Finite Group Databasebeta

← #46822360

Subgroups — #46822360

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
2C2×D8×D269C211yesmaximal, characteristic, complemented
3#31771250C211yesmaximal, characteristic, complemented
4#31771240C211yesmaximal, characteristic, complemented
5#31770992C211yesmaximal, characteristic, complemented
6#31770857C211yesmaximal, characteristic, complemented
7#31770833C211yesmaximal, characteristic, complemented
8#31770821C211yesmaximal, characteristic, complemented
9#31770757C211yesmaximal, characteristic, complemented
10#31770747C211yesmaximal, characteristic, complemented
11#31770740C211yesmaximal, characteristic, complemented
12#31770737C211yesmaximal, characteristic, complemented
13#31770715C211yesmaximal, characteristic, complemented
14#31770675C211yesmaximal, characteristic, complemented
15#459×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic, complemented
17C2×D4×D269—21no—
18C2×D4×D269C2211yescharacteristic
19C2×D4×D269C2211yescharacteristic, complemented
20C2×D4×D269C2211yescharacteristic, complemented
21#22575477—24no—
22C2×#16864391C2211yescharacteristic, complemented
23C2×#16864390C2211yescharacteristic, complemented
24C2×#16864390C2211yescharacteristic, complemented
25C2×D2152C2211yescharacteristic
26C2×#16864387C2211yescharacteristic
27C2×#16864387C2211yescharacteristic, complemented
28C2×#16864387C2211yescharacteristic, complemented
29D8×D269—24no—
30C2×#16864385C2211yescharacteristic, complemented
31C2×C8×D269C2211yescharacteristic, complemented
32#22575441C2211yescharacteristic, complemented
33#22575437—21no—
34#22575433C2211yescharacteristic, complemented
35#22575428C2211yescharacteristic, complemented
36#22575427—21no—
37#22575424—21no—
38#22575424C2211yescharacteristic
39#22575424C2211yescharacteristic, complemented
40#22575423—21no—
41#22575421—21no—
42#22575421—21no—
43C4×D1076C2211yescharacteristic
44C4×D1076C2211yescharacteristic, complemented
45C2×#16864379—21no—
46#22575384C2211yescharacteristic, complemented
47#22575378C2211yescharacteristic, complemented
48#22575375C2211yescharacteristic, complemented
49#22575374C2211yescharacteristic, complemented
50#22575373C2211yescharacteristic, complemented
51#22575365C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53#22575364C2211yescharacteristic, complemented
54C2×D8×C269C2211yescharacteristic, complemented
55C2×D8×C269C2211yescharacteristic, complemented
56#127×C269C2211yescharacteristic, complemented
57C4×D4×C269C2211yescharacteristic, complemented
58C4⋊C8×C269C2211yescharacteristic, complemented
59#102×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62C23×D269—21no—
63C23×D269—21no—
64#16864401—22no—
65#16864401—41no—
66C2×Dic538C2311yescharacteristic
67C2×D1076—21no—
68C2×D1076D411yescharacteristic
69C2×D1076C2311yescharacteristic
70C2×D1076C2311yescharacteristic
71C2×C22⋊D269—21no—
72C2×C22⋊D269—21no—
73C2×C22⋊D269—21no—
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76C2×C22⋊D269—21no—
77D4×D269—22no—
78D4×D269—22no—
79D4×D269—22no—
80D4×D269—41no—
81D4×D269—41no—
82D4×D269—41no—
83D4×D269—44no—
84C22×Dic269—21no—
85C2×C4×D269—21no—
86C2×C4×D269C2311yescharacteristic
87C2×C4×D269C2311yescharacteristic
88#16864391—22no—
89#16864390—22no—
90#16864390—41no—
91D2152—22no—
92#16864387—22no—
93#16864387—41no—
94#16864387D414yescomplemented
95#16864385—22no—
96C2×#14524130C2311yescharacteristic
97C2×#14524130C2311yescharacteristic
98C8×D269—22no—
99C4⋊Dic269—21no—
100C4⋊Dic269D411yescharacteristic
101C4⋊Dic269C2311yescharacteristic
102C22⋊Dic269—21no—
103C22⋊Dic269—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379—21no—
108#16864379—21no—
109#16864379D411yescharacteristic
110#16864379D411yescharacteristic
111C2×D4×C269—21no—
112C2×D4×C269—21no—
113C2×D4×C269C2311yescharacteristic
114C2×D4×C269C2311yescharacteristic
115C2×D4×C269C2311yescharacteristic
116C22×C4×C269—21no—
117D8×C269—22no—
118D8×C269—22no—
119C2×C8×C269C2311yescharacteristic
120C2×C8×C269C2311yescharacteristic
121C4⋊C4×C269C2311yescharacteristic
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
137Dic538—21no—
138Dic538—21no—
139D1076—22no—
140D1076—21no—
141D1076—21no—
142D1076—21no—
143D1076—81no—
144D1076C2×D412yes—
145C22⋊D269—42no—
146C22⋊D269—42no—
147C22⋊D269—42no—
148C22⋊D269—42no—
149C22⋊D269—42no—
150C22⋊D269—42no—
151C2×Dic269—21no—
152C2×Dic269—21no—
153C2×Dic269—42no—
154C2×Dic269C2×D411yescharacteristic
155C2×Dic269C2×D411yescharacteristic
156C4×D269—21no—
157C4×D269—21no—
158C4×D269—21no—
159C4×D269—21no—
160C4×D269—42no—
161#14524130—21no—
162#14524130C2×D412yescomplemented
163C23×C269—21no—
164C23×C269—21no—
165C23×C269—21no—
166D4×C269—21no—
167D4×C269—21no—
168D4×C269—21no—
169D4×C269—21no—
170D4×C269—21no—
171D4×C269—42no—
172D4×C269—81no—
173D4×C269C2×D412yescomplemented
174C2×C4×C269—21no—
175C2×C4×C269—81no—
176C2×C4×C269C2×D411yescharacteristic
177C2×C4×C269C2×D411yescharacteristic
178C2×C4×C269C2411yescharacteristic
179C8×C269—21no—
180C8×C269—21no—
181C22×C269—22no—
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—21no—
196C2×D269—21no—
197C2×D269—41no—
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—42no—
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—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#459D26911yescharacteristic, complemented
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×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#127C2×D26911yescharacteristic, complemented
261#124—2691no—
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×D4C2×D26911yescharacteristic, complemented
275C2×C22⋊C4—5381no—
276#106—2691no—
277C4⋊C8C2×D26911yescharacteristic, complemented
278#103—2691no—
279#102—2691no—
280#102—2691no—
281#102—2691no—
282#102C2×D26911yescharacteristic, complemented
283#102C2×D26911yescharacteristic, complemented
284C22⋊C8—2691no—
285C22⋊C8—2691no—
286#97—2691no—
287C24—5381no—
288C24—5381no—
289C24—5381no—
290#41—5382no—
291#41—10761no—
292C2×Q8—2691no—
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—21no—
316C22×C4—2691no—
317C22×C4—2691no—
318C22×C4—5381no—
319C22×C4—5381no—
320SD16—5382no—
321SD16—5382no—
322SD16—10761no—
323D8—22no—
324D8—22no—
325D8—2694no—
326D8—5382no—
327D8—5382no—
328D8—10761no—
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—
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—5381no—
367Q8—5381no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—2692no—
376D4—5382no—
377D4—5381no—
378D4—5381no—
379D4—5381no—
380D4—10762no—
381D4—10762no—
382D4—10762no—
383D4—10762no—
384D4—10762no—
385D4—10762no—
386D4—21521no—
387D4D4×D26912yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—2691no—
391C2×C4—2691no—
392C2×C4—5381no—
393C2×C4—5381no—
394C2×C4—5381no—
395C2×C4—5381no—
396C2×C4—5381no—
397C2×C4—5381no—
398C2×C4—10762no—
399C2×C4—10762no—
400C2×C4D4×D26911yescharacteristic
401C2×C4D4×D26911yescharacteristic
402C2×C4C23×D26911yescharacteristic
403C8—21no—
404C8—21no—
405C8—2692no—
406C8—5381no—
407C22—22no—
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—5381no—
423C22—10761no—
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#2257549911yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—5382no—
445C2—10761no—
446C2—10762no—
447C2—21521no—
448C2#3177123911yescharacteristic
449C2AGL(1,ℤ/8)×D26911yescharacteristic
450C2#3177140611yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#459×C269C2
socleC22×C269C22×D4
hypercenter#459D269
hypocenterC269#3264
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22575477C2semidirect
2. C2×D8×D269C2semidirect
3. #31771250C2semidirect
4. #31771240C2semidirect
5. #31770992C2semidirect
6. #31770857C2semidirect
7. #31770833C2semidirect
8. #31770821C2semidirect
9. #31770757C2semidirect
10. #31770747C2semidirect
11. #31770740C2semidirect
12. #31770737C2semidirect
13. #31770715C2semidirect
14. #31770675C2semidirect
15. #459×C269C2semidirect
16. C2×#16864401C22semidirect
19. C2×D4×D269C22semidirect
20. C2×D4×D269C22semidirect
22. C2×#16864391C22semidirect
23. C2×#16864390C22semidirect
24. C2×#16864390C22semidirect
27. C2×#16864387C22semidirect
28. C2×#16864387C22semidirect
30. C2×#16864385C22semidirect
31. C2×C8×D269C22semidirect
32. #22575441C22semidirect
34. #22575433C22semidirect
35. #22575428C22semidirect
39. #22575424C22semidirect
44. C4×D1076C22semidirect
46. #22575384C22semidirect
47. #22575378C22semidirect
48. #22575375C22semidirect
49. #22575374C22semidirect
50. #22575373C22semidirect
51. #22575365C22semidirect
52. #22575365C22semidirect
53. #22575364C22semidirect
54. C2×D8×C269C22semidirect
55. C2×D8×C269C22semidirect
56. #127×C269C22semidirect
57. C4×D4×C269C22semidirect
58. C4⋊C8×C269C22semidirect
59. #102×C269C22semidirect
60. #102×C269C22semidirect
94. #16864387D4semidirect
162. #14524130C2×D4semidirect
173. D4×C269C2×D4semidirect
205. C2×D269AGL(1,ℤ/8)semidirect
225. C269#3264semidirect
235. #459D269semidirect
256. C2×D8C2×D269semidirect
257. C2×D8C2×D269semidirect
260. #127C2×D269semidirect
274. C4×D4C2×D269semidirect
277. C4⋊C8C2×D269semidirect
282. #102C2×D269semidirect
283. #102C2×D269semidirect
387. D4D4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754772
C2×D8×D2692
#317712502
#317712402
#317709922
#317708572
#317708332
#317708212
#317707572
#317707472
#317707402
#317707372
#317707152
#317706752
#459×C2692
#3264269