Finite Group Databasebeta

← #46822364

Subgroups — #46822364

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×#22575480C211yesmaximal, characteristic, complemented
2C2×D8×D269C211yesmaximal, characteristic, complemented
3#31771248C211yesmaximal, characteristic, complemented
4#31771240C211yesmaximal, characteristic, complemented
5#31770855C211yesmaximal, characteristic, complemented
6#31770831C211yesmaximal, characteristic, complemented
7#31770817C211yesmaximal, characteristic, complemented
8#31770797C211yesmaximal, characteristic, complemented
9#31770754C211yesmaximal, characteristic, complemented
10#31770749C211yesmaximal, characteristic, complemented
11#31770737C211yesmaximal, characteristic, complemented
12#31770737C211yesmaximal, characteristic, complemented
13#31770718C211yesmaximal, characteristic, complemented
14#31770678C211yesmaximal, characteristic, complemented
15#496×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic, complemented
17C2×D4×D269—21no—
18C2×D4×D269C2211yescharacteristic
19C2×D4×D269C2211yescharacteristic, complemented
20C2×D4×D269C2211yescharacteristic, complemented
21#22575480—24no—
22C2×#16864389C2211yescharacteristic, complemented
23C2×D2152C2211yescharacteristic
24C2×#16864387C2211yescharacteristic
25C2×#16864387C2211yescharacteristic, complemented
26C2×#16864387C2211yescharacteristic, complemented
27D8×D269—24no—
28C2×#16864385C2211yescharacteristic, complemented
29C2×C8×D269C2211yescharacteristic, complemented
30#22575435—21no—
31#22575429C2211yescharacteristic, complemented
32#22575426—21no—
33#22575424—21no—
34#22575424C2211yescharacteristic
35#22575424C2211yescharacteristic, complemented
36#22575424C2211yescharacteristic, complemented
37#22575423—21no—
38#22575421—21no—
39#22575421—21no—
40C4×D1076C2211yescharacteristic
41C4×D1076C2211yescharacteristic, complemented
42C4×D1076C2211yescharacteristic, complemented
43C22⋊C4×D269—21no—
44#22575385C2211yescharacteristic, complemented
45C4⋊#14524130C2211yescharacteristic, complemented
46#22575377C2211yescharacteristic, complemented
47#22575375C2211yescharacteristic, complemented
48#22575375C2211yescharacteristic, complemented
49#22575374C2211yescharacteristic, complemented
50#22575374C2211yescharacteristic, complemented
51#22575374C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53#22575365C2211yescharacteristic, complemented
54C2×SD16×C269C2211yescharacteristic, complemented
55C2×SD16×C269C2211yescharacteristic, complemented
56C2×D8×C269C2211yescharacteristic, complemented
57C2×D8×C269C2211yescharacteristic, complemented
58#127×C269C2211yescharacteristic, complemented
59#124×C269C2211yescharacteristic, complemented
60#97×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62C23×D269—21no—
63C23×D269—21no—
64#16864403—22no—
65#16864403—41no—
66C2×D1076—21no—
67C2×D1076D411yescharacteristic
68C2×D1076C2311yescharacteristic
69C2×D1076C2311yescharacteristic
70C2×C22⋊D269—21no—
71C2×C22⋊D269—21no—
72C2×C22⋊D269—21no—
73C2×C22⋊D269—21no—
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76D4×D269—22no—
77D4×D269—22no—
78D4×D269—22no—
79D4×D269—41no—
80D4×D269—41no—
81D4×D269—41no—
82D4×D269—44no—
83C2×C4×D269—21no—
84C2×C4×D269—21no—
85C2×C4×D269—21no—
86C2×C4×D269C2311yescharacteristic
87C2×C4×D269C2311yescharacteristic
88#16864389—22no—
89D2152D414yescomplemented
90#16864387—22no—
91#16864387—22no—
92#16864387—22no—
93#16864385—22no—
94C2×#14524130C2311yescharacteristic
95C2×#14524130C2311yescharacteristic
96C8×D269—22no—
97C4⋊Dic269D411yescharacteristic
98C4⋊Dic269C2311yescharacteristic
99C4⋊Dic269C2311yescharacteristic
100#16864381—21no—
101C22⋊Dic269—21no—
102C22⋊Dic269—21no—
103C22⋊Dic269—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379D411yescharacteristic
108#16864379D411yescharacteristic
109C2×Q8×C269C2311yescharacteristic
110C2×D4×C269—21no—
111C2×D4×C269—21no—
112C2×D4×C269C2311yescharacteristic
113C2×D4×C269C2311yescharacteristic
114C2×D4×C269C2311yescharacteristic
115SD16×C269—22no—
116SD16×C269—41no—
117D8×C269—22no—
118D8×C269—41no—
119C2×C8×C269C2311yescharacteristic
120C2×C8×C269C2311yescharacteristic
121C22⋊C4×C269—21no—
122C22⋊C4×C269—21no—
123C42×C269C2311yescharacteristic
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—42no—
127C22×D269—42no—
128C22×D269—42no—
129C22×D269—42no—
130C22×D269—42no—
131C22×D269—42no—
132C22×D269—81no—
133C22×D269—81no—
134C22×D269—81no—
135C22×D269C2×D411yescharacteristic
136C22×D269C2×D411yescharacteristic
137D1076—22no—
138D1076—21no—
139D1076—21no—
140D1076—42no—
141D1076C2×D412yes—
142D1076C2×D412yescomplemented
143C22⋊D269—42no—
144C22⋊D269—42no—
145C22⋊D269—42no—
146C22⋊D269—42no—
147C22⋊D269—42no—
148C22⋊D269—81no—
149C2×Dic269—21no—
150C2×Dic269—21no—
151C2×Dic269C2×D411yescharacteristic
152C2×Dic269C2×D411yescharacteristic
153C4×D269—21no—
154C4×D269—21no—
155C4×D269—21no—
156C4×D269—21no—
157C4×D269—42no—
158C4×D269—42no—
159C4×D269—81no—
160#14524130—21no—
161#14524130—21no—
162C23×C269—21no—
163C23×C269—21no—
164C23×C269—21no—
165Q8×C269—21no—
166Q8×C269—21no—
167D4×C269—21no—
168D4×C269—21no—
169D4×C269—21no—
170D4×C269—21no—
171D4×C269—21no—
172D4×C269—21no—
173D4×C269—42no—
174D4×C269—81no—
175C2×C4×C269—21no—
176C2×C4×C269C2×D411yescharacteristic
177C2×C4×C269C2×D411yescharacteristic
178C2×C4×C269C2411yescharacteristic
179C8×C269—21no—
180C8×C269C2×D412yescomplemented
181C22×C269—41no—
182C22×C269—41no—
183C22×C269—41no—
184C22×C269—41no—
185C22×C269—41no—
186C22×C269—41no—
187C22×C269—41no—
188C22×C269—41no—
189C22×C269—41no—
190C22×C269C22×D411yescharacteristic
191C2×D269—21no—
192C2×D269—21no—
193C2×D269—21no—
194C2×D269—22no—
195C2×D269—22no—
196C2×D269—21no—
197C2×D269—21no—
198C2×D269—41no—
199C2×D269—41no—
200C2×D269—41no—
201C2×D269—41no—
202C2×D269—84no—
203C2×D269—84no—
204C2×D269—84no—
205C2×D269AGL(1,ℤ/8)12yescomplemented
206C4×C269—21no—
207C4×C269—21no—
208C4×C269—41no—
209C4×C269C22×D411yescharacteristic
210C4×C269H(5,𝔽2)11yescharacteristic
211Dic269—21no—
212Dic269—21no—
213Dic269—41no—
214Dic269—41no—
215C2×C269—81no—
216C2×C269—81no—
217C2×C269—81no—
218C2×C269D4211yescharacteristic
219C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C269#57611yescharacteristic
221D269—22no—
222D269—41no—
223D269—42no—
224D269—42no—
225C269#326411yescharacteristic, complemented, Sylow
226#3264—2691nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2691no—
228C22×D8—2691no—
229#546—2691no—
230D42—2691no—
231#496D26911yescharacteristic, complemented
232#469—2691no—
233#465—2691no—
234#463—2691no—
235#459—2691no—
236#450—2691no—
237#449—2691no—
238#447—2691no—
239#447—2691no—
240#442—2691no—
241#435—2691no—
242C2×#41—2691no—
243C22×D4—2691no—
244C22×D4—2691no—
245C22×D4—2691no—
246C22×D4—5381no—
247AGL(1,ℤ/8)—5384no—
248C2×SD16—2691no—
249C2×SD16C2×D26911yescharacteristic, complemented
250C2×SD16C2×D26911yescharacteristic, complemented
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#127C2×D26911yescharacteristic, complemented
261#124C2×D26911yescharacteristic, complemented
262#123—5381no—
263C22⋊Q8—2691no—
264#121—2691no—
265#121—2691no—
266#121—2691no—
267#121—5381no—
268#121—5381no—
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#102—2691no—
283#102—2691no—
284C22⋊C8—2691no—
285C22⋊C8—2691no—
286#97C2×D26911yescharacteristic, complemented
287C24—5381no—
288C24—5381no—
289C24—5381no—
290#41—5382no—
291#41—10761no—
292C2×Q8C22×D26911yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2691no—
296C2×D4—2691no—
297C2×D4—2691no—
298C2×D4—5381no—
299C2×D4—5381no—
300C2×D4—5381no—
301C2×D4—5381no—
302C2×D4—5382no—
303C2×D4—5382no—
304C2×D4—5382no—
305C2×D4—5381no—
306C2×D4—5381no—
307C2×D4—5381no—
308C2×D4—10761no—
309C2×D4—10761no—
310C2×D4—10761no—
311C2×D4—10764no—
312C2×D4C22×D26911yescharacteristic
313C2×D4C22×D26911yescharacteristic
314C2×D4C22×D26911yescharacteristic
315C22×C4—2691no—
316C22×C4—2691no—
317C22×C4—5381no—
318C22×C4—5381no—
319C22×C4—5381no—
320SD16—22no—
321SD16—41no—
322SD16—5382no—
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—2691no—
338C4⋊C4—5381no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2691no—
342C22⋊C4—2691no—
343C22⋊C4—5381no—
344C22⋊C4—5381no—
345C22⋊C4—5381no—
346C22⋊C4—5381no—
347C22⋊C4—5381no—
348C22⋊C4—5381no—
349C42C22×D26911yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—2691no—
354C23—2691no—
355C23—5381no—
356C23—5381no—
357C23—10762no—
358C23—10762no—
359C23—10762no—
360C23—10762no—
361C23—10762no—
362C23—10762no—
363C23—21521no—
364C23—21521no—
365C23—21521no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—2692no—
377D4—2692no—
378D4—5382no—
379D4—5381no—
380D4—5381no—
381D4—10762no—
382D4—10762no—
383D4—10762no—
384D4—10762no—
385D4—10762no—
386D4—10762no—
387D4—21521no—
388C2×C4—21no—
389C2×C4—2691no—
390C2×C4—2691no—
391C2×C4—5381no—
392C2×C4—5381no—
393C2×C4—5381no—
394C2×C4—5381no—
395C2×C4—5381no—
396C2×C4—5381no—
397C2×C4—10762no—
398C2×C4—10762no—
399C2×C4—21521no—
400C2×C4D4×D26911yescharacteristic
401C2×C4D4×D26911yescharacteristic
402C2×C4C23×D26911yescharacteristic
403C8—21no—
404C8—5381no—
405C8—5381no—
406C8D4×D26912yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2692no—
417C22—5381no—
418C22—5381no—
419C22—5381no—
420C22—5381no—
421C22—5382no—
422C22—5382no—
423C22—5381no—
424C22—10761no—
425C22—10761no—
426C22—10761no—
427C22—10761no—
428C22—21524no—
429C22—21524no—
430C22—21524no—
431C22C2×D4×D26911yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—5381no—
436C4—5381no—
437C4—10761no—
438C4—10761no—
439C4C2×D4×D26911yescharacteristic
440C4#2257549811yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—5382no—
445C2—10761no—
446C2—10762no—
447C2—10762no—
448C2#3177124011yescharacteristic
449C2AGL(1,ℤ/8)×D26911yescharacteristic
450C2#3177140911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#496×C269C2
socleC22×C269C22×D4
hypercenter#496D269
hypocenterC269#3264
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22575480C2semidirect
2. C2×D8×D269C2semidirect
3. #31771248C2semidirect
4. #31771240C2semidirect
5. #31770855C2semidirect
6. #31770831C2semidirect
7. #31770817C2semidirect
8. #31770797C2semidirect
9. #31770754C2semidirect
10. #31770749C2semidirect
11. #31770737C2semidirect
12. #31770737C2semidirect
13. #31770718C2semidirect
14. #31770678C2semidirect
15. #496×C269C2semidirect
16. C2×#16864403C22semidirect
19. C2×D4×D269C22semidirect
20. C2×D4×D269C22semidirect
22. C2×#16864389C22semidirect
25. C2×#16864387C22semidirect
26. C2×#16864387C22semidirect
28. C2×#16864385C22semidirect
29. C2×C8×D269C22semidirect
31. #22575429C22semidirect
35. #22575424C22semidirect
36. #22575424C22semidirect
41. C4×D1076C22semidirect
42. C4×D1076C22semidirect
44. #22575385C22semidirect
45. C4⋊#14524130C22semidirect
46. #22575377C22semidirect
47. #22575375C22semidirect
48. #22575375C22semidirect
49. #22575374C22semidirect
50. #22575374C22semidirect
51. #22575374C22semidirect
52. #22575365C22semidirect
53. #22575365C22semidirect
54. C2×SD16×C269C22semidirect
55. C2×SD16×C269C22semidirect
56. C2×D8×C269C22semidirect
57. C2×D8×C269C22semidirect
58. #127×C269C22semidirect
59. #124×C269C22semidirect
60. #97×C269C22semidirect
89. D2152D4semidirect
142. D1076C2×D4semidirect
180. C8×C269C2×D4semidirect
205. C2×D269AGL(1,ℤ/8)semidirect
225. C269#3264semidirect
231. #496D269semidirect
249. C2×SD16C2×D269semidirect
250. C2×SD16C2×D269semidirect
256. C2×D8C2×D269semidirect
257. C2×D8C2×D269semidirect
260. #127C2×D269semidirect
261. #124C2×D269semidirect
286. #97C2×D269semidirect
406. C8D4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754802
C2×D8×D2692
#317712482
#317712402
#317708552
#317708312
#317708172
#317707972
#317707542
#317707492
#317707372
#317707182
#317706782
#496×C2692
#3264269