Finite Group Databasebeta

← #46822365

Subgroups — #46822365

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
2#31771249C211yesmaximal, characteristic, complemented
3D4×C22⋊D269C211yesmaximal, characteristic, complemented
4#31770993C211yesmaximal, characteristic, complemented
5#31770855C211yesmaximal, characteristic, complemented
6#31770834C211yesmaximal, characteristic, complemented
7#31770818C211yesmaximal, characteristic, complemented
8#31770798C211yesmaximal, characteristic, complemented
9#31770756C211yesmaximal, characteristic, complemented
10#31770747C211yesmaximal, characteristic, complemented
11#31770739C211yesmaximal, characteristic, complemented
12#31770737C211yesmaximal, characteristic, complemented
13#31770717C211yesmaximal, characteristic, complemented
14#31770676C211yesmaximal, characteristic, complemented
15C22×D8×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic, complemented
17C22×C22⋊D269—21no—
18C2×D4×D269C2211yescharacteristic, complemented
19#22575477—24no—
20C2×#16864391C2211yescharacteristic, complemented
21C2×#16864390C2211yescharacteristic, complemented
22C2×#16864390C2211yescharacteristic, complemented
23C2×#16864387C2211yescharacteristic, complemented
24C2×#16864387C2211yescharacteristic, complemented
25C2×#16864385C2211yescharacteristic, complemented
26#22575449C2211yescharacteristic, complemented
27#22575440C2211yescharacteristic, complemented
28#22575434—21no—
29#22575432C2211yescharacteristic, complemented
30#22575427C2211yescharacteristic
31#22575425—21no—
32#22575425—21no—
33#22575424C2211yescharacteristic, complemented
34#22575424C2211yescharacteristic, complemented
35#22575422—21no—
36#22575422—21no—
37#22575421—21no—
38D4×Dic269C2211yescharacteristic, complemented
39D4×Dic269C2211yescharacteristic, complemented
40C4×C22⋊D269C2211yescharacteristic
41C2×C22⋊Dic269—21no—
42#22575385C2211yescharacteristic, complemented
43#22575382C2211yescharacteristic, complemented
44#22575378C2211yescharacteristic, complemented
45#22575375C2211yescharacteristic, complemented
46#22575374C2211yescharacteristic, complemented
47#22575374C2211yescharacteristic, complemented
48#22575374C2211yescharacteristic, complemented
49#22575374C2211yescharacteristic, complemented
50C22⋊#14524130C2211yescharacteristic, complemented
51#22575365C2211yescharacteristic, complemented
52#22575363C2211yescharacteristic, complemented
53C22×D4×C269C2211yescharacteristic
54C22×D4×C269C2211yescharacteristic, complemented
55C2×D8×C269—24no—
56C2×D8×C269C2211yescharacteristic
57C2×D8×C269C2211yescharacteristic
58C2×D8×C269C2211yescharacteristic, complemented
59C2×D8×C269C2211yescharacteristic, complemented
60C22×C8×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62#16864401—22no—
63#16864401—41no—
64C2×Dic538C2311yescharacteristic
65C2×D1076C2311yescharacteristic
66C2×C22⋊D269—21no—
67C2×C22⋊D269—21no—
68C2×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C2×C22⋊D269—21no—
72C2×C22⋊D269—21no—
73C2×C22⋊D269—44no—
74C2×C22⋊D269D411yescharacteristic
75D4×D269—22no—
76D4×D269—41no—
77C22×Dic269—21no—
78C22×Dic269—21no—
79C22×Dic269—21no—
80C2×C4×D269C2311yescharacteristic
81#16864391—22no—
82#16864390—22no—
83#16864390—41no—
84#16864387—22no—
85#16864387—41no—
86#16864385—22no—
87C2×#14524130C2311yescharacteristic
88C2×#14524130C2311yescharacteristic
89C4⋊Dic269C2311yescharacteristic
90C4⋊Dic269C2311yescharacteristic
91#16864381—21no—
92#16864381D411yescharacteristic
93C22⋊Dic269—21no—
94C22⋊Dic269—21no—
95C22⋊Dic269—21no—
96C22⋊Dic269—21no—
97C22⋊Dic269—21no—
98C22⋊Dic269—21no—
99C22⋊Dic269—21no—
100C22⋊Dic269D411yescharacteristic
101#16864379—21no—
102#16864379D411yescharacteristic
103C4×Dic269C2311yescharacteristic
104C24×C269—21no—
105C24×C269—21no—
106C2×D4×C269—22no—
107C2×D4×C269—22no—
108C2×D4×C269—21no—
109C2×D4×C269—21no—
110C2×D4×C269—41no—
111C2×D4×C269—41no—
112C2×D4×C269C2311yescharacteristic
113C2×D4×C269C2311yescharacteristic
114C2×D4×C269C2311yescharacteristic
115C2×D4×C269C2311yescharacteristic
116C22×C4×C269C2311yescharacteristic
117D8×C269—22no—
118D8×C269—22no—
119D8×C269—22no—
120D8×C269D414yescomplemented
121C2×C8×C269—22no—
122C2×C8×C269C2311yescharacteristic
123C2×C8×C269C2311yescharacteristic
124C22×D269—21no—
125C22×D269—42no—
126C22×D269—42no—
127C22×D269—81no—
128C22×D269C2×D411yescharacteristic
129Dic538—21no—
130Dic538—21no—
131D1076—21no—
132D1076—21no—
133C22⋊D269—22no—
134C22⋊D269—42no—
135C22⋊D269—42no—
136C22⋊D269—42no—
137C22⋊D269—42no—
138C22⋊D269—42no—
139C22⋊D269—81no—
140C22⋊D269—81no—
141C2×Dic269—21no—
142C2×Dic269—21no—
143C2×Dic269—21no—
144C2×Dic269—42no—
145C2×Dic269—42no—
146C2×Dic269—81no—
147C2×Dic269C2×D411yescharacteristic
148C2×Dic269C2×D411yescharacteristic
149C2×Dic269C2×D411yescharacteristic
150C4×D269—21no—
151C4×D269—21no—
152#14524130—21no—
153#14524130—21no—
154C23×C269—21no—
155C23×C269—21no—
156C23×C269—21no—
157C23×C269—21no—
158C23×C269—42no—
159C23×C269—42no—
160C23×C269—42no—
161C23×C269—42no—
162C23×C269—81no—
163C23×C269—81no—
164C23×C269C2×D411yescharacteristic
165D4×C269—21no—
166D4×C269—21no—
167D4×C269—21no—
168D4×C269—21no—
169D4×C269—21no—
170D4×C269—21no—
171D4×C269—42no—
172D4×C269—42no—
173D4×C269C2×D412yes—
174D4×C269C2×D412yescomplemented
175C2×C4×C269—21no—
176C2×C4×C269—21no—
177C2×C4×C269C2×D411yescharacteristic
178C2×C4×C269C2411yescharacteristic
179C8×C269—21no—
180C8×C269C2×D412yescomplemented
181C22×C269—21no—
182C22×C269—21no—
183C22×C269—22no—
184C22×C269—22no—
185C22×C269—41no—
186C22×C269—41no—
187C22×C269—41no—
188C22×C269—41no—
189C22×C269—41no—
190C22×C269—41no—
191C22×C269—41no—
192C22×C269—41no—
193C22×C269—41no—
194C22×C269—41no—
195C22×C269—84no—
196C22×C269—84no—
197C22×C269AGL(1,ℤ/8)12yescomplemented
198C22×C269C22×D411yescharacteristic
199C2×D269—21no—
200C2×D269—21no—
201C2×D269—21no—
202C2×D269—41no—
203C2×D269—41no—
204C2×D269—41no—
205C2×D269—84no—
206C4×C269—21no—
207C4×C269C22×D411yescharacteristic
208C4×C269H(5,𝔽2)11yescharacteristic
209Dic269—21no—
210Dic269—21no—
211Dic269—21no—
212Dic269—41no—
213Dic269—41no—
214Dic269—41no—
215C2×C269—22no—
216C2×C269—42no—
217C2×C269—42no—
218C2×C269—81no—
219C2×C269—81no—
220C2×C269D4211yescharacteristic
221C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C269#57611yescharacteristic
223D269—41no—
224D269—81no—
225C269#326411yescharacteristic, complemented, Sylow
226#3264—2691nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2691no—
228C22×D8D26911yescharacteristic, complemented
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#447—2691no—
240#442—2691no—
241#435—2691no—
242C2×#41—2691no—
243C22×D4—2691no—
244C22×D4—5381no—
245C22×D4C2×D26911yescharacteristic
246C22×D4C2×D26911yescharacteristic, complemented
247AGL(1,ℤ/8)—5384no—
248C2×SD16—2691no—
249C2×SD16—2691no—
250C2×SD16—2691no—
251C2×D8—24no—
252C2×D8—2691no—
253C2×D8—2691no—
254C2×D8C2×D26911yescharacteristic
255C2×D8C2×D26911yescharacteristic
256C2×D8C2×D26911yescharacteristic, complemented
257C2×D8C2×D26911yescharacteristic, complemented
258C2×M16—2691no—
259C22×C8C2×D26911yescharacteristic, complemented
260#127—2691no—
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×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#97—2691no—
287C24—21no—
288C24—21no—
289C24—5381no—
290#41—5382no—
291#41—10761no—
292C2×Q8—2691no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—2691no—
300C2×D4—2691no—
301C2×D4—5381no—
302C2×D4—5381no—
303C2×D4—5381no—
304C2×D4—5382no—
305C2×D4—5381no—
306C2×D4—5381no—
307C2×D4—5381no—
308C2×D4—5381no—
309C2×D4—10761no—
310C2×D4—10764no—
311C2×D4C22×D26911yescharacteristic
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—22no—
325D8—22no—
326D8—5382no—
327D8—10761no—
328D8C22⋊D26914yescomplemented
329M16—5382no—
330C2×C8—22no—
331C2×C8—2691no—
332C2×C8—2691no—
333C2×C8C22×D26911yescharacteristic
334C2×C8C22×D26911yescharacteristic
335C4⋊C4—2691no—
336C4⋊C4—2691no—
337C4⋊C4—2691no—
338C4⋊C4—5381no—
339C22⋊C4—2691no—
340C22⋊C4—2691no—
341C22⋊C4—5381no—
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—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—2691no—
361C23—5381no—
362C23—10762no—
363C23—10762no—
364C23—21521no—
365C23D4×D26911yescharacteristic
366Q8—5381no—
367Q8—5381no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—5382no—
377D4—5381no—
378D4—5381no—
379D4—10762no—
380D4—10762no—
381D4—10762no—
382D4—10762no—
383D4—10762no—
384D4—21521no—
385D4—21521no—
386D4C2×C22⋊D26912yes—
387D4C2×C22⋊D26912yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—2691no—
391C2×C4—2691no—
392C2×C4—2691no—
393C2×C4—5381no—
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—5381no—
405C8—5381no—
406C8C2×C22⋊D26912yescomplemented
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—5381no—
424C22—5381no—
425C22—5381no—
426C22—10761no—
427C22—10761no—
428C22—10761no—
429C22—21524no—
430C22#2257547712yescomplemented
431C22C2×D4×D26911yescharacteristic
432C4—21no—
433C4—5381no—
434C4—5381no—
435C4—5381no—
436C4—10761no—
437C4—10761no—
438C4—10761no—
439C4C22×C22⋊D26911yescharacteristic
440C4#2257549811yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—10761no—
447C2—21521no—
448C2D4×C22⋊D26911yescharacteristic
449C2C2×#2257547711yescharacteristic
450C2#3177140611yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC22×D8×C269C2
socleC22×C269C22×D4
hypercenterC22×D8D269
hypocenterC269#3264
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#22575477C2semidirect
2. #31771249C2semidirect
3. D4×C22⋊D269C2semidirect
4. #31770993C2semidirect
5. #31770855C2semidirect
6. #31770834C2semidirect
7. #31770818C2semidirect
8. #31770798C2semidirect
9. #31770756C2semidirect
10. #31770747C2semidirect
11. #31770739C2semidirect
12. #31770737C2semidirect
13. #31770717C2semidirect
14. #31770676C2semidirect
15. C22×D8×C269C2semidirect
16. C2×#16864401C22semidirect
18. C2×D4×D269C22semidirect
20. C2×#16864391C22semidirect
21. C2×#16864390C22semidirect
22. C2×#16864390C22semidirect
23. C2×#16864387C22semidirect
24. C2×#16864387C22semidirect
25. C2×#16864385C22semidirect
26. #22575449C22semidirect
27. #22575440C22semidirect
29. #22575432C22semidirect
33. #22575424C22semidirect
34. #22575424C22semidirect
38. D4×Dic269C22semidirect
39. D4×Dic269C22semidirect
42. #22575385C22semidirect
43. #22575382C22semidirect
44. #22575378C22semidirect
45. #22575375C22semidirect
46. #22575374C22semidirect
47. #22575374C22semidirect
48. #22575374C22semidirect
49. #22575374C22semidirect
50. C22⋊#14524130C22semidirect
51. #22575365C22semidirect
52. #22575363C22semidirect
54. C22×D4×C269C22semidirect
58. C2×D8×C269C22semidirect
59. C2×D8×C269C22semidirect
60. C22×C8×C269C22semidirect
120. D8×C269D4semidirect
174. D4×C269C2×D4semidirect
180. C8×C269C2×D4semidirect
197. C22×C269AGL(1,ℤ/8)semidirect
225. C269#3264semidirect
228. C22×D8D269semidirect
246. C22×D4C2×D269semidirect
256. C2×D8C2×D269semidirect
257. C2×D8C2×D269semidirect
259. C22×C8C2×D269semidirect
328. D8C22⋊D269semidirect
387. D4C2×C22⋊D269semidirect
406. C8C2×C22⋊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
#317712492
D4×C22⋊D2692
#317709932
#317708552
#317708342
#317708182
#317707982
#317707562
#317707472
#317707392
#317707372
#317707172
#317706762
C22×D8×C2692
#3264269