Finite Group Databasebeta

← #28347961

Subgroups — #28347961

Order 13696.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812137C211yesmaximal, characteristic, complemented
2C22×#13133807C211yesmaximal, characteristic, complemented
3#20293286C211yesmaximal, characteristic, complemented
4D4×D428C211yesmaximal, characteristic, complemented
5#20293016C211yesmaximal, characteristic, complemented
6#20292873C211yesmaximal, characteristic, complemented
7#20292863C211yesmaximal, characteristic, complemented
8#20292843C211yesmaximal, characteristic, complemented
9#20292828C211yesmaximal, characteristic, complemented
10#20292780C211yesmaximal, characteristic, complemented
11#20292774C211yesmaximal, characteristic, complemented
12#20292768C211yesmaximal, characteristic, complemented
13#20292763C211yesmaximal, characteristic, complemented
14#20292729C211yesmaximal, characteristic, complemented
15#435×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic, complemented
17C22×Dic214C2211yescharacteristic, complemented
18C22×D428C2211yescharacteristic
19C22×D428C2211yescharacteristic
20C2×D4×D107—21no—
21#16812137—24no—
22C2×#13133807—24no—
23C2×#13133807C2211yescharacteristic
24C2×#13133807C2211yescharacteristic
25C2×#13133807C2211yescharacteristic, complemented
26C2×#13133807C2211yescharacteristic, complemented
27C2×#13133807C2211yescharacteristic, complemented
28C2×#13133807C2211yescharacteristic, complemented
29C2×D856C2211yescharacteristic
30C2×D856C2211yescharacteristic, complemented
31C2×D856C2211yescharacteristic, complemented
32#16812106C2211yescharacteristic
33#16812097C2211yescharacteristic, complemented
34#16812095—21no—
35#16812088C2211yescharacteristic, complemented
36#16812087—21no—
37#16812084—21no—
38#16812083C2211yescharacteristic
39#16812083C2211yescharacteristic
40#16812083C2211yescharacteristic, complemented
41#16812081—21no—
42#16812079—21no—
43#16812079—21no—
44C4×Dic214C2211yescharacteristic, complemented
45C4×D428C2211yescharacteristic
46(C22⋊D107)⋊C4—21no—
47#16812043C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812034C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic
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—
63#13133818—22no—
64#13133818—41no—
65C2×Dic214—22no—
66C2×Dic214—21no—
67C2×Dic214—41no—
68C2×Dic214C2311yescharacteristic
69C2×Dic214C2311yescharacteristic
70C2×D428—22no—
71C2×D428—22no—
72C2×D428—21no—
73C2×D428—21no—
74C2×D428—21no—
75C2×D428—21no—
76C2×D428—41no—
77C2×D428—41no—
78C2×D428C2311yescharacteristic
79C2×D428C2311yescharacteristic
80C2×D428C2311yescharacteristic
81C2×D428C2311yescharacteristic
82C2×C22⋊D107—21no—
83C2×C22⋊D107—21no—
84C2×C22⋊D107—21no—
85C2×C22⋊D107—21no—
86D4×D107—44no—
87C22×Dic107—21no—
88C2×C4×D107—21no—
89C2×C4×D107—21no—
90#13133807—22no—
91#13133807—22no—
92#13133807—22no—
93#13133807—22no—
94#13133807—41no—
95#13133807D414yescomplemented
96D856—22no—
97D856—22no—
98D856—41no—
99C4⋊Dic107C2311yescharacteristic
100C4⋊Dic107C2311yescharacteristic
101Dic107⋊C4—21no—
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—
110C4×Dic107—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—42no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—81no—
135C22×D107—81no—
136D4×C107—22no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107C2×D411yescharacteristic
142C2×C4×C107C2×D411yescharacteristic
143C2×C4×C107C2×D411yescharacteristic
144C2×C4×C107C2×D411yescharacteristic
145C2×C4×C107C2411yescharacteristic
146C22⋊D107—42no—
147C22⋊D107—42no—
148C22⋊D107—42no—
149C22⋊D107—42no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107—21no—
153C2×Dic107—21no—
154C2×Dic107—42no—
155D428—21no—
156D428—21no—
157D428—21no—
158D428—21no—
159D428—21no—
160D428—21no—
161D428—21no—
162D428—42no—
163D428—42no—
164D428—42no—
165D428—81no—
166D428C2×D412yes—
167C4×D107—42no—
168C4×D107—42no—
169Dic214—21no—
170Dic214—21no—
171Dic214—21no—
172Dic214—42no—
173Dic214C2×D412yescomplemented
174C8×C107—21no—
175C8×C107—21no—
176C8×C107—21no—
177C8×C107C2×D412yescomplemented
178C22×C107—21no—
179C22×C107—21no—
180C22×C107—21no—
181C22×C107—21no—
182C22×C107—21no—
183C22×C107AGL(1,ℤ/8)12yescomplemented
184C22×C107C22×D411yescharacteristic
185C2×D107—22no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—41no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—84no—
198C2×D107—84no—
199C4×C107—21no—
200C4×C107—21no—
201C4×C107—21no—
202C4×C107—21no—
203C4×C107C22×D411yescharacteristic
204C4×C107H(5,𝔽2)11yescharacteristic
205Dic107—22no—
206Dic107—41no—
207Dic107—41no—
208Dic107—41no—
209C2×C107—22no—
210C2×C107—41no—
211C2×C107D4211yescharacteristic
212C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C107#57711yescharacteristic
214D107—42no—
215D107—81no—
216D107—81no—
217D107—81no—
218#3265—1071nomaximal, Sylow
219C107#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1071no—
221C22×SD16—1071no—
222#548—1071no—
223D42—1071no—
224#496—1071no—
225#468—1071no—
226#466—1071no—
227#463—1071no—
228#460—1071no—
229#450—1071no—
230#449—1071no—
231#448—1071no—
232#447—1071no—
233#440—1071no—
234#435D10711yescharacteristic, complemented
235C2×#41—1071no—
236C22×Q8—1071no—
237C22×D4—1071no—
238C22×D4—1071no—
239C22×D4—2141no—
240AGL(1,ℤ/8)—2144no—
241C2×SD16—1071no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—1071no—
245C2×SD16—1071no—
246C2×SD16—1071no—
247C2×SD16—2144no—
248C2×D8—1071no—
249C2×D8—1071no—
250C2×D8—1071no—
251C2×M16C2×D10711yescharacteristic, complemented
252C22×C8C2×D10711yescharacteristic, complemented
253#127—1071no—
254#124—1071no—
255#124—2141no—
256C22⋊Q8—1071no—
257C22⋊Q8—2141no—
258#121—1071no—
259#121—1071no—
260#121—1071no—
261#121—2141no—
262#121—2141no—
263#120—2141no—
264#120—2141no—
265C4×Q8—1071no—
266C4×D4—1071no—
267C4×D4—2141no—
268C4×D4C2×D10711yescharacteristic
269#107—1071no—
270C4⋊C8C2×D10711yescharacteristic, complemented
271#103—1071no—
272#103—1071no—
273#102—1071no—
274#102—1071no—
275#102—1071no—
276#102—1071no—
277C22⋊C8C2×D10711yescharacteristic, complemented
278C22⋊C8C2×D10711yescharacteristic, complemented
279#97C2×D10711yescharacteristic, complemented
280C24—2141no—
281C24—2141no—
282#41—4281no—
283#41—2142no—
284C2×Q8—1071no—
285C2×Q8—1071no—
286C2×Q8—4281no—
287C2×Q8—2142no—
288C2×Q8—2141no—
289C2×D4—1071no—
290C2×D4—1071no—
291C2×D4—1071no—
292C2×D4—1071no—
293C2×D4—4281no—
294C2×D4—4281no—
295C2×D4—4284no—
296C2×D4—2142no—
297C2×D4—2142no—
298C2×D4—2141no—
299C2×D4—2141no—
300C2×D4—2141no—
301C2×D4—2141no—
302C2×D4—2141no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4D42811yescharacteristic
307C22×C4—2141no—
308C22×C4—2141no—
309C22×C4—2141no—
310C22×C4C22×D10711yescharacteristic
311C22×C4C22×D10711yescharacteristic
312SD16—1074no—
313SD16—4281no—
314SD16—2142no—
315SD16—2142no—
316SD16—2142no—
317SD16—2142no—
318D8—4281no—
319D8—2142no—
320D8—2142no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D10711yescharacteristic
324C2×C8C22×D10711yescharacteristic
325C2×C8C22×D10711yescharacteristic
326C2×C8C22×D10711yescharacteristic
327C4⋊C4—1071no—
328C4⋊C4—1071no—
329C4⋊C4—2141no—
330C4⋊C4—2141no—
331C4⋊C4D42811yescharacteristic
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C22⋊C4—2141no—
339C22⋊C4D42811yescharacteristic
340C22⋊C4D42811yescharacteristic
341C42—2141no—
342C42C22×D10711yescharacteristic
343C23—8561no—
344C23—8561no—
345C23—4282no—
346C23—4282no—
347C23—4282no—
348C23—4282no—
349C23—2141no—
350C23—2141no—
351C23—2141no—
352C23—2141no—
353C23C2×D42811yescharacteristic
354C23C2×D42811yescharacteristic
355Q8—1072no—
356Q8—4282no—
357Q8—2141no—
358Q8—2141no—
359Q8—2141no—
360D4—22no—
361D4—8561no—
362D4—1072no—
363D4—4282no—
364D4—4282no—
365D4—4282no—
366D4—4282no—
367D4—4282no—
368D4—4282no—
369D4—4282no—
370D4—2141no—
371D4—2141no—
372D4—2141no—
373D4—2141no—
374D4—2141no—
375D4—2141no—
376D4—2141no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—4282no—
382C2×C4—4282no—
383C2×C4—4282no—
384C2×C4—2141no—
385C2×C4—2141no—
386C2×C4—2141no—
387C2×C4—2141no—
388C2×C4C2×D42811yescharacteristic
389C2×C4C2×D42811yescharacteristic
390C2×C4C2×D42811yescharacteristic
391C2×C4C2×D42811yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D10712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—8564no—
403C22—8564no—
404C22—4281no—
405C22—4281no—
406C22—4281no—
407C22—4281no—
408C22—4281no—
409C22—4281no—
410C22—4281no—
411C22—4281no—
412C22—4281no—
413C22—4281no—
414C22—4281no—
415C22—2142no—
416C22#1681213712yescomplemented
417C22C22×D42811yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—4281no—
423C4—4281no—
424C4—4281no—
425C4—2142no—
426C4C2×D4×D10711yescharacteristic
427C4#1681215511yescharacteristic
428C2—22no—
429C2—41no—
430C2—8561no—
431C2—8561no—
432C2—8561no—
433C2—4282no—
434C2D4×D42811yescharacteristic
435C2C2×#1681213711yescharacteristic
436C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#435×C107C2
socleC22×C107C22×D4
hypercenter#435D107
hypocenterC107#3265
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×#13133807C2semidirect
3. #20293286C2semidirect
4. D4×D428C2semidirect
5. #20293016C2semidirect
6. #20292873C2semidirect
7. #20292863C2semidirect
8. #20292843C2semidirect
9. #20292828C2semidirect
10. #20292780C2semidirect
11. #20292774C2semidirect
12. #20292768C2semidirect
13. #20292763C2semidirect
14. #20292729C2semidirect
15. #435×C107C2semidirect
16. C2×#13133818C22semidirect
17. C22×Dic214C22semidirect
25. C2×#13133807C22semidirect
26. C2×#13133807C22semidirect
27. C2×#13133807C22semidirect
28. C2×#13133807C22semidirect
30. C2×D856C22semidirect
31. C2×D856C22semidirect
33. #16812097C22semidirect
35. #16812088C22semidirect
40. #16812083C22semidirect
44. C4×Dic214C22semidirect
47. #16812043C22semidirect
48. #16812034C22semidirect
49. #16812034C22semidirect
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
95. #13133807D4semidirect
173. Dic214C2×D4semidirect
177. C8×C107C2×D4semidirect
183. C22×C107AGL(1,ℤ/8)semidirect
219. C107#3265semidirect
234. #435D107semidirect
251. C2×M16C2×D107semidirect
252. C22×C8C2×D107semidirect
270. C4⋊C8C2×D107semidirect
277. C22⋊C8C2×D107semidirect
278. C22⋊C8C2×D107semidirect
279. #97C2×D107semidirect
396. C8D4×D107semidirect
416. 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×#131338072
#202932862
D4×D4282
#202930162
#202928732
#202928632
#202928432
#202928282
#202927802
#202927742
#202927682
#202927632
#202927292
#435×C1072
#3265107