Finite Group Databasebeta

← #24919548

Subgroups — #24919548

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#19128409C211yesmaximal, characteristic, complemented
2#19128345C211yesmaximal, characteristic
3#19128059C211yesmaximal, characteristic, complemented
4#19128037C211yesmaximal, characteristic, complemented
5#19128013C211yesmaximal, characteristic, complemented
6#19127979C211yesmaximal, characteristic, complemented
7#19127944C211yesmaximal, characteristic, complemented
8#19127940C211yesmaximal, characteristic, complemented
9#19127921C211yesmaximal, characteristic, complemented
10#19127860C211yesmaximal, characteristic, complemented
11#19127802C211yesmaximal, characteristic, complemented
12#19127769C211yesmaximal, characteristic, complemented
13#19127699C211yesmaximal, characteristic, complemented
14C2×#14581713C211yesmaximal, characteristic, complemented
15#450×C83C211yesmaximal, characteristic, complemented
16C2×#13050589C2211yescharacteristic
17C2×#13050580C2211yescharacteristic, complemented
18C2×#13050573C2211yescharacteristic, complemented
19C22×C83⋊C8C2211yescharacteristic
20#14581791C2211yescharacteristic, complemented
21#14581782—21no—
22#14581779C2211yescharacteristic
23#14581773—21no—
24#14581772—21no—
25#14581771C2211yescharacteristic
26#14581769—21no—
27#14581768C2211yescharacteristic, complemented
28#14581764—21no—
29#14581763C2211yescharacteristic
30#14581761—21no—
31Q8×Dic83C2211yescharacteristic, complemented
32D4×Dic83C2211yescharacteristic
33C4×C22⋊D83C2211yescharacteristic
34(C22⋊D83)⋊C4—21no—
35C2×C4⋊Dic83C2211yescharacteristic
36C4⋊C4×D83C2211yescharacteristic, complemented
37C2×C22⋊Dic83—21no—
38#14581726C2211yescharacteristic, complemented
39#14581726C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, complemented
41Dic83⋊C8C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581716C2211yescharacteristic, complemented
45#14581715C2211yescharacteristic, complemented
46#14581713C2211yescharacteristic
47#14581713C2211yescharacteristic
48#14581713C2211yescharacteristic
49#14581713C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51#14581702C2211yescharacteristic, complemented
52C22×D4×C83C2211yescharacteristic
53C2×SD16×C83C2211yescharacteristic
54C2×SD16×C83C2211yescharacteristic
55C22⋊Q8×C83C2211yescharacteristic
56#102×C83C2211yescharacteristic
57#102×C83C2211yescharacteristic
58C22⋊C8×C83C2211yescharacteristic
59#13050589—22no—
60#13050589—41no—
61C2×Dic166—21no—
62C2×Dic166C2311yescharacteristic
63C2×C22⋊D83—21no—
64C2×C22⋊D83—21no—
65C2×C22⋊D83D411yescharacteristic
66C22×Dic83—21no—
67C22×Dic83—21no—
68C22×Dic83—21no—
69C2×C4×D83—21no—
70C2×C4×D83C2311yescharacteristic
71#13050580—41no—
72#13050573—41no—
73C2×C83⋊C8—41no—
74C2×C83⋊C8C2311yescharacteristic
75C2×C83⋊C8C2311yescharacteristic
76C4⋊Dic83—21no—
77C4⋊Dic83—21no—
78C4⋊Dic83C2311yescharacteristic
79C4⋊Dic83C2311yescharacteristic
80C4⋊Dic83C2311yescharacteristic
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83Dic83⋊C4—21no—
84Dic83⋊C4—21no—
85Dic83⋊C4D411yescharacteristic
86C22⋊Dic83—21no—
87C22⋊Dic83—21no—
88C22⋊Dic83—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83—21no—
91C22⋊Dic83D411yescharacteristic
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4D411yescharacteristic
94C4×Dic83—21no—
95C4×Dic83C2311yescharacteristic
96C24×C83—21no—
97C2×Q8×C83C2311yescharacteristic
98C2×D4×C83—22no—
99C2×D4×C83—21no—
100C2×D4×C83—41no—
101C2×D4×C83C2311yescharacteristic
102C2×D4×C83C2311yescharacteristic
103C22×C4×C83C2311yescharacteristic
104SD16×C83—22no—
105SD16×C83—41no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—42no—
114C23×C83—42no—
115C23×C83—81no—
116C23×C83C2×D411yescharacteristic
117C22×D83C2×D411yescharacteristic
118Q8×C83—21no—
119Q8×C83—21no—
120D4×C83—21no—
121D4×C83—21no—
122D4×C83—21no—
123D4×C83—42no—
124D4×C83#4112yes—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2411yescharacteristic
131C22⋊D83—22no—
132C22⋊D83—42no—
133C22⋊D83—81no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—42no—
139C2×Dic83—81no—
140C2×Dic83—81no—
141C2×Dic83C2×D411yescharacteristic
142C2×Dic83C2×D411yescharacteristic
143C2×Dic83C2×D411yescharacteristic
144C4×D83—21no—
145C4×D83—21no—
146C4×D83—81no—
147Dic166—21no—
148Dic166—21no—
149Dic166—81no—
150C8×C83—21no—
151C8×C83—21no—
152C83⋊C8—21no—
153C83⋊C8—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—22no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—84no—
163C22×C83#13712yes—
164C22×C83C22×D411yescharacteristic
165C2×D83—21no—
166C2×D83—21no—
167C2×D83—21no—
168C4×C83—21no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83C2×#4111yescharacteristic
172C4×C83H(5,𝔽2)11yescharacteristic
173Dic83—21no—
174Dic83—21no—
175Dic83—21no—
176Dic83—41no—
177Dic83—41no—
178Dic83—41no—
179Dic83—41no—
180C2×C83—22no—
181C2×C83—42no—
182C2×C83—81no—
183C2×C83#54611yescharacteristic
184C2×C83C2×#13711yescharacteristic
185C2×C83#57611yescharacteristic
186D83—41no—
187#3314—831nomaximal, Sylow
188C83#331411yescharacteristic, complemented, Sylow
189#554—831no—
190#546—831no—
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#450D8311yescharacteristic, complemented
200#440—831no—
201#435—831no—
202#417—831no—
203C2×#102—831no—
204C2×#41—831no—
205C22×D4C2×D8311yescharacteristic
206C2×Q16—831no—
207C2×SD16C2×D8311yescharacteristic
208C2×SD16C2×D8311yescharacteristic
209C2×M16—831no—
210C22×C8—831no—
211C4⋊Q8—831no—
212#125—1661no—
213#124—831no—
214#123—1661no—
215C22⋊Q8—831no—
216C22⋊Q8—831no—
217C22⋊Q8—1661no—
218C22⋊Q8—1661no—
219C22⋊Q8C2×D8311yescharacteristic
220#121—831no—
221#121—1661no—
222#120—1661no—
223C4×Q8—831no—
224C4×D4—831no—
225C4×D4—831no—
226C4×D4—1661no—
227C2×C4⋊C4—831no—
228C2×C4⋊C4—831no—
229C2×C22⋊C4—1661no—
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#102C2×D8311yescharacteristic
243#102C2×D8311yescharacteristic
244C22⋊C8—831no—
245C22⋊C8C2×D8311yescharacteristic
246#97—831no—
247C24—21no—
248#41—3321no—
249#41—1662no—
250C2×Q8—831no—
251C2×Q8—1661no—
252C2×Q8C22×D8311yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—831no—
257C2×D4—1661no—
258C2×D4—1661no—
259C2×D4C22×D8311yescharacteristic
260C2×D4C22×D8311yescharacteristic
261C22×C4—831no—
262C22×C4—1661no—
263C22×C4—1661no—
264C22×C4—1661no—
265C22×C4—1661no—
266C22×C4C22×D8311yescharacteristic
267Q16—3321no—
268SD16—22no—
269SD16—41no—
270M16—3321no—
271C2×C8—3321no—
272C2×C8—831no—
273C2×C8—831no—
274C2×C8C22×D8311yescharacteristic
275C2×C8C22×D8311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—831no—
278C4⋊C4—831no—
279C4⋊C4—831no—
280C4⋊C4—831no—
281C4⋊C4—1661no—
282C4⋊C4—1661no—
283C4⋊C4—1661no—
284C4⋊C4—1661no—
285C4⋊C4—1661no—
286C4⋊C4—1661no—
287C4⋊C4C22×D8311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—831no—
290C22⋊C4—831no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4—1661no—
296C22⋊C4—1661no—
297C42—831no—
298C42—1661no—
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23—831no—
305C23D4×D8311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—6641no—
309Q8—1661no—
310Q8—1661no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—3322no—
316D4—6641no—
317D4—1662no—
318D4#1305058912yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—3322no—
324C2×C4—831no—
325C2×C4—831no—
326C2×C4—831no—
327C2×C4—6641no—
328C2×C4—6641no—
329C2×C4—6641no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4—1661no—
334C2×C4—1661no—
335C2×C4—1661no—
336C2×C4D4×D8311yescharacteristic
337C2×C4C23×D8311yescharacteristic
338C8—21no—
339C8—21no—
340C8—1661no—
341C8—1661no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22—1661no—
352C22—1661no—
353C22—1661no—
354C22#1458182112yes—
355C22C2×D4×D8311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—3321no—
360C4—3321no—
361C4—3321no—
362C4—3321no—
363C4—1661no—
364C4—1661no—
365C4—1661no—
366C4C2×#1305058911yescharacteristic
367C4#1458183711yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—3321no—
372C2#1912833811yescharacteristic
373C2C2×#1458182111yescharacteristic
374C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
prime radical#19128345C2
hypercenter#450D83
hypocenterC83#3314
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #19128409C2semidirect
3. #19128059C2semidirect
4. #19128037C2semidirect
5. #19128013C2semidirect
6. #19127979C2semidirect
7. #19127944C2semidirect
8. #19127940C2semidirect
9. #19127921C2semidirect
10. #19127860C2semidirect
11. #19127802C2semidirect
12. #19127769C2semidirect
13. #19127699C2semidirect
14. C2×#14581713C2semidirect
15. #450×C83C2semidirect
17. C2×#13050580C22semidirect
18. C2×#13050573C22semidirect
20. #14581791C22semidirect
27. #14581768C22semidirect
31. Q8×Dic83C22semidirect
36. C4⋊C4×D83C22semidirect
38. #14581726C22semidirect
39. #14581726C22semidirect
40. #14581725C22semidirect
41. Dic83⋊C8C22semidirect
42. #14581717C22semidirect
43. #14581717C22semidirect
44. #14581716C22semidirect
45. #14581715C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. #14581702C22semidirect
188. C83#3314semidirect
199. #450D83semidirect

Maximal subgroups

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

subgroupindex
#191284092
#191283452
#191280592
#191280372
#191280132
#191279792
#191279442
#191279402
#191279212
#191278602
#191278022
#191277692
#191276992
C2×#145817132
#450×C832
#331483