Finite Group Databasebeta

← #46822379

Subgroups — #46822379

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×#22575477C211yesmaximal, characteristic, complemented
2C2×SD16×D269C211yesmaximal, characteristic, complemented
3#31771263C211yesmaximal, characteristic, complemented
4#31771240C211yesmaximal, characteristic, complemented
5#31770848C211yesmaximal, characteristic, complemented
6#31770838C211yesmaximal, characteristic, complemented
7#31770817C211yesmaximal, characteristic, complemented
8#31770802C211yesmaximal, characteristic, complemented
9#31770754C211yesmaximal, characteristic, complemented
10#31770747C211yesmaximal, characteristic, complemented
11#31770742C211yesmaximal, characteristic, complemented
12#31770737C211yesmaximal, characteristic, complemented
13#31770707C211yesmaximal, characteristic, complemented
14#31770678C211yesmaximal, characteristic, complemented
15#496×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic, complemented
17C2×Q8×D269C2211yescharacteristic, complemented
18C2×D4×D269—21no—
19C2×D4×D269C2211yescharacteristic
20C2×D4×D269C2211yescharacteristic
21#22575477—24no—
22C2×#16864391C2211yescharacteristic
23C2×#16864390C2211yescharacteristic, complemented
24C2×#16864390C2211yescharacteristic, complemented
25C2×#16864389C2211yescharacteristic
26SD16×D269—24no—
27C2×#16864387C2211yescharacteristic
28C2×#16864385C2211yescharacteristic, complemented
29C2×C8×D269C2211yescharacteristic, complemented
30#22575439—21no—
31#22575430—21no—
32#22575429C2211yescharacteristic, complemented
33#22575426—21no—
34#22575424—21no—
35#22575424C2211yescharacteristic
36#22575424C2211yescharacteristic
37#22575424C2211yescharacteristic, complemented
38#22575421—21no—
39#22575421—21no—
40C4×Dic538C2211yescharacteristic, complemented
41C4×D1076C2211yescharacteristic
42C4×D1076C2211yescharacteristic
43#22575413—21no—
44#22575387C2211yescharacteristic, complemented
45C4⋊#14524130C2211yescharacteristic, complemented
46#22575378C2211yescharacteristic, complemented
47#22575377C2211yescharacteristic, complemented
48#22575375C2211yescharacteristic
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
59#124×C269C2211yescharacteristic, complemented
60#97×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62C23×D269—21no—
63#16864401—22no—
64#16864401—41no—
65C2×Dic538—21no—
66C2×Dic538C2311yescharacteristic
67Q8×D269—22no—
68Q8×D269—41no—
69C2×D1076D411yescharacteristic
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—41no—
80D4×D269—41no—
81D4×D269—44no—
82C22×Dic269—21no—
83C2×C4×D269—21no—
84C2×C4×D269—21no—
85C2×C4×D269C2311yescharacteristic
86C2×C4×D269C2311yescharacteristic
87#16864391D414yescomplemented
88#16864390—22no—
89#16864390—22no—
90#16864389—22no—
91#16864387—22no—
92#16864385—22no—
93C2×#14524130C2311yescharacteristic
94C2×#14524130C2311yescharacteristic
95C8×D269—22no—
96C4⋊Dic269D411yescharacteristic
97C4⋊Dic269C2311yescharacteristic
98C4⋊Dic269C2311yescharacteristic
99#16864381—21no—
100#16864381—21no—
101C22⋊Dic269—21no—
102C22⋊Dic269—21no—
103C22⋊Dic269—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379D411yescharacteristic
107#16864379D411yescharacteristic
108C4×Dic269—21no—
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—42no—
126C22×D269—42no—
127C22×D269—42no—
128C22×D269—42no—
129C22×D269—81no—
130C22×D269—81no—
131C22×D269C2×D411yescharacteristic
132C22×D269C2×D411yescharacteristic
133Dic538—21no—
134Dic538—42no—
135Dic538C2×D412yescomplemented
136D1076—22no—
137D1076—21no—
138D1076C2×D412yes—
139C22⋊D269—42no—
140C22⋊D269—42no—
141C22⋊D269—42no—
142C22⋊D269—42no—
143C22⋊D269—42no—
144C22⋊D269—42no—
145C2×Dic269—21no—
146C2×Dic269—21no—
147C2×Dic269—21no—
148C2×Dic269—42no—
149C2×Dic269C2×D411yescharacteristic
150C2×Dic269C2×D411yescharacteristic
151C4×D269—21no—
152C4×D269—21no—
153C4×D269—21no—
154C4×D269—21no—
155C4×D269—42no—
156C4×D269—42no—
157#14524130—21no—
158#14524130—21no—
159C23×C269—21no—
160C23×C269—21no—
161C23×C269—21no—
162Q8×C269—21no—
163Q8×C269—21no—
164D4×C269—21no—
165D4×C269—21no—
166D4×C269—21no—
167D4×C269—21no—
168D4×C269—21no—
169D4×C269—21no—
170D4×C269—42no—
171D4×C269—81no—
172C2×C4×C269—21no—
173C2×C4×C269C2×D411yescharacteristic
174C2×C4×C269C2×D411yescharacteristic
175C2×C4×C269C2411yescharacteristic
176C8×C269—21no—
177C8×C269C2×D412yescomplemented
178C22×C269—41no—
179C22×C269—41no—
180C22×C269—41no—
181C22×C269—41no—
182C22×C269—41no—
183C22×C269—41no—
184C22×C269—41no—
185C22×C269—41no—
186C22×C269—41no—
187C22×C269C22×D411yescharacteristic
188C2×D269—21no—
189C2×D269—21no—
190C2×D269—21no—
191C2×D269—22no—
192C2×D269—21no—
193C2×D269—21no—
194C2×D269—41no—
195C2×D269—41no—
196C2×D269—84no—
197C2×D269—84no—
198C2×D269AGL(1,ℤ/8)12yescomplemented
199C4×C269—21no—
200C4×C269—21no—
201C4×C269—41no—
202C4×C269C22×D411yescharacteristic
203C4×C269H(5,𝔽2)11yescharacteristic
204Dic269—21no—
205Dic269—21no—
206Dic269—22no—
207Dic269—41no—
208Dic269—41no—
209C2×C269—81no—
210C2×C269—81no—
211C2×C269—81no—
212C2×C269D4211yescharacteristic
213C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C269#57711yescharacteristic
215D269—22no—
216D269—41no—
217D269—42no—
218C269#326511yescharacteristic, complemented, Sylow
219#3265—2691nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2691no—
221C22×SD16—2691no—
222#548—2691no—
223D42—2691no—
224#496D26911yescharacteristic, complemented
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×#41—2691no—
236C22×Q8—2691no—
237C22×D4—2691no—
238C22×D4—2691no—
239C22×D4—5381no—
240AGL(1,ℤ/8)—5384no—
241C2×SD16—2691no—
242C2×SD16—2691no—
243C2×SD16—2691no—
244C2×SD16—2691no—
245C2×SD16—5384no—
246C2×SD16C2×D26911yescharacteristic, complemented
247C2×SD16C2×D26911yescharacteristic, complemented
248C2×D8—2691no—
249C2×D8C2×D26911yescharacteristic, complemented
250C2×D8C2×D26911yescharacteristic, complemented
251C2×M16—2691no—
252C22×C8—2691no—
253#127C2×D26911yescharacteristic
254#124—5381no—
255#124C2×D26911yescharacteristic, complemented
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#97C2×D26911yescharacteristic, complemented
280C24—5381no—
281C24—5381no—
282#41—5382no—
283#41—10761no—
284C2×Q8—2691no—
285C2×Q8—5381no—
286C2×Q8—5382no—
287C2×Q8—10761no—
288C2×Q8C22×D26911yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2691no—
292C2×D4—2691no—
293C2×D4—5381no—
294C2×D4—5381no—
295C2×D4—5382no—
296C2×D4—5382no—
297C2×D4—5381no—
298C2×D4—5381no—
299C2×D4—5381no—
300C2×D4—5381no—
301C2×D4—10761no—
302C2×D4—10761no—
303C2×D4—10764no—
304C2×D4C22×D26911yescharacteristic
305C2×D4C22×D26911yescharacteristic
306C2×D4C22×D26911yescharacteristic
307C22×C4—2691no—
308C22×C4—2691no—
309C22×C4—5381no—
310C22×C4—5381no—
311C22×C4—5381no—
312SD16—22no—
313SD16—41no—
314SD16—2694no—
315SD16—5382no—
316SD16—5382no—
317SD16—5382no—
318D8—22no—
319D8—41no—
320D8—5382no—
321M16—5382no—
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—21no—
333C22⋊C4—21no—
334C22⋊C4—2691no—
335C22⋊C4—2691no—
336C22⋊C4—5381no—
337C22⋊C4—5381no—
338C22⋊C4—5381no—
339C22⋊C4—5381no—
340C22⋊C4—5381no—
341C42—5381no—
342C42C22×D26911yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—2691no—
347C23—2691no—
348C23—5381no—
349C23—10762no—
350C23—10762no—
351C23—10762no—
352C23—10762no—
353C23—21521no—
354C23—21521no—
355Q8—21no—
356Q8—21no—
357Q8—2692no—
358Q8—5381no—
359Q8—10762no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—2692no—
369D4—5382no—
370D4—5381no—
371D4—10762no—
372D4—10762no—
373D4—10762no—
374D4—10762no—
375D4—10762no—
376D4—10762no—
377C2×C4—21no—
378C2×C4—2691no—
379C2×C4—2691no—
380C2×C4—5381no—
381C2×C4—5381no—
382C2×C4—5381no—
383C2×C4—5381no—
384C2×C4—5381no—
385C2×C4—5381no—
386C2×C4—5381no—
387C2×C4—10762no—
388C2×C4—10762no—
389C2×C4—10762no—
390C2×C4D4×D26911yescharacteristic
391C2×C4D4×D26911yescharacteristic
392C2×C4C23×D26911yescharacteristic
393C8—21no—
394C8—5381no—
395C8—5381no—
396C8D4×D26912yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—2692no—
407C22—5381no—
408C22—5381no—
409C22—5381no—
410C22—5381no—
411C22—5382no—
412C22—5381no—
413C22—10761no—
414C22—10761no—
415C22—21524no—
416C22—21524no—
417C22C2×D4×D26911yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—5381no—
422C4—5381no—
423C4—5382no—
424C4—10761no—
425C4—10761no—
426C4C2×D4×D26911yescharacteristic
427C4#2257549811yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—5382no—
432C2—10761no—
433C2—10762no—
434C2#3177124011yescharacteristic
435C2AGL(1,ℤ/8)×D26911yescharacteristic
436C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#496×C269C2
socleC22×C269C22×D4
hypercenter#496D269
hypocenterC269#3265
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×#22575477C2semidirect
2. C2×SD16×D269C2semidirect
3. #31771263C2semidirect
4. #31771240C2semidirect
5. #31770848C2semidirect
6. #31770838C2semidirect
7. #31770817C2semidirect
8. #31770802C2semidirect
9. #31770754C2semidirect
10. #31770747C2semidirect
11. #31770742C2semidirect
12. #31770737C2semidirect
13. #31770707C2semidirect
14. #31770678C2semidirect
15. #496×C269C2semidirect
16. C2×#16864401C22semidirect
17. C2×Q8×D269C22semidirect
23. C2×#16864390C22semidirect
24. C2×#16864390C22semidirect
28. C2×#16864385C22semidirect
29. C2×C8×D269C22semidirect
32. #22575429C22semidirect
37. #22575424C22semidirect
40. C4×Dic538C22semidirect
44. #22575387C22semidirect
45. C4⋊#14524130C22semidirect
46. #22575378C22semidirect
47. #22575377C22semidirect
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
59. #124×C269C22semidirect
60. #97×C269C22semidirect
87. #16864391D4semidirect
135. Dic538C2×D4semidirect
177. C8×C269C2×D4semidirect
198. C2×D269AGL(1,ℤ/8)semidirect
218. C269#3265semidirect
224. #496D269semidirect
246. C2×SD16C2×D269semidirect
247. C2×SD16C2×D269semidirect
249. C2×D8C2×D269semidirect
250. C2×D8C2×D269semidirect
255. #124C2×D269semidirect
279. #97C2×D269semidirect
396. C8D4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754772
C2×SD16×D2692
#317712632
#317712402
#317708482
#317708382
#317708172
#317708022
#317707542
#317707472
#317707422
#317707372
#317707072
#317706782
#496×C2692
#3265269