Finite Group Databasebeta

← #46822652

Subgroups — #46822652

Order 34432.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771417C211yesmaximal, characteristic, complemented
2#31771412C211yesmaximal, characteristic, complemented
3#31771410C211yesmaximal, characteristic, complemented
4#31771407C211yesmaximal, characteristic, complemented
5#31770994C211yesmaximal, characteristic, complemented
6#31770876C211yesmaximal, characteristic, complemented
7#31770875C211yesmaximal, characteristic, complemented
8#31770869C211yesmaximal, characteristic, complemented
9#31770868C211yesmaximal, characteristic, complemented
10#31770784C211yesmaximal, characteristic, complemented
11#31770779C211yesmaximal, characteristic, complemented
12#31770774C211yesmaximal, characteristic, complemented
13#31770774C211yesmaximal, characteristic, complemented
14#31770728C211yesmaximal, characteristic, complemented
15#444×C269C211yesmaximal, characteristic, complemented
16#22575500C2211yescharacteristic, complemented
17#22575499C2211yescharacteristic
18#22575499C2211yescharacteristic, complemented
19#22575499C2211yescharacteristic, complemented
20#22575486—21no—
21#22575486C2211yescharacteristic, complemented
22#22575485—21no—
23#22575483C2211yescharacteristic, complemented
24#22575481—21no—
25#22575481—21no—
26#22575481—21no—
27#22575481C2211yescharacteristic
28#22575481C2211yescharacteristic, complemented
29#22575481C2211yescharacteristic, complemented
30#22575480—21no—
31#22575480—21no—
32#22575478C2211yescharacteristic
33#22575478C2211yescharacteristic, complemented
34#22575478C2211yescharacteristic, complemented
35#22575477—21no—
36#22575476—21no—
37#22575475—21no—
38#22575475—21no—
39#22575475C2211yescharacteristic
40#22575474—21no—
41C2×#16864391—21no—
42C2×#16864391C2211yescharacteristic, complemented
43C2×#16864391C2211yescharacteristic, complemented
44SD16×D269—21no—
45C2×D2152—21no—
46C2×D2152C2211yescharacteristic, complemented
47C2×D2152C2211yescharacteristic, complemented
48D8×D269—21no—
49#22575459C2211yescharacteristic, complemented
50#22575459C2211yescharacteristic, complemented
51#22575450C2211yescharacteristic, complemented
52#22575441C2211yescharacteristic, complemented
53#22575389C2211yescharacteristic, complemented
54#22575381C2211yescharacteristic, complemented
55#22575381C2211yescharacteristic, complemented
56#22575371C2211yescharacteristic, complemented
57#22575369C2211yescharacteristic, complemented
58#22575369C2211yescharacteristic, complemented
59#22575369C2211yescharacteristic, complemented
60#135×C269C2211yescharacteristic
61#131×C269C2211yescharacteristic, complemented
62#131×C269C2211yescharacteristic, complemented
63#108×C269C2211yescharacteristic, complemented
64#104×C269C2211yescharacteristic, complemented
65#104×C269C2211yescharacteristic, complemented
66#97×C269C2211yescharacteristic, complemented
67#16864403—21no—
68#16864403—21no—
69#16864403—21no—
70#16864402—21no—
71#16864402—21no—
72#16864402—21no—
73#16864402—21no—
74#16864402C2311yescharacteristic
75#16864402C2311yescharacteristic
76#16864401—21no—
77#16864401—41no—
78C2×Dic538—21no—
79C2×Dic538C2311yescharacteristic
80Q8×D269—21no—
81C2×D1076—21no—
82C2×D1076—21no—
83C2×D1076—21no—
84C2×D1076—41no—
85C2×D1076C2311yescharacteristic
86C2×D1076C2311yescharacteristic
87C2×D1076C2311yescharacteristic
88D4×D269—21no—
89D4×D269—21no—
90D4×D269—21no—
91D4×D269—41no—
92D4×D269—41no—
93D4×D269—41no—
94Dic1076—21no—
95Dic1076D411yescharacteristic, complemented
96#16864392—21no—
97#16864391—21no—
98#16864391—21no—
99#16864391—21no—
100#16864391—21no—
101#16864391—21no—
102#16864391—21no—
103#16864391—21no—
104#16864391—21no—
105#16864391D411yescharacteristic, complemented
106#16864391D411yescharacteristic, complemented
107#16864390—21no—
108#16864389—21no—
109#16864389—21no—
110#16864389—21no—
111D2152—21no—
112D2152—21no—
113D2152—21no—
114D2152—21no—
115D2152—21no—
116D2152—21no—
117D2152—21no—
118D2152D411yescharacteristic, complemented
119#16864387—21no—
120#16864387—21no—
121#16864387—21no—
122#16864386C2311yescharacteristic
123#16864386C2311yescharacteristic
124#16864385—21no—
125#16864385—21no—
126C8×D269—21no—
127C8×D269—21no—
128#16864379—41no—
129#41×C269C2311yescharacteristic
130#41×C269C2311yescharacteristic
131Q16×C269D411yescharacteristic, complemented
132SD16×C269D411yescharacteristic, complemented
133SD16×C269D411yescharacteristic, complemented
134D8×C269D411yescharacteristic, complemented
135M16×C269—21no—
136M16×C269—21no—
137M16×C269C2311yescharacteristic
138M16×C269C2311yescharacteristic
139C2×C8×C269—21no—
140C2×C8×C269—21no—
141C2×C8×C269C2311yescharacteristic
142C2×C8×C269C2311yescharacteristic
143C42×C269C2311yescharacteristic
144C22×D269—21no—
145C22×D269—21no—
146C22×D269—21no—
147C22×D269—41no—
148C22×D269—41no—
149C22×D269—41no—
150Dic538—21no—
151Dic538—21no—
152Dic538—41no—
153Dic538C2×D411yescharacteristic
154Dic538C2×D411yescharacteristic
155D1076—21no—
156D1076—21no—
157D1076—21no—
158D1076—21no—
159D1076—21no—
160D1076—21no—
161D1076—41no—
162D1076—41no—
163D1076—41no—
164D1076—82no—
165D1076C2×D411yescharacteristic
166D1076C2×D411yescharacteristic
167C22⋊D269—21no—
168C22⋊D269—21no—
169C22⋊D269—41no—
170C22⋊D269—41no—
171C22⋊D269—41no—
172C22⋊D269—41no—
173C2×Dic269—21no—
174C2×Dic269—41no—
175C4×D269—21no—
176C4×D269—21no—
177C4×D269—41no—
178C4×D269—41no—
179C4×D269—41no—
180C4×D269—41no—
181#14524130—21no—
182#14524130—21no—
183Q8×C269C2×D411yescharacteristic
184Q8×C269C2×D411yescharacteristic
185D4×C269—21no—
186D4×C269—21no—
187D4×C269C2×D411yescharacteristic
188D4×C269C2×D411yescharacteristic
189C2×C4×C269—21no—
190C2×C4×C269—21no—
191C2×C4×C269—21no—
192C2×C4×C269C2411yescharacteristic
193C8×C269—21no—
194C8×C269—21no—
195C8×C269C2×D411yescharacteristic
196C8×C269C2×D411yescharacteristic
197C8×C269C2×D411yescharacteristic
198C8×C269C2×D411yescharacteristic
199C22×C269—21no—
200C22×C269—21no—
201C22×C269H(5,𝔽2)11yescharacteristic
202C2×D269—21no—
203C2×D269—21no—
204C2×D269—41no—
205C2×D269—41no—
206C2×D269—41no—
207C2×D269—81no—
208C2×D269—81no—
209C2×D269—81no—
210C2×D269—161no—
211C2×D269—161no—
212C2×D269—161no—
213C4×C269—21no—
214C4×C269—21no—
215C4×C269—41no—
216C4×C269C22×D411yescharacteristic
217C4×C269C22×D411yescharacteristic
218Dic269—21no—
219Dic269—21no—
220Dic269—41no—
221C2×C269—21no—
222C2×C269—41no—
223C2×C269—41no—
224C2×C269D4211yescharacteristic
225D269—41no—
226D269—41no—
227D269—81no—
228D269—81no—
229D269—81no—
230C269#328811yescharacteristic, complemented, Sylow
231#3288—2691nomaximal, Sylow
232#577—2691no—
233#577—2691no—
234#576—2691no—
235#576—2691no—
236#496—2691no—
237#472—2691no—
238#472—2691no—
239#471—2691no—
240#471—2691no—
241#455—2691no—
242#454—2691no—
243#453—2691no—
244#453—2691no—
245#444—2691no—
246#444D26911yescharacteristic, complemented
2472−1+4—2691no—
248H(5,𝔽2)—2691no—
249H(5,𝔽2)—2691no—
250H(5,𝔽2)—2691no—
251#137—2691no—
252#137—2691no—
253#137—5381no—
254#137—5381no—
255AGL(1,ℤ/8)—2691no—
256AGL(1,ℤ/8)—2691no—
257AGL(1,ℤ/8)—2691no—
258AGL(1,ℤ/8)—2691no—
259AGL(1,ℤ/8)—2691no—
260AGL(1,ℤ/8)—2691no—
261AGL(1,ℤ/8)—5381no—
262AGL(1,ℤ/8)—5381no—
263AGL(1,ℤ/8)—5381no—
264AGL(1,ℤ/8)—5381no—
265AGL(1,ℤ/8)—5381no—
266AGL(1,ℤ/8)—5381no—
267#135—2691no—
268#135—5381no—
269#135—5381no—
270#135—5381no—
271#135—5381no—
272#135C2×D26911yescharacteristic
273C2×SD16—2691no—
274C2×SD16—2691no—
275C2×SD16—5381no—
276C2×SD16—5381no—
277C2×D8—2691no—
278C2×D8—2691no—
279C2×D8—5381no—
280C2×D8—5381no—
281#131—2691no—
282#131—2691no—
283#131C2×D26911yescharacteristic, complemented
284#131C2×D26911yescharacteristic, complemented
285#127—2691no—
286#124—2691no—
287#108—2691no—
288#108C2×D26911yescharacteristic, complemented
289#104—2691no—
290#104—2691no—
291#104C2×D26911yescharacteristic, complemented
292#104C2×D26911yescharacteristic, complemented
293#101—2691no—
294#100—2691no—
295#100—2691no—
296#100—2691no—
297#97C2×D26911yescharacteristic, complemented
298#41—2691no—
299#41—2691no—
300#41—5381no—
301#41—5381no—
302#41—5381no—
303#41—5381no—
304#41—5381no—
305#41—5381no—
306#41—5381no—
307#41—5381no—
308#41—10761no—
309#41C22×D26911yescharacteristic
310#41C22×D26911yescharacteristic
311C2×Q8—2691no—
312C2×Q8—5381no—
313C2×Q8—5381no—
314C2×D4—2691no—
315C2×D4—2691no—
316C2×D4—2691no—
317C2×D4—5381no—
318C2×D4—5381no—
319C2×D4—5381no—
320C2×D4—5381no—
321C2×D4—5381no—
322C2×D4—5381no—
323C2×D4—10761no—
324C2×D4—10761no—
325C2×D4—10761no—
326C2×D4—10761no—
327Q16—2691no—
328Q16—5381no—
329Q16—5381no—
330Q16D107611yescharacteristic, complemented
331SD16—2691no—
332SD16—2691no—
333SD16—5381no—
334SD16—5381no—
335SD16—5381no—
336SD16—5381no—
337SD16—5381no—
338SD16—5381no—
339SD16—5381no—
340SD16—5381no—
341SD16—5381no—
342SD16—5381no—
343SD16—5381no—
344SD16—5381no—
345SD16D107611yescharacteristic, complemented
346SD16D107611yescharacteristic, complemented
347D8—2691no—
348D8—5381no—
349D8—5381no—
350D8—5381no—
351D8—5381no—
352D8—5381no—
353D8—5381no—
354D8—5381no—
355D8—5381no—
356D8—5381no—
357D8—5381no—
358D8D107611yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—2691no—
362M16—2691no—
363M16—5381no—
364M16—5381no—
365M16C22×D26911yescharacteristic
366M16C22×D26911yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—5381no—
370C2×C8—5381no—
371C2×C8C22×D26911yescharacteristic
372C2×C8C22×D26911yescharacteristic
373C22⋊C4—10761no—
374C42C22×D26911yescharacteristic
375C23—5381no—
376C23—5381no—
377C23—5381no—
378C23—10761no—
379C23—10761no—
380C23—10761no—
381Q8—2691no—
382Q8—2691no—
383Q8—5381no—
384Q8—5381no—
385Q8—10761no—
386Q8C2×D107611yescharacteristic
387Q8C2×D107611yescharacteristic
388D4—21no—
389D4—21no—
390D4—2691no—
391D4—2691no—
392D4—5381no—
393D4—5381no—
394D4—5381no—
395D4—5381no—
396D4—5381no—
397D4—5381no—
398D4—5381no—
399D4—5381no—
400D4—10761no—
401D4—10761no—
402D4—10761no—
403D4—10761no—
404D4—10761no—
405D4—10761no—
406D4—10761no—
407D4—21522no—
408D4C2×D107611yescharacteristic
409D4C2×D107611yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—5381no—
414C2×C4—5381no—
415C2×C4—5381no—
416C2×C4—10761no—
417C2×C4—10761no—
418C2×C4—10761no—
419C2×C4—10761no—
420C2×C4—10761no—
421C2×C4C23×D26911yescharacteristic
422C8—21no—
423C8—21no—
424C8—5381no—
425C8—5381no—
426C8D4×D26911yescharacteristic
427C8D4×D26911yescharacteristic
428C8C2×D107611yescharacteristic
429C8C2×D107611yescharacteristic
430C22—21no—
431C22—21no—
432C22—5381no—
433C22—5381no—
434C22—10761no—
435C22—10761no—
436C22—10761no—
437C22—21521no—
438C22—21521no—
439C22—21521no—
440C22—43041no—
441C22—43041no—
442C22—43041no—
443C22#2257549911yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—5381no—
448C4—5381no—
449C4—10761no—
450C4C2×D4×D26911yescharacteristic
451C4C22×D107611yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—10761no—
456C2—10761no—
457C2—21521no—
458C2—21521no—
459C2—21521no—
460C2D4×D107611yescharacteristic
subgroupG/H
centerC2D4×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#444×C269C2
socleC2×C269D42
hypercenter#444D269
hypocenterC269#3288
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #31771417C2semidirect
2. #31771412C2semidirect
3. #31771410C2semidirect
4. #31771407C2semidirect
5. #31770994C2semidirect
6. #31770876C2semidirect
7. #31770875C2semidirect
8. #31770869C2semidirect
9. #31770868C2semidirect
10. #31770784C2semidirect
11. #31770779C2semidirect
12. #31770774C2semidirect
13. #31770774C2semidirect
14. #31770728C2semidirect
15. #444×C269C2semidirect
16. #22575500C22semidirect
18. #22575499C22semidirect
19. #22575499C22semidirect
21. #22575486C22semidirect
23. #22575483C22semidirect
28. #22575481C22semidirect
29. #22575481C22semidirect
33. #22575478C22semidirect
34. #22575478C22semidirect
42. C2×#16864391C22semidirect
43. C2×#16864391C22semidirect
46. C2×D2152C22semidirect
47. C2×D2152C22semidirect
49. #22575459C22semidirect
50. #22575459C22semidirect
51. #22575450C22semidirect
52. #22575441C22semidirect
53. #22575389C22semidirect
54. #22575381C22semidirect
55. #22575381C22semidirect
56. #22575371C22semidirect
57. #22575369C22semidirect
58. #22575369C22semidirect
59. #22575369C22semidirect
61. #131×C269C22semidirect
62. #131×C269C22semidirect
63. #108×C269C22semidirect
64. #104×C269C22semidirect
65. #104×C269C22semidirect
66. #97×C269C22semidirect
95. Dic1076D4semidirect
105. #16864391D4semidirect
106. #16864391D4semidirect
118. D2152D4semidirect
131. Q16×C269D4semidirect
132. SD16×C269D4semidirect
133. SD16×C269D4semidirect
134. D8×C269D4semidirect
230. C269#3288semidirect
246. #444D269semidirect
283. #131C2×D269semidirect
284. #131C2×D269semidirect
288. #108C2×D269semidirect
291. #104C2×D269semidirect
292. #104C2×D269semidirect
297. #97C2×D269semidirect
330. Q16D1076semidirect
345. SD16D1076semidirect
346. SD16D1076semidirect
358. D8D1076semidirect

Maximal subgroups

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

subgroupindex
#317714172
#317714122
#317714102
#317714072
#317709942
#317708762
#317708752
#317708692
#317708682
#317707842
#317707792
#317707742
#317707282
#444×C2692
#3288269