Finite Group Databasebeta

← #24918309

Subgroups — #24918309

Order 10624.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128265C211yesmaximal, characteristic, complemented
2#19128039C211yesmaximal, characteristic, complemented
3#19128022C211yesmaximal, characteristic, complemented
4#19128019C211yesmaximal, characteristic, complemented
5#19128014C211yesmaximal, characteristic, complemented
6#19127952C211yesmaximal, characteristic, complemented
7#19127949C211yesmaximal, characteristic, complemented
8#19127935C211yesmaximal, characteristic, complemented
9#19127932C211yesmaximal, characteristic, complemented
10#19127851C211yesmaximal, characteristic, complemented
11#19127846C211yesmaximal, characteristic, complemented
12#19127846C211yesmaximal, characteristic, complemented
13#19127834C211yesmaximal, characteristic, complemented
14#19127668C211yesmaximal, characteristic, complemented
15#537×C83C211yesmaximal, characteristic, complemented
16C22×D332C2211yescharacteristic
17C2×#13050577C2211yescharacteristic
18C2×#13050577C2211yescharacteristic
19C2×#13050575C2211yescharacteristic
20C2×#13050575C2211yescharacteristic
21C2×#13050574C2211yescharacteristic, complemented
22C22×C83⋊C8C2211yescharacteristic, complemented
23#14581788—21no—
24#14581781—21no—
25#14581766C2211yescharacteristic
26#14581766C2211yescharacteristic
27#14581766C2211yescharacteristic
28#14581766C2211yescharacteristic
29#14581760—21no—
30#14581760—21no—
31#14581751C2211yescharacteristic
32#14581725C2211yescharacteristic, complemented
33#14581725C2211yescharacteristic, complemented
34#14581723C2211yescharacteristic, complemented
35#14581723C2211yescharacteristic, complemented
36#14581716C2211yescharacteristic, complemented
37#14581716C2211yescharacteristic, complemented
38#14581713C2211yescharacteristic, complemented
39#14581713C2211yescharacteristic, complemented
40#14581712C2211yescharacteristic
41#14581712C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44C22⋊(C83⋊C8)C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic, complemented
47C22⋊(C83⋊C8)C2211yescharacteristic, complemented
48C2×#41×C83C2211yescharacteristic
49C2×#41×C83C2211yescharacteristic, complemented
50C22⋊Q8×C83—21no—
51C22⋊Q8×C83C2211yescharacteristic
52C22⋊Q8×C83C2211yescharacteristic, complemented
53#121×C83—21no—
54#121×C83—21no—
55#121×C83—21no—
56#121×C83C2211yescharacteristic
57#121×C83C2211yescharacteristic, complemented
58C2×C4⋊C4×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60C2×D332—21no—
61C2×D332—41no—
62C2×D332—41no—
63C2×D332C2311yescharacteristic
64C2×D332C2311yescharacteristic
65C2×C22⋊D83—21no—
66C2×C22⋊D83—21no—
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69#13050577—41no—
70#13050577—41no—
71#13050575—41no—
72#13050575—41no—
73#13050574—41no—
74C2×C83⋊C8—41no—
75C2×C83⋊C8C2311yescharacteristic
76C2×C83⋊C8C2311yescharacteristic
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C4⋊Dic83C2311yescharacteristic
80C4⋊Dic83C2311yescharacteristic
81C22⋊Dic83—21no—
82(C2×D83)⋊C4—21no—
83(C2×D83)⋊C4—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86C4×Dic83—21no—
87#41×C83—41no—
88#41×C83—41no—
89#41×C83—41no—
90#41×C83—41no—
91C2×Q8×C83D411yescharacteristic
92C2×Q8×C83C2311yescharacteristic
93C2×D4×C83—21no—
94C2×D4×C83—21no—
95C2×D4×C83—21no—
96C2×D4×C83D411yescharacteristic
97C2×D4×C83D411yescharacteristic
98C2×D4×C83D411yescharacteristic
99C2×D4×C83C2311yescharacteristic
100C22×C4×C83—21no—
101C22×C4×C83—21no—
102C22×C4×C83C2311yescharacteristic
103C22×C4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105C4⋊C4×C83—21no—
106C4⋊C4×C83—21no—
107C4⋊C4×C83—21no—
108C4⋊C4×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C22⋊C4×C83—21no—
112C22⋊C4×C83—21no—
113C22⋊C4×C83—21no—
114C23×C83—21no—
115C23×C83C2×D411yescharacteristic
116C23×C83C2×D411yescharacteristic
117C23×C83C2×D411yescharacteristic
118C22×D83—21no—
119C22×D83—21no—
120C22×D83—42no—
121C22×D83—81no—
122C22×D83—81no—
123Q8×C83—21no—
124Q8×C83—21no—
125Q8×C83—41no—
126D4×C83—21no—
127D4×C83—21no—
128D4×C83—41no—
129D4×C83—41no—
130D4×C83—41no—
131D4×C83—42no—
132D4×C83—81no—
133D4×C83—81no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—42no—
144C2×C4×C83—81no—
145C2×C4×C83C2×D411yescharacteristic
146C2×C4×C83C2×D411yescharacteristic
147C2×C4×C83C2×D411yescharacteristic
148C2×C4×C83C2411yescharacteristic
149C22⋊D83—81no—
150C22⋊D83—81no—
151C22⋊D83—81no—
152C22⋊D83—81no—
153C2×Dic83—21no—
154C2×Dic83—21no—
155D332—21no—
156D332—21no—
157D332—21no—
158D332—21no—
159D332—42no—
160C83⋊C8—21no—
161C83⋊C8—21no—
162C83⋊C8—21no—
163C83⋊C8—21no—
164C22×C83—21no—
165C22×C83—21no—
166C22×C83—21no—
167C22×C83—21no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—41no—
174C22×C83—41no—
175C22×C83—41no—
176C22×C83C22×D411yescharacteristic
177C2×D83—41no—
178C2×D83—41no—
179C2×D83—41no—
180C2×D83—41no—
181C2×D83—41no—
182C2×D83—41no—
183C2×D83—84no—
184C4×C83—21no—
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—41no—
188C4×C83—41no—
189C4×C83—41no—
190C4×C83H(5,𝔽2)11yescharacteristic
191C4×C83H(5,𝔽2)11yescharacteristic
192Dic83—41no—
193Dic83—41no—
194C2×C83—41no—
195C2×C83—41no—
196C2×C83—41no—
197C2×C83—81no—
198C2×C83#53411yescharacteristic
199C2×C83#57611yescharacteristic
200C2×C83#57711yescharacteristic
201D83—81no—
202D83—81no—
203#3198—831nomaximal, Sylow
204C83#319811yescharacteristic, complemented, Sylow
205#537D8311yescharacteristic, complemented
206#535—831no—
207#485—831no—
208#482—831no—
209#481—831no—
210#480—831no—
211#469—831no—
212#468—831no—
213#466—831no—
214#465—831no—
215#450—831no—
216#449—831no—
217#449—831no—
218#447—831no—
219#408—831no—
220C2×#41C2×D8311yescharacteristic
221C2×#41C2×D8311yescharacteristic, complemented
222C22×D4—831no—
223C2×SD16—831no—
224C2×SD16—831no—
225C2×D8—831no—
226C2×D8—831no—
227C2×M16—831no—
228C22×C8—831no—
229#127—1661no—
230#124—1661no—
231C22⋊Q8—21no—
232C22⋊Q8C2×D8311yescharacteristic
233C22⋊Q8C2×D8311yescharacteristic, complemented
234#121—21no—
235#121—21no—
236#121—21no—
237#121—831no—
238#121—831no—
239#121—831no—
240#121—831no—
241#121C2×D8311yescharacteristic
242#121C2×D8311yescharacteristic, complemented
243#120—1661no—
244#120—1661no—
245#117—831no—
246C2×C4⋊C4C2×D8311yescharacteristic, complemented
247#107—831no—
248#107—831no—
249#106—831no—
250#106—831no—
251#103—831no—
252#103—831no—
253#102—831no—
254#102—831no—
255#102—831no—
256#102—831no—
257#102—831no—
258#102—831no—
259C22⋊C8—831no—
260C22⋊C8—831no—
261C22⋊C8—831no—
262C22⋊C8—831no—
263C24—1661no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8C22⋊D8311yescharacteristic
269C2×Q8C22×D8311yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—3321no—
274C2×D4—3321no—
275C2×D4—831no—
276C2×D4—831no—
277C2×D4—1661no—
278C2×D4—1661no—
279C2×D4—1661no—
280C2×D4—1661no—
281C2×D4—1661no—
282C2×D4C22⋊D8311yescharacteristic
283C2×D4C22⋊D8311yescharacteristic
284C2×D4C22⋊D8311yescharacteristic
285C2×D4C22×D8311yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4C22×D8311yescharacteristic
289C22×C4C22×D8311yescharacteristic
290C22×C4C22×D8311yescharacteristic
291SD16—3321no—
292SD16—3321no—
293D8—3321no—
294D8—3321no—
295M16—3321no—
296C2×C8—3321no—
297C2×C8—831no—
298C2×C8—831no—
299C2×C8—831no—
300C2×C8—831no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—831no—
305C4⋊C4—831no—
306C4⋊C4C22×D8311yescharacteristic
307C4⋊C4C22×D8311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—1661no—
313C22⋊C4—1661no—
314C22⋊C4—1661no—
315C22⋊C4—1661no—
316C22⋊C4—1661no—
317C42—1661no—
318C23—21no—
319C23—3322no—
320C23—6641no—
321C23—6641no—
322C23—1661no—
323C23—1661no—
324C23C2×C22⋊D8311yescharacteristic
325C23C2×C22⋊D8311yescharacteristic
326C23C2×C22⋊D8311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—41no—
330D4—21no—
331D4—21no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—42no—
336D4—81no—
337D4—81no—
338D4—3322no—
339D4—6641no—
340D4—6641no—
341D4—6641no—
342D4—6641no—
343D4—1661no—
344D4—1661no—
345D4—1661no—
346D4—1661no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—1661no—
359C2×C4—1661no—
360C2×C4C2×C22⋊D8311yescharacteristic
361C2×C4C2×C22⋊D8311yescharacteristic
362C2×C4C2×C22⋊D8311yescharacteristic
363C2×C4C23×D8311yescharacteristic
364C8—1661no—
365C8—1661no—
366C8—1661no—
367C8—1661no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—3321no—
381C22—3321no—
382C22—3321no—
383C22—3321no—
384C22—3321no—
385C22—3321no—
386C22—6644no—
387C22C22×C22⋊D8311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—3321no—
395C4—3321no—
396C4#1458183711yescharacteristic
397C4#1458183711yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—81no—
402C2—6641no—
403C2—6641no—
404C2#1912826011yescharacteristic
405C2#1912850411yescharacteristic
406C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#537×C83C2
socleC22×C83C22×D4
hypercenter#537D83
hypocenterC83#3198
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #19128265C2semidirect
2. #19128039C2semidirect
3. #19128022C2semidirect
4. #19128019C2semidirect
5. #19128014C2semidirect
6. #19127952C2semidirect
7. #19127949C2semidirect
8. #19127935C2semidirect
9. #19127932C2semidirect
10. #19127851C2semidirect
11. #19127846C2semidirect
12. #19127846C2semidirect
13. #19127834C2semidirect
14. #19127668C2semidirect
15. #537×C83C2semidirect
21. C2×#13050574C22semidirect
22. C22×C83⋊C8C22semidirect
32. #14581725C22semidirect
33. #14581725C22semidirect
34. #14581723C22semidirect
35. #14581723C22semidirect
36. #14581716C22semidirect
37. #14581716C22semidirect
38. #14581713C22semidirect
39. #14581713C22semidirect
46. C22⋊(C83⋊C8)C22semidirect
47. C22⋊(C83⋊C8)C22semidirect
49. C2×#41×C83C22semidirect
52. C22⋊Q8×C83C22semidirect
57. #121×C83C22semidirect
58. C2×C4⋊C4×C83C22semidirect
204. C83#3198semidirect
205. #537D83semidirect
221. C2×#41C2×D83semidirect
233. C22⋊Q8C2×D83semidirect
242. #121C2×D83semidirect
246. C2×C4⋊C4C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282652
#191280392
#191280222
#191280192
#191280142
#191279522
#191279492
#191279352
#191279322
#191278512
#191278462
#191278342
#191276682
#537×C832
#319883