Finite Group Databasebeta

← #24919424

Subgroups — #24919424

Order 10624.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128355C211yesmaximal, characteristic, complemented
2#19128060C211yesmaximal, characteristic, complemented
3#19128029C211yesmaximal, characteristic, complemented
4#19128017C211yesmaximal, characteristic, complemented
5#19127991C211yesmaximal, characteristic, complemented
6#19127952C211yesmaximal, characteristic, complemented
7#19127932C211yesmaximal, characteristic, complemented
8#19127913C211yesmaximal, characteristic, complemented
9#19127846C211yesmaximal, characteristic, complemented
10#19127835C211yesmaximal, characteristic, complemented
11#19127811C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127702C211yesmaximal, characteristic, complemented
14C2×#14581713C211yesmaximal, characteristic, complemented
15#546×C83C211yesmaximal, characteristic, complemented
16C2×#13050589—21no—
17C2×#13050577C2211yescharacteristic, complemented
18C2×#13050575C2211yescharacteristic
19C2×#13050575C2211yescharacteristic
20C2×#13050574C2211yescharacteristic, complemented
21C22×C83⋊C8C2211yescharacteristic
22#14581792C2211yescharacteristic, complemented
23#14581776—21no—
24#14581776—21no—
25#14581772—21no—
26#14581766C2211yescharacteristic
27#14581766C2211yescharacteristic, complemented
28C4×D332C2211yescharacteristic, complemented
29C2×C4⋊Dic83C2211yescharacteristic
30C2×C4⋊Dic83C2211yescharacteristic, complemented
31#14581723C2211yescharacteristic, complemented
32#14581723C2211yescharacteristic, complemented
33#14581723C2211yescharacteristic, complemented
34C4⋊(C83⋊C8)C2211yescharacteristic, complemented
35#14581716C2211yescharacteristic, complemented
36#14581716C2211yescharacteristic, complemented
37#14581716C2211yescharacteristic, complemented
38#14581713C2211yescharacteristic
39#14581713C2211yescharacteristic
40#14581713C2211yescharacteristic
41#14581713C2211yescharacteristic
42#14581713C2211yescharacteristic
43#14581713C2211yescharacteristic
44#14581712C2211yescharacteristic, complemented
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic, complemented
47#14581703C2211yescharacteristic, complemented
48C2×#41×C83C2211yescharacteristic
49C22×D4×C83C2211yescharacteristic
50#124×C83C2211yescharacteristic
51#123×C83—21no—
52C22⋊Q8×C83C2211yescharacteristic
53#121×C83—21no—
54#121×C83C2211yescharacteristic
55#120×C83—21no—
56C4×D4×C83C2211yescharacteristic
57C4×D4×C83C2211yescharacteristic
58C2×C22⋊C4×C83—21no—
59#13050589—44no—
60C2×Dic166—21no—
61C2×D332C2311yescharacteristic
62C2×C22⋊D83—21no—
63C2×C22⋊D83—21no—
64C22×Dic83—21no—
65C22×Dic83—21no—
66C2×C4×D83—21no—
67#13050577—41no—
68#13050575—22no—
69#13050575—41no—
70#13050574—41no—
71C2×C83⋊C8—41no—
72C2×C83⋊C8C2311yescharacteristic
73C2×C83⋊C8C2311yescharacteristic
74C2×C83⋊C8C2311yescharacteristic
75C2×C83⋊C8C2311yescharacteristic
76C4⋊Dic83—21no—
77C4⋊Dic83—21no—
78C4⋊Dic83—21no—
79C4⋊Dic83C2311yescharacteristic
80C4⋊Dic83C2311yescharacteristic
81C4⋊Dic83C2311yescharacteristic
82(C2×D83)⋊C4—21no—
83(C2×D83)⋊C4—21no—
84(C2×D83)⋊C4—21no—
85C24×C83—21no—
86#41×C83—22no—
87#41×C83—41no—
88C2×Q8×C83C2311yescharacteristic
89C2×D4×C83—22no—
90C2×D4×C83—21no—
91C2×D4×C83—21no—
92C2×D4×C83—21no—
93C2×D4×C83—41no—
94C2×D4×C83D411yescharacteristic
95C2×D4×C83C2311yescharacteristic
96C2×D4×C83C2311yescharacteristic
97C22×C4×C83—21no—
98C22×C4×C83—21no—
99C22×C4×C83C2311yescharacteristic
100C22×C4×C83C2311yescharacteristic
101C4⋊C4×C83—21no—
102C4⋊C4×C83D411yescharacteristic
103C4⋊C4×C83C2311yescharacteristic
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C22⋊C4×C83—21no—
108C22⋊C4×C83—21no—
109C22⋊C4×C83D411yescharacteristic
110C22⋊C4×C83D411yescharacteristic
111C42×C83C2311yescharacteristic
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—42no—
115C23×C83—42no—
116C23×C83—81no—
117C23×C83C2×D411yescharacteristic
118C23×C83C2×D411yescharacteristic
119C22×D83—21no—
120Q8×C83—21no—
121Q8×C83—21no—
122D4×C83—22no—
123D4×C83—21no—
124D4×C83—21no—
125D4×C83—21no—
126D4×C83—42no—
127D4×C83—42no—
128D4×C83—81no—
129D4×C83#4112yescomplemented
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—42no—
137C2×C4×C83—81no—
138C2×C4×C83C2×D411yescharacteristic
139C2×C4×C83C2×D411yescharacteristic
140C2×C4×C83C2×D411yescharacteristic
141C2×C4×C83C2×D411yescharacteristic
142C2×C4×C83C2411yescharacteristic
143C22⋊D83—42no—
144C22⋊D83—42no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—42no—
149C2×Dic83—42no—
150D332—21no—
151D332—21no—
152C4×D83—42no—
153Dic166—42no—
154C83⋊C8—21no—
155C83⋊C8—21no—
156C83⋊C8—21no—
157C83⋊C8—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—22no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83—84no—
170C22×C83AGL(1,ℤ/8)12yes—
171C22×C83C22×D411yescharacteristic
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83C2×#4111yescharacteristic
182C4×C83H(5,𝔽2)11yescharacteristic
183Dic83—41no—
184Dic83—41no—
185Dic83—41no—
186C2×C83—22no—
187C2×C83—41no—
188C2×C83—42no—
189C2×C83—81no—
190C2×C83#54611yescharacteristic
191C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C83#57711yescharacteristic
193D83—81no—
194#3306—831nomaximal, Sylow
195C83#330611yescharacteristic, complemented, Sylow
196#547—831no—
197#546D8311yescharacteristic, complemented
198#489—831no—
199#483—831no—
200#481—831no—
201#476—831no—
202#469—831no—
203#465—831no—
204#463—831no—
205#449—831no—
206#447—831no—
207#442—831no—
208#435—831no—
209#417—831no—
210C2×#102—831no—
211C2×#41—1661no—
212C2×#41C2×D8311yescharacteristic
213C22×D4C2×D8311yescharacteristic
214C2×SD16—831no—
215C2×D8—831no—
216C2×D8—831no—
217C2×M16—831no—
218C22×C8—831no—
219C4⋊Q8—831no—
220#124C2×D8311yescharacteristic
221#123—21no—
222#123—1661no—
223#123—1661no—
224C22⋊Q8—1661no—
225C22⋊Q8C2×D8311yescharacteristic
226#121—21no—
227#121—831no—
228#121—831no—
229#121C2×D8311yescharacteristic
230#120—21no—
231C4×D4—831no—
232C4×D4C2×D8311yescharacteristic
233C4×D4C2×D8311yescharacteristic
234C2×C4⋊C4—831no—
235C2×C4⋊C4—831no—
236C2×C22⋊C4—21no—
237#106—831no—
238#106—831no—
239#106—831no—
240C4⋊C8—831no—
241#103—831no—
242#103—831no—
243#103—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247#102—831no—
248#102—831no—
249#102—831no—
250#102—831no—
251C22⋊C8—831no—
252C22⋊C8—831no—
253#97—831no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—3324no—
258C2×Q8—1661no—
259C2×Q8C22×D8311yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—831no—
266C2×D4—1661no—
267C2×D4—1661no—
268C2×D4C22⋊D8311yescharacteristic
269C2×D4C22×D8311yescharacteristic
270C2×D4C22×D8311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—1661no—
274C22×C4—1661no—
275C22×C4—1661no—
276C22×C4C22×D8311yescharacteristic
277C22×C4C22×D8311yescharacteristic
278SD16—3321no—
279D8—3321no—
280D8—1662no—
281M16—3321no—
282C2×C8—3321no—
283C2×C8—831no—
284C2×C8—831no—
285C2×C8—831no—
286C2×C8—831no—
287C4⋊C4—21no—
288C4⋊C4—831no—
289C4⋊C4—831no—
290C4⋊C4—831no—
291C4⋊C4—1661no—
292C4⋊C4—1661no—
293C4⋊C4—1661no—
294C4⋊C4C22⋊D8311yescharacteristic
295C4⋊C4C22×D8311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4C22⋊D8311yescharacteristic
305C22⋊C4C22⋊D8311yescharacteristic
306C42C22×D8311yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—1661no—
313C23C2×C22⋊D8311yescharacteristic
314C23C2×C22⋊D8311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—3322no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—3322no—
326D4—3322no—
327D4—1661no—
328D4—1661no—
329D4#1305058912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—3322no—
339C2×C4—3322no—
340C2×C4—3322no—
341C2×C4—1661no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4C2×C22⋊D8311yescharacteristic
345C2×C4C2×C22⋊D8311yescharacteristic
346C2×C4C2×C22⋊D8311yescharacteristic
347C2×C4C2×C22⋊D8311yescharacteristic
348C2×C4C23×D8311yescharacteristic
349C8—1661no—
350C8—1661no—
351C8—1661no—
352C8—1661no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—3321no—
366C22—3321no—
367C22—3321no—
368C22#1458181712yes—
369C22C22×C22⋊D8311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—3321no—
377C4—3321no—
378C4—3321no—
379C4C2×#1305058911yescharacteristic
380C4#1458183711yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—6641no—
386C2#1912834511yescharacteristic
387C2C2×#1458181711yescharacteristic
388C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#546×C83C2
socleC22×C83C22×D4
hypercenter#546D83
hypocenterC83#3306
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. #19128355C2semidirect
2. #19128060C2semidirect
3. #19128029C2semidirect
4. #19128017C2semidirect
5. #19127991C2semidirect
6. #19127952C2semidirect
7. #19127932C2semidirect
8. #19127913C2semidirect
9. #19127846C2semidirect
10. #19127835C2semidirect
11. #19127811C2semidirect
12. #19127773C2semidirect
13. #19127702C2semidirect
14. C2×#14581713C2semidirect
15. #546×C83C2semidirect
17. C2×#13050577C22semidirect
20. C2×#13050574C22semidirect
22. #14581792C22semidirect
27. #14581766C22semidirect
28. C4×D332C22semidirect
30. C2×C4⋊Dic83C22semidirect
31. #14581723C22semidirect
32. #14581723C22semidirect
33. #14581723C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. #14581716C22semidirect
36. #14581716C22semidirect
37. #14581716C22semidirect
44. #14581712C22semidirect
46. C22⋊(C83⋊C8)C22semidirect
47. #14581703C22semidirect
129. D4×C83#41semidirect
195. C83#3306semidirect
197. #546D83semidirect
329. D4#13050589semidirect

Maximal subgroups

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

subgroupindex
#191283552
#191280602
#191280292
#191280172
#191279912
#191279522
#191279322
#191279132
#191278462
#191278352
#191278112
#191277732
#191277022
C2×#145817132
#546×C832
#330683