Finite Group Databasebeta

← #28347946

Subgroups — #28347946

Order 13696.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812137C211yesmaximal, characteristic, complemented
2C22×D856C211yesmaximal, characteristic, complemented
3#20293273C211yesmaximal, characteristic, complemented
4D4×D428C211yesmaximal, characteristic, complemented
5#20293016C211yesmaximal, characteristic, complemented
6#20292880C211yesmaximal, characteristic, complemented
7#20292856C211yesmaximal, characteristic, complemented
8#20292843C211yesmaximal, characteristic, complemented
9#20292823C211yesmaximal, characteristic, complemented
10#20292780C211yesmaximal, characteristic, complemented
11#20292774C211yesmaximal, characteristic, complemented
12#20292763C211yesmaximal, characteristic, complemented
13#20292763C211yesmaximal, characteristic, complemented
14#20292740C211yesmaximal, characteristic, complemented
15#435×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic, complemented
17C22×D428C2211yescharacteristic
18C22×D428C2211yescharacteristic, complemented
19C22×D428C2211yescharacteristic, complemented
20C2×D4×D107—21no—
21#16812137—24no—
22C2×#13133807C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic, complemented
24C2×#13133807C2211yescharacteristic, complemented
25C2×D856—24no—
26C2×D856C2211yescharacteristic
27C2×D856C2211yescharacteristic
28C2×D856C2211yescharacteristic, complemented
29C2×D856C2211yescharacteristic, complemented
30C2×D856C2211yescharacteristic, complemented
31C2×D856C2211yescharacteristic, complemented
32#16812106C2211yescharacteristic, complemented
33#16812097C2211yescharacteristic, complemented
34#16812094—21no—
35#16812088C2211yescharacteristic, complemented
36#16812084—21no—
37#16812083C2211yescharacteristic
38#16812083C2211yescharacteristic, complemented
39#16812083C2211yescharacteristic, complemented
40#16812082—21no—
41#16812079—21no—
42#16812079—21no—
43#16812077—21no—
44C4×D428C2211yescharacteristic, complemented
45C4×D428C2211yescharacteristic, complemented
46C22⋊C4×D107—21no—
47#16812041C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic, complemented
51#16812031C2211yescharacteristic, complemented
52#16812031C2211yescharacteristic, complemented
53#16812031C2211yescharacteristic, complemented
54C2×M16×C107C2211yescharacteristic, complemented
55C22×C8×C107C2211yescharacteristic, complemented
56C4×D4×C107C2211yescharacteristic
57C4⋊C8×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic, complemented
59C22⋊C8×C107C2211yescharacteristic, complemented
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C23×D107—21no—
64#13133818—22no—
65#13133818—41no—
66C2×Dic214C2311yescharacteristic
67C2×D428—22no—
68C2×D428—22no—
69C2×D428—22no—
70C2×D428—21no—
71C2×D428—21no—
72C2×D428—21no—
73C2×D428—21no—
74C2×D428—21no—
75C2×D428—41no—
76C2×D428—41no—
77C2×D428—41no—
78C2×D428C2311yescharacteristic
79C2×D428C2311yescharacteristic
80C2×D428C2311yescharacteristic
81C2×D428C2311yescharacteristic
82C2×D428C2311yescharacteristic
83C2×C22⋊D107—21no—
84C2×C22⋊D107—21no—
85C2×C22⋊D107—21no—
86C2×C22⋊D107—21no—
87D4×D107—44no—
88C2×C4×D107—21no—
89C2×C4×D107—21no—
90C2×C4×D107—21no—
91#13133807—22no—
92#13133807—22no—
93#13133807—41no—
94D856—22no—
95D856—22no—
96D856—22no—
97D856—22no—
98D856—41no—
99D856D414yescomplemented
100C4⋊Dic107C2311yescharacteristic
101C4⋊Dic107C2311yescharacteristic
102Dic107⋊C4—21no—
103C22⋊Dic107—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4—21no—
109(C2×D107)⋊C4—21no—
110(C2×D107)⋊C4—21no—
111C2×D4×C107D411yescharacteristic
112C22×C4×C107C2311yescharacteristic
113C22×C4×C107C2311yescharacteristic
114M16×C107—22no—
115C2×C8×C107—22no—
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C4⋊C4×C107D411yescharacteristic
121C22⋊C4×C107D411yescharacteristic
122C22⋊C4×C107D411yescharacteristic
123C42×C107C2311yescharacteristic
124C23×C107C2×D411yescharacteristic
125C23×C107C2×D411yescharacteristic
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—21no—
130C22×D107—21no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—42no—
135C22×D107—42no—
136C22×D107—42no—
137C22×D107—81no—
138C22×D107—81no—
139C22×D107—81no—
140D4×C107—22no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107C2×D411yescharacteristic
146C2×C4×C107C2×D411yescharacteristic
147C2×C4×C107C2×D411yescharacteristic
148C2×C4×C107C2×D411yescharacteristic
149C2×C4×C107C2411yescharacteristic
150C22⋊D107—42no—
151C22⋊D107—42no—
152C22⋊D107—42no—
153C22⋊D107—81no—
154C2×Dic107—21no—
155C2×Dic107—21no—
156C2×Dic107—21no—
157D428—21no—
158D428—21no—
159D428—21no—
160D428—21no—
161D428—21no—
162D428—21no—
163D428—21no—
164D428—21no—
165D428—42no—
166D428—42no—
167D428—42no—
168D428—42no—
169D428—81no—
170D428C2×D412yes—
171D428C2×D412yescomplemented
172C4×D107—42no—
173C4×D107—42no—
174C4×D107—81no—
175Dic214—21no—
176Dic214—21no—
177C8×C107—21no—
178C8×C107—21no—
179C8×C107—21no—
180C8×C107C2×D412yescomplemented
181C22×C107—21no—
182C22×C107—21no—
183C22×C107—21no—
184C22×C107—21no—
185C22×C107—21no—
186C22×C107AGL(1,ℤ/8)12yescomplemented
187C22×C107C22×D411yescharacteristic
188C2×D107—22no—
189C2×D107—22no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—41no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107—41no—
203C2×D107—84no—
204C2×D107—84no—
205C2×D107—84no—
206C4×C107—21no—
207C4×C107—21no—
208C4×C107—21no—
209C4×C107—21no—
210C4×C107C22×D411yescharacteristic
211C4×C107H(5,𝔽2)11yescharacteristic
212Dic107—41no—
213Dic107—41no—
214Dic107—41no—
215C2×C107—22no—
216C2×C107—41no—
217C2×C107D4211yescharacteristic
218C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C107#57611yescharacteristic
220D107—42no—
221D107—42no—
222D107—81no—
223D107—81no—
224D107—81no—
225#3264—1071nomaximal, Sylow
226C107#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1071no—
228C22×D8—1071no—
229#546—1071no—
230D42—1071no—
231#496—1071no—
232#469—1071no—
233#465—1071no—
234#463—1071no—
235#459—1071no—
236#450—1071no—
237#449—1071no—
238#447—1071no—
239#447—1071no—
240#442—1071no—
241#435D10711yescharacteristic, complemented
242C2×#41—1071no—
243C22×D4—1071no—
244C22×D4—1071no—
245C22×D4—1071no—
246C22×D4—2141no—
247AGL(1,ℤ/8)—2144no—
248C2×SD16—1071no—
249C2×SD16—1071no—
250C2×SD16—1071no—
251C2×D8—1071no—
252C2×D8—1071no—
253C2×D8—1071no—
254C2×D8—1071no—
255C2×D8—1071no—
256C2×D8—1071no—
257C2×D8—2144no—
258C2×M16C2×D10711yescharacteristic, complemented
259C22×C8C2×D10711yescharacteristic, complemented
260#127—1071no—
261#124—1071no—
262#123—2141no—
263C22⋊Q8—1071no—
264#121—1071no—
265#121—1071no—
266#121—1071no—
267#121—2141no—
268#121—2141no—
269#120—2141no—
270#120—2141no—
271#120—2141no—
272C4×D4—1071no—
273C4×D4—1071no—
274C4×D4C2×D10711yescharacteristic
275C2×C22⋊C4—2141no—
276#106—1071no—
277C4⋊C8C2×D10711yescharacteristic, complemented
278#103—1071no—
279#102—1071no—
280#102—1071no—
281#102—1071no—
282#102—1071no—
283#102—1071no—
284C22⋊C8C2×D10711yescharacteristic, complemented
285C22⋊C8C2×D10711yescharacteristic, complemented
286#97C2×D10711yescharacteristic, complemented
287C24—2141no—
288C24—2141no—
289C24—2141no—
290#41—4281no—
291#41—2142no—
292C2×Q8—1071no—
293C2×D4—1071no—
294C2×D4—1071no—
295C2×D4—1071no—
296C2×D4—1071no—
297C2×D4—1071no—
298C2×D4—4281no—
299C2×D4—4281no—
300C2×D4—4281no—
301C2×D4—4284no—
302C2×D4—2142no—
303C2×D4—2142no—
304C2×D4—2142no—
305C2×D4—2141no—
306C2×D4—2141no—
307C2×D4—2141no—
308C2×D4—2141no—
309C2×D4—2141no—
310C2×D4—2141no—
311C2×D4—2141no—
312C2×D4—2141no—
313C2×D4—2141no—
314C2×D4D42811yescharacteristic
315C22×C4—2141no—
316C22×C4—2141no—
317C22×C4—2141no—
318C22×C4C22×D10711yescharacteristic
319C22×C4C22×D10711yescharacteristic
320SD16—4281no—
321SD16—2142no—
322SD16—2142no—
323D8—1074no—
324D8—4281no—
325D8—2142no—
326D8—2142no—
327D8—2142no—
328D8—2142no—
329M16—22no—
330C2×C8—22no—
331C2×C8C22×D10711yescharacteristic
332C2×C8C22×D10711yescharacteristic
333C2×C8C22×D10711yescharacteristic
334C2×C8C22×D10711yescharacteristic
335C4⋊C4—1071no—
336C4⋊C4—1071no—
337C4⋊C4—2141no—
338C4⋊C4D42811yescharacteristic
339C22⋊C4—2141no—
340C22⋊C4—2141no—
341C22⋊C4—2141no—
342C22⋊C4—2141no—
343C22⋊C4—2141no—
344C22⋊C4—2141no—
345C22⋊C4—2141no—
346C22⋊C4—2141no—
347C22⋊C4D42811yescharacteristic
348C22⋊C4D42811yescharacteristic
349C42C22×D10711yescharacteristic
350C23—8561no—
351C23—8561no—
352C23—8561no—
353C23—4282no—
354C23—4282no—
355C23—4282no—
356C23—4282no—
357C23—4282no—
358C23—4282no—
359C23—2141no—
360C23—2141no—
361C23—2141no—
362C23—2141no—
363C23—2141no—
364C23C2×D42811yescharacteristic
365C23C2×D42811yescharacteristic
366Q8—2141no—
367Q8—2141no—
368D4—22no—
369D4—8561no—
370D4—8561no—
371D4—1072no—
372D4—1072no—
373D4—4282no—
374D4—4282no—
375D4—4282no—
376D4—4282no—
377D4—4282no—
378D4—4282no—
379D4—4282no—
380D4—2141no—
381D4—2141no—
382D4—2141no—
383D4—2141no—
384D4—2141no—
385D4—2141no—
386D4—2141no—
387D4—2141no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—8561no—
393C2×C4—4282no—
394C2×C4—4282no—
395C2×C4—2141no—
396C2×C4—2141no—
397C2×C4—2141no—
398C2×C4C2×D42811yescharacteristic
399C2×C4C2×D42811yescharacteristic
400C2×C4C2×D42811yescharacteristic
401C2×C4C2×D42811yescharacteristic
402C2×C4C23×D10711yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D10712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—8564no—
413C22—8564no—
414C22—8564no—
415C22—4281no—
416C22—4281no—
417C22—4281no—
418C22—4281no—
419C22—4281no—
420C22—4281no—
421C22—4281no—
422C22—4281no—
423C22—4281no—
424C22—4281no—
425C22—4281no—
426C22—4281no—
427C22—4281no—
428C22—2142no—
429C22—2142no—
430C22#1681213712yescomplemented
431C22C22×D42811yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—4281no—
437C4—4281no—
438C4—4281no—
439C4C2×D4×D10711yescharacteristic
440C4#1681215511yescharacteristic
441C2—22no—
442C2—41no—
443C2—8561no—
444C2—8561no—
445C2—8561no—
446C2—4282no—
447C2—4282no—
448C2D4×D42811yescharacteristic
449C2C2×#1681213711yescharacteristic
450C2#2029343211yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#435×C107C2
socleC22×C107C22×D4
hypercenter#435D107
hypocenterC107#3264
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C22×D856C2semidirect
3. #20293273C2semidirect
4. D4×D428C2semidirect
5. #20293016C2semidirect
6. #20292880C2semidirect
7. #20292856C2semidirect
8. #20292843C2semidirect
9. #20292823C2semidirect
10. #20292780C2semidirect
11. #20292774C2semidirect
12. #20292763C2semidirect
13. #20292763C2semidirect
14. #20292740C2semidirect
15. #435×C107C2semidirect
16. C2×#13133818C22semidirect
18. C22×D428C22semidirect
19. C22×D428C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133807C22semidirect
24. C2×#13133807C22semidirect
28. C2×D856C22semidirect
29. C2×D856C22semidirect
30. C2×D856C22semidirect
31. C2×D856C22semidirect
32. #16812106C22semidirect
33. #16812097C22semidirect
35. #16812088C22semidirect
38. #16812083C22semidirect
39. #16812083C22semidirect
44. C4×D428C22semidirect
45. C4×D428C22semidirect
47. #16812041C22semidirect
48. #16812034C22semidirect
49. #16812031C22semidirect
50. #16812031C22semidirect
51. #16812031C22semidirect
52. #16812031C22semidirect
53. #16812031C22semidirect
54. C2×M16×C107C22semidirect
55. C22×C8×C107C22semidirect
57. C4⋊C8×C107C22semidirect
58. C22⋊C8×C107C22semidirect
59. C22⋊C8×C107C22semidirect
60. #97×C107C22semidirect
99. D856D4semidirect
171. D428C2×D4semidirect
180. C8×C107C2×D4semidirect
186. C22×C107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
241. #435D107semidirect
258. C2×M16C2×D107semidirect
259. C22×C8C2×D107semidirect
277. C4⋊C8C2×D107semidirect
284. C22⋊C8C2×D107semidirect
285. C22⋊C8C2×D107semidirect
286. #97C2×D107semidirect
406. C8D4×D107semidirect
430. C22#16812137semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C22×D8562
#202932732
D4×D4282
#202930162
#202928802
#202928562
#202928432
#202928232
#202927802
#202927742
#202927632
#202927402
#435×C1072
#3264107