Finite Group Databasebeta

← #24919537

Subgroups — #24919537

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#19128339C211yesmaximal, characteristic
3#19128062C211yesmaximal, characteristic, complemented
4#19128035C211yesmaximal, characteristic, complemented
5#19128015C211yesmaximal, characteristic, complemented
6#19127979C211yesmaximal, characteristic, complemented
7#19127949C211yesmaximal, characteristic, complemented
8#19127939C211yesmaximal, characteristic, complemented
9#19127925C211yesmaximal, characteristic, complemented
10#19127850C211yesmaximal, characteristic, complemented
11#19127802C211yesmaximal, characteristic, complemented
12#19127775C211yesmaximal, characteristic, complemented
13#19127702C211yesmaximal, characteristic, complemented
14#102×D83C211yesmaximal, characteristic, complemented
15#452×C83C211yesmaximal, characteristic, complemented
16C2×D4×D83C2211yescharacteristic
17C2×#13050577C2211yescharacteristic
18C2×#13050574C2211yescharacteristic, complemented
19C2×C8×D83C2211yescharacteristic
20#14581791C2211yescharacteristic, complemented
21#14581782—21no—
22#14581781C2211yescharacteristic
23#14581775—21no—
24#14581772—21no—
25#14581769—21no—
26#14581768C2211yescharacteristic
27#14581768C2211yescharacteristic
28#14581766C2211yescharacteristic
29#14581764—21no—
30#14581760—21no—
31Q8×Dic83C2211yescharacteristic, complemented
32D4×Dic83C2211yescharacteristic
33C4×C22⋊D83C2211yescharacteristic
34(C22⋊D83)⋊C4—21no—
35C2×C4⋊Dic83C2211yescharacteristic, complemented
36C4⋊C4×D83C2211yescharacteristic
37C2×(C2×D83)⋊C4—21no—
38#14581726C2211yescharacteristic, complemented
39#14581726C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, complemented
41Dic83⋊C8C2211yescharacteristic, complemented
42#14581716C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581716C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic
46#14581714C2211yescharacteristic
47#14581713C2211yescharacteristic
48#14581713C2211yescharacteristic
49#14581712C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51#14581702C2211yescharacteristic, complemented
52C2×#41×C83C2211yescharacteristic
53C2×Q16×C83C2211yescharacteristic, complemented
54C2×SD16×C83C2211yescharacteristic
55C22⋊Q8×C83C2211yescharacteristic, complemented
56#103×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic
58C22⋊C8×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60C2×Dic166—21no—
61C2×D332C2311yescharacteristic
62C2×C22⋊D83—21no—
63C2×C22⋊D83—21no—
64C2×C22⋊D83D411yescharacteristic
65D4×D83—22no—
66D4×D83—41no—
67C22×Dic83—21no—
68C22×Dic83—21no—
69C2×C4×D83—21no—
70C2×C4×D83C2311yescharacteristic
71#13050577—41no—
72#13050574—41no—
73C2×C83⋊C8C2311yescharacteristic
74C2×C83⋊C8C2311yescharacteristic
75C8×D83—41no—
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⋊Dic83D411yescharacteristic
88(C2×D83)⋊C4—21no—
89(C2×D83)⋊C4—21no—
90(C2×D83)⋊C4—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4D411yescharacteristic
94C4×Dic83—21no—
95C4×Dic83C2311yescharacteristic
96#41×C83—22no—
97#41×C83—41no—
98C2×Q8×C83C2311yescharacteristic
99C2×Q8×C83C2311yescharacteristic
100C2×D4×C83—21no—
101C2×D4×C83C2311yescharacteristic
102C22×C4×C83—21no—
103C22×C4×C83C2311yescharacteristic
104Q16×C83—41no—
105SD16×C83—22no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C23×C83—21no—
112C23×C83C2×D411yescharacteristic
113C22×D83—21no—
114C22×D83—42no—
115C22×D83—42no—
116C22×D83—81no—
117C22×D83C2×D411yescharacteristic
118Q8×C83—21no—
119Q8×C83—21no—
120Q8×C83—21no—
121Q8×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×C83—21no—
130C2×C4×C83—42no—
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2411yescharacteristic
133C22⋊D83—22no—
134C22⋊D83—42no—
135C22⋊D83—81no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—81no—
140C2×Dic83—81no—
141C2×Dic83C2×D411yescharacteristic
142C2×Dic83C2×D411yescharacteristic
143C2×Dic83C2×D411yescharacteristic
144D332—21no—
145D332—21no—
146C4×D83—21no—
147C4×D83—21no—
148C4×D83—81no—
149Dic166—81no—
150C8×C83—21no—
151C8×C83—21no—
152C83⋊C8—21no—
153C83⋊C8—21no—
154C22×C83—22no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83C22×D411yescharacteristic
161C2×D83—21no—
162C2×D83—21no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—41no—
166C2×D83—84no—
167C2×D83#13712yes—
168C4×C83—21no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83C2×#4111yescharacteristic
173C4×C83H(5,𝔽2)11yescharacteristic
174Dic83—21no—
175Dic83—21no—
176Dic83—21no—
177Dic83—41no—
178Dic83—41no—
179Dic83—41no—
180C2×C83—42no—
181C2×C83—41no—
182C2×C83#54611yescharacteristic
183C2×C83C2×#13711yescharacteristic
184C2×C83#57611yescharacteristic
185D83—22no—
186D83—81no—
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#452D8311yescharacteristic, complemented
199#450—831no—
200#440—831no—
201#435—831no—
202#417—831no—
203C2×#102—831no—
204C2×#41C2×D8311yescharacteristic
205C22×D4—831no—
206C2×Q16C2×D8311yescharacteristic, complemented
207C2×SD16—831no—
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, complemented
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#103C2×D8311yescharacteristic, complemented
238#102—831no—
239#102—831no—
240#102—831no—
241#102—831no—
242#102—831no—
243#102C2×D8311yescharacteristic
244C22⋊C8—831no—
245C22⋊C8C2×D8311yescharacteristic, complemented
246#97—831no—
247C24—1661no—
248#41—22no—
249#41—41no—
250C2×Q8—1661no—
251C2×Q8C22×D8311yescharacteristic
252C2×Q8C22×D8311yescharacteristic
253C2×D4—21no—
254C2×D4—3321no—
255C2×D4—831no—
256C2×D4—831no—
257C2×D4—1661no—
258C2×D4—1662no—
259C2×D4—1661no—
260C2×D4C22×D8311yescharacteristic
261C22×C4—21no—
262C22×C4—831no—
263C22×C4—1661no—
264C22×C4—1661no—
265C22×C4—1661no—
266C22×C4C22×D8311yescharacteristic
267Q16—41no—
268SD16—22no—
269SD16—3321no—
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—3322no—
301C23—3322no—
302C23—831no—
303C23—6641no—
304C23—1661no—
305C23D4×D8311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—6641no—
311D4—21no—
312D4—42no—
313D4—3322no—
314D4—6641no—
315D4—1662no—
316D4—1661no—
317D4—1661no—
318D4#1305058912yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—831no—
326C2×C4—831no—
327C2×C4—831no—
328C2×C4—6641no—
329C2×C4—6641no—
330C2×C4—6641no—
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—22no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—3321no—
349C22—3321no—
350C22—3321no—
351C22—832no—
352C22—6644no—
353C22—1661no—
354C22—1661no—
355C22C2×D4×D8311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3321no—
361C4—3321no—
362C4—3321no—
363C4—1661no—
364C4—1661no—
365C4—1661no—
366C4C2×#1305058911yescharacteristic
367C4#1458183711yescharacteristic
368C2—41no—
369C2—42no—
370C2—6641no—
371C2—1662no—
372C2#1912833811yescharacteristic
373C2#137×D8311yescharacteristic
374C2#1912850311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#452×C83C2
socleC22×C83C22×D4
prime radical#19128339C2
hypercenter#452D83
hypocenterC83#3314
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #19128409C2semidirect
3. #19128062C2semidirect
4. #19128035C2semidirect
5. #19128015C2semidirect
6. #19127979C2semidirect
7. #19127949C2semidirect
8. #19127939C2semidirect
9. #19127925C2semidirect
10. #19127850C2semidirect
11. #19127802C2semidirect
12. #19127775C2semidirect
13. #19127702C2semidirect
14. #102×D83C2semidirect
15. #452×C83C2semidirect
18. C2×#13050574C22semidirect
20. #14581791C22semidirect
31. Q8×Dic83C22semidirect
35. C2×C4⋊Dic83C22semidirect
38. #14581726C22semidirect
39. #14581726C22semidirect
40. #14581725C22semidirect
41. Dic83⋊C8C22semidirect
42. #14581716C22semidirect
43. #14581716C22semidirect
44. #14581716C22semidirect
51. #14581702C22semidirect
53. C2×Q16×C83C22semidirect
55. C22⋊Q8×C83C22semidirect
56. #103×C83C22semidirect
58. C22⋊C8×C83C22semidirect
188. C83#3314semidirect
198. #452D83semidirect
206. C2×Q16C2×D83semidirect
219. C22⋊Q8C2×D83semidirect
237. #103C2×D83semidirect
245. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284092
#191283392
#191280622
#191280352
#191280152
#191279792
#191279492
#191279392
#191279252
#191278502
#191278022
#191277752
#191277022
#102×D832
#452×C832
#331483