Finite Group Databasebeta

← #24919234

Subgroups — #24919234

Order 10624.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128516C211yesmaximal, characteristic, complemented
2#19128509C211yesmaximal, characteristic, complemented
3#19128508C211yesmaximal, characteristic, complemented
4#19128094C211yesmaximal, characteristic, complemented
5#19127974C211yesmaximal, characteristic, complemented
6#19127973C211yesmaximal, characteristic, complemented
7#19127966C211yesmaximal, characteristic, complemented
8#19127964C211yesmaximal, characteristic, complemented
9#19127884C211yesmaximal, characteristic, complemented
10#19127882C211yesmaximal, characteristic, complemented
11#19127877C211yesmaximal, characteristic, complemented
12#19127874C211yesmaximal, characteristic, complemented
13#19127823C211yesmaximal, characteristic, complemented
14#19127821C211yesmaximal, characteristic, complemented
15#578×C83C211yesmaximal, characteristic, complemented
16#14581839C2211yescharacteristic
17#14581838C2211yescharacteristic
18#14581824—21no—
19#14581823—21no—
20#14581823—21no—
21#14581823C2211yescharacteristic
22#14581823C2211yescharacteristic
23#14581822—21no—
24#14581822C2211yescharacteristic
25#14581822C2211yescharacteristic, complemented
26#14581817—21no—
27#14581817C2211yescharacteristic
28#14581817C2211yescharacteristic
29#14581816—21no—
30#14581813—21no—
31#14581812—21no—
32#14581812—21no—
33#14581812C2211yescharacteristic
34C2×#13050580—21no—
35C2×#13050580C2211yescharacteristic, complemented
36C2×#13050579C2211yescharacteristic
37C2×#13050577—21no—
38C2×#13050577C2211yescharacteristic
39SD16×D83—21no—
40#14581799C2211yescharacteristic
41#14581799C2211yescharacteristic, complemented
42#14581798C2211yescharacteristic
43#14581790C2211yescharacteristic, complemented
44#14581781C2211yescharacteristic
45#14581728C2211yescharacteristic, complemented
46#14581727C2211yescharacteristic, complemented
47#14581719C2211yescharacteristic
48#14581719C2211yescharacteristic, complemented
49#14581719C2211yescharacteristic, complemented
50#14581718C2211yescharacteristic
51#14581711C2211yescharacteristic, complemented
52#14581711C2211yescharacteristic, complemented
53#14581710C2211yescharacteristic, complemented
54#14581708C2211yescharacteristic
55#14581702C2211yescharacteristic, complemented
562−1+4×C83C2211yescharacteristic
572−1+4×C83C2211yescharacteristic, complemented
58#137×C83—21no—
59#137×C83—21no—
60#137×C83C2211yescharacteristic
61#137×C83C2211yescharacteristic, complemented
62#135×C83—21no—
63#135×C83C2211yescharacteristic
64C2×Q16×C83—21no—
65C2×Q16×C83C2211yescharacteristic, complemented
66#131×C83C2211yescharacteristic, complemented
67#13050591—21no—
68#13050590—21no—
69#13050590—21no—
70#13050590C2311yescharacteristic
71#13050589—21no—
72#13050589—41no—
73C2×Dic166—21no—
74C2×Dic166C2311yescharacteristic
75Q8×D83—21no—
76C2×D332—21no—
77C2×D332C2311yescharacteristic
78D4×D83—21no—
79D4×D83—41no—
80#13050580—21no—
81#13050580—21no—
82#13050580—21no—
83#13050580—21no—
84#13050580—21no—
85#13050580—21no—
86#13050580D411yescharacteristic
87#13050579—21no—
88#13050579—21no—
89#13050578—21no—
90#13050578—21no—
91#13050578D411yescharacteristic
92#13050577—21no—
93#13050577—21no—
94#13050577—21no—
95#13050577—21no—
96#13050577—21no—
97#13050577—21no—
98#13050577D411yescharacteristic
99#13050575—21no—
100#13050575—21no—
101#13050575D411yescharacteristic
102#13050574—21no—
103#13050574—21no—
104#13050574C2311yescharacteristic
105#13050574C2311yescharacteristic
106#13050574C2311yescharacteristic
107#13050573—21no—
108C2×C83⋊C8—21no—
109C2×C83⋊C8—21no—
110C2×C83⋊C8C2311yescharacteristic
111C8×D83—21no—
112Dic83⋊C4—41no—
113(C2×D83)⋊C4—41no—
114C4×Dic83C2311yescharacteristic
115#41×C83—21no—
116#41×C83—21no—
117#41×C83—21no—
118#41×C83—21no—
119#41×C83—41no—
120#41×C83—41no—
121#41×C83C2311yescharacteristic
122#41×C83C2311yescharacteristic
123#41×C83C2311yescharacteristic
124C2×Q8×C83—21no—
125C2×Q8×C83—21no—
126C2×Q8×C83—21no—
127C2×Q8×C83—21no—
128C2×Q8×C83C2311yescharacteristic
129C2×Q8×C83C2311yescharacteristic
130Q16×C83—21no—
131Q16×C83—21no—
132Q16×C83—21no—
133Q16×C83—21no—
134Q16×C83D411yescharacteristic
135SD16×C83—21no—
136SD16×C83—21no—
137SD16×C83D411yescharacteristic
138SD16×C83D411yescharacteristic
139D8×C83D411yescharacteristic
140M16×C83—21no—
141M16×C83C2311yescharacteristic
142C2×C8×C83—21no—
143C2×C8×C83C2311yescharacteristic
144C22×D83—21no—
145C22×D83—41no—
146Q8×C83—21no—
147Q8×C83—21no—
148Q8×C83—21no—
149Q8×C83—21no—
150Q8×C83—41no—
151Q8×C83—41no—
152Q8×C83C2×D411yescharacteristic
153Q8×C83C2×D411yescharacteristic
154Q8×C83C2×D411yescharacteristic
155D4×C83—21no—
156D4×C83—21no—
157D4×C83—21no—
158D4×C83—41no—
159D4×C83—41no—
160D4×C83C2×D411yescharacteristic
161D4×C83C2×D411yescharacteristic
162D4×C83C2×D411yescharacteristic
163C2×C4×C83—21no—
164C2×C4×C83—21no—
165C2×C4×C83—21no—
166C2×C4×C83—21no—
167C2×C4×C83—21no—
168C2×C4×C83—41no—
169C2×C4×C83—41no—
170C2×C4×C83—41no—
171C2×C4×C83—41no—
172C2×C4×C83C2411yescharacteristic
173C22⋊D83—21no—
174C22⋊D83—41no—
175C22⋊D83—41no—
176C2×Dic83—21no—
177C2×Dic83—21no—
178C2×Dic83—41no—
179D332—21no—
180D332—21no—
181D332—41no—
182D332C2×D411yescharacteristic
183C4×D83—21no—
184C4×D83—41no—
185C4×D83—41no—
186Dic166—21no—
187Dic166—21no—
188Dic166—41no—
189Dic166C2×D411yescharacteristic
190C8×C83—21no—
191C8×C83C2×D411yescharacteristic
192C8×C83C2×D411yescharacteristic
193C83⋊C8—21no—
194C83⋊C8—21no—
195C83⋊C8—21no—
196C83⋊C8C2×D411yescharacteristic
197C83⋊C8C2×D411yescharacteristic
198C22×C83—21no—
199C22×C83—21no—
200C22×C83—21no—
201C22×C83H(5,𝔽2)11yescharacteristic
202C2×D83—21no—
203C2×D83—41no—
204C2×D83—81no—
205C2×D83—161no—
206C4×C83—21no—
207C4×C83—21no—
208C4×C83—21no—
209C4×C83—41no—
210C4×C83—41no—
211C4×C83C22×D411yescharacteristic
212C4×C83C22×D411yescharacteristic
213Dic83—21no—
214Dic83—41no—
215Dic83—41no—
216C2×C83—21no—
217C2×C83—41no—
218C2×C83—41no—
219C2×C83—41no—
220C2×C83D4211yescharacteristic
221D83—41no—
222D83—81no—
223#3289—831nomaximal, Sylow
224C83#328911yescharacteristic, complemented, Sylow
225#578—831no—
226#578D8311yescharacteristic, complemented
227#577—831no—
228#577—831no—
229#497—831no—
230#473—831no—
231#473—831no—
232#471—831no—
233#471—831no—
234#456—831no—
235#456—831no—
236#455—831no—
237#454—831no—
238#444—831no—
239#444—831no—
2402−1+4—831no—
2412−1+4C2×D8311yescharacteristic
2422−1+4C2×D8311yescharacteristic, complemented
243H(5,𝔽2)—831no—
244#137—21no—
245#137—21no—
246#137—831no—
247#137—831no—
248#137—831no—
249#137—831no—
250#137—1661no—
251#137—1661no—
252#137—1661no—
253#137—1661no—
254#137C2×D8311yescharacteristic
255#137C2×D8311yescharacteristic, complemented
256AGL(1,ℤ/8)—831no—
257AGL(1,ℤ/8)—831no—
258AGL(1,ℤ/8)—1661no—
259AGL(1,ℤ/8)—1661no—
260#135—21no—
261#135—831no—
262#135—1661no—
263#135—1661no—
264#135—1661no—
265#135C2×D8311yescharacteristic
266C2×Q16—21no—
267C2×Q16—831no—
268C2×Q16—1661no—
269C2×Q16C2×D8311yescharacteristic, complemented
270C2×SD16—831no—
271C2×SD16—831no—
272C2×SD16—1661no—
273C2×SD16—1661no—
274#131—831no—
275#131—831no—
276#131—831no—
277#131C2×D8311yescharacteristic, complemented
278C4⋊Q8—831no—
279#124—831no—
280#108—831no—
281#108—831no—
282#104—831no—
283#104—831no—
284#104—831no—
285#104—831no—
286#101—831no—
287#101—831no—
288#101—831no—
289#100—831no—
290#97—831no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—41no—
296#41—41no—
297#41—3321no—
298#41—831no—
299#41—1661no—
300#41—1661no—
301#41—1661no—
302#41—1661no—
303#41C22×D8311yescharacteristic
304#41C22×D8311yescharacteristic
305#41C22×D8311yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—831no—
311C2×Q8—1661no—
312C2×Q8—1661no—
313C2×Q8C22×D8311yescharacteristic
314C2×Q8C22×D8311yescharacteristic
315C2×D4—3321no—
316C2×D4—831no—
317C2×D4—1661no—
318C2×D4—1661no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—831no—
324Q16—1661no—
325Q16—1661no—
326Q16—1661no—
327Q16—1661no—
328Q16—1661no—
329Q16—1661no—
330Q16C22⋊D8311yescharacteristic
331SD16—21no—
332SD16—21no—
333SD16—831no—
334SD16—831no—
335SD16—1661no—
336SD16—1661no—
337SD16—1661no—
338SD16—1661no—
339SD16—1661no—
340SD16—1661no—
341SD16—1661no—
342SD16—1661no—
343SD16—1661no—
344SD16—1661no—
345SD16C22⋊D8311yescharacteristic
346SD16C22⋊D8311yescharacteristic
347D8—831no—
348D8—1661no—
349D8—1661no—
350D8C22⋊D8311yescharacteristic
351M16—21no—
352M16—831no—
353M16—831no—
354M16—831no—
355M16—1661no—
356M16—1661no—
357M16—1661no—
358M16C22×D8311yescharacteristic
359C2×C8—21no—
360C2×C8—831no—
361C2×C8—1661no—
362C2×C8—1661no—
363C2×C8—1661no—
364C2×C8C22×D8311yescharacteristic
365C4⋊C4—3321no—
366C22⋊C4—3321no—
367C42—831no—
368C23—3321no—
369C23—1661no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—41no—
376Q8—3321no—
377Q8—831no—
378Q8—1661no—
379Q8—1661no—
380Q8D4×D8311yescharacteristic
381Q8C2×C22⋊D8311yescharacteristic
382Q8C2×C22⋊D8311yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—41no—
387D4—41no—
388D4—3321no—
389D4—3321no—
390D4—3321no—
391D4—831no—
392D4—1661no—
393D4—1661no—
394D4—1661no—
395D4D4×D8311yescharacteristic
396D4C2×C22⋊D8311yescharacteristic
397D4C2×C22⋊D8311yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—3321no—
408C2×C4—3321no—
409C2×C4—3321no—
410C2×C4—1661no—
411C2×C4—1661no—
412C2×C4—1661no—
413C2×C4C23×D8311yescharacteristic
414C8—21no—
415C8—831no—
416C8—831no—
417C8—1661no—
418C8—1661no—
419C8—1661no—
420C8C2×C22⋊D8311yescharacteristic
421C8C2×C22⋊D8311yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—13281no—
426C22—3321no—
427C22—6641no—
428C22—1661no—
429C22#1458183711yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3321no—
436C4—3321no—
437C4—1661no—
438C4C2×D4×D8311yescharacteristic
439C4C22×C22⋊D8311yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—3321no—
445C2—6641no—
446C2D4×C22⋊D8311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#578×C83C2
socleC2×C83D42
hypercenter#578D83
hypocenterC83#3289
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #19128516C2semidirect
2. #19128509C2semidirect
3. #19128508C2semidirect
4. #19128094C2semidirect
5. #19127974C2semidirect
6. #19127973C2semidirect
7. #19127966C2semidirect
8. #19127964C2semidirect
9. #19127884C2semidirect
10. #19127882C2semidirect
11. #19127877C2semidirect
12. #19127874C2semidirect
13. #19127823C2semidirect
14. #19127821C2semidirect
15. #578×C83C2semidirect
25. #14581822C22semidirect
35. C2×#13050580C22semidirect
41. #14581799C22semidirect
43. #14581790C22semidirect
45. #14581728C22semidirect
46. #14581727C22semidirect
48. #14581719C22semidirect
49. #14581719C22semidirect
51. #14581711C22semidirect
52. #14581711C22semidirect
53. #14581710C22semidirect
55. #14581702C22semidirect
57. 2−1+4×C83C22semidirect
61. #137×C83C22semidirect
65. C2×Q16×C83C22semidirect
66. #131×C83C22semidirect
224. C83#3289semidirect
226. #578D83semidirect
242. 2−1+4C2×D83semidirect
255. #137C2×D83semidirect
269. C2×Q16C2×D83semidirect
277. #131C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285162
#191285092
#191285082
#191280942
#191279742
#191279732
#191279662
#191279642
#191278842
#191278822
#191278772
#191278742
#191278232
#191278212
#578×C832
#328983