Finite Group Databasebeta

← #46822676

Subgroups — #46822676

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#31771416C211yesmaximal, characteristic, complemented
2#31771413C211yesmaximal, characteristic, complemented
3#31771412C211yesmaximal, characteristic, complemented
4#31770976C211yesmaximal, characteristic, complemented
5#31770870C211yesmaximal, characteristic, complemented
6#31770868C211yesmaximal, characteristic, complemented
7#31770866C211yesmaximal, characteristic, complemented
8#31770865C211yesmaximal, characteristic, complemented
9#31770788C211yesmaximal, characteristic, complemented
10#31770786C211yesmaximal, characteristic, complemented
11#31770773C211yesmaximal, characteristic, complemented
12#31770771C211yesmaximal, characteristic, complemented
13#31770722C211yesmaximal, characteristic, complemented
14#31770720C211yesmaximal, characteristic, complemented
15#577×C269C211yesmaximal, characteristic, complemented
16#22575500C2211yescharacteristic
17#22575499C2211yescharacteristic, complemented
18#22575485—21no—
19#22575484—21no—
20#22575484C2211yescharacteristic, complemented
21#22575483—21no—
22#22575483C2211yescharacteristic, complemented
23#22575483C2211yescharacteristic, complemented
24#22575479—21no—
25#22575479C2211yescharacteristic
26#22575478—21no—
27#22575478C2211yescharacteristic
28#22575478C2211yescharacteristic
29#22575477—21no—
30#22575474—21no—
31#22575473—21no—
32#22575473—21no—
33#22575473C2211yescharacteristic
34C2×#16864391C2211yescharacteristic, complemented
35C2×#16864390—21no—
36C2×#16864390C2211yescharacteristic, complemented
37C2×#16864389—21no—
38C2×#16864389C2211yescharacteristic, complemented
39SD16×D269—21no—
40#22575460C2211yescharacteristic, complemented
41#22575460C2211yescharacteristic, complemented
42#22575459C2211yescharacteristic, complemented
43#22575451C2211yescharacteristic, complemented
44#22575449C2211yescharacteristic
45#22575389C2211yescharacteristic, complemented
46#22575388C2211yescharacteristic, complemented
47#22575380C2211yescharacteristic, complemented
48#22575380C2211yescharacteristic, complemented
49#22575380C2211yescharacteristic, complemented
50#22575379C2211yescharacteristic, complemented
51#22575372C2211yescharacteristic, complemented
52#22575371C2211yescharacteristic, complemented
53#22575370C2211yescharacteristic, complemented
54#22575369C2211yescharacteristic, complemented
55C8×Dic269C2211yescharacteristic, complemented
562−1+4×C269C2211yescharacteristic
57H(5,𝔽2)×C269C2211yescharacteristic, complemented
58#137×C269—21no—
59#137×C269C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C269—21no—
61AGL(1,ℤ/8)×C269C2211yescharacteristic
62#135×C269—21no—
63#135×C269C2211yescharacteristic
64C2×SD16×C269—21no—
65C2×SD16×C269C2211yescharacteristic, complemented
66#131×C269C2211yescharacteristic, complemented
67#16864403—21no—
68#16864402—21no—
69#16864402—21no—
70#16864402C2311yescharacteristic
71#16864401—21no—
72#16864401—41no—
73C2×Dic538—21no—
74C2×Dic538C2311yescharacteristic
75Q8×D269—21no—
76C2×D1076—21no—
77C2×D1076C2311yescharacteristic
78C2×C22⋊D269—41no—
79D4×D269—21no—
80D4×D269—41no—
81#16864392—21no—
82#16864392—21no—
83#16864392—21no—
84#16864392D411yescharacteristic, complemented
85#16864391—21no—
86#16864391—21no—
87#16864390—21no—
88#16864390—21no—
89#16864390—21no—
90#16864390—21no—
91#16864390—21no—
92#16864390D411yescharacteristic, complemented
93#16864389—21no—
94#16864389—21no—
95#16864389—21no—
96#16864389—21no—
97#16864389—21no—
98#16864389D411yescharacteristic, complemented
99#16864387—21no—
100#16864387—21no—
101#16864387—21no—
102#16864387D411yescharacteristic, complemented
103#16864386—21no—
104#16864386—21no—
105#16864386C2311yescharacteristic
106#16864386C2311yescharacteristic
107#16864386C2311yescharacteristic
108#16864385—21no—
109C2×#14524130—21no—
110C2×#14524130—21no—
111C2×#14524130C2311yescharacteristic
112C8×D269—21no—
113#16864381—41no—
114C4×Dic269C2311yescharacteristic
115#41×C269—21no—
116#41×C269—21no—
117#41×C269—21no—
118#41×C269—21no—
119#41×C269—41no—
120#41×C269C2311yescharacteristic
121#41×C269C2311yescharacteristic
122#41×C269C2311yescharacteristic
123C2×Q8×C269—21no—
124C2×Q8×C269—21no—
125C2×Q8×C269C2311yescharacteristic
126C2×D4×C269—21no—
127C2×D4×C269—21no—
128C2×D4×C269—41no—
129C2×D4×C269C2311yescharacteristic
130Q16×C269—21no—
131Q16×C269D411yescharacteristic, complemented
132SD16×C269—21no—
133SD16×C269—21no—
134SD16×C269—21no—
135SD16×C269—21no—
136SD16×C269D411yescharacteristic, complemented
137SD16×C269D411yescharacteristic, complemented
138D8×C269—21no—
139D8×C269D411yescharacteristic, complemented
140M16×C269—21no—
141M16×C269C2311yescharacteristic
142C2×C8×C269—21no—
143C2×C8×C269C2311yescharacteristic
144C22×D269—21no—
145C22×D269—41no—
146Dic538—21no—
147Dic538—21no—
148Dic538—41no—
149Dic538C2×D411yescharacteristic
150D1076—21no—
151D1076—21no—
152D1076—41no—
153D1076C2×D411yescharacteristic
154C22⋊D269—21no—
155C22⋊D269—41no—
156C22⋊D269—41no—
157C22⋊D269—82no—
158C2×Dic269—21no—
159C2×Dic269—21no—
160C2×Dic269—41no—
161C4×D269—21no—
162C4×D269—41no—
163C4×D269—41no—
164#14524130—21no—
165#14524130—21no—
166#14524130—21no—
167#14524130C2×D411yescharacteristic
168#14524130C2×D411yescharacteristic
169C23×C269—21no—
170C23×C269—41no—
171Q8×C269—21no—
172Q8×C269—21no—
173Q8×C269—41no—
174Q8×C269C2×D411yescharacteristic
175Q8×C269C2×D411yescharacteristic
176Q8×C269C2×D411yescharacteristic
177D4×C269—21no—
178D4×C269—21no—
179D4×C269—21no—
180D4×C269—21no—
181D4×C269—21no—
182D4×C269—41no—
183D4×C269—41no—
184D4×C269—41no—
185D4×C269C2×D411yescharacteristic
186D4×C269C2×D411yescharacteristic
187D4×C269C2×D411yescharacteristic
188C2×C4×C269—21no—
189C2×C4×C269—21no—
190C2×C4×C269—21no—
191C2×C4×C269—21no—
192C2×C4×C269—41no—
193C2×C4×C269—41no—
194C2×C4×C269—41no—
195C2×C4×C269C2411yescharacteristic
196C8×C269—21no—
197C8×C269C2×D411yescharacteristic
198C8×C269C2×D411yescharacteristic
199C22×C269—21no—
200C22×C269—21no—
201C22×C269—21no—
202C22×C269—41no—
203C22×C269—81no—
204C22×C269—161no—
205C22×C269H(5,𝔽2)11yescharacteristic
206C2×D269—21no—
207C2×D269—41no—
208C2×D269—81no—
209C2×D269—161no—
210C4×C269—21no—
211C4×C269—21no—
212C4×C269—21no—
213C4×C269—41no—
214C4×C269C22×D411yescharacteristic
215C4×C269C22×D411yescharacteristic
216Dic269—22no—
217Dic269—21no—
218Dic269—41no—
219C2×C269—21no—
220C2×C269—41no—
221C2×C269—41no—
222C2×C269—41no—
223C2×C269—81no—
224C2×C269D4211yescharacteristic
225D269—41no—
226D269—81no—
227C269#329111yescharacteristic, complemented, Sylow
228#3291—2691nomaximal, Sylow
229#577—2691no—
230#577—2691no—
231#577—2691no—
232#577D26911yescharacteristic, complemented
233#492—2691no—
234#471—2691no—
235#471—2691no—
236#471—2691no—
237#471—2691no—
238#456—2691no—
239#456—2691no—
240#453—2691no—
241#453—2691no—
242#443—2691no—
243#443—2691no—
2442−1+4—2691no—
2452−1+4C2×D26911yescharacteristic
246H(5,𝔽2)—2691no—
247H(5,𝔽2)C2×D26911yescharacteristic, complemented
248#137—21no—
249#137—2691no—
250#137—2691no—
251#137—2691no—
252#137—5381no—
253#137—5381no—
254#137—5381no—
255#137C2×D26911yescharacteristic, complemented
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—2691no—
258AGL(1,ℤ/8)—2691no—
259AGL(1,ℤ/8)—2691no—
260AGL(1,ℤ/8)—5381no—
261AGL(1,ℤ/8)—5381no—
262AGL(1,ℤ/8)—5381no—
263AGL(1,ℤ/8)C2×D26911yescharacteristic
264#135—21no—
265#135—2691no—
266#135—5381no—
267#135—5381no—
268#135—5381no—
269#135C2×D26911yescharacteristic
270C2×SD16—21no—
271C2×SD16—2691no—
272C2×SD16—2691no—
273C2×SD16—2691no—
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#108—2691no—
286#104—2691no—
287#104—2691no—
288#104—2691no—
289#104—2691no—
290#101—2691no—
291#101—2691no—
292#100—2691no—
293#100—2691no—
294C4×C8—2691no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—2691no—
301#41—5381no—
302#41—5381no—
303#41—5381no—
304#41—5381no—
305#41—10761no—
306#41C22×D26911yescharacteristic
307#41C22×D26911yescharacteristic
308#41C22×D26911yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—2691no—
312C2×Q8—5381no—
313C2×Q8—5381no—
314C2×Q8C22×D26911yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—2691no—
319C2×D4—5381no—
320C2×D4—5381no—
321C2×D4—10761no—
322C2×D4—10761no—
323C2×D4C22×D26911yescharacteristic
324Q16—21no—
325Q16—2691no—
326Q16—5381no—
327Q16—5381no—
328Q16—5381no—
329Q16C22⋊D26911yescharacteristic, complemented
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—2691no—
335SD16—2691no—
336SD16—5381no—
337SD16—5381no—
338SD16—5381no—
339SD16—5381no—
340SD16—5381no—
341SD16—5381no—
342SD16—5381no—
343SD16—5381no—
344SD16—5381no—
345SD16—5381no—
346SD16—5381no—
347SD16—5381no—
348SD16C22⋊D26911yescharacteristic, complemented
349SD16C22⋊D26911yescharacteristic, complemented
350D8—21no—
351D8—2691no—
352D8—5381no—
353D8—5381no—
354D8—5381no—
355D8C22⋊D26911yescharacteristic, complemented
356M16—21no—
357M16—2691no—
358M16—2691no—
359M16—2691no—
360M16—5381no—
361M16—5381no—
362M16—5381no—
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—41no—
374C23—5381no—
375C23—10761no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—2691no—
380Q8—5381no—
381Q8—5381no—
382Q8—10761no—
383Q8D4×D26911yescharacteristic
384Q8C2×C22⋊D26911yescharacteristic
385Q8C2×C22⋊D26911yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—2691no—
395D4—5381no—
396D4—5381no—
397D4—5381no—
398D4—10761no—
399D4—10761no—
400D4—10761no—
401D4—21522no—
402D4D4×D26911yescharacteristic
403D4C2×C22⋊D26911yescharacteristic
404D4C2×C22⋊D26911yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—5381no—
413C2×C4—5381no—
414C2×C4—5381no—
415C2×C4—10761no—
416C2×C4—10761no—
417C2×C4—10761no—
418C2×C4C23×D26911yescharacteristic
419C8—21no—
420C8—2691no—
421C8—2691no—
422C8—5381no—
423C8—5381no—
424C8—5381no—
425C8C2×C22⋊D26911yescharacteristic
426C8C2×C22⋊D26911yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—5381no—
434C22—10761no—
435C22—21521no—
436C22—43041no—
437C22#2257549811yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—5382no—
443C4—5381no—
444C4—10761no—
445C4C2×D4×D26911yescharacteristic
446C4C22×C22⋊D26911yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—10761no—
453C2—21521no—
454C2D4×C22⋊D26911yescharacteristic
subgroupG/H
centerC2D4×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#577×C269C2
socleC2×C269D42
hypercenter#577D269
hypocenterC269#3291
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #31771416C2semidirect
2. #31771413C2semidirect
3. #31771412C2semidirect
4. #31770976C2semidirect
5. #31770870C2semidirect
6. #31770868C2semidirect
7. #31770866C2semidirect
8. #31770865C2semidirect
9. #31770788C2semidirect
10. #31770786C2semidirect
11. #31770773C2semidirect
12. #31770771C2semidirect
13. #31770722C2semidirect
14. #31770720C2semidirect
15. #577×C269C2semidirect
17. #22575499C22semidirect
20. #22575484C22semidirect
22. #22575483C22semidirect
23. #22575483C22semidirect
34. C2×#16864391C22semidirect
36. C2×#16864390C22semidirect
38. C2×#16864389C22semidirect
40. #22575460C22semidirect
41. #22575460C22semidirect
42. #22575459C22semidirect
43. #22575451C22semidirect
45. #22575389C22semidirect
46. #22575388C22semidirect
47. #22575380C22semidirect
48. #22575380C22semidirect
49. #22575380C22semidirect
50. #22575379C22semidirect
51. #22575372C22semidirect
52. #22575371C22semidirect
53. #22575370C22semidirect
54. #22575369C22semidirect
55. C8×Dic269C22semidirect
57. H(5,𝔽2)×C269C22semidirect
59. #137×C269C22semidirect
65. C2×SD16×C269C22semidirect
66. #131×C269C22semidirect
84. #16864392D4semidirect
92. #16864390D4semidirect
98. #16864389D4semidirect
102. #16864387D4semidirect
131. Q16×C269D4semidirect
136. SD16×C269D4semidirect
137. SD16×C269D4semidirect
139. D8×C269D4semidirect
227. C269#3291semidirect
232. #577D269semidirect
247. H(5,𝔽2)C2×D269semidirect
255. #137C2×D269semidirect
277. C2×SD16C2×D269semidirect
281. #131C2×D269semidirect
329. Q16C22⋊D269semidirect
348. SD16C22⋊D269semidirect
349. SD16C22⋊D269semidirect
355. D8C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
#317714162
#317714132
#317714122
#317709762
#317708702
#317708682
#317708662
#317708652
#317707882
#317707862
#317707732
#317707712
#317707222
#317707202
#577×C2692
#3291269