Finite Group Databasebeta

← #24919090

Subgroups — #24919090

Order 10624.

498 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×#14581822C211yesmaximal, characteristic, complemented
2C2×#14581818C211yesmaximal, characteristic, complemented
3#19128360C211yesmaximal, characteristic, complemented
4#19128083C211yesmaximal, characteristic, complemented
5#19127957C211yesmaximal, characteristic, complemented
6#19127952C211yesmaximal, characteristic, complemented
7#19127920C211yesmaximal, characteristic, complemented
8#19127916C211yesmaximal, characteristic, complemented
9#19127861C211yesmaximal, characteristic, complemented
10#19127852C211yesmaximal, characteristic, complemented
11#19127846C211yesmaximal, characteristic, complemented
12#19127839C211yesmaximal, characteristic, complemented
13C4×#13050578C211yesmaximal, characteristic, complemented
14#19127779C211yesmaximal, characteristic, complemented
15#546×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×Dic166C2211yescharacteristic, complemented
18#14581822—24no—
19#14581818—24no—
20C2×#13050580C2211yescharacteristic, complemented
21C2×#13050580C2211yescharacteristic, complemented
22C2×#13050578C2211yescharacteristic
23C2×#13050578C2211yescharacteristic
24C2×#13050578C2211yescharacteristic
25C2×#13050578C2211yescharacteristic
26C2×#13050577C2211yescharacteristic, complemented
27C2×#13050575C2211yescharacteristic
28C2×#13050575C2211yescharacteristic
29C2×#13050574C2211yescharacteristic
30C2×#13050574C2211yescharacteristic, complemented
31#14581780C2211yescharacteristic, complemented
32#14581778—21no—
33#14581771C2211yescharacteristic
34#14581770—21no—
35#14581766C2211yescharacteristic, complemented
36#14581764—21no—
37C4×Dic166C2211yescharacteristic, complemented
38(C22⋊D83)⋊C4—21no—
39#14581723C2211yescharacteristic, complemented
40C4⋊(C83⋊C8)C2211yescharacteristic, complemented
41#14581716C2211yescharacteristic, complemented
42#14581715C2211yescharacteristic, complemented
43#14581715C2211yescharacteristic, complemented
44#14581713C2211yescharacteristic
45#14581713C2211yescharacteristic
46#14581712C2211yescharacteristic, complemented
47C22⋊(C83⋊C8)C2211yescharacteristic
48C22⋊(C83⋊C8)C2211yescharacteristic, complemented
49C4×C83⋊C8C2211yescharacteristic, complemented
50C2×#41×C83C2211yescharacteristic
51C22×D4×C83C2211yescharacteristic
52#124×C83C2211yescharacteristic
53#123×C83—21no—
54C22⋊Q8×C83C2211yescharacteristic
55#121×C83—21no—
56#121×C83C2211yescharacteristic
57#120×C83—21no—
58C4×D4×C83C2211yescharacteristic
59C4×D4×C83C2211yescharacteristic
60C2×C22⋊C4×C83—21no—
61#13050590—22no—
62#13050590—41no—
63C2×Dic166—22no—
64C2×Dic166—21no—
65C2×Dic166—41no—
66C2×Dic166C2311yescharacteristic
67C2×Dic166C2311yescharacteristic
68C2×D332C2311yescharacteristic
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71C22×Dic83—21no—
72C2×C4×D83—21no—
73#13050580—22no—
74#13050580—22no—
75#13050578—22no—
76#13050578—22no—
77#13050578—41no—
78#13050578D414yescomplemented
79#13050577—41no—
80#13050575—22no—
81#13050575—22no—
82#13050574—22no—
83#13050574—22no—
84C2×C83⋊C8C2311yescharacteristic
85C2×C83⋊C8C2311yescharacteristic
86C2×C83⋊C8C2311yescharacteristic
87C2×C83⋊C8C2311yescharacteristic
88C4⋊Dic83C2311yescharacteristic
89Dic83⋊C4—21no—
90Dic83⋊C4—21no—
91C22⋊Dic83—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95C4×Dic83—21no—
96C24×C83—21no—
97#41×C83—22no—
98#41×C83—41no—
99C2×Q8×C83C2311yescharacteristic
100C2×D4×C83—22no—
101C2×D4×C83—21no—
102C2×D4×C83—21no—
103C2×D4×C83—21no—
104C2×D4×C83—41no—
105C2×D4×C83D411yescharacteristic
106C2×D4×C83C2311yescharacteristic
107C2×D4×C83C2311yescharacteristic
108C22×C4×C83—21no—
109C22×C4×C83—21no—
110C22×C4×C83C2311yescharacteristic
111C22×C4×C83C2311yescharacteristic
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×C83—21no—
120C22⋊C4×C83D411yescharacteristic
121C22⋊C4×C83D411yescharacteristic
122C42×C83C2311yescharacteristic
123C23×C83—21no—
124C23×C83—21no—
125C23×C83—42no—
126C23×C83—42no—
127C23×C83—81no—
128C23×C83C2×D411yescharacteristic
129C23×C83C2×D411yescharacteristic
130C22×D83—21no—
131Q8×C83—21no—
132Q8×C83—21no—
133D4×C83—21no—
134D4×C83—21no—
135D4×C83—21no—
136D4×C83—41no—
137D4×C83—42no—
138D4×C83—42no—
139D4×C83—81no—
140D4×C83C2×D412yes—
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—42no—
148C2×C4×C83—81no—
149C2×C4×C83C2×D411yescharacteristic
150C2×C4×C83C2×D411yescharacteristic
151C2×C4×C83C2×D411yescharacteristic
152C2×C4×C83C2×D411yescharacteristic
153C2×C4×C83C2411yescharacteristic
154C22⋊D83—42no—
155C22⋊D83—42no—
156C2×Dic83—21no—
157C2×Dic83—21no—
158C2×Dic83—21no—
159C2×Dic83—42no—
160D332—21no—
161D332—21no—
162C4×D83—42no—
163Dic166—21no—
164Dic166—21no—
165Dic166—21no—
166Dic166—42no—
167Dic166C2×D412yes—
168C83⋊C8—21no—
169C83⋊C8—21no—
170C83⋊C8—21no—
171C83⋊C8C2×D412yes—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—22no—
175C22×C83—21no—
176C22×C83—21no—
177C22×C83—21no—
178C22×C83—21no—
179C22×C83—41no—
180C22×C83—41no—
181C22×C83—41no—
182C22×C83—41no—
183C22×C83—41no—
184C22×C83—84no—
185C22×C83C22×D411yescharacteristic
186C2×D83—41no—
187C2×D83—41no—
188C2×D83—41no—
189C4×C83—21no—
190C4×C83—21no—
191C4×C83—21no—
192C4×C83—21no—
193C4×C83—41no—
194C4×C83—41no—
195C4×C83C22×D411yescharacteristic
196C4×C83H(5,𝔽2)11yescharacteristic
197Dic83—22no—
198Dic83—41no—
199Dic83—41no—
200C2×C83—41no—
201C2×C83—41no—
202C2×C83—42no—
203C2×C83—81no—
204C2×C83D4211yescharacteristic
205C2×C83#57511yescharacteristic
206C2×C83#57711yescharacteristic
207D83—81no—
208#3275—831nomaximal, Sylow
209C83#327511yescharacteristic, complemented, Sylow
210C2×#137—831no—
211C2×AGL(1,ℤ/8)—831no—
212#548—831no—
213#546D8311yescharacteristic, complemented
214#495—831no—
215#470—831no—
216#469—831no—
217#464—831no—
218#463—831no—
219#452—831no—
220#450—831no—
221#449—831no—
222#448—831no—
223C4×SD16—831no—
224#436—831no—
225C2×#41—831no—
226C2×#41C2×D8311yescharacteristic
227C22×Q8—831no—
228C22×D4C2×D8311yescharacteristic
229#137—1664no—
230AGL(1,ℤ/8)—1664no—
231C2×Q16—831no—
232C2×Q16—831no—
233C2×SD16—831no—
234C2×SD16—831no—
235C2×SD16—831no—
236C2×SD16—831no—
237C2×SD16—831no—
238C2×D8—831no—
239C2×D8—831no—
240C2×M16—831no—
241C2×M16—831no—
242#124—831no—
243#124—1661no—
244#124C2×D8311yescharacteristic
245#123—21no—
246C22⋊Q8—831no—
247C22⋊Q8—1661no—
248C22⋊Q8C2×D8311yescharacteristic
249#121—21no—
250#121—831no—
251#121—1661no—
252#121C2×D8311yescharacteristic
253#120—21no—
254C4×Q8—831no—
255C4×D4—1661no—
256C4×D4C2×D8311yescharacteristic
257C4×D4C2×D8311yescharacteristic
258C2×C22⋊C4—21no—
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—21no—
271#41—22no—
272#41—41no—
273#41—3321no—
274#41—1662no—
275C2×Q8—3321no—
276C2×Q8—831no—
277C2×Q8—831no—
278C2×Q8—1662no—
279C2×Q8—1661no—
280C2×Q8C22×D8311yescharacteristic
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—831no—
287C2×D4—1661no—
288C2×D4—1661no—
289C2×D4C22⋊D8311yescharacteristic
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—1662no—
299Q16—1662no—
300SD16—3321no—
301SD16—3321no—
302SD16—834no—
303SD16—1662no—
304SD16—1662no—
305D8—1662no—
306D8—1662no—
307M16—1662no—
308M16—1662no—
309C2×C8—831no—
310C2×C8—831no—
311C2×C8—831no—
312C2×C8—831no—
313C4⋊C4—21no—
314C4⋊C4—831no—
315C4⋊C4—1661no—
316C4⋊C4—1661no—
317C4⋊C4C22⋊D8311yescharacteristic
318C4⋊C4C22×D8311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—1661no—
325C22⋊C4—1661no—
326C22⋊C4—1661no—
327C22⋊C4—1661no—
328C22⋊C4C22⋊D8311yescharacteristic
329C22⋊C4C22⋊D8311yescharacteristic
330C42—1661no—
331C42C22×D8311yescharacteristic
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—1661no—
338C23C2×C22⋊D8311yescharacteristic
339C23C2×C22⋊D8311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3322no—
343Q8—832no—
344Q8—1661no—
345Q8—1661no—
346Q8—1661no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—41no—
351D4—42no—
352D4—42no—
353D4—81no—
354D4—3322no—
355D4—3322no—
356D4—1661no—
357D4—1661no—
358D4D4×D8312yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—3322no—
368C2×C4—3322no—
369C2×C4—1661no—
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—22no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—3321no—
395C22—3321no—
396C22—3321no—
397C22C22×C22⋊D8311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—3321no—
405C4—3321no—
406C4—1662no—
407C4C2×D4×D8311yescharacteristic
408C4#1458183711yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—81no—
413C2—6641no—
414C2D4×C22⋊D8311yescharacteristic
415C2#1912849911yescharacteristic
416C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#546×C83C2
socleC22×C83C22×D4
hypercenter#546D83
hypocenterC83#3275
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#14581822C2semidirect
2. C2×#14581818C2semidirect
3. #19128360C2semidirect
4. #19128083C2semidirect
5. #19127957C2semidirect
6. #19127952C2semidirect
7. #19127920C2semidirect
8. #19127916C2semidirect
9. #19127861C2semidirect
10. #19127852C2semidirect
11. #19127846C2semidirect
12. #19127839C2semidirect
13. C4×#13050578C2semidirect
14. #19127779C2semidirect
15. #546×C83C2semidirect
17. C22×Dic166C22semidirect
20. C2×#13050580C22semidirect
21. C2×#13050580C22semidirect
26. C2×#13050577C22semidirect
30. C2×#13050574C22semidirect
31. #14581780C22semidirect
35. #14581766C22semidirect
37. C4×Dic166C22semidirect
39. #14581723C22semidirect
40. C4⋊(C83⋊C8)C22semidirect
41. #14581716C22semidirect
42. #14581715C22semidirect
43. #14581715C22semidirect
46. #14581712C22semidirect
48. C22⋊(C83⋊C8)C22semidirect
49. C4×C83⋊C8C22semidirect
78. #13050578D4semidirect
209. C83#3275semidirect
213. #546D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818222
C2×#145818182
#191283602
#191280832
#191279572
#191279522
#191279202
#191279162
#191278612
#191278522
#191278462
#191278392
C4×#130505782
#191277792
#546×C832
#327583