Finite Group Databasebeta

← #28348228

Subgroups — #28348228

Order 13696.

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#20293439C211yesmaximal, characteristic, complemented
2#20293434C211yesmaximal, characteristic, complemented
3#20293432C211yesmaximal, characteristic, complemented
4#20293429C211yesmaximal, characteristic, complemented
5#20293016C211yesmaximal, characteristic, complemented
6#20292898C211yesmaximal, characteristic, complemented
7#20292897C211yesmaximal, characteristic, complemented
8#20292891C211yesmaximal, characteristic, complemented
9#20292890C211yesmaximal, characteristic, complemented
10#20292806C211yesmaximal, characteristic, complemented
11#20292801C211yesmaximal, characteristic, complemented
12#20292796C211yesmaximal, characteristic, complemented
13#20292796C211yesmaximal, characteristic, complemented
14#20292750C211yesmaximal, characteristic, complemented
15#444×C107C211yesmaximal, characteristic, complemented
16#16812156C2211yescharacteristic, complemented
17#16812155C2211yescharacteristic
18#16812155C2211yescharacteristic, complemented
19#16812155C2211yescharacteristic, complemented
20#16812142—21no—
21#16812142C2211yescharacteristic, complemented
22#16812141—21no—
23#16812139C2211yescharacteristic, complemented
24#16812137—21no—
25#16812137—21no—
26#16812137—21no—
27#16812137C2211yescharacteristic
28#16812137C2211yescharacteristic, complemented
29#16812137C2211yescharacteristic, complemented
30#16812136—21no—
31#16812136—21no—
32#16812134C2211yescharacteristic
33#16812134C2211yescharacteristic, complemented
34#16812134C2211yescharacteristic, complemented
35#16812133—21no—
36#16812132—21no—
37#16812131—21no—
38#16812131—21no—
39#16812131C2211yescharacteristic
40#16812130—21no—
41C2×#13133807—21no—
42C2×#13133807C2211yescharacteristic, complemented
43C2×#13133807C2211yescharacteristic, complemented
44SD16×D107—21no—
45C2×D856—21no—
46C2×D856C2211yescharacteristic, complemented
47C2×D856C2211yescharacteristic, complemented
48D8×D107—21no—
49#16812115C2211yescharacteristic, complemented
50#16812115C2211yescharacteristic, complemented
51#16812106C2211yescharacteristic, complemented
52#16812097C2211yescharacteristic, complemented
53#16812045C2211yescharacteristic, complemented
54#16812037C2211yescharacteristic, complemented
55#16812037C2211yescharacteristic, complemented
56#16812027C2211yescharacteristic, complemented
57#16812025C2211yescharacteristic, complemented
58#16812025C2211yescharacteristic, complemented
59#16812025C2211yescharacteristic, complemented
60#135×C107C2211yescharacteristic
61#131×C107C2211yescharacteristic, complemented
62#131×C107C2211yescharacteristic, complemented
63#108×C107C2211yescharacteristic, complemented
64#104×C107C2211yescharacteristic, complemented
65#104×C107C2211yescharacteristic, complemented
66#97×C107C2211yescharacteristic, complemented
67#13133819—21no—
68#13133819—21no—
69#13133819—21no—
70#13133818—21no—
71#13133818—21no—
72#13133818—21no—
73#13133818—21no—
74#13133818C2311yescharacteristic
75#13133818C2311yescharacteristic
76#13133817—21no—
77#13133817—41no—
78C2×Dic214—21no—
79C2×Dic214C2311yescharacteristic
80Q8×D107—21no—
81C2×D428—21no—
82C2×D428—21no—
83C2×D428—21no—
84C2×D428—41no—
85C2×D428C2311yescharacteristic
86C2×D428C2311yescharacteristic
87C2×D428C2311yescharacteristic
88D4×D107—21no—
89D4×D107—21no—
90D4×D107—21no—
91D4×D107—41no—
92D4×D107—41no—
93D4×D107—41no—
94Dic428—21no—
95Dic428D411yescharacteristic, complemented
96#13133808—21no—
97#13133807—21no—
98#13133807—21no—
99#13133807—21no—
100#13133807—21no—
101#13133807—21no—
102#13133807—21no—
103#13133807—21no—
104#13133807—21no—
105#13133807D411yescharacteristic, complemented
106#13133807D411yescharacteristic, complemented
107#13133806—21no—
108#13133805—21no—
109#13133805—21no—
110#13133805—21no—
111D856—21no—
112D856—21no—
113D856—21no—
114D856—21no—
115D856—21no—
116D856—21no—
117D856—21no—
118D856D411yescharacteristic, complemented
119#13133803—21no—
120#13133803—21no—
121#13133803—21no—
122#13133802C2311yescharacteristic
123#13133802C2311yescharacteristic
124#13133801—21no—
125#13133801—21no—
126C8×D107—21no—
127C8×D107—21no—
128(C2×D107)⋊C4—41no—
129#41×C107C2311yescharacteristic
130#41×C107C2311yescharacteristic
131Q16×C107D411yescharacteristic, complemented
132SD16×C107D411yescharacteristic, complemented
133SD16×C107D411yescharacteristic, complemented
134D8×C107D411yescharacteristic, complemented
135M16×C107—21no—
136M16×C107—21no—
137M16×C107C2311yescharacteristic
138M16×C107C2311yescharacteristic
139C2×C8×C107—21no—
140C2×C8×C107—21no—
141C2×C8×C107C2311yescharacteristic
142C2×C8×C107C2311yescharacteristic
143C42×C107C2311yescharacteristic
144C22×D107—21no—
145C22×D107—21no—
146C22×D107—21no—
147C22×D107—41no—
148C22×D107—41no—
149C22×D107—41no—
150Q8×C107C2×D411yescharacteristic
151Q8×C107C2×D411yescharacteristic
152D4×C107—21no—
153D4×C107—21no—
154D4×C107C2×D411yescharacteristic
155D4×C107C2×D411yescharacteristic
156C2×C4×C107—21no—
157C2×C4×C107—21no—
158C2×C4×C107—21no—
159C2×C4×C107C2411yescharacteristic
160C22⋊D107—21no—
161C22⋊D107—21no—
162C22⋊D107—41no—
163C22⋊D107—41no—
164C22⋊D107—41no—
165C22⋊D107—41no—
166C2×Dic107—21no—
167C2×Dic107—41no—
168D428—21no—
169D428—21no—
170D428—21no—
171D428—21no—
172D428—21no—
173D428—21no—
174D428—41no—
175D428—41no—
176D428—41no—
177D428—82no—
178D428C2×D411yescharacteristic
179D428C2×D411yescharacteristic
180C4×D107—21no—
181C4×D107—21no—
182C4×D107—41no—
183C4×D107—41no—
184C4×D107—41no—
185C4×D107—41no—
186Dic214—21no—
187Dic214—21no—
188Dic214—41no—
189Dic214C2×D411yescharacteristic
190Dic214C2×D411yescharacteristic
191C8×C107—21no—
192C8×C107—21no—
193C8×C107C2×D411yescharacteristic
194C8×C107C2×D411yescharacteristic
195C8×C107C2×D411yescharacteristic
196C8×C107C2×D411yescharacteristic
197C107⋊C8—21no—
198C107⋊C8—21no—
199C22×C107—21no—
200C22×C107—21no—
201C22×C107H(5,𝔽2)11yescharacteristic
202C2×D107—21no—
203C2×D107—21no—
204C2×D107—41no—
205C2×D107—41no—
206C2×D107—41no—
207C2×D107—81no—
208C2×D107—81no—
209C2×D107—81no—
210C2×D107—161no—
211C2×D107—161no—
212C2×D107—161no—
213C4×C107—21no—
214C4×C107—21no—
215C4×C107—41no—
216C4×C107C22×D411yescharacteristic
217C4×C107C22×D411yescharacteristic
218Dic107—21no—
219Dic107—21no—
220Dic107—41no—
221C2×C107—21no—
222C2×C107—41no—
223C2×C107—41no—
224C2×C107D4211yescharacteristic
225D107—41no—
226D107—41no—
227D107—81no—
228D107—81no—
229D107—81no—
230#3288—1071nomaximal, Sylow
231C107#328811yescharacteristic, complemented, Sylow
232#577—1071no—
233#577—1071no—
234#576—1071no—
235#576—1071no—
236#496—1071no—
237#472—1071no—
238#472—1071no—
239#471—1071no—
240#471—1071no—
241#455—1071no—
242#454—1071no—
243#453—1071no—
244#453—1071no—
245#444—1071no—
246#444D10711yescharacteristic, complemented
2472−1+4—1071no—
248H(5,𝔽2)—1071no—
249H(5,𝔽2)—1071no—
250H(5,𝔽2)—1071no—
251#137—1071no—
252#137—1071no—
253#137—2141no—
254#137—2141no—
255AGL(1,ℤ/8)—1071no—
256AGL(1,ℤ/8)—1071no—
257AGL(1,ℤ/8)—1071no—
258AGL(1,ℤ/8)—1071no—
259AGL(1,ℤ/8)—1071no—
260AGL(1,ℤ/8)—1071no—
261AGL(1,ℤ/8)—2141no—
262AGL(1,ℤ/8)—2141no—
263AGL(1,ℤ/8)—2141no—
264AGL(1,ℤ/8)—2141no—
265AGL(1,ℤ/8)—2141no—
266AGL(1,ℤ/8)—2141no—
267#135—1071no—
268#135—2141no—
269#135—2141no—
270#135—2141no—
271#135—2141no—
272#135C2×D10711yescharacteristic
273C2×SD16—1071no—
274C2×SD16—1071no—
275C2×SD16—2141no—
276C2×SD16—2141no—
277C2×D8—1071no—
278C2×D8—1071no—
279C2×D8—2141no—
280C2×D8—2141no—
281#131—1071no—
282#131—1071no—
283#131C2×D10711yescharacteristic, complemented
284#131C2×D10711yescharacteristic, complemented
285#127—1071no—
286#124—1071no—
287#108—1071no—
288#108C2×D10711yescharacteristic, complemented
289#104—1071no—
290#104—1071no—
291#104C2×D10711yescharacteristic, complemented
292#104C2×D10711yescharacteristic, complemented
293#101—1071no—
294#100—1071no—
295#100—1071no—
296#100—1071no—
297#97C2×D10711yescharacteristic, complemented
298#41—1071no—
299#41—1071no—
300#41—4281no—
301#41—2141no—
302#41—2141no—
303#41—2141no—
304#41—2141no—
305#41—2141no—
306#41—2141no—
307#41—2141no—
308#41—2141no—
309#41C22×D10711yescharacteristic
310#41C22×D10711yescharacteristic
311C2×Q8—1071no—
312C2×Q8—2141no—
313C2×Q8—2141no—
314C2×D4—1071no—
315C2×D4—1071no—
316C2×D4—1071no—
317C2×D4—4281no—
318C2×D4—4281no—
319C2×D4—4281no—
320C2×D4—4281no—
321C2×D4—2141no—
322C2×D4—2141no—
323C2×D4—2141no—
324C2×D4—2141no—
325C2×D4—2141no—
326C2×D4—2141no—
327Q16—1071no—
328Q16—2141no—
329Q16—2141no—
330Q16D42811yescharacteristic, complemented
331SD16—1071no—
332SD16—1071no—
333SD16—2141no—
334SD16—2141no—
335SD16—2141no—
336SD16—2141no—
337SD16—2141no—
338SD16—2141no—
339SD16—2141no—
340SD16—2141no—
341SD16—2141no—
342SD16—2141no—
343SD16—2141no—
344SD16—2141no—
345SD16D42811yescharacteristic, complemented
346SD16D42811yescharacteristic, complemented
347D8—1071no—
348D8—2141no—
349D8—2141no—
350D8—2141no—
351D8—2141no—
352D8—2141no—
353D8—2141no—
354D8—2141no—
355D8—2141no—
356D8—2141no—
357D8—2141no—
358D8D42811yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—1071no—
362M16—1071no—
363M16—2141no—
364M16—2141no—
365M16C22×D10711yescharacteristic
366M16C22×D10711yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—2141no—
370C2×C8—2141no—
371C2×C8C22×D10711yescharacteristic
372C2×C8C22×D10711yescharacteristic
373C22⋊C4—4281no—
374C42C22×D10711yescharacteristic
375C23—4281no—
376C23—4281no—
377C23—4281no—
378C23—2141no—
379C23—2141no—
380C23—2141no—
381Q8—1071no—
382Q8—1071no—
383Q8—4281no—
384Q8—2141no—
385Q8—2141no—
386Q8C2×D42811yescharacteristic
387Q8C2×D42811yescharacteristic
388D4—21no—
389D4—21no—
390D4—8562no—
391D4—1071no—
392D4—1071no—
393D4—4281no—
394D4—4281no—
395D4—4281no—
396D4—4281no—
397D4—4281no—
398D4—4281no—
399D4—4281no—
400D4—2141no—
401D4—2141no—
402D4—2141no—
403D4—2141no—
404D4—2141no—
405D4—2141no—
406D4—2141no—
407D4—2141no—
408D4C2×D42811yescharacteristic
409D4C2×D42811yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—4281no—
414C2×C4—4281no—
415C2×C4—4281no—
416C2×C4—4281no—
417C2×C4—4281no—
418C2×C4—2141no—
419C2×C4—2141no—
420C2×C4—2141no—
421C2×C4C23×D10711yescharacteristic
422C8—21no—
423C8—21no—
424C8—2141no—
425C8—2141no—
426C8D4×D10711yescharacteristic
427C8D4×D10711yescharacteristic
428C8C2×D42811yescharacteristic
429C8C2×D42811yescharacteristic
430C22—21no—
431C22—21no—
432C22—8561no—
433C22—8561no—
434C22—8561no—
435C22—4281no—
436C22—4281no—
437C22—4281no—
438C22—17121no—
439C22—17121no—
440C22—17121no—
441C22—2141no—
442C22—2141no—
443C22#1681215511yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—4281no—
448C4—2141no—
449C4—2141no—
450C4C2×D4×D10711yescharacteristic
451C4C22×D42811yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—8561no—
456C2—8561no—
457C2—8561no—
458C2—4281no—
459C2—4281no—
460C2D4×D42811yescharacteristic
subgroupG/H
centerC2D4×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#444×C107C2
socleC2×C107D42
hypercenter#444D107
hypocenterC107#3288
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #20293439C2semidirect
2. #20293434C2semidirect
3. #20293432C2semidirect
4. #20293429C2semidirect
5. #20293016C2semidirect
6. #20292898C2semidirect
7. #20292897C2semidirect
8. #20292891C2semidirect
9. #20292890C2semidirect
10. #20292806C2semidirect
11. #20292801C2semidirect
12. #20292796C2semidirect
13. #20292796C2semidirect
14. #20292750C2semidirect
15. #444×C107C2semidirect
16. #16812156C22semidirect
18. #16812155C22semidirect
19. #16812155C22semidirect
21. #16812142C22semidirect
23. #16812139C22semidirect
28. #16812137C22semidirect
29. #16812137C22semidirect
33. #16812134C22semidirect
34. #16812134C22semidirect
42. C2×#13133807C22semidirect
43. C2×#13133807C22semidirect
46. C2×D856C22semidirect
47. C2×D856C22semidirect
49. #16812115C22semidirect
50. #16812115C22semidirect
51. #16812106C22semidirect
52. #16812097C22semidirect
53. #16812045C22semidirect
54. #16812037C22semidirect
55. #16812037C22semidirect
56. #16812027C22semidirect
57. #16812025C22semidirect
58. #16812025C22semidirect
59. #16812025C22semidirect
61. #131×C107C22semidirect
62. #131×C107C22semidirect
63. #108×C107C22semidirect
64. #104×C107C22semidirect
65. #104×C107C22semidirect
66. #97×C107C22semidirect
95. Dic428D4semidirect
105. #13133807D4semidirect
106. #13133807D4semidirect
118. D856D4semidirect
131. Q16×C107D4semidirect
132. SD16×C107D4semidirect
133. SD16×C107D4semidirect
134. D8×C107D4semidirect
231. C107#3288semidirect
246. #444D107semidirect
283. #131C2×D107semidirect
284. #131C2×D107semidirect
288. #108C2×D107semidirect
291. #104C2×D107semidirect
292. #104C2×D107semidirect
297. #97C2×D107semidirect
330. Q16D428semidirect
345. SD16D428semidirect
346. SD16D428semidirect
358. D8D428semidirect

Maximal subgroups

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

subgroupindex
#202934392
#202934342
#202934322
#202934292
#202930162
#202928982
#202928972
#202928912
#202928902
#202928062
#202928012
#202927962
#202927502
#444×C1072
#3288107