Finite Group Databasebeta

← #24919544

Subgroups — #24919544

Order 10624.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128408C211yesmaximal, characteristic, complemented
2#19128060C211yesmaximal, characteristic, complemented
3#19128034C211yesmaximal, characteristic, complemented
4#19128012C211yesmaximal, characteristic, complemented
5#19127981C211yesmaximal, characteristic, complemented
6#19127945C211yesmaximal, characteristic, complemented
7#19127940C211yesmaximal, characteristic, complemented
8#19127920C211yesmaximal, characteristic, complemented
9#19127863C211yesmaximal, characteristic, complemented
10#19127852C211yesmaximal, characteristic, complemented
11#19127801C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127702C211yesmaximal, characteristic, complemented
14C2×#14581713C211yesmaximal, characteristic, complemented
15#546×C83C211yesmaximal, characteristic
16C2×#13050580C2211yescharacteristic, complemented
17C2×#13050578C2211yescharacteristic
18C2×#13050578C2211yescharacteristic
19C2×#13050574C2211yescharacteristic, complemented
20C22×C83⋊C8C2211yescharacteristic
21#14581792C2211yescharacteristic, complemented
22#14581783—21no—
23#14581771C2211yescharacteristic
24#14581771C2211yescharacteristic, complemented
25#14581769—21no—
26#14581769—21no—
27C4×Dic166C2211yescharacteristic, complemented
28D4×Dic83—21no—
29C2×C4⋊Dic83C2211yescharacteristic
30C2×C4⋊Dic83C2211yescharacteristic, complemented
31#14581725C2211yescharacteristic, complemented
32#14581725C2211yescharacteristic, complemented
33#14581725C2211yescharacteristic, complemented
34C4⋊(C83⋊C8)C2211yescharacteristic, complemented
35#14581716C2211yescharacteristic, complemented
36#14581716C2211yescharacteristic, complemented
37#14581716C2211yescharacteristic, complemented
38#14581715C2211yescharacteristic, complemented
39#14581713C2211yescharacteristic
40#14581713C2211yescharacteristic
41#14581713C2211yescharacteristic
42#14581713C2211yescharacteristic
43#14581713C2211yescharacteristic
44#14581713C2211yescharacteristic
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—
59C2×Dic166—21no—
60C2×Dic166C2311yescharacteristic
61C22×Dic83—21no—
62C22×Dic83—21no—
63#13050580—41no—
64#13050578—22no—
65#13050578—41no—
66#13050574—41no—
67C2×C83⋊C8—41no—
68C2×C83⋊C8C2311yescharacteristic
69C2×C83⋊C8C2311yescharacteristic
70C2×C83⋊C8C2311yescharacteristic
71C2×C83⋊C8C2311yescharacteristic
72C4⋊Dic83—21no—
73C4⋊Dic83—21no—
74C4⋊Dic83—21no—
75C4⋊Dic83C2311yescharacteristic
76C4⋊Dic83C2311yescharacteristic
77C4⋊Dic83C2311yescharacteristic
78Dic83⋊C4—21no—
79Dic83⋊C4—21no—
80Dic83⋊C4—21no—
81C22⋊Dic83—21no—
82C22⋊Dic83—21no—
83C4×Dic83—21no—
84C24×C83—21no—
85#41×C83—22no—
86#41×C83—41no—
87C2×Q8×C83C2311yescharacteristic
88C2×D4×C83—22no—
89C2×D4×C83—21no—
90C2×D4×C83—21no—
91C2×D4×C83—21no—
92C2×D4×C83—41no—
93C2×D4×C83D411yescharacteristic
94C2×D4×C83C2311yescharacteristic
95C2×D4×C83C2311yescharacteristic
96C22×C4×C83—21no—
97C22×C4×C83—21no—
98C22×C4×C83C2311yescharacteristic
99C22×C4×C83C2311yescharacteristic
100C4⋊C4×C83—21no—
101C4⋊C4×C83D411yescharacteristic
102C4⋊C4×C83C2311yescharacteristic
103C22⋊C4×C83—21no—
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C22⋊C4×C83—21no—
108C22⋊C4×C83D411yescharacteristic
109C22⋊C4×C83D411yescharacteristic
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—42no—
114C23×C83—42no—
115C23×C83—81no—
116C23×C83C2×D411yescharacteristic
117C23×C83C2×D411yescharacteristic
118Q8×C83—21no—
119Q8×C83—21no—
120D4×C83—22no—
121D4×C83—21no—
122D4×C83—21no—
123D4×C83—21no—
124D4×C83—42no—
125D4×C83—42no—
126D4×C83—81no—
127D4×C83#4112yes—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—42no—
135C2×C4×C83—81no—
136C2×C4×C83C2×D411yescharacteristic
137C2×C4×C83C2×D411yescharacteristic
138C2×C4×C83C2×D411yescharacteristic
139C2×C4×C83C2×D411yescharacteristic
140C2×C4×C83C2411yescharacteristic
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—81no—
146C2×Dic83—81no—
147Dic166—21no—
148Dic166—21no—
149Dic166—81no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C83⋊C8—21no—
153C83⋊C8—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—22no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—84no—
166C22×C83#13712yes—
167C22×C83C22×D411yescharacteristic
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—21no—
171C4×C83—21no—
172C4×C83—41no—
173C4×C83—41no—
174C4×C83C2×#4111yescharacteristic
175C4×C83H(5,𝔽2)11yescharacteristic
176Dic83—41no—
177Dic83—41no—
178Dic83—41no—
179Dic83—41no—
180C2×C83—22no—
181C2×C83—41no—
182C2×C83—42no—
183C2×C83—81no—
184C2×C83#54611yescharacteristic
185C2×C83C2×#13711yescharacteristic
186C2×C83#57611yescharacteristic
187#3314—831nomaximal, Sylow
188C83#331411yescharacteristic, complemented, Sylow
189#554—831no—
190#546D8311yescharacteristic
191#489—831no—
192#484—831no—
193#480—831no—
194#474—831no—
195#468—831no—
196#467—831no—
197#464—831no—
198#452—831no—
199#450—831no—
200#440—831no—
201#435—831no—
202#417—831no—
203C2×#102—831no—
204C2×#41C2×D8311yescharacteristic
205C22×D4C2×D8311yescharacteristic
206C2×Q16—831no—
207C2×SD16—831no—
208C2×SD16—831no—
209C2×M16—831no—
210C22×C8—831no—
211C4⋊Q8—831no—
212#125—1661no—
213#124C2×D8311yescharacteristic
214#123—21no—
215C22⋊Q8—831no—
216C22⋊Q8—831no—
217C22⋊Q8—1661no—
218C22⋊Q8—1661no—
219C22⋊Q8C2×D8311yescharacteristic
220#121—21no—
221#121C2×D8311yescharacteristic
222#120—21no—
223C4×Q8—831no—
224C4×D4—1661no—
225C4×D4C2×D8311yescharacteristic
226C4×D4C2×D8311yescharacteristic
227C2×C4⋊C4—831no—
228C2×C4⋊C4—831no—
229C2×C22⋊C4—21no—
230#107—831no—
231#107—831no—
232#107—831no—
233C4⋊C8—831no—
234#103—831no—
235#103—831no—
236#103—831no—
237#103—831no—
238#102—831no—
239#102—831no—
240#102—831no—
241#102—831no—
242#102—831no—
243#102—831no—
244C22⋊C8—831no—
245C22⋊C8—831no—
246#97—831no—
247C24—21no—
248#41—22no—
249#41—41no—
250C2×Q8—831no—
251C2×Q8—1661no—
252C2×Q8C22×D8311yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4C22⋊D8311yescharacteristic
259C2×D4C22×D8311yescharacteristic
260C2×D4C22×D8311yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—1661no—
264C22×C4—1661no—
265C22×C4C22×D8311yescharacteristic
266C22×C4C22×D8311yescharacteristic
267Q16—3321no—
268SD16—3321no—
269SD16—1662no—
270M16—3321no—
271C2×C8—3321no—
272C2×C8—831no—
273C2×C8—831no—
274C2×C8—831no—
275C2×C8—831no—
276C4⋊C4—21no—
277C4⋊C4—831no—
278C4⋊C4—831no—
279C4⋊C4—831no—
280C4⋊C4—1661no—
281C4⋊C4—1661no—
282C4⋊C4—1661no—
283C4⋊C4—1661no—
284C4⋊C4—1661no—
285C4⋊C4—1661no—
286C4⋊C4C22⋊D8311yescharacteristic
287C4⋊C4C22×D8311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4C22⋊D8311yescharacteristic
296C22⋊C4C22⋊D8311yescharacteristic
297C42—1661no—
298C42C22×D8311yescharacteristic
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23C2×C22⋊D8311yescharacteristic
305C23C2×C22⋊D8311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—6641no—
309Q8—1661no—
310Q8—1661no—
311D4—22no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4#1305058912yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—6641no—
328C2×C4—6641no—
329C2×C4—1661no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4C2×C22⋊D8311yescharacteristic
334C2×C4C2×C22⋊D8311yescharacteristic
335C2×C4C2×C22⋊D8311yescharacteristic
336C2×C4C2×C22⋊D8311yescharacteristic
337C2×C4C23×D8311yescharacteristic
338C8—1661no—
339C8—1661no—
340C8—1661no—
341C8—1661no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—84no—
354C22#1458182212yes—
355C22C22×C22⋊D8311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—3321no—
363C4—3321no—
364C4—3321no—
365C4—3321no—
366C4C2×#1305058911yescharacteristic
367C4#1458183711yescharacteristic
368C2—22no—
369C2—41no—
370C2—42no—
371C2—81no—
372C2#1912834511yescharacteristic
373C2C2×#1458182211yescharacteristic
374C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#546×C83C2
socleC22×C83C22×D4
hypercenter#546D83
hypocenterC83#3314
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. #19128408C2semidirect
2. #19128060C2semidirect
3. #19128034C2semidirect
4. #19128012C2semidirect
5. #19127981C2semidirect
6. #19127945C2semidirect
7. #19127940C2semidirect
8. #19127920C2semidirect
9. #19127863C2semidirect
10. #19127852C2semidirect
11. #19127801C2semidirect
12. #19127773C2semidirect
13. #19127702C2semidirect
14. C2×#14581713C2semidirect
16. C2×#13050580C22semidirect
19. C2×#13050574C22semidirect
21. #14581792C22semidirect
24. #14581771C22semidirect
27. C4×Dic166C22semidirect
30. C2×C4⋊Dic83C22semidirect
31. #14581725C22semidirect
32. #14581725C22semidirect
33. #14581725C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. #14581716C22semidirect
36. #14581716C22semidirect
37. #14581716C22semidirect
38. #14581715C22semidirect
46. C22⋊(C83⋊C8)C22semidirect
47. #14581703C22semidirect
188. C83#3314semidirect

Maximal subgroups

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

subgroupindex
#191284082
#191280602
#191280342
#191280122
#191279812
#191279452
#191279402
#191279202
#191278632
#191278522
#191278012
#191277732
#191277022
C2×#145817132
#546×C832
#331483