Finite Group Databasebeta

← #46822385

Subgroups — #46822385

Order 34432.

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×#22575481C211yesmaximal, characteristic, complemented
2C22×#16864391C211yesmaximal, characteristic, complemented
3#31771264C211yesmaximal, characteristic, complemented
4D4×D1076C211yesmaximal, characteristic, complemented
5#31770994C211yesmaximal, characteristic, complemented
6#31770851C211yesmaximal, characteristic, complemented
7#31770841C211yesmaximal, characteristic, complemented
8#31770821C211yesmaximal, characteristic, complemented
9#31770806C211yesmaximal, characteristic, complemented
10#31770758C211yesmaximal, characteristic, complemented
11#31770752C211yesmaximal, characteristic, complemented
12#31770746C211yesmaximal, characteristic, complemented
13#31770741C211yesmaximal, characteristic, complemented
14#31770707C211yesmaximal, characteristic, complemented
15#435×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic, complemented
17C22×Dic538C2211yescharacteristic, complemented
18C22×D1076C2211yescharacteristic
19C22×D1076C2211yescharacteristic
20C2×D4×D269—21no—
21#22575481—24no—
22C2×#16864391—24no—
23C2×#16864391C2211yescharacteristic
24C2×#16864391C2211yescharacteristic
25C2×#16864391C2211yescharacteristic, complemented
26C2×#16864391C2211yescharacteristic, complemented
27C2×#16864391C2211yescharacteristic, complemented
28C2×#16864391C2211yescharacteristic, complemented
29C2×D2152C2211yescharacteristic
30C2×D2152C2211yescharacteristic, complemented
31C2×D2152C2211yescharacteristic, complemented
32#22575450C2211yescharacteristic
33#22575441C2211yescharacteristic, complemented
34#22575439—21no—
35#22575432C2211yescharacteristic, complemented
36#22575431—21no—
37#22575428—21no—
38#22575427C2211yescharacteristic
39#22575427C2211yescharacteristic
40#22575427C2211yescharacteristic, complemented
41#22575425—21no—
42#22575423—21no—
43#22575423—21no—
44C4×Dic538C2211yescharacteristic, complemented
45C4×D1076C2211yescharacteristic
46#22575413—21no—
47#22575387C2211yescharacteristic, complemented
48#22575378C2211yescharacteristic, complemented
49#22575378C2211yescharacteristic, complemented
50#22575375C2211yescharacteristic
51#22575375C2211yescharacteristic, complemented
52#22575375C2211yescharacteristic, complemented
53#22575375C2211yescharacteristic, complemented
54C2×M16×C269C2211yescharacteristic, complemented
55C22×C8×C269C2211yescharacteristic, complemented
56C4×D4×C269C2211yescharacteristic
57C4⋊C8×C269C2211yescharacteristic, complemented
58C22⋊C8×C269C2211yescharacteristic, complemented
59C22⋊C8×C269C2211yescharacteristic, complemented
60#97×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62C23×D269—21no—
63#16864402—22no—
64#16864402—41no—
65C2×Dic538—22no—
66C2×Dic538—21no—
67C2×Dic538—41no—
68C2×Dic538C2311yescharacteristic
69C2×Dic538C2311yescharacteristic
70C2×D1076—22no—
71C2×D1076—22no—
72C2×D1076—21no—
73C2×D1076—21no—
74C2×D1076—21no—
75C2×D1076—21no—
76C2×D1076—41no—
77C2×D1076—41no—
78C2×D1076C2311yescharacteristic
79C2×D1076C2311yescharacteristic
80C2×D1076C2311yescharacteristic
81C2×D1076C2311yescharacteristic
82C2×C22⋊D269—21no—
83C2×C22⋊D269—21no—
84C2×C22⋊D269—21no—
85C2×C22⋊D269—21no—
86D4×D269—44no—
87C22×Dic269—21no—
88C2×C4×D269—21no—
89C2×C4×D269—21no—
90#16864391—22no—
91#16864391—22no—
92#16864391—22no—
93#16864391—22no—
94#16864391—41no—
95#16864391D414yescomplemented
96D2152—22no—
97D2152—22no—
98D2152—41no—
99C4⋊Dic269C2311yescharacteristic
100C4⋊Dic269C2311yescharacteristic
101#16864381—21no—
102#16864381—21no—
103C22⋊Dic269—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379—21no—
108#16864379—21no—
109#16864379—21no—
110C4×Dic269—21no—
111C2×D4×C269D411yescharacteristic
112C22×C4×C269C2311yescharacteristic
113C22×C4×C269C2311yescharacteristic
114M16×C269—22no—
115C2×C8×C269—22no—
116C2×C8×C269C2311yescharacteristic
117C2×C8×C269C2311yescharacteristic
118C2×C8×C269C2311yescharacteristic
119C2×C8×C269C2311yescharacteristic
120C4⋊C4×C269D411yescharacteristic
121C22⋊C4×C269D411yescharacteristic
122C22⋊C4×C269D411yescharacteristic
123C42×C269C2311yescharacteristic
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—21no—
127C22×D269—21no—
128C22×D269—42no—
129C22×D269—42no—
130C22×D269—42no—
131C22×D269—42no—
132C22×D269—81no—
133C22×D269—81no—
134Dic538—21no—
135Dic538—21no—
136Dic538—21no—
137Dic538—42no—
138Dic538C2×D412yescomplemented
139D1076—21no—
140D1076—21no—
141D1076—21no—
142D1076—21no—
143D1076—21no—
144D1076—21no—
145D1076—21no—
146D1076—42no—
147D1076—42no—
148D1076—42no—
149D1076—81no—
150D1076C2×D412yes—
151C22⋊D269—42no—
152C22⋊D269—42no—
153C22⋊D269—42no—
154C22⋊D269—42no—
155C2×Dic269—21no—
156C2×Dic269—21no—
157C2×Dic269—21no—
158C2×Dic269—21no—
159C2×Dic269—42no—
160C4×D269—42no—
161C4×D269—42no—
162C23×C269C2×D411yescharacteristic
163C23×C269C2×D411yescharacteristic
164D4×C269—22no—
165C2×C4×C269—21no—
166C2×C4×C269—21no—
167C2×C4×C269—21no—
168C2×C4×C269—21no—
169C2×C4×C269C2×D411yescharacteristic
170C2×C4×C269C2×D411yescharacteristic
171C2×C4×C269C2×D411yescharacteristic
172C2×C4×C269C2×D411yescharacteristic
173C2×C4×C269C2411yescharacteristic
174C8×C269—21no—
175C8×C269—21no—
176C8×C269—21no—
177C8×C269C2×D412yescomplemented
178C22×C269—21no—
179C22×C269—21no—
180C22×C269—21no—
181C22×C269—21no—
182C22×C269—21no—
183C22×C269AGL(1,ℤ/8)12yescomplemented
184C22×C269C22×D411yescharacteristic
185C2×D269—22no—
186C2×D269—41no—
187C2×D269—41no—
188C2×D269—41no—
189C2×D269—41no—
190C2×D269—41no—
191C2×D269—41no—
192C2×D269—41no—
193C2×D269—41no—
194C2×D269—41no—
195C2×D269—41no—
196C2×D269—41no—
197C2×D269—84no—
198C2×D269—84no—
199C4×C269—21no—
200C4×C269—21no—
201C4×C269—21no—
202C4×C269—21no—
203C4×C269C22×D411yescharacteristic
204C4×C269H(5,𝔽2)11yescharacteristic
205Dic269—22no—
206Dic269—41no—
207Dic269—41no—
208Dic269—41no—
209C2×C269—22no—
210C2×C269—41no—
211C2×C269D4211yescharacteristic
212C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C269#57711yescharacteristic
214D269—42no—
215D269—81no—
216D269—81no—
217D269—81no—
218C269#326511yescharacteristic, complemented, Sylow
219#3265—2691nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2691no—
221C22×SD16—2691no—
222#548—2691no—
223D42—2691no—
224#496—2691no—
225#468—2691no—
226#466—2691no—
227#463—2691no—
228#460—2691no—
229#450—2691no—
230#449—2691no—
231#448—2691no—
232#447—2691no—
233#440—2691no—
234#435D26911yescharacteristic, complemented
235C2×#41—2691no—
236C22×Q8—2691no—
237C22×D4—2691no—
238C22×D4—2691no—
239C22×D4—5381no—
240AGL(1,ℤ/8)—5384no—
241C2×SD16—2691no—
242C2×SD16—2691no—
243C2×SD16—2691no—
244C2×SD16—2691no—
245C2×SD16—2691no—
246C2×SD16—2691no—
247C2×SD16—5384no—
248C2×D8—2691no—
249C2×D8—2691no—
250C2×D8—2691no—
251C2×M16C2×D26911yescharacteristic, complemented
252C22×C8C2×D26911yescharacteristic, complemented
253#127—2691no—
254#124—2691no—
255#124—5381no—
256C22⋊Q8—2691no—
257C22⋊Q8—5381no—
258#121—2691no—
259#121—2691no—
260#121—2691no—
261#121—5381no—
262#121—5381no—
263#120—5381no—
264#120—5381no—
265C4×Q8—2691no—
266C4×D4—2691no—
267C4×D4—5381no—
268C4×D4C2×D26911yescharacteristic
269#107—2691no—
270C4⋊C8C2×D26911yescharacteristic, complemented
271#103—2691no—
272#103—2691no—
273#102—2691no—
274#102—2691no—
275#102—2691no—
276#102—2691no—
277C22⋊C8C2×D26911yescharacteristic, complemented
278C22⋊C8C2×D26911yescharacteristic, complemented
279#97C2×D26911yescharacteristic, complemented
280C24—5381no—
281C24—5381no—
282#41—5382no—
283#41—10761no—
284C2×Q8—2691no—
285C2×Q8—2691no—
286C2×Q8—5382no—
287C2×Q8—5381no—
288C2×Q8—10761no—
289C2×D4—2691no—
290C2×D4—2691no—
291C2×D4—2691no—
292C2×D4—2691no—
293C2×D4—5382no—
294C2×D4—5382no—
295C2×D4—5381no—
296C2×D4—5381no—
297C2×D4—5381no—
298C2×D4—5381no—
299C2×D4—5381no—
300C2×D4—5381no—
301C2×D4—5381no—
302C2×D4—5381no—
303C2×D4—10761no—
304C2×D4—10761no—
305C2×D4—10764no—
306C2×D4D107611yescharacteristic
307C22×C4—5381no—
308C22×C4—5381no—
309C22×C4—5381no—
310C22×C4C22×D26911yescharacteristic
311C22×C4C22×D26911yescharacteristic
312SD16—2694no—
313SD16—5382no—
314SD16—5382no—
315SD16—5382no—
316SD16—5382no—
317SD16—10761no—
318D8—5382no—
319D8—5382no—
320D8—10761no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D26911yescharacteristic
324C2×C8C22×D26911yescharacteristic
325C2×C8C22×D26911yescharacteristic
326C2×C8C22×D26911yescharacteristic
327C4⋊C4—2691no—
328C4⋊C4—2691no—
329C4⋊C4—5381no—
330C4⋊C4—5381no—
331C4⋊C4D107611yescharacteristic
332C22⋊C4—5381no—
333C22⋊C4—5381no—
334C22⋊C4—5381no—
335C22⋊C4—5381no—
336C22⋊C4—5381no—
337C22⋊C4—5381no—
338C22⋊C4—5381no—
339C22⋊C4D107611yescharacteristic
340C22⋊C4D107611yescharacteristic
341C42—5381no—
342C42C22×D26911yescharacteristic
343C23—5381no—
344C23—5381no—
345C23—5381no—
346C23—5381no—
347C23—10762no—
348C23—10762no—
349C23—10762no—
350C23—10762no—
351C23—21521no—
352C23—21521no—
353C23C2×D107611yescharacteristic
354C23C2×D107611yescharacteristic
355Q8—2692no—
356Q8—5381no—
357Q8—5381no—
358Q8—5381no—
359Q8—10762no—
360D4—22no—
361D4—2692no—
362D4—5381no—
363D4—5381no—
364D4—5381no—
365D4—5381no—
366D4—5381no—
367D4—5381no—
368D4—5381no—
369D4—10762no—
370D4—10762no—
371D4—10762no—
372D4—10762no—
373D4—10762no—
374D4—10762no—
375D4—10762no—
376D4—21521no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—5381no—
382C2×C4—5381no—
383C2×C4—5381no—
384C2×C4—5381no—
385C2×C4—10762no—
386C2×C4—10762no—
387C2×C4—10762no—
388C2×C4C2×D107611yescharacteristic
389C2×C4C2×D107611yescharacteristic
390C2×C4C2×D107611yescharacteristic
391C2×C4C2×D107611yescharacteristic
392C2×C4C23×D26911yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D26912yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—5382no—
403C22—10761no—
404C22—10761no—
405C22—10761no—
406C22—10761no—
407C22—10761no—
408C22—10761no—
409C22—10761no—
410C22—10761no—
411C22—10761no—
412C22—10761no—
413C22—10761no—
414C22—21524no—
415C22—21524no—
416C22#2257548112yescomplemented
417C22C22×D107611yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—5382no—
423C4—10761no—
424C4—10761no—
425C4—10761no—
426C4C2×D4×D26911yescharacteristic
427C4#2257549911yescharacteristic
428C2—22no—
429C2—41no—
430C2—10762no—
431C2—21521no—
432C2—21521no—
433C2—21521no—
434C2D4×D107611yescharacteristic
435C2C2×#2257548111yescharacteristic
436C2#3177141711yescharacteristic
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#435×C269C2
socleC22×C269C22×D4
hypercenter#435D269
hypocenterC269#3265
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22575481C2semidirect
2. C22×#16864391C2semidirect
3. #31771264C2semidirect
4. D4×D1076C2semidirect
5. #31770994C2semidirect
6. #31770851C2semidirect
7. #31770841C2semidirect
8. #31770821C2semidirect
9. #31770806C2semidirect
10. #31770758C2semidirect
11. #31770752C2semidirect
12. #31770746C2semidirect
13. #31770741C2semidirect
14. #31770707C2semidirect
15. #435×C269C2semidirect
16. C2×#16864402C22semidirect
17. C22×Dic538C22semidirect
25. C2×#16864391C22semidirect
26. C2×#16864391C22semidirect
27. C2×#16864391C22semidirect
28. C2×#16864391C22semidirect
30. C2×D2152C22semidirect
31. C2×D2152C22semidirect
33. #22575441C22semidirect
35. #22575432C22semidirect
40. #22575427C22semidirect
44. C4×Dic538C22semidirect
47. #22575387C22semidirect
48. #22575378C22semidirect
49. #22575378C22semidirect
51. #22575375C22semidirect
52. #22575375C22semidirect
53. #22575375C22semidirect
54. C2×M16×C269C22semidirect
55. C22×C8×C269C22semidirect
57. C4⋊C8×C269C22semidirect
58. C22⋊C8×C269C22semidirect
59. C22⋊C8×C269C22semidirect
60. #97×C269C22semidirect
95. #16864391D4semidirect
138. Dic538C2×D4semidirect
177. C8×C269C2×D4semidirect
183. C22×C269AGL(1,ℤ/8)semidirect
218. C269#3265semidirect
234. #435D269semidirect
251. C2×M16C2×D269semidirect
252. C22×C8C2×D269semidirect
270. C4⋊C8C2×D269semidirect
277. C22⋊C8C2×D269semidirect
278. C22⋊C8C2×D269semidirect
279. #97C2×D269semidirect
396. C8D4×D269semidirect
416. C22#22575481semidirect

Maximal subgroups

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

subgroupindex
C2×#225754812
C22×#168643912
#317712642
D4×D10762
#317709942
#317708512
#317708412
#317708212
#317708062
#317707582
#317707522
#317707462
#317707412
#317707072
#435×C2692
#3265269