Finite Group Databasebeta

← #24919171

Subgroups — #24919171

Order 10624.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581812C211yesmaximal, characteristic, complemented
2C22×#13050577C211yesmaximal, characteristic, complemented
3#19128347C211yesmaximal, characteristic, complemented
4#19128083C211yesmaximal, characteristic, complemented
5#19127932C211yesmaximal, characteristic, complemented
6#19127928C211yesmaximal, characteristic, complemented
7#19127923C211yesmaximal, characteristic, complemented
8#19127913C211yesmaximal, characteristic, complemented
9#19127861C211yesmaximal, characteristic, complemented
10#19127857C211yesmaximal, characteristic, complemented
11#19127846C211yesmaximal, characteristic, complemented
12#19127834C211yesmaximal, characteristic, complemented
13C4×#13050577C211yesmaximal, characteristic, complemented
14D4×C83⋊C8C211yesmaximal, characteristic, complemented
15#548×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×D332C2211yescharacteristic
18#14581812—24no—
19C2×#13050580C2211yescharacteristic, complemented
20C2×#13050580C2211yescharacteristic, complemented
21C2×#13050578C2211yescharacteristic, complemented
22C2×#13050577—24no—
23C2×#13050577C2211yescharacteristic
24C2×#13050577C2211yescharacteristic
25C2×#13050577C2211yescharacteristic
26C2×#13050577C2211yescharacteristic
27C2×#13050575C2211yescharacteristic
28C2×#13050575C2211yescharacteristic
29C22×C83⋊C8C2211yescharacteristic
30C22×C83⋊C8C2211yescharacteristic, complemented
31#14581780C2211yescharacteristic
32#14581777—21no—
33#14581771C2211yescharacteristic
34#14581766C2211yescharacteristic
35#14581765—21no—
36#14581760—21no—
37C4×D332C2211yescharacteristic
38C22⋊C4×D83—21no—
39#14581723C2211yescharacteristic, complemented
40C4⋊(C83⋊C8)C2211yescharacteristic, complemented
41#14581716C2211yescharacteristic, complemented
42#14581716C2211yescharacteristic, complemented
43#14581715C2211yescharacteristic, complemented
44#14581713C2211yescharacteristic, complemented
45#14581712C2211yescharacteristic
46#14581712C2211yescharacteristic
47C22⋊(C83⋊C8)C2211yescharacteristic
48C22⋊(C83⋊C8)C2211yescharacteristic, complemented
49C4×C83⋊C8C2211yescharacteristic, complemented
50C2×#41×C83C2211yescharacteristic, complemented
51C22×Q8×C83C2211yescharacteristic
52#124×C83—21no—
53#124×C83C2211yescharacteristic, complemented
54C22⋊Q8×C83—21no—
55C22⋊Q8×C83C2211yescharacteristic, complemented
56#121×C83—21no—
57#121×C83C2211yescharacteristic
58C4×Q8×C83C2211yescharacteristic, complemented
59C4×D4×C83—21no—
60C4×D4×C83C2211yescharacteristic
61C23×D83—21no—
62#13050590—22no—
63#13050590—41no—
64C2×Dic166C2311yescharacteristic
65C2×D332—22no—
66C2×D332—21no—
67C2×D332—41no—
68C2×D332C2311yescharacteristic
69C2×D332C2311yescharacteristic
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72C2×C4×D83—21no—
73C2×C4×D83—21no—
74#13050580—22no—
75#13050580—41no—
76#13050578—22no—
77#13050577—22no—
78#13050577—22no—
79#13050577—22no—
80#13050577D414yescomplemented
81#13050575—22no—
82#13050575—41no—
83C2×C83⋊C8—22no—
84C2×C83⋊C8—22no—
85C2×C83⋊C8C2311yescharacteristic
86C2×C83⋊C8C2311yescharacteristic
87C2×C83⋊C8C2311yescharacteristic
88C2×C83⋊C8C2311yescharacteristic
89C4⋊Dic83C2311yescharacteristic
90Dic83⋊C4—21no—
91C22⋊Dic83—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96#41×C83—22no—
97#41×C83—41no—
98C2×Q8×C83—21no—
99C2×Q8×C83—22no—
100C2×Q8×C83—41no—
101C2×Q8×C83C2311yescharacteristic
102C2×Q8×C83C2311yescharacteristic
103C2×D4×C83—21no—
104C2×D4×C83—21no—
105C2×D4×C83D411yescharacteristic
106C2×D4×C83C2311yescharacteristic
107C22×C4×C83—21no—
108C22×C4×C83—21no—
109C22×C4×C83C2311yescharacteristic
110C22×C4×C83C2311yescharacteristic
111C4⋊C4×C83—21no—
112C4⋊C4×C83—21no—
113C4⋊C4×C83D411yescharacteristic
114C4⋊C4×C83C2311yescharacteristic
115C22⋊C4×C83—21no—
116C22⋊C4×C83—21no—
117C22⋊C4×C83—21no—
118C22⋊C4×C83—21no—
119C22⋊C4×C83D411yescharacteristic
120C22⋊C4×C83D411yescharacteristic
121C42×C83—21no—
122C42×C83C2311yescharacteristic
123C23×C83—21no—
124C23×C83C2×D411yescharacteristic
125C23×C83C2×D411yescharacteristic
126C22×D83—21no—
127C22×D83—21no—
128C22×D83—42no—
129C22×D83—42no—
130C22×D83—81no—
131Q8×C83—21no—
132Q8×C83—21no—
133Q8×C83—21no—
134Q8×C83—42no—
135Q8×C83C2×D412yes—
136D4×C83—22no—
137D4×C83—21no—
138D4×C83—21no—
139D4×C83—42no—
140D4×C83—42no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—21no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83—21no—
147C2×C4×C83—21no—
148C2×C4×C83—42no—
149C2×C4×C83—42no—
150C2×C4×C83C2×D411yescharacteristic
151C2×C4×C83C2×D411yescharacteristic
152C2×C4×C83C2×D411yescharacteristic
153C2×C4×C83C2×D411yescharacteristic
154C2×C4×C83C2411yescharacteristic
155C22⋊D83—42no—
156C22⋊D83—81no—
157C2×Dic83—21no—
158C2×Dic83—21no—
159D332—21no—
160D332—21no—
161D332—21no—
162D332—42no—
163D332C2×D412yes—
164C4×D83—42no—
165C4×D83—81no—
166Dic166—21no—
167Dic166—21no—
168C83⋊C8—21no—
169C83⋊C8—21no—
170C83⋊C8—21no—
171C83⋊C8C2×D412yes—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—21no—
175C22×C83—21no—
176C22×C83—21no—
177C22×C83—41no—
178C22×C83—41no—
179C22×C83—41no—
180C22×C83#13512yescomplemented
181C22×C83C22×D411yescharacteristic
182C2×D83—22no—
183C2×D83—41no—
184C2×D83—41no—
185C2×D83—41no—
186C2×D83—41no—
187C2×D83—41no—
188C2×D83—84no—
189C4×C83—21no—
190C4×C83—21no—
191C4×C83—22no—
192C4×C83—21no—
193C4×C83—21no—
194C4×C83—41no—
195C4×C83—41no—
196C4×C83C22×D411yescharacteristic
197C4×C83H(5,𝔽2)11yescharacteristic
198Dic83—41no—
199Dic83—41no—
200C2×C83—22no—
201C2×C83—41no—
202C2×C83—81no—
203C2×C83D4211yescharacteristic
204C2×C83C2×#13511yescharacteristic
205C2×C83#57711yescharacteristic
206D83—42no—
207D83—81no—
208#3282—831nomaximal, Sylow
209C83#328211yescharacteristic, complemented, Sylow
210C2×#135—831no—
211C22×SD16—831no—
212#548D8311yescharacteristic, complemented
213#546—831no—
214#495—831no—
215#465—831no—
216#465—831no—
217#464—831no—
218#463—831no—
219#452—831no—
220#451—831no—
221#449—831no—
222#447—831no—
223C4×SD16—831no—
224C8×D4—831no—
225C2×#41—831no—
226C2×#41C2×D8311yescharacteristic, complemented
227C22×Q8C2×D8311yescharacteristic
228C22×D4—831no—
229#135—1664no—
230C2×Q16—831no—
231C2×Q16—831no—
232C2×SD16—831no—
233C2×SD16—831no—
234C2×SD16—831no—
235C2×SD16—831no—
236C2×SD16—831no—
237C2×SD16—1664no—
238C2×D8—831no—
239C2×D8—831no—
240C22×C8—831no—
241C22×C8—831no—
242#124—21no—
243#124—831no—
244#124C2×D8311yescharacteristic, complemented
245#123—1661no—
246C22⋊Q8—21no—
247C22⋊Q8—831no—
248C22⋊Q8C2×D8311yescharacteristic, complemented
249#121—21no—
250#121—831no—
251#121—1661no—
252#121C2×D8311yescharacteristic
253#120—1661no—
254C4×Q8C2×D8311yescharacteristic, complemented
255C4×D4—21no—
256C4×D4—831no—
257C4×D4C2×D8311yescharacteristic
258C2×C22⋊C4—1661no—
259#106—831no—
260C4⋊C8—831no—
261#103—831no—
262#103—831no—
263#103—831no—
264#102—831no—
265#102—831no—
266#102—831no—
267C22⋊C8—831no—
268C22⋊C8—831no—
269C4×C8—831no—
270C24—1661no—
271#41—22no—
272#41—41no—
273#41—3321no—
274#41—1662no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—831no—
279C2×Q8C22×D8311yescharacteristic
280C2×Q8C22×D8311yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3321no—
284C2×D4—831no—
285C2×D4—831no—
286C2×D4—1662no—
287C2×D4—1661no—
288C2×D4—1661no—
289C2×D4—1661no—
290C2×D4C22⋊D8311yescharacteristic
291C2×D4C22×D8311yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—1661no—
295C22×C4—1661no—
296C22×C4C22×D8311yescharacteristic
297C22×C4C22×D8311yescharacteristic
298Q16—3321no—
299Q16—1662no—
300SD16—834no—
301SD16—1662no—
302SD16—1662no—
303SD16—1662no—
304SD16—1662no—
305D8—3321no—
306D8—1662no—
307C2×C8—831no—
308C2×C8—831no—
309C2×C8—831no—
310C2×C8—831no—
311C2×C8—1662no—
312C2×C8—1662no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—831no—
316C4⋊C4—1661no—
317C4⋊C4C22⋊D8311yescharacteristic
318C4⋊C4C22×D8311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—1661no—
324C22⋊C4—1661no—
325C22⋊C4—1661no—
326C22⋊C4—1661no—
327C22⋊C4—1661no—
328C22⋊C4C22⋊D8311yescharacteristic
329C22⋊C4C22⋊D8311yescharacteristic
330C42—21no—
331C42C22×D8311yescharacteristic
332C23—21no—
333C23—3322no—
334C23—3322no—
335C23—6641no—
336C23—1661no—
337C23—1661no—
338C23C2×C22⋊D8311yescharacteristic
339C23C2×C22⋊D8311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—42no—
344Q8—1661no—
345Q8—1661no—
346Q8D4×D8312yes—
347D4—22no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—3322no—
353D4—3322no—
354D4—832no—
355D4—6641no—
356D4—1661no—
357D4—1661no—
358D4—1661no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—3322no—
369C2×C4—6641no—
370C2×C4—1661no—
371C2×C4—1661no—
372C2×C4C2×C22⋊D8311yescharacteristic
373C2×C4C2×C22⋊D8311yescharacteristic
374C2×C4C2×C22⋊D8311yescharacteristic
375C2×C4C2×C22⋊D8311yescharacteristic
376C2×C4C23×D8311yescharacteristic
377C8—832no—
378C8—1661no—
379C8—1661no—
380C8—1661no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3321no—
390C22—3321no—
391C22—3321no—
392C22—3321no—
393C22—3321no—
394C22—6644no—
395C22—1662no—
396C22#1458181212yescomplemented
397C22C22×C22⋊D8311yescharacteristic
398C4—21no—
399C4—21no—
400C4—22no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—3321no—
406C4—3321no—
407C4C2×D4×D8311yescharacteristic
408C4#1458183711yescharacteristic
409C2—22no—
410C2—41no—
411C2—81no—
412C2—3322no—
413C2—6641no—
414C2D4×C22⋊D8311yescharacteristic
415C2C2×#1458181211yescharacteristic
416C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#548×C83C2
socleC22×C83C22×D4
hypercenter#548D83
hypocenterC83#3282
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#14581812C2semidirect
2. C22×#13050577C2semidirect
3. #19128347C2semidirect
4. #19128083C2semidirect
5. #19127932C2semidirect
6. #19127928C2semidirect
7. #19127923C2semidirect
8. #19127913C2semidirect
9. #19127861C2semidirect
10. #19127857C2semidirect
11. #19127846C2semidirect
12. #19127834C2semidirect
13. C4×#13050577C2semidirect
14. D4×C83⋊C8C2semidirect
15. #548×C83C2semidirect
19. C2×#13050580C22semidirect
20. C2×#13050580C22semidirect
21. C2×#13050578C22semidirect
30. C22×C83⋊C8C22semidirect
39. #14581723C22semidirect
40. C4⋊(C83⋊C8)C22semidirect
41. #14581716C22semidirect
42. #14581716C22semidirect
43. #14581715C22semidirect
44. #14581713C22semidirect
48. C22⋊(C83⋊C8)C22semidirect
49. C4×C83⋊C8C22semidirect
50. C2×#41×C83C22semidirect
53. #124×C83C22semidirect
55. C22⋊Q8×C83C22semidirect
58. C4×Q8×C83C22semidirect
80. #13050577D4semidirect
180. C22×C83#135semidirect
209. C83#3282semidirect
212. #548D83semidirect
226. C2×#41C2×D83semidirect
244. #124C2×D83semidirect
248. C22⋊Q8C2×D83semidirect
254. C4×Q8C2×D83semidirect
396. C22#14581812semidirect

Maximal subgroups

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

subgroupindex
C2×#145818122
C22×#130505772
#191283472
#191280832
#191279322
#191279282
#191279232
#191279132
#191278612
#191278572
#191278462
#191278342
C4×#130505772
D4×C83⋊C82
#548×C832
#328283