Finite Group Databasebeta

← #46822677

Subgroups — #46822677

Order 34432.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771417C211yesmaximal, characteristic, complemented
2#31771415C211yesmaximal, characteristic, complemented
3#31771414C211yesmaximal, characteristic, complemented
4#31771411C211yesmaximal, characteristic, complemented
5#31770976C211yesmaximal, characteristic, complemented
6#31770871C211yesmaximal, characteristic, complemented
7#31770869C211yesmaximal, characteristic, complemented
8#31770867C211yesmaximal, characteristic, complemented
9#31770790C211yesmaximal, characteristic, complemented
10#31770786C211yesmaximal, characteristic, complemented
11#31770775C211yesmaximal, characteristic, complemented
12#31770771C211yesmaximal, characteristic, complemented
13#31770721C211yesmaximal, characteristic, complemented
14#31770720C211yesmaximal, characteristic, complemented
15#471×C269C211yesmaximal, characteristic, complemented
16#22575501C2211yescharacteristic
17#22575500C2211yescharacteristic
18#22575499C2211yescharacteristic, complemented
19#22575498C2211yescharacteristic, complemented
20#22575486—21no—
21#22575486C2211yescharacteristic, complemented
22#22575486C2211yescharacteristic, complemented
23#22575485—21no—
24#22575484C2211yescharacteristic, complemented
25#22575482—21no—
26#22575482—21no—
27#22575481—21no—
28#22575481C2211yescharacteristic
29#22575481C2211yescharacteristic
30#22575480—21no—
31#22575480—21no—
32#22575479C2211yescharacteristic
33#22575477—21no—
34#22575476—21no—
35#22575475—21no—
36#22575475C2211yescharacteristic
37#22575474—21no—
38#22575473C2211yescharacteristic
39#22575472—21no—
40C2×#16864391—21no—
41C2×#16864391C2211yescharacteristic, complemented
42C2×#16864390C2211yescharacteristic, complemented
43C2×#16864389C2211yescharacteristic, complemented
44SD16×D269—21no—
45SD16×D269—21no—
46SD16×D269—21no—
47#22575461C2211yescharacteristic, complemented
48#22575461C2211yescharacteristic, complemented
49#22575459C2211yescharacteristic, complemented
50#22575451C2211yescharacteristic, complemented
51#22575449C2211yescharacteristic
52#22575388C2211yescharacteristic, complemented
53#22575380C2211yescharacteristic, complemented
54#22575379C2211yescharacteristic, complemented
55#22575379C2211yescharacteristic, complemented
56#22575379C2211yescharacteristic, complemented
57#22575371C2211yescharacteristic, complemented
58#22575369C2211yescharacteristic, complemented
59C8×Dic269C2211yescharacteristic, complemented
60#137×C269C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C269C2211yescharacteristic
62C2×SD16×C269C2211yescharacteristic, complemented
63#131×C269C2211yescharacteristic, complemented
64#108×C269C2211yescharacteristic, complemented
65#101×C269C2211yescharacteristic, complemented
66#100×C269C2211yescharacteristic, complemented
67#16864403—21no—
68#16864403—21no—
69#16864403—21no—
70#16864402—21no—
71#16864402—21no—
72#16864402—41no—
73#16864402C2311yescharacteristic
74#16864402C2311yescharacteristic
75#16864402C2311yescharacteristic
76#16864401—21no—
77#16864401—21no—
78#16864401—21no—
79#16864401—41no—
80C2×Dic538—21no—
81C2×Dic538C2311yescharacteristic
82Q8×D269—21no—
83Q8×D269—21no—
84Q8×D269—21no—
85C2×D1076—21no—
86C2×D1076C2311yescharacteristic
87C2×C22⋊D269—41no—
88C2×C22⋊D269—41no—
89D4×D269—21no—
90D4×D269—21no—
91D4×D269—21no—
92D4×D269—41no—
93Dic1076—21no—
94Dic1076—21no—
95Dic1076D411yescharacteristic, complemented
96#16864392—21no—
97#16864392D411yescharacteristic, complemented
98#16864391—21no—
99#16864391—21no—
100#16864391—21no—
101#16864391—21no—
102#16864391—21no—
103#16864391—21no—
104#16864391D411yescharacteristic, complemented
105#16864391D411yescharacteristic, complemented
106#16864390—21no—
107#16864390—21no—
108#16864390—21no—
109#16864390D411yescharacteristic, complemented
110#16864389—21no—
111#16864389—21no—
112#16864389—21no—
113#16864389D411yescharacteristic, complemented
114D2152—21no—
115D2152—21no—
116D2152D411yescharacteristic, complemented
117#16864387—21no—
118#16864387D411yescharacteristic, complemented
119#16864386C2311yescharacteristic
120#16864385—21no—
121#16864385—21no—
122#16864385—21no—
123C2×#14524130C2311yescharacteristic
124C8×D269—21no—
125C8×D269—21no—
126C8×D269—21no—
127#16864381—41no—
128C4×Dic269C2311yescharacteristic
129#41×C269C2311yescharacteristic
130C2×Q8×C269C2311yescharacteristic
131C2×D4×C269C2311yescharacteristic
132Q16×C269—21no—
133SD16×C269—21no—
134SD16×C269—21no—
135SD16×C269—21no—
136SD16×C269—21no—
137D8×C269—21no—
138M16×C269—21no—
139M16×C269C2311yescharacteristic
140M16×C269C2311yescharacteristic
141M16×C269C2311yescharacteristic
142C2×C8×C269—21no—
143C2×C8×C269C2311yescharacteristic
144C22×D269—21no—
145C22×D269—41no—
146C22×D269—41no—
147Dic538—21no—
148Dic538—21no—
149Dic538—41no—
150Dic538—41no—
151Dic538C2×D411yescharacteristic
152Dic538C2×D411yescharacteristic
153Dic538C2×D411yescharacteristic
154D1076—21no—
155D1076—21no—
156D1076—41no—
157D1076—41no—
158D1076C2×D411yescharacteristic
159D1076C2×D411yescharacteristic
160D1076C2×D411yescharacteristic
161C22⋊D269—21no—
162C22⋊D269—21no—
163C22⋊D269—21no—
164C22⋊D269—41no—
165C22⋊D269—41no—
166C22⋊D269—41no—
167C22⋊D269—41no—
168C22⋊D269—82no—
169C2×Dic269—21no—
170C2×Dic269—21no—
171C2×Dic269—41no—
172C2×Dic269—41no—
173C4×D269—21no—
174C4×D269—21no—
175C4×D269—21no—
176C4×D269—41no—
177C4×D269—41no—
178C4×D269—41no—
179C4×D269—41no—
180#14524130—21no—
181#14524130C2×D411yescharacteristic
182#14524130C2×D411yescharacteristic
183C23×C269—21no—
184Q8×C269—21no—
185Q8×C269—21no—
186Q8×C269C2×D411yescharacteristic
187D4×C269—21no—
188D4×C269—21no—
189D4×C269—21no—
190D4×C269C2×D411yescharacteristic
191C2×C4×C269—21no—
192C2×C4×C269—21no—
193C2×C4×C269C2411yescharacteristic
194C8×C269—21no—
195C8×C269—21no—
196C8×C269—21no—
197C8×C269C2×D411yescharacteristic
198C8×C269C2×D411yescharacteristic
199C22×C269—21no—
200C22×C269—41no—
201C22×C269—81no—
202C22×C269H(5,𝔽2)11yescharacteristic
203C2×D269—21no—
204C2×D269—21no—
205C2×D269—21no—
206C2×D269—41no—
207C2×D269—81no—
208C2×D269—161no—
209C2×D269—161no—
210C4×C269—21no—
211C4×C269—41no—
212C4×C269C22×D411yescharacteristic
213C4×C269C22×D411yescharacteristic
214Dic269—22no—
215Dic269—21no—
216Dic269—21no—
217Dic269—21no—
218Dic269—41no—
219C2×C269—21no—
220C2×C269—41no—
221C2×C269—81no—
222C2×C269D4211yescharacteristic
223D269—41no—
224D269—41no—
225D269—41no—
226D269—81no—
227C269#329111yescharacteristic, complemented, Sylow
228#3291—2691nomaximal, Sylow
229#577—2691no—
230#577—2691no—
231#577—2691no—
232#577—2691no—
233#492—2691no—
234#471—2691no—
235#471—2691no—
236#471—2691no—
237#471D26911yescharacteristic, complemented
238#456—2691no—
239#456—2691no—
240#453—2691no—
241#453—2691no—
242#443—2691no—
243#443—2691no—
2442−1+4—2691no—
2452−1+4—2691no—
246H(5,𝔽2)—2691no—
247H(5,𝔽2)—2691no—
248#137—2691no—
249#137—2691no—
250#137—2691no—
251#137—5381no—
252#137—5381no—
253#137—5381no—
254#137—5381no—
255#137C2×D26911yescharacteristic, complemented
256AGL(1,ℤ/8)—2691no—
257AGL(1,ℤ/8)—2691no—
258AGL(1,ℤ/8)—2691no—
259AGL(1,ℤ/8)—5381no—
260AGL(1,ℤ/8)—5381no—
261AGL(1,ℤ/8)—5381no—
262AGL(1,ℤ/8)—5381no—
263AGL(1,ℤ/8)C2×D26911yescharacteristic
264#135—2691no—
265#135—2691no—
266#135—5381no—
267#135—5381no—
268#135—5381no—
269#135—5381no—
270C2×SD16—2691no—
271C2×SD16—2691no—
272C2×SD16—2691no—
273C2×SD16—5381no—
274C2×SD16—5381no—
275C2×SD16—5381no—
276C2×SD16—5381no—
277C2×SD16C2×D26911yescharacteristic, complemented
278#131—2691no—
279#131—2691no—
280#131—2691no—
281#131C2×D26911yescharacteristic, complemented
282C4⋊Q8—2691no—
283#127—2691no—
284#108—2691no—
285#108C2×D26911yescharacteristic, complemented
286#104—2691no—
287#104—2691no—
288#104—2691no—
289#104—2691no—
290#101—2691no—
291#101C2×D26911yescharacteristic, complemented
292#100—2691no—
293#100C2×D26911yescharacteristic, complemented
294C4×C8—2691no—
295#41—2691no—
296#41—2691no—
297#41—2691no—
298#41—5381no—
299#41—5381no—
300#41—5381no—
301#41—5381no—
302#41—5381no—
303#41—5381no—
304#41—5381no—
305#41—5381no—
306#41—10761no—
307#41—10761no—
308#41C22×D26911yescharacteristic
309C2×Q8—2691no—
310C2×Q8—5381no—
311C2×Q8—5381no—
312C2×Q8—5381no—
313C2×Q8—5381no—
314C2×Q8C22×D26911yescharacteristic
315C2×D4—2691no—
316C2×D4—5381no—
317C2×D4—5381no—
318C2×D4—5381no—
319C2×D4—5381no—
320C2×D4—10761no—
321C2×D4—10761no—
322C2×D4—10761no—
323C2×D4C22×D26911yescharacteristic
324Q16—21no—
325Q16—2691no—
326Q16—2691no—
327Q16—5381no—
328Q16—5381no—
329Q16—5381no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—2691no—
335SD16—2691no—
336SD16—2691no—
337SD16—2691no—
338SD16—5381no—
339SD16—5381no—
340SD16—5381no—
341SD16—5381no—
342SD16—5381no—
343SD16—5381no—
344SD16—5381no—
345SD16—5381no—
346SD16—5381no—
347SD16—5381no—
348SD16—5381no—
349SD16—5381no—
350D8—21no—
351D8—2691no—
352D8—2691no—
353D8—5381no—
354D8—5381no—
355D8—5381no—
356M16—21no—
357M16—2691no—
358M16—5381no—
359M16—5381no—
360M16—5381no—
361M16C22×D26911yescharacteristic
362M16C22×D26911yescharacteristic
363M16C22×D26911yescharacteristic
364C2×C8—21no—
365C2×C8—2691no—
366C2×C8—5381no—
367C2×C8—5381no—
368C2×C8—5381no—
369C2×C8C22×D26911yescharacteristic
370C4⋊C4—10761no—
371C42—2691no—
372C23—21no—
373C23—5381no—
374C23—10761no—
375C23—10761no—
376Q8—21no—
377Q8—21no—
378Q8—2691no—
379Q8—2691no—
380Q8—2691no—
381Q8—5381no—
382Q8—5381no—
383Q8—10761no—
384Q8—10761no—
385Q8D4×D26911yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—2691no—
390D4—2691no—
391D4—2691no—
392D4—5381no—
393D4—5381no—
394D4—5381no—
395D4—5381no—
396D4—5381no—
397D4—10761no—
398D4—10761no—
399D4—10761no—
400D4—10761no—
401D4—10761no—
402D4—10761no—
403D4—21522no—
404D4D4×D26911yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—5381no—
408C2×C4—5381no—
409C2×C4—5381no—
410C2×C4—5381no—
411C2×C4—5381no—
412C2×C4—10761no—
413C2×C4—10761no—
414C2×C4—10761no—
415C2×C4—10761no—
416C2×C4—10761no—
417C2×C4—10761no—
418C2×C4C23×D26911yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—2691no—
423C8—2691no—
424C8—5381no—
425C8D4×D26911yescharacteristic
426C8D4×D26911yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—5381no—
431C22—5381no—
432C22—5381no—
433C22—10761no—
434C22—21521no—
435C22—43041no—
436C22—43041no—
437C22#2257549911yescharacteristic
438C4—21no—
439C4—41no—
440C4—5382no—
441C4—5381no—
442C4—5381no—
443C4—5381no—
444C4—10761no—
445C4C2×D4×D26911yescharacteristic
446C4C2×D4×D26911yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—10761no—
451C2—10761no—
452C2—10761no—
453C2—21521no—
454C2#3177123911yescharacteristic
subgroupG/H
centerC2#31771239
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#471×C269C2
socleC2×C269D42
hypercenter#471D269
hypocenterC269#3291
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #31771417C2semidirect
2. #31771415C2semidirect
3. #31771414C2semidirect
4. #31771411C2semidirect
5. #31770976C2semidirect
6. #31770871C2semidirect
7. #31770869C2semidirect
8. #31770867C2semidirect
9. #31770790C2semidirect
10. #31770786C2semidirect
11. #31770775C2semidirect
12. #31770771C2semidirect
13. #31770721C2semidirect
14. #31770720C2semidirect
15. #471×C269C2semidirect
18. #22575499C22semidirect
19. #22575498C22semidirect
21. #22575486C22semidirect
22. #22575486C22semidirect
24. #22575484C22semidirect
41. C2×#16864391C22semidirect
42. C2×#16864390C22semidirect
43. C2×#16864389C22semidirect
47. #22575461C22semidirect
48. #22575461C22semidirect
49. #22575459C22semidirect
50. #22575451C22semidirect
52. #22575388C22semidirect
53. #22575380C22semidirect
54. #22575379C22semidirect
55. #22575379C22semidirect
56. #22575379C22semidirect
57. #22575371C22semidirect
58. #22575369C22semidirect
59. C8×Dic269C22semidirect
60. #137×C269C22semidirect
62. C2×SD16×C269C22semidirect
63. #131×C269C22semidirect
64. #108×C269C22semidirect
65. #101×C269C22semidirect
66. #100×C269C22semidirect
95. Dic1076D4semidirect
97. #16864392D4semidirect
104. #16864391D4semidirect
105. #16864391D4semidirect
109. #16864390D4semidirect
113. #16864389D4semidirect
116. D2152D4semidirect
118. #16864387D4semidirect
227. C269#3291semidirect
237. #471D269semidirect
255. #137C2×D269semidirect
277. C2×SD16C2×D269semidirect
281. #131C2×D269semidirect
285. #108C2×D269semidirect
291. #101C2×D269semidirect
293. #100C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714172
#317714152
#317714142
#317714112
#317709762
#317708712
#317708692
#317708672
#317707902
#317707862
#317707752
#317707712
#317707212
#317707202
#471×C2692
#3291269