Finite Group Databasebeta

← #46822374

Subgroups — #46822374

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×SD16×D269C211yesmaximal, characteristic, complemented
2#31771262C211yesmaximal, characteristic, complemented
3D4×C22⋊D269C211yesmaximal, characteristic, complemented
4#31770993C211yesmaximal, characteristic, complemented
5#31770851C211yesmaximal, characteristic, complemented
6#31770838C211yesmaximal, characteristic, complemented
7#31770818C211yesmaximal, characteristic, complemented
8#31770803C211yesmaximal, characteristic, complemented
9#31770756C211yesmaximal, characteristic, complemented
10#31770750C211yesmaximal, characteristic, complemented
11#31770742C211yesmaximal, characteristic, complemented
12#31770737C211yesmaximal, characteristic, complemented
13#31770706C211yesmaximal, characteristic, complemented
14#31770674C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
16C2×Q8×D269C2211yescharacteristic, complemented
17C22×C22⋊D269—21no—
18C2×D4×D269C2211yescharacteristic
19C2×#16864391C2211yescharacteristic, complemented
20C2×#16864390C2211yescharacteristic
21C2×#16864390C2211yescharacteristic, complemented
22C2×#16864389C2211yescharacteristic, complemented
23SD16×D269—24no—
24C2×#16864387C2211yescharacteristic, complemented
25C2×C8×D269C2211yescharacteristic, complemented
26#22575449C2211yescharacteristic
27#22575440—21no—
28#22575440C2211yescharacteristic, complemented
29#22575432—21no—
30#22575432C2211yescharacteristic, complemented
31#22575427C2211yescharacteristic
32#22575425—21no—
33#22575424—21no—
34#22575424C2211yescharacteristic
35#22575424C2211yescharacteristic, complemented
36#22575422—21no—
37#22575421—21no—
38Q8×Dic269C2211yescharacteristic, complemented
39D4×Dic269C2211yescharacteristic
40C4×C22⋊D269—21no—
41C4×C22⋊D269C2211yescharacteristic
42#22575387C2211yescharacteristic, complemented
43#22575382C2211yescharacteristic, complemented
44#22575378C2211yescharacteristic, complemented
45#22575377C2211yescharacteristic, complemented
46#22575375C2211yescharacteristic, complemented
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic, complemented
49#22575374C2211yescharacteristic, complemented
50C22⋊#14524130C2211yescharacteristic, complemented
51#22575365C2211yescharacteristic, complemented
52#22575363C2211yescharacteristic, complemented
53C2×#41×C269C2211yescharacteristic, complemented
54C22×D4×C269C2211yescharacteristic
55AGL(1,ℤ/8)×C269—24no—
56C2×SD16×C269C2211yescharacteristic
57C2×SD16×C269C2211yescharacteristic, complemented
58C2×D8×C269C2211yescharacteristic
59C2×D8×C269C2211yescharacteristic, complemented
60C2×M16×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62C2×Dic538—21no—
63C2×Dic538C2311yescharacteristic
64Q8×D269—22no—
65Q8×D269—41no—
66C2×D1076C2311yescharacteristic
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—
78C2×C4×D269—21no—
79C2×C4×D269C2311yescharacteristic
80#16864391—22no—
81#16864390—22no—
82#16864390—41no—
83#16864389—22no—
84#16864387—41no—
85C2×#14524130C2311yescharacteristic
86C2×#14524130C2311yescharacteristic
87C8×D269—22no—
88C4⋊Dic269—21no—
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⋊Dic269D411yescharacteristic
100#16864379—21no—
101#16864379D411yescharacteristic
102C4×Dic269—21no—
103C4×Dic269C2311yescharacteristic
104C24×C269—21no—
105#41×C269—22no—
106#41×C269—41no—
107C2×Q8×C269C2311yescharacteristic
108C2×D4×C269—22no—
109C2×D4×C269—21no—
110C2×D4×C269—21no—
111C2×D4×C269—41no—
112C2×D4×C269C2311yescharacteristic
113C2×D4×C269C2311yescharacteristic
114C2×D4×C269C2311yescharacteristic
115C22×C4×C269—21no—
116C22×C4×C269C2311yescharacteristic
117SD16×C269—22no—
118SD16×C269D414yescomplemented
119D8×C269—22no—
120D8×C269—22no—
121M16×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—
131Dic538—42no—
132D1076—21no—
133D1076—21no—
134C22⋊D269—22no—
135C22⋊D269—42no—
136C22⋊D269—42no—
137C22⋊D269—42no—
138C22⋊D269—42no—
139C22⋊D269—42no—
140C22⋊D269—81no—
141C2×Dic269—21no—
142C2×Dic269—21no—
143C2×Dic269—21no—
144C2×Dic269—42no—
145C2×Dic269C2×D411yescharacteristic
146C2×Dic269C2×D411yescharacteristic
147C2×Dic269C2×D411yescharacteristic
148C4×D269—21no—
149C4×D269—21no—
150C4×D269—42no—
151#14524130—21no—
152#14524130—21no—
153C23×C269—21no—
154C23×C269—21no—
155C23×C269—21no—
156C23×C269—42no—
157C23×C269—42no—
158C23×C269—81no—
159C23×C269C2×D411yescharacteristic
160Q8×C269—21no—
161Q8×C269C2×D412yescomplemented
162D4×C269—21no—
163D4×C269—21no—
164D4×C269—21no—
165D4×C269—21no—
166D4×C269—21no—
167D4×C269—42no—
168D4×C269—42no—
169D4×C269C2×D412yes—
170C2×C4×C269—21no—
171C2×C4×C269—21no—
172C2×C4×C269—21no—
173C2×C4×C269—42no—
174C2×C4×C269C2×D411yescharacteristic
175C2×C4×C269C2411yescharacteristic
176C8×C269—21no—
177C8×C269C2×D412yescomplemented
178C22×C269—21no—
179C22×C269—22no—
180C22×C269—21no—
181C22×C269—21no—
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×C269—84no—
191C22×C269C22×D411yescharacteristic
192C2×D269—21no—
193C2×D269—21no—
194C2×D269—41no—
195C2×D269—41no—
196C2×D269—41no—
197C2×D269—84no—
198C2×D269AGL(1,ℤ/8)12yescomplemented
199C4×C269—21no—
200C4×C269—22no—
201C4×C269C22×D411yescharacteristic
202C4×C269H(5,𝔽2)11yescharacteristic
203Dic269—21no—
204Dic269—21no—
205Dic269—21no—
206Dic269—41no—
207Dic269—41no—
208Dic269—41no—
209C2×C269—42no—
210C2×C269—41no—
211C2×C269—81no—
212C2×C269—81no—
213C2×C269D4211yescharacteristic
214C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C269#57711yescharacteristic
216D269—22no—
217D269—81no—
218C269#326511yescharacteristic, complemented, Sylow
219#3265—2691nomaximal, Sylow
220C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
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#435—2691no—
235C2×#41C2×D26911yescharacteristic, complemented
236C22×Q8—2691no—
237C22×D4—2691no—
238C22×D4—5381no—
239C22×D4C2×D26911yescharacteristic
240AGL(1,ℤ/8)—24no—
241C2×SD16—2691no—
242C2×SD16—2691no—
243C2×SD16—2691no—
244C2×SD16—2691no—
245C2×SD16—5384no—
246C2×SD16C2×D26911yescharacteristic
247C2×SD16C2×D26911yescharacteristic, complemented
248C2×D8—2691no—
249C2×D8C2×D26911yescharacteristic
250C2×D8C2×D26911yescharacteristic, complemented
251C2×M16C2×D26911yescharacteristic, complemented
252C22×C8—2691no—
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—2691no—
268C4×D4—5381no—
269#107—2691no—
270C4⋊C8—2691no—
271#103—2691no—
272#103—2691no—
273#102—2691no—
274#102—2691no—
275#102—2691no—
276#102—2691no—
277C22⋊C8—2691no—
278C22⋊C8—2691no—
279#97—2691no—
280C24—21no—
281C24—5381no—
282#41—22no—
283#41—41no—
284C2×Q8—2691no—
285C2×Q8—5381no—
286C2×Q8—5382no—
287C2×Q8—10761no—
288C2×Q8C22×D26911yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—2691no—
294C2×D4—2691no—
295C2×D4—5381no—
296C2×D4—5381no—
297C2×D4—5381no—
298C2×D4—5382no—
299C2×D4—5381no—
300C2×D4—5381no—
301C2×D4—5381no—
302C2×D4—10761no—
303C2×D4—10764no—
304C2×D4C22×D26911yescharacteristic
305C2×D4C22×D26911yescharacteristic
306C2×D4C22×D26911yescharacteristic
307C22×C4—21no—
308C22×C4—2691no—
309C22×C4—5381no—
310C22×C4—5381no—
311C22×C4C22×D26911yescharacteristic
312SD16—22no—
313SD16—5382no—
314SD16—5382no—
315SD16—5382no—
316SD16—10761no—
317SD16C22⋊D26914yescomplemented
318D8—22no—
319D8—22no—
320D8—10761no—
321M16—22no—
322C2×C8—2691no—
323C2×C8—2691no—
324C2×C8—5382no—
325C2×C8C22×D26911yescharacteristic
326C2×C8C22×D26911yescharacteristic
327C4⋊C4—2691no—
328C4⋊C4—2691no—
329C4⋊C4—2691no—
330C4⋊C4—5381no—
331C4⋊C4—5381no—
332C22⋊C4—2691no—
333C22⋊C4—2691no—
334C22⋊C4—5381no—
335C22⋊C4—5381no—
336C22⋊C4—5381no—
337C22⋊C4—5381no—
338C22⋊C4—5381no—
339C22⋊C4—5381no—
340C22⋊C4—5381no—
341C42—2691no—
342C42—5381no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—2691no—
350C23—5381no—
351C23—10762no—
352C23—10762no—
353C23—21521no—
354C23D4×D26911yescharacteristic
355Q8—21no—
356Q8—5381no—
357Q8—5381no—
358Q8—10762no—
359Q8C2×C22⋊D26912yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—5382no—
368D4—5381no—
369D4—5381no—
370D4—10762no—
371D4—10762no—
372D4—10762no—
373D4—10762no—
374D4—10762no—
375D4—21521no—
376D4C2×C22⋊D26912yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—2691no—
382C2×C4—2691no—
383C2×C4—2691no—
384C2×C4—5381no—
385C2×C4—5381no—
386C2×C4—5381no—
387C2×C4—5381no—
388C2×C4—5381no—
389C2×C4—10762no—
390C2×C4—10762no—
391C2×C4D4×D26911yescharacteristic
392C2×C4C23×D26911yescharacteristic
393C8—21no—
394C8—5381no—
395C8—5381no—
396C8C2×C22⋊D26912yescomplemented
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—2692no—
411C22—5381no—
412C22—5381no—
413C22—10761no—
414C22—10761no—
415C22—10761no—
416C22—21524no—
417C22C2×D4×D26911yescharacteristic
418C4—21no—
419C4—22no—
420C4—5381no—
421C4—5381no—
422C4—5381no—
423C4—10761no—
424C4—10761no—
425C4—10761no—
426C4C22×C22⋊D26911yescharacteristic
427C4#2257549811yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—5382no—
433C2—21521no—
434C2D4×C22⋊D26911yescharacteristic
435C2AGL(1,ℤ/8)×D26911yescharacteristic
436C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×AGL(1,ℤ/8)×C269C2
socleC22×C269C22×D4
hypercenterC2×AGL(1,ℤ/8)D269
hypocenterC269#3265
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D269C2semidirect
2. #31771262C2semidirect
3. D4×C22⋊D269C2semidirect
4. #31770993C2semidirect
5. #31770851C2semidirect
6. #31770838C2semidirect
7. #31770818C2semidirect
8. #31770803C2semidirect
9. #31770756C2semidirect
10. #31770750C2semidirect
11. #31770742C2semidirect
12. #31770737C2semidirect
13. #31770706C2semidirect
14. #31770674C2semidirect
15. C2×AGL(1,ℤ/8)×C269C2semidirect
16. C2×Q8×D269C22semidirect
19. C2×#16864391C22semidirect
21. C2×#16864390C22semidirect
22. C2×#16864389C22semidirect
24. C2×#16864387C22semidirect
25. C2×C8×D269C22semidirect
28. #22575440C22semidirect
30. #22575432C22semidirect
35. #22575424C22semidirect
38. Q8×Dic269C22semidirect
42. #22575387C22semidirect
43. #22575382C22semidirect
44. #22575378C22semidirect
45. #22575377C22semidirect
46. #22575375C22semidirect
48. #22575374C22semidirect
49. #22575374C22semidirect
50. C22⋊#14524130C22semidirect
51. #22575365C22semidirect
52. #22575363C22semidirect
53. C2×#41×C269C22semidirect
57. C2×SD16×C269C22semidirect
59. C2×D8×C269C22semidirect
60. C2×M16×C269C22semidirect
118. SD16×C269D4semidirect
161. Q8×C269C2×D4semidirect
177. C8×C269C2×D4semidirect
198. C2×D269AGL(1,ℤ/8)semidirect
218. C269#3265semidirect
220. C2×AGL(1,ℤ/8)D269semidirect
235. C2×#41C2×D269semidirect
247. C2×SD16C2×D269semidirect
250. C2×D8C2×D269semidirect
251. C2×M16C2×D269semidirect
317. SD16C22⋊D269semidirect
359. Q8C2×C22⋊D269semidirect
396. C8C2×C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2692
#317712622
D4×C22⋊D2692
#317709932
#317708512
#317708382
#317708182
#317708032
#317707562
#317707502
#317707422
#317707372
#317707062
#317706742
C2×AGL(1,ℤ/8)×C2692
#3265269