Finite Group Databasebeta

← #24919499

Subgroups — #24919499

Order 10624.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128421C211yesmaximal, characteristic, complemented
2#19128060C211yesmaximal, characteristic, complemented
3#19128039C211yesmaximal, characteristic, complemented
4#19128022C211yesmaximal, characteristic, complemented
5#19127991C211yesmaximal, characteristic, complemented
6#19127957C211yesmaximal, characteristic, complemented
7#19127928C211yesmaximal, characteristic, complemented
8#19127920C211yesmaximal, characteristic, complemented
9#19127863C211yesmaximal, characteristic, complemented
10#19127852C211yesmaximal, characteristic, complemented
11#19127801C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127706C211yesmaximal, characteristic, complemented
14#19127697C211yesmaximal, characteristic, complemented
15#546×C83C211yesmaximal, characteristic
16C2×#13050580C2211yescharacteristic, complemented
17C2×#13050578C2211yescharacteristic
18C2×#13050578C2211yescharacteristic
19C2×#13050574C2211yescharacteristic
20C22×C83⋊C8C2211yescharacteristic, complemented
21#14581792C2211yescharacteristic, complemented
22#14581785—21no—
23#14581785—21no—
24#14581779—21no—
25#14581771C2211yescharacteristic
26#14581771C2211yescharacteristic, complemented
27C4×Dic166C2211yescharacteristic, complemented
28#14581757—21no—
29#14581751C2211yescharacteristic
30#14581751C2211yescharacteristic, complemented
31#14581725C2211yescharacteristic, complemented
32#14581723C2211yescharacteristic, complemented
33#14581723C2211yescharacteristic, 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
61#13050580—41no—
62#13050578—22no—
63#13050578—41no—
64#13050574—41no—
65C2×C83⋊C8—41no—
66C2×C83⋊C8C2311yescharacteristic
67C2×C83⋊C8C2311yescharacteristic
68C2×C83⋊C8C2311yescharacteristic
69C2×C83⋊C8C2311yescharacteristic
70C4⋊Dic83—21no—
71C4⋊Dic83C2311yescharacteristic
72C4⋊Dic83C2311yescharacteristic
73C4⋊Dic83C2311yescharacteristic
74Dic83⋊C4—21no—
75Dic83⋊C4—21no—
76Dic83⋊C4—21no—
77C22⋊Dic83—21no—
78C22⋊Dic83—21no—
79C22⋊Dic83—21no—
80C22⋊Dic83—21no—
81C4×Dic83—21no—
82C4×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—21no—
121D4×C83—21no—
122D4×C83—21no—
123D4×C83—41no—
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—
145Dic166—21no—
146Dic166—21no—
147Dic166—42no—
148C83⋊C8—21no—
149C83⋊C8—21no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—22no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—84no—
165C22×C83C22×D411yescharacteristic
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83C2×#4111yescharacteristic
173C4×C83H(5,𝔽2)11yescharacteristic
174Dic83—41no—
175Dic83—41no—
176Dic83—41no—
177Dic83—41no—
178C2×C83—41no—
179C2×C83—41no—
180C2×C83—42no—
181C2×C83—81no—
182C2×C83#54611yescharacteristic
183C2×C83#57511yescharacteristic
184C2×C83#57711yescharacteristic
185#3311—831nomaximal, Sylow
186C83#331111yescharacteristic, complemented, Sylow
187#556—831no—
188#546D8311yescharacteristic
189#489—831no—
190#485—831no—
191#482—831no—
192#476—831no—
193#470—831no—
194#465—831no—
195#464—831no—
196#452—831no—
197#450—831no—
198#440—831no—
199#435—831no—
200#418—831no—
201#416—831no—
202C2×#41C2×D8311yescharacteristic
203C22×D4C2×D8311yescharacteristic
204C2×Q16—831no—
205C2×SD16—831no—
206C2×SD16—831no—
207C2×M16—831no—
208C22×C8—831no—
209C4⋊Q8—831no—
210#126—1661no—
211#126—1661no—
212#124—1661no—
213#124C2×D8311yescharacteristic
214#123—21no—
215C22⋊Q8—831no—
216C22⋊Q8—831no—
217C22⋊Q8C2×D8311yescharacteristic
218#121—21no—
219#121C2×D8311yescharacteristic
220#120—21no—
221C4×Q8—831no—
222C4×Q8—1661no—
223C4×D4C2×D8311yescharacteristic
224C4×D4C2×D8311yescharacteristic
225#117—831no—
226#117—831no—
227C2×C22⋊C4—21no—
228#107—831no—
229#106—831no—
230#106—831no—
231C4⋊C8—831no—
232#103—831no—
233#103—831no—
234#103—831no—
235#103—831no—
236#102—831no—
237#102—831no—
238#102—831no—
239#102—831no—
240#102—831no—
241#102—831no—
242C22⋊C8—831no—
243C22⋊C8—831no—
244#97—831no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—831no—
249C2×Q8—1661no—
250C2×Q8C22×D8311yescharacteristic
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4C22⋊D8311yescharacteristic
257C2×D4C22×D8311yescharacteristic
258C2×D4C22×D8311yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4C22×D8311yescharacteristic
262C22×C4C22×D8311yescharacteristic
263Q16—3321no—
264SD16—3321no—
265SD16—1662no—
266M16—3321no—
267C2×C8—3321no—
268C2×C8—831no—
269C2×C8—831no—
270C2×C8—831no—
271C2×C8—831no—
272C4⋊C4—21no—
273C4⋊C4—831no—
274C4⋊C4—831no—
275C4⋊C4—831no—
276C4⋊C4—1661no—
277C4⋊C4—1661no—
278C4⋊C4—1661no—
279C4⋊C4—1661no—
280C4⋊C4C22⋊D8311yescharacteristic
281C4⋊C4C22×D8311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4C22⋊D8311yescharacteristic
292C22⋊C4C22⋊D8311yescharacteristic
293C42—1661no—
294C42—1661no—
295C42—1661no—
296C42C22×D8311yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D8311yescharacteristic
303C23C2×C22⋊D8311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3322no—
307Q8—1661no—
308Q8—1661no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—81no—
316D4#1305058912yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4—1661no—
329C2×C4C2×C22⋊D8311yescharacteristic
330C2×C4C2×C22⋊D8311yescharacteristic
331C2×C4C2×C22⋊D8311yescharacteristic
332C2×C4C2×C22⋊D8311yescharacteristic
333C2×C4C23×D8311yescharacteristic
334C8—1661no—
335C8—1661no—
336C8—1661no—
337C8—1661no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22C22×C22⋊D8311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—3321no—
359C4—3321no—
360C4—3321no—
361C4—3321no—
362C4C2×#1305058911yescharacteristic
363C4#1458183711yescharacteristic
364C2—41no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#1912834511yescharacteristic
369C2#1912849911yescharacteristic
370C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#546×C83C2
socleC22×C83C22×D4
hypercenter#546D83
hypocenterC83#3311
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. #19128421C2semidirect
2. #19128060C2semidirect
3. #19128039C2semidirect
4. #19128022C2semidirect
5. #19127991C2semidirect
6. #19127957C2semidirect
7. #19127928C2semidirect
8. #19127920C2semidirect
9. #19127863C2semidirect
10. #19127852C2semidirect
11. #19127801C2semidirect
12. #19127773C2semidirect
13. #19127706C2semidirect
14. #19127697C2semidirect
16. C2×#13050580C22semidirect
20. C22×C83⋊C8C22semidirect
21. #14581792C22semidirect
26. #14581771C22semidirect
27. C4×Dic166C22semidirect
30. #14581751C22semidirect
31. #14581725C22semidirect
32. #14581723C22semidirect
33. #14581723C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. #14581716C22semidirect
36. #14581716C22semidirect
37. #14581716C22semidirect
38. #14581715C22semidirect
46. C22⋊(C83⋊C8)C22semidirect
47. #14581703C22semidirect
186. C83#3311semidirect

Maximal subgroups

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

subgroupindex
#191284212
#191280602
#191280392
#191280222
#191279912
#191279572
#191279282
#191279202
#191278632
#191278522
#191278012
#191277732
#191277062
#191276972
#546×C832
#331183