Finite Group Databasebeta

← #46822358

Subgroups — #46822358

Order 34432.

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×#22575477C211yesmaximal, characteristic, complemented
2C22×#16864387C211yesmaximal, characteristic, complemented
3#31771249C211yesmaximal, characteristic, complemented
4#31771239C211yesmaximal, characteristic, complemented
5#31770993C211yesmaximal, characteristic, complemented
6#31770857C211yesmaximal, characteristic, complemented
7#31770832C211yesmaximal, characteristic, complemented
8#31770819C211yesmaximal, characteristic, complemented
9#31770798C211yesmaximal, characteristic, complemented
10#31770758C211yesmaximal, characteristic, complemented
11#31770751C211yesmaximal, characteristic, complemented
12#31770737C211yesmaximal, characteristic, complemented
13#31770714C211yesmaximal, characteristic, complemented
14#31770673C211yesmaximal, characteristic, complemented
15#447×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic, complemented
17C22×D1076C2211yescharacteristic
18C2×D4×D269—21no—
19C2×D4×D269C2211yescharacteristic, complemented
20#22575477—24no—
21C2×#16864391C2211yescharacteristic, complemented
22C2×#16864391C2211yescharacteristic, complemented
23C2×#16864390C2211yescharacteristic, complemented
24C2×#16864387—24no—
25C2×#16864387C2211yescharacteristic
26C2×#16864387C2211yescharacteristic
27C2×#16864387C2211yescharacteristic, complemented
28C2×#16864387C2211yescharacteristic, complemented
29C2×#16864385C2211yescharacteristic, complemented
30C22×#14524130C2211yescharacteristic, complemented
31#22575449C2211yescharacteristic, complemented
32#22575440C2211yescharacteristic, complemented
33#22575434—21no—
34#22575432C2211yescharacteristic, complemented
35#22575428C2211yescharacteristic, complemented
36#22575426—21no—
37#22575425—21no—
38#22575424C2211yescharacteristic, complemented
39#22575423—21no—
40#22575422—21no—
41#22575421—21no—
42D4×Dic269C2211yescharacteristic, complemented
43#22575415C2211yescharacteristic, complemented
44C4×C22⋊D269C2211yescharacteristic
45C2×C22⋊Dic269—21no—
46#22575384C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575376C2211yescharacteristic, complemented
49#22575375C2211yescharacteristic, complemented
50#22575375C2211yescharacteristic, complemented
51#22575374C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53#22575363C2211yescharacteristic, complemented
54C22×D4×C269C2211yescharacteristic, complemented
55C2×D8×C269C2211yescharacteristic, complemented
56C2×D8×C269C2211yescharacteristic, complemented
57#121×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic, complemented
59#102×C269C2211yescharacteristic, complemented
60C22⋊C8×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62C23×D269—21no—
63#16864401—22no—
64#16864401—41no—
65C2×Dic538C2311yescharacteristic
66C2×D1076—22no—
67C2×D1076—21no—
68C2×D1076—21no—
69C2×D1076—41no—
70C2×D1076C2311yescharacteristic
71C2×D1076C2311yescharacteristic
72C2×C22⋊D269—21no—
73C2×C22⋊D269—21no—
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76C2×C22⋊D269—21no—
77C2×C22⋊D269D411yescharacteristic
78D4×D269—22no—
79D4×D269—41no—
80D4×D269—44no—
81C22×Dic269—21no—
82C22×Dic269—21no—
83C2×C4×D269—21no—
84C2×C4×D269C2311yescharacteristic
85#16864391—22no—
86#16864391—41no—
87#16864390—22no—
88#16864387—22no—
89#16864387—22no—
90#16864387—22no—
91#16864387D414yescomplemented
92#16864385—22no—
93C2×#14524130—22no—
94C2×#14524130C2311yescharacteristic
95C2×#14524130C2311yescharacteristic
96C4⋊Dic269C2311yescharacteristic
97#16864381—21no—
98#16864381D411yescharacteristic
99C22⋊Dic269—21no—
100C22⋊Dic269—21no—
101C22⋊Dic269—21no—
102C22⋊Dic269—21no—
103C22⋊Dic269—21no—
104C22⋊Dic269D411yescharacteristic
105#16864379—21no—
106#16864379—21no—
107#16864379D411yescharacteristic
108C4×Dic269C2311yescharacteristic
109C24×C269—21no—
110C2×D4×C269—22no—
111C2×D4×C269—21no—
112C2×D4×C269—21no—
113C2×D4×C269—41no—
114C2×D4×C269C2311yescharacteristic
115C2×D4×C269C2311yescharacteristic
116C2×D4×C269C2311yescharacteristic
117C22×C4×C269C2311yescharacteristic
118D8×C269—22no—
119D8×C269—41no—
120C2×C8×C269C2311yescharacteristic
121C2×C8×C269C2311yescharacteristic
122C4⋊C4×C269C2311yescharacteristic
123C22⋊C4×C269—21no—
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—42no—
127C22×D269—42no—
128C22×D269—42no—
129C22×D269—42no—
130C22×D269—81no—
131C22×D269—81no—
132C22×D269C2×D411yescharacteristic
133Dic538—21no—
134Dic538—21no—
135D1076—21no—
136D1076—21no—
137D1076—21no—
138D1076—42no—
139D1076—42no—
140D1076C2×D412yes—
141C22⋊D269—22no—
142C22⋊D269—42no—
143C22⋊D269—42no—
144C22⋊D269—42no—
145C22⋊D269—81no—
146C22⋊D269—81no—
147C2×Dic269—21no—
148C2×Dic269—21no—
149C2×Dic269—42no—
150C2×Dic269—81no—
151C2×Dic269C2×D411yescharacteristic
152C2×Dic269C2×D411yescharacteristic
153C2×Dic269C2×D411yescharacteristic
154C4×D269—21no—
155C4×D269—21no—
156C4×D269—42no—
157#14524130—21no—
158#14524130C2×D412yescomplemented
159C23×C269—21no—
160C23×C269—21no—
161C23×C269—21no—
162C23×C269—42no—
163C23×C269—42no—
164C23×C269—81no—
165C23×C269C2×D411yescharacteristic
166D4×C269—21no—
167D4×C269—21no—
168D4×C269—21no—
169D4×C269—21no—
170D4×C269—21no—
171D4×C269—42no—
172D4×C269—42no—
173D4×C269C2×D412yescomplemented
174C2×C4×C269—21no—
175C2×C4×C269—21no—
176C2×C4×C269—21no—
177C2×C4×C269C2×D411yescharacteristic
178C2×C4×C269C2411yescharacteristic
179C8×C269—21no—
180C8×C269—21no—
181C22×C269—21no—
182C22×C269—21no—
183C22×C269—22no—
184C22×C269—41no—
185C22×C269—41no—
186C22×C269—41no—
187C22×C269—41no—
188C22×C269—41no—
189C22×C269—41no—
190C22×C269—41no—
191C22×C269—41no—
192C22×C269—84no—
193C22×C269AGL(1,ℤ/8)12yescomplemented
194C22×C269C22×D411yescharacteristic
195C2×D269—21no—
196C2×D269—22no—
197C2×D269—21no—
198C2×D269—21no—
199C2×D269—41no—
200C2×D269—41no—
201C2×D269—41no—
202C2×D269—41no—
203C2×D269—41no—
204C2×D269—84no—
205C2×D269—84no—
206C4×C269—21no—
207C4×C269—41no—
208C4×C269C22×D411yescharacteristic
209C4×C269H(5,𝔽2)11yescharacteristic
210Dic269—21no—
211Dic269—21no—
212Dic269—21no—
213Dic269—41no—
214Dic269—41no—
215C2×C269—22no—
216C2×C269—42no—
217C2×C269—81no—
218C2×C269—81no—
219C2×C269D4211yescharacteristic
220C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C269#57611yescharacteristic
222D269—41no—
223D269—42no—
224D269—81no—
225C269#326411yescharacteristic, complemented, Sylow
226#3264—2691nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2691no—
228C22×D8—2691no—
229#546—2691no—
230D42—2691no—
231#496—2691no—
232#469—2691no—
233#465—2691no—
234#463—2691no—
235#459—2691no—
236#450—2691no—
237#449—2691no—
238#447—2691no—
239#447D26911yescharacteristic, complemented
240#442—2691no—
241#435—2691no—
242C2×#41—2691no—
243C22×D4—2691no—
244C22×D4—2691no—
245C22×D4—5381no—
246C22×D4C2×D26911yescharacteristic, complemented
247AGL(1,ℤ/8)—5384no—
248C2×SD16—2691no—
249C2×SD16—2691no—
250C2×SD16—2691no—
251C2×D8—2691no—
252C2×D8—2691no—
253C2×D8—2691no—
254C2×D8—2691no—
255C2×D8—5384no—
256C2×D8C2×D26911yescharacteristic, complemented
257C2×D8C2×D26911yescharacteristic, complemented
258C2×M16—2691no—
259C22×C8—2691no—
260#127—2691no—
261#124—2691no—
262#123—5381no—
263C22⋊Q8—2691no—
264#121—2691no—
265#121—2691no—
266#121—5381no—
267#121—5381no—
268#121C2×D26911yescharacteristic
269#120—5381no—
270#120—5381no—
271#120—5381no—
272C4×D4—2691no—
273C4×D4—2691no—
274C4×D4—2691no—
275C2×C22⋊C4—5381no—
276#106—2691no—
277C4⋊C8—2691no—
278#103—2691no—
279#102—2691no—
280#102—2691no—
281#102—2691no—
282#102C2×D26911yescharacteristic, complemented
283#102C2×D26911yescharacteristic, complemented
284C22⋊C8—2691no—
285C22⋊C8C2×D26911yescharacteristic, complemented
286#97—2691no—
287C24—21no—
288C24—5381no—
289C24—5381no—
290#41—5382no—
291#41—10761no—
292C2×Q8—2691no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—2691no—
298C2×D4—2691no—
299C2×D4—2691no—
300C2×D4—5382no—
301C2×D4—5381no—
302C2×D4—5381no—
303C2×D4—5381no—
304C2×D4—5381no—
305C2×D4—5382no—
306C2×D4—5381no—
307C2×D4—5381no—
308C2×D4—5381no—
309C2×D4—10761no—
310C2×D4—10761no—
311C2×D4—10764no—
312C2×D4C22×D26911yescharacteristic
313C2×D4C22×D26911yescharacteristic
314C2×D4C22×D26911yescharacteristic
315C22×C4—2691no—
316C22×C4—5381no—
317C22×C4—5381no—
318C22×C4—5381no—
319C22×C4C22×D26911yescharacteristic
320SD16—5382no—
321SD16—5382no—
322SD16—10761no—
323D8—22no—
324D8—41no—
325D8—2694no—
326D8—5382no—
327D8—5382no—
328D8—5382no—
329M16—5382no—
330C2×C8—2691no—
331C2×C8—2691no—
332C2×C8—5382no—
333C2×C8C22×D26911yescharacteristic
334C2×C8C22×D26911yescharacteristic
335C4⋊C4—2691no—
336C4⋊C4—2691no—
337C4⋊C4—5381no—
338C4⋊C4C22×D26911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—2691no—
341C22⋊C4—2691no—
342C22⋊C4—5381no—
343C22⋊C4—5381no—
344C22⋊C4—5381no—
345C22⋊C4—5381no—
346C22⋊C4—5381no—
347C22⋊C4—5381no—
348C22⋊C4—5381no—
349C42—2691no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—2691no—
357C23—5381no—
358C23—5381no—
359C23—10762no—
360C23—10762no—
361C23—10762no—
362C23—10762no—
363C23—21521no—
364C23—21521no—
365C23D4×D26911yescharacteristic
366Q8—5381no—
367Q8—5381no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—42no—
375D4—2692no—
376D4—5382no—
377D4—5381no—
378D4—5381no—
379D4—5381no—
380D4—10762no—
381D4—10762no—
382D4—10762no—
383D4—10762no—
384D4—10762no—
385D4—21521no—
386D4—21521no—
387D4D4×D26912yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—2691no—
392C2×C4—2691no—
393C2×C4—2691no—
394C2×C4—5381no—
395C2×C4—5381no—
396C2×C4—5381no—
397C2×C4—5381no—
398C2×C4—10762no—
399C2×C4—10762no—
400C2×C4—21521no—
401C2×C4D4×D26911yescharacteristic
402C2×C4C23×D26911yescharacteristic
403C8—21no—
404C8—21no—
405C8—2692no—
406C8—5381no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—5381no—
420C22—5381no—
421C22—5382no—
422C22—5381no—
423C22—10761no—
424C22—10761no—
425C22—10761no—
426C22—10761no—
427C22—10761no—
428C22—21524no—
429C22—21524no—
430C22#2257547712yescomplemented
431C22C2×D4×D26911yescharacteristic
432C4—21no—
433C4—41no—
434C4—5381no—
435C4—5381no—
436C4—5381no—
437C4—10761no—
438C4—10761no—
439C4C2×D4×D26911yescharacteristic
440C4#2257549811yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—81no—
445C2—10761no—
446C2—10762no—
447C2—21521no—
448C2#3177123711yescharacteristic
449C2C2×#2257547711yescharacteristic
450C2#3177140911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#447×C269C2
socleC22×C269C22×D4
hypercenter#447D269
hypocenterC269#3264
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#22575477C2semidirect
2. C22×#16864387C2semidirect
3. #31771249C2semidirect
4. #31771239C2semidirect
5. #31770993C2semidirect
6. #31770857C2semidirect
7. #31770832C2semidirect
8. #31770819C2semidirect
9. #31770798C2semidirect
10. #31770758C2semidirect
11. #31770751C2semidirect
12. #31770737C2semidirect
13. #31770714C2semidirect
14. #31770673C2semidirect
15. #447×C269C2semidirect
16. C2×#16864401C22semidirect
19. C2×D4×D269C22semidirect
21. C2×#16864391C22semidirect
22. C2×#16864391C22semidirect
23. C2×#16864390C22semidirect
27. C2×#16864387C22semidirect
28. C2×#16864387C22semidirect
29. C2×#16864385C22semidirect
30. C22×#14524130C22semidirect
31. #22575449C22semidirect
32. #22575440C22semidirect
34. #22575432C22semidirect
35. #22575428C22semidirect
38. #22575424C22semidirect
42. D4×Dic269C22semidirect
43. #22575415C22semidirect
46. #22575384C22semidirect
47. #22575382C22semidirect
48. #22575376C22semidirect
49. #22575375C22semidirect
50. #22575375C22semidirect
51. #22575374C22semidirect
52. #22575365C22semidirect
53. #22575363C22semidirect
54. C22×D4×C269C22semidirect
55. C2×D8×C269C22semidirect
56. C2×D8×C269C22semidirect
58. #102×C269C22semidirect
59. #102×C269C22semidirect
60. C22⋊C8×C269C22semidirect
91. #16864387D4semidirect
158. #14524130C2×D4semidirect
173. D4×C269C2×D4semidirect
193. C22×C269AGL(1,ℤ/8)semidirect
225. C269#3264semidirect
239. #447D269semidirect
246. C22×D4C2×D269semidirect
256. C2×D8C2×D269semidirect
257. C2×D8C2×D269semidirect
282. #102C2×D269semidirect
283. #102C2×D269semidirect
285. C22⋊C8C2×D269semidirect
387. D4D4×D269semidirect
430. C22#22575477semidirect

Maximal subgroups

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

subgroupindex
C2×#225754772
C22×#168643872
#317712492
#317712392
#317709932
#317708572
#317708322
#317708192
#317707982
#317707582
#317707512
#317707372
#317707142
#317706732
#447×C2692
#3264269