Finite Group Databasebeta

← #28374799

Subgroups — #28374799

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#20295838C211yesmaximal, characteristic, complemented
2#20295833C211yesmaximal, characteristic, complemented
3#20295831C211yesmaximal, characteristic, complemented
4#20295828C211yesmaximal, characteristic, complemented
5#20295415C211yesmaximal, characteristic, complemented
6#20295297C211yesmaximal, characteristic, complemented
7#20295296C211yesmaximal, characteristic, complemented
8#20295290C211yesmaximal, characteristic, complemented
9#20295289C211yesmaximal, characteristic, complemented
10#20295205C211yesmaximal, characteristic, complemented
11#20295200C211yesmaximal, characteristic, complemented
12#20295195C211yesmaximal, characteristic, complemented
13#20295195C211yesmaximal, characteristic, complemented
14#20295149C211yesmaximal, characteristic, complemented
15#444×C109C211yesmaximal, characteristic, complemented
16#16812746C2211yescharacteristic, complemented
17#16812745C2211yescharacteristic
18#16812745C2211yescharacteristic, complemented
19#16812745C2211yescharacteristic, complemented
20#16812732—21no—
21#16812732C2211yescharacteristic, complemented
22#16812731—21no—
23#16812729C2211yescharacteristic, complemented
24#16812727—21no—
25#16812727—21no—
26#16812727—21no—
27#16812727C2211yescharacteristic
28#16812727C2211yescharacteristic, complemented
29#16812727C2211yescharacteristic, complemented
30#16812726—21no—
31#16812726—21no—
32#16812724C2211yescharacteristic
33#16812724C2211yescharacteristic, complemented
34#16812724C2211yescharacteristic, complemented
35#16812723—21no—
36#16812722—21no—
37#16812721—21no—
38#16812721—21no—
39#16812721C2211yescharacteristic
40#16812720—21no—
41C2×#13182018—21no—
42C2×#13182018C2211yescharacteristic, complemented
43C2×#13182018C2211yescharacteristic, complemented
44SD16×D109—21no—
45C2×D872—21no—
46C2×D872C2211yescharacteristic, complemented
47C2×D872C2211yescharacteristic, complemented
48D8×D109—21no—
49#16812705C2211yescharacteristic, complemented
50#16812705C2211yescharacteristic, complemented
51#16812696C2211yescharacteristic, complemented
52#16812687C2211yescharacteristic, complemented
53#16812635C2211yescharacteristic, complemented
54#16812627C2211yescharacteristic, complemented
55#16812627C2211yescharacteristic, complemented
56#16812617C2211yescharacteristic, complemented
57#16812615C2211yescharacteristic, complemented
58#16812615C2211yescharacteristic, complemented
59#16812615C2211yescharacteristic, complemented
60#135×C109C2211yescharacteristic
61#131×C109C2211yescharacteristic, complemented
62#131×C109C2211yescharacteristic, complemented
63#108×C109C2211yescharacteristic, complemented
64#104×C109C2211yescharacteristic, complemented
65#104×C109C2211yescharacteristic, complemented
66#97×C109C2211yescharacteristic, complemented
67#13182030—21no—
68#13182030—21no—
69#13182030—21no—
70#13182029—21no—
71#13182029—21no—
72#13182029—21no—
73#13182029—21no—
74#13182029C2311yescharacteristic
75#13182029C2311yescharacteristic
76#13182028—21no—
77#13182028—41no—
78C2×Dic218—21no—
79C2×Dic218C2311yescharacteristic
80Q8×D109—21no—
81C2×D436—21no—
82C2×D436—21no—
83C2×D436—21no—
84C2×D436—41no—
85C2×D436C2311yescharacteristic
86C2×D436C2311yescharacteristic
87C2×D436C2311yescharacteristic
88D4×D109—21no—
89D4×D109—21no—
90D4×D109—21no—
91D4×D109—41no—
92D4×D109—41no—
93D4×D109—41no—
94Dic436—21no—
95Dic436D411yescharacteristic, complemented
96#13182019—21no—
97#13182018—21no—
98#13182018—21no—
99#13182018—21no—
100#13182018—21no—
101#13182018—21no—
102#13182018—21no—
103#13182018—21no—
104#13182018—21no—
105#13182018D411yescharacteristic, complemented
106#13182018D411yescharacteristic, complemented
107#13182017—21no—
108#13182016—21no—
109#13182016—21no—
110#13182016—21no—
111D872—21no—
112D872—21no—
113D872—21no—
114D872—21no—
115D872—21no—
116D872—21no—
117D872—21no—
118D872D411yescharacteristic, complemented
119#13182014—21no—
120#13182014—21no—
121#13182014—21no—
122#13182013C2311yescharacteristic
123#13182013C2311yescharacteristic
124#13182012—21no—
125#13182012—21no—
126C8×D109—21no—
127C8×D109—21no—
128#13182006—41no—
129#41×C109C2311yescharacteristic
130#41×C109C2311yescharacteristic
131Q16×C109D411yescharacteristic, complemented
132SD16×C109D411yescharacteristic, complemented
133SD16×C109D411yescharacteristic, complemented
134D8×C109D411yescharacteristic, complemented
135M16×C109—21no—
136M16×C109—21no—
137M16×C109C2311yescharacteristic
138M16×C109C2311yescharacteristic
139C2×C8×C109—21no—
140C2×C8×C109—21no—
141C2×C8×C109C2311yescharacteristic
142C2×C8×C109C2311yescharacteristic
143C42×C109C2311yescharacteristic
144C22×D109—21no—
145C22×D109—21no—
146C22×D109—21no—
147C22×D109—41no—
148C22×D109—41no—
149C22×D109—41no—
150Q8×C109C2×D411yescharacteristic
151Q8×C109C2×D411yescharacteristic
152D4×C109—21no—
153D4×C109—21no—
154D4×C109C2×D411yescharacteristic
155D4×C109C2×D411yescharacteristic
156C2×C4×C109—21no—
157C2×C4×C109—21no—
158C2×C4×C109—21no—
159C2×C4×C109C2411yescharacteristic
160C22⋊D109—21no—
161C22⋊D109—21no—
162C22⋊D109—41no—
163C22⋊D109—41no—
164C22⋊D109—41no—
165C22⋊D109—41no—
166C2×Dic109—21no—
167C2×Dic109—41no—
168D436—21no—
169D436—21no—
170D436—21no—
171D436—21no—
172D436—21no—
173D436—21no—
174D436—41no—
175D436—41no—
176D436—41no—
177D436—82no—
178D436C2×D411yescharacteristic
179D436C2×D411yescharacteristic
180C4×D109—21no—
181C4×D109—21no—
182C4×D109—41no—
183C4×D109—41no—
184C4×D109—41no—
185C4×D109—41no—
186Dic218—21no—
187Dic218—21no—
188Dic218—41no—
189Dic218C2×D411yescharacteristic
190Dic218C2×D411yescharacteristic
191C8×C109—21no—
192C8×C109—21no—
193C8×C109C2×D411yescharacteristic
194C8×C109C2×D411yescharacteristic
195C8×C109C2×D411yescharacteristic
196C8×C109C2×D411yescharacteristic
197#11724784—21no—
198#11724784—21no—
199C22×C109—21no—
200C22×C109—21no—
201C22×C109H(5,𝔽2)11yescharacteristic
202C2×D109—21no—
203C2×D109—21no—
204C2×D109—41no—
205C2×D109—41no—
206C2×D109—41no—
207C2×D109—81no—
208C2×D109—81no—
209C2×D109—81no—
210C2×D109—161no—
211C2×D109—161no—
212C2×D109—161no—
213C4×C109—21no—
214C4×C109—21no—
215C4×C109—41no—
216C4×C109C22×D411yescharacteristic
217C4×C109C22×D411yescharacteristic
218Dic109—21no—
219Dic109—21no—
220Dic109—41no—
221C2×C109—21no—
222C2×C109—41no—
223C2×C109—41no—
224C2×C109D4211yescharacteristic
225D109—41no—
226D109—41no—
227D109—81no—
228D109—81no—
229D109—81no—
230#3288—1091nomaximal, Sylow
231C109#328811yescharacteristic, complemented, Sylow
232#577—1091no—
233#577—1091no—
234#576—1091no—
235#576—1091no—
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#444D10911yescharacteristic, complemented
2472−1+4—1091no—
248H(5,𝔽2)—1091no—
249H(5,𝔽2)—1091no—
250H(5,𝔽2)—1091no—
251#137—1091no—
252#137—1091no—
253#137—2181no—
254#137—2181no—
255AGL(1,ℤ/8)—1091no—
256AGL(1,ℤ/8)—1091no—
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)—2181no—
266AGL(1,ℤ/8)—2181no—
267#135—1091no—
268#135—2181no—
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—1091no—
278C2×D8—1091no—
279C2×D8—2181no—
280C2×D8—2181no—
281#131—1091no—
282#131—1091no—
283#131C2×D10911yescharacteristic, complemented
284#131C2×D10911yescharacteristic, complemented
285#127—1091no—
286#124—1091no—
287#108—1091no—
288#108C2×D10911yescharacteristic, complemented
289#104—1091no—
290#104—1091no—
291#104C2×D10911yescharacteristic, complemented
292#104C2×D10911yescharacteristic, complemented
293#101—1091no—
294#100—1091no—
295#100—1091no—
296#100—1091no—
297#97C2×D10911yescharacteristic, complemented
298#41—1091no—
299#41—1091no—
300#41—4361no—
301#41—2181no—
302#41—2181no—
303#41—2181no—
304#41—2181no—
305#41—2181no—
306#41—2181no—
307#41—2181no—
308#41—2181no—
309#41C22×D10911yescharacteristic
310#41C22×D10911yescharacteristic
311C2×Q8—1091no—
312C2×Q8—2181no—
313C2×Q8—2181no—
314C2×D4—1091no—
315C2×D4—1091no—
316C2×D4—1091no—
317C2×D4—4361no—
318C2×D4—4361no—
319C2×D4—4361no—
320C2×D4—4361no—
321C2×D4—2181no—
322C2×D4—2181no—
323C2×D4—2181no—
324C2×D4—2181no—
325C2×D4—2181no—
326C2×D4—2181no—
327Q16—1091no—
328Q16—2181no—
329Q16—2181no—
330Q16D43611yescharacteristic, complemented
331SD16—1091no—
332SD16—1091no—
333SD16—2181no—
334SD16—2181no—
335SD16—2181no—
336SD16—2181no—
337SD16—2181no—
338SD16—2181no—
339SD16—2181no—
340SD16—2181no—
341SD16—2181no—
342SD16—2181no—
343SD16—2181no—
344SD16—2181no—
345SD16D43611yescharacteristic, complemented
346SD16D43611yescharacteristic, complemented
347D8—1091no—
348D8—2181no—
349D8—2181no—
350D8—2181no—
351D8—2181no—
352D8—2181no—
353D8—2181no—
354D8—2181no—
355D8—2181no—
356D8—2181no—
357D8—2181no—
358D8D43611yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—1091no—
362M16—1091no—
363M16—2181no—
364M16—2181no—
365M16C22×D10911yescharacteristic
366M16C22×D10911yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—2181no—
370C2×C8—2181no—
371C2×C8C22×D10911yescharacteristic
372C2×C8C22×D10911yescharacteristic
373C22⋊C4—4361no—
374C42C22×D10911yescharacteristic
375C23—4361no—
376C23—4361no—
377C23—4361no—
378C23—2181no—
379C23—2181no—
380C23—2181no—
381Q8—1091no—
382Q8—1091no—
383Q8—4361no—
384Q8—2181no—
385Q8—2181no—
386Q8C2×D43611yescharacteristic
387Q8C2×D43611yescharacteristic
388D4—21no—
389D4—21no—
390D4—8722no—
391D4—1091no—
392D4—1091no—
393D4—4361no—
394D4—4361no—
395D4—4361no—
396D4—4361no—
397D4—4361no—
398D4—4361no—
399D4—4361no—
400D4—2181no—
401D4—2181no—
402D4—2181no—
403D4—2181no—
404D4—2181no—
405D4—2181no—
406D4—2181no—
407D4—2181no—
408D4C2×D43611yescharacteristic
409D4C2×D43611yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—4361no—
414C2×C4—4361no—
415C2×C4—4361no—
416C2×C4—4361no—
417C2×C4—4361no—
418C2×C4—2181no—
419C2×C4—2181no—
420C2×C4—2181no—
421C2×C4C23×D10911yescharacteristic
422C8—21no—
423C8—21no—
424C8—2181no—
425C8—2181no—
426C8D4×D10911yescharacteristic
427C8D4×D10911yescharacteristic
428C8C2×D43611yescharacteristic
429C8C2×D43611yescharacteristic
430C22—21no—
431C22—21no—
432C22—8721no—
433C22—8721no—
434C22—8721no—
435C22—4361no—
436C22—4361no—
437C22—4361no—
438C22—17441no—
439C22—17441no—
440C22—17441no—
441C22—2181no—
442C22—2181no—
443C22#1681274511yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—4361no—
448C4—2181no—
449C4—2181no—
450C4C2×D4×D10911yescharacteristic
451C4C22×D43611yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—8721no—
456C2—8721no—
457C2—8721no—
458C2—4361no—
459C2—4361no—
460C2D4×D43611yescharacteristic
subgroupG/H
centerC2D4×D436
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#444×C109C2
socleC2×C109D42
hypercenter#444D109
hypocenterC109#3288
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #20295838C2semidirect
2. #20295833C2semidirect
3. #20295831C2semidirect
4. #20295828C2semidirect
5. #20295415C2semidirect
6. #20295297C2semidirect
7. #20295296C2semidirect
8. #20295290C2semidirect
9. #20295289C2semidirect
10. #20295205C2semidirect
11. #20295200C2semidirect
12. #20295195C2semidirect
13. #20295195C2semidirect
14. #20295149C2semidirect
15. #444×C109C2semidirect
16. #16812746C22semidirect
18. #16812745C22semidirect
19. #16812745C22semidirect
21. #16812732C22semidirect
23. #16812729C22semidirect
28. #16812727C22semidirect
29. #16812727C22semidirect
33. #16812724C22semidirect
34. #16812724C22semidirect
42. C2×#13182018C22semidirect
43. C2×#13182018C22semidirect
46. C2×D872C22semidirect
47. C2×D872C22semidirect
49. #16812705C22semidirect
50. #16812705C22semidirect
51. #16812696C22semidirect
52. #16812687C22semidirect
53. #16812635C22semidirect
54. #16812627C22semidirect
55. #16812627C22semidirect
56. #16812617C22semidirect
57. #16812615C22semidirect
58. #16812615C22semidirect
59. #16812615C22semidirect
61. #131×C109C22semidirect
62. #131×C109C22semidirect
63. #108×C109C22semidirect
64. #104×C109C22semidirect
65. #104×C109C22semidirect
66. #97×C109C22semidirect
95. Dic436D4semidirect
105. #13182018D4semidirect
106. #13182018D4semidirect
118. D872D4semidirect
131. Q16×C109D4semidirect
132. SD16×C109D4semidirect
133. SD16×C109D4semidirect
134. D8×C109D4semidirect
231. C109#3288semidirect
246. #444D109semidirect
283. #131C2×D109semidirect
284. #131C2×D109semidirect
288. #108C2×D109semidirect
291. #104C2×D109semidirect
292. #104C2×D109semidirect
297. #97C2×D109semidirect
330. Q16D436semidirect
345. SD16D436semidirect
346. SD16D436semidirect
358. D8D436semidirect

Maximal subgroups

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

subgroupindex
#202958382
#202958332
#202958312
#202958282
#202954152
#202952972
#202952962
#202952902
#202952892
#202952052
#202952002
#202951952
#202951492
#444×C1092
#3288109