091no—
424C8—1091no—
425C8—2181no—
426C8—2181no—
427C8—2181no—
428C8C2×C22⋊D10911yescharacteristic
429C8C2×C22⋊D10911yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—8721no—
440C22—4361no—
441C22—17441no—
442C22—2181no—
443C22#1681274411yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—4361no—
448C4—4361no—
449C4—2181no—
450C4C2×D4×D10911yescharacteristic
451C4C22×C22⋊D10911yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—8721no—
459C2—4361no—
460C2D4×C22⋊D10911yescharacteristic
subgroupG/H
centerC2D4×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#576×C109C2
socleC2×C109D42
hypercenter#576D109
hypocenterC109#3288
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20295837C2semidirect
2. #20295833C2semidirect
3. #20295829C2semidirect
4. #20295414C2semidirect
5. #20295294C2semidirect
6. #20295293C2semidirect
7. #20295289C2semidirect
8. #20295287C2semidirect
9. #20295204C2semidirect
10. #20295197C2semidirect
11. #20295194C2semidirect
12. #20295192C2semidirect
13. #20295148C2semidirect
14. #20295146C2semidirect
15. #576×C109C2semidirect
16. #16812746C22semidirect
17. #16812745C22semidirect
20. #16812729C22semidirect
21. #16812729C22semidirect
24. #16812725C22semidirect
25. #16812725C22semidirect
28. #16812724C22semidirect
34. C2×#13182018C22semidirect
36. C2×#13182017C22semidirect
39. C2×#13182014C22semidirect
40. #16812706C22semidirect
41. #16812706C22semidirect
42. #16812705C22semidirect
43. #16812695C22semidirect
44. #16812686C22semidirect
45. #16812635C22semidirect
46. #16812634C22semidirect
47. #16812626C22semidirect
48. #16812626C22semidirect
49. #16812626C22semidirect
50. #16812625C22semidirect
51. #16812617C22semidirect
52. #16812616C22semidirect
53. #16812616C22semidirect
54. #16812615C22semidirect
55. #16812609C22semidirect
57. H(5,𝔽2)×C109C22semidirect
61. AGL(1,ℤ/8)×C109C22semidirect
65. C2×D8×C109C22semidirect
66. #131×C109C22semidirect
83. #13182019D4semidirect
92. #13182017D4semidirect
95. #13182016D4semidirect
102. #13182014D4semidirect
130. Q16×C109D4semidirect
133. SD16×C109D4semidirect
134. SD16×C109D4semidirect
139. D8×C109D4semidirect
231. C109#3288semidirect
235. #576D109semidirect
250. H(5,𝔽2)C2×D109semidirect
266. AGL(1,ℤ/8)C2×D109semidirect
280. C2×D8C2×D109semidirect
284. #131C2×D109semidirect
330. Q16C22⋊D109semidirect
345. SD16C22⋊D109semidirect
346. SD16C22⋊D109semidirect
358. D8C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202958372
#202958332
#202958292
#202954142
#202952942
#202952932
#202952892
#202952872
#202952042
#202951972
#202951942
#202951922
#202951482
#202951462
#576×C1092
#3288109