Finite Group Databasebeta

← #24918317

Subgroups — #24918317

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#19128279C211yesmaximal, characteristic, complemented
2#19128265C211yesmaximal, characteristic, complemented
3#19128043C211yesmaximal, characteristic, complemented
4#19128026C211yesmaximal, characteristic, complemented
5#19128020C211yesmaximal, characteristic, complemented
6#19128015C211yesmaximal, characteristic, complemented
7#19127954C211yesmaximal, characteristic, complemented
8#19127947C211yesmaximal, characteristic, complemented
9#19127937C211yesmaximal, characteristic, complemented
10#19127930C211yesmaximal, characteristic, complemented
11#19127854C211yesmaximal, characteristic, complemented
12#19127848C211yesmaximal, characteristic, complemented
13#19127848C211yesmaximal, characteristic, complemented
14#19127837C211yesmaximal, characteristic, complemented
15#408×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic, complemented
17C22×D332C2211yescharacteristic
18C2×#13050579C2211yescharacteristic
19C2×#13050579C2211yescharacteristic
20C2×D664C2211yescharacteristic
21C2×D664C2211yescharacteristic
22#14581788—21no—
23#14581781—21no—
24#14581771C2211yescharacteristic
25#14581771C2211yescharacteristic, complemented
26#14581768—21no—
27#14581766C2211yescharacteristic
28#14581766C2211yescharacteristic
29#14581766C2211yescharacteristic
30#14581766C2211yescharacteristic
31#14581766C2211yescharacteristic
32#14581766C2211yescharacteristic, complemented
33#14581764—21no—
34#14581764—21no—
35#14581763—21no—
36#14581760—21no—
37#14581760—21no—
38#14581751C2211yescharacteristic
39C2×C4⋊Dic83C2211yescharacteristic, complemented
40#14581726C2211yescharacteristic, complemented
41#14581726C2211yescharacteristic, complemented
42#14581724C2211yescharacteristic, complemented
43#14581724C2211yescharacteristic, complemented
44#14581717C2211yescharacteristic, complemented
45#14581717C2211yescharacteristic, complemented
46#14581714C2211yescharacteristic
47#14581714C2211yescharacteristic
48#14581714C2211yescharacteristic
49#14581714C2211yescharacteristic
50#14581714C2211yescharacteristic, complemented
51#14581714C2211yescharacteristic, complemented
52C2×#41×C83C2211yescharacteristic
53C2×M16×C83C2211yescharacteristic, complemented
54C22×C8×C83C2211yescharacteristic, complemented
55C22⋊C8×C83C2211yescharacteristic
56C22⋊C8×C83C2211yescharacteristic
57C22⋊C8×C83C2211yescharacteristic, complemented
58C22⋊C8×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050590—41no—
61#13050590—41no—
62C2×Dic166C2311yescharacteristic
63C2×D332—21no—
64C2×D332—41no—
65C2×D332—41no—
66C2×D332C2311yescharacteristic
67C2×D332C2311yescharacteristic
68C2×D332C2311yescharacteristic
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72C2×C22⋊D83—21no—
73C2×C22⋊D83—21no—
74C2×C22⋊D83—21no—
75C2×C22⋊D83—21no—
76C22×Dic83—21no—
77C2×C4×D83—21no—
78#13050579—41no—
79#13050579—41no—
80D664—41no—
81D664—41no—
82C4⋊Dic83—21no—
83C4⋊Dic83C2311yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85C4⋊Dic83C2311yescharacteristic
86C4⋊Dic83C2311yescharacteristic
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83—21no—
91C22⋊Dic83—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97(C2×D83)⋊C4—21no—
98C4×Dic83—21no—
99#41×C83—41no—
100#41×C83—41no—
101C2×Q8×C83D411yescharacteristic
102C2×D4×C83D411yescharacteristic
103C2×D4×C83D411yescharacteristic
104C2×D4×C83D411yescharacteristic
105C22×C4×C83C2311yescharacteristic
106C22×C4×C83C2311yescharacteristic
107C22×C4×C83C2311yescharacteristic
108M16×C83—41no—
109C2×C8×C83—41no—
110C2×C8×C83C2311yescharacteristic
111C2×C8×C83C2311yescharacteristic
112C2×C8×C83C2311yescharacteristic
113C2×C8×C83C2311yescharacteristic
114C23×C83C2×D411yescharacteristic
115C23×C83C2×D411yescharacteristic
116C23×C83C2×D411yescharacteristic
117C22×D83—21no—
118C22×D83—21no—
119C22×D83—21no—
120C22×D83—42no—
121C22×D83—81no—
122C22×D83—81no—
123Q8×C83—41no—
124D4×C83—41no—
125D4×C83—41no—
126D4×C83—41no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2×D411yescharacteristic
136C2×C4×C83C2411yescharacteristic
137C22⋊D83—42no—
138C22⋊D83—81no—
139C22⋊D83—81no—
140C22⋊D83—81no—
141C22⋊D83—81no—
142C22⋊D83—81no—
143C22⋊D83—81no—
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—81no—
150D332—21no—
151D332—21no—
152D332—21no—
153D332—21no—
154D332—21no—
155D332—21no—
156D332—42no—
157C4×D83—42no—
158Dic166—21no—
159Dic166—21no—
160C8×C83—21no—
161C8×C83—21no—
162C8×C83—21no—
163C8×C83—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×C83C22×D411yescharacteristic
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
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×C83H(5,𝔽2)11yescharacteristic
188C4×C83H(5,𝔽2)11yescharacteristic
189Dic83—41no—
190Dic83—41no—
191Dic83—41no—
192Dic83—41no—
193Dic83—41no—
194C2×C83—41no—
195C2×C83—41no—
196C2×C83—41no—
197C2×C83#53411yescharacteristic
198C2×C83#57611yescharacteristic
199C2×C83#57711yescharacteristic
200D83—81no—
201D83—81no—
202D83—81no—
203#3198—831nomaximal, Sylow
204C83#319811yescharacteristic, complemented, Sylow
205#537—831no—
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#408D8311yescharacteristic, complemented
220C2×#41—831no—
221C2×#41C2×D8311yescharacteristic
222C22×D4—831no—
223C2×SD16—831no—
224C2×SD16—831no—
225C2×D8—831no—
226C2×D8—831no—
227C2×M16C2×D8311yescharacteristic, complemented
228C22×C8C2×D8311yescharacteristic, complemented
229#127—1661no—
230#124—1661no—
231C22⋊Q8—831no—
232C22⋊Q8—831no—
233C22⋊Q8—1661no—
234#121—831no—
235#121—831no—
236#121—831no—
237#121—831no—
238#121—831no—
239#121—831no—
240#121—1661no—
241#121—1661no—
242#121—1661no—
243#120—1661no—
244#120—1661no—
245#117—831no—
246C2×C4⋊C4—831no—
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⋊C8C2×D8311yescharacteristic
260C22⋊C8C2×D8311yescharacteristic
261C22⋊C8C2×D8311yescharacteristic, complemented
262C22⋊C8C2×D8311yescharacteristic, complemented
263C24—1661no—
264#41—41no—
265#41—41no—
266#41—3321no—
267#41—3321no—
268C2×Q8—831no—
269C2×Q8D33211yescharacteristic
270C2×D4—3321no—
271C2×D4—3321no—
272C2×D4—831no—
273C2×D4—831no—
274C2×D4—831no—
275C2×D4—1661no—
276C2×D4—1661no—
277C2×D4—1661no—
278C2×D4—1661no—
279C2×D4—1661no—
280C2×D4—1661no—
281C2×D4—1661no—
282C2×D4—1661no—
283C2×D4D33211yescharacteristic
284C2×D4D33211yescharacteristic
285C2×D4D33211yescharacteristic
286C22×C4—1661no—
287C22×C4—1661no—
288C22×C4C22×D8311yescharacteristic
289C22×C4C22×D8311yescharacteristic
290C22×C4C22×D8311yescharacteristic
291SD16—3321no—
292SD16—3321no—
293D8—3321no—
294D8—3321no—
295M16—41no—
296C2×C8—41no—
297C2×C8C22×D8311yescharacteristic
298C2×C8C22×D8311yescharacteristic
299C2×C8C22×D8311yescharacteristic
300C2×C8C22×D8311yescharacteristic
301C4⋊C4—831no—
302C4⋊C4—831no—
303C4⋊C4—831no—
304C4⋊C4—831no—
305C4⋊C4—1661no—
306C4⋊C4—1661no—
307C4⋊C4—1661no—
308C22⋊C4—1661no—
309C22⋊C4—1661no—
310C22⋊C4—1661no—
311C22⋊C4—1661no—
312C22⋊C4—1661no—
313C22⋊C4—1661no—
314C22⋊C4—1661no—
315C22⋊C4—1661no—
316C22⋊C4—1661no—
317C42—1661no—
318C23—3322no—
319C23—6641no—
320C23—6641no—
321C23—1661no—
322C23—1661no—
323C23—1661no—
324C23C2×D33211yescharacteristic
325C23C2×D33211yescharacteristic
326C23C2×D33211yescharacteristic
327Q8—41no—
328Q8—1661no—
329Q8—1661no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—3322no—
334D4—3322no—
335D4—6641no—
336D4—6641no—
337D4—6641no—
338D4—6641no—
339D4—6641no—
340D4—6641no—
341D4—1661no—
342D4—1661no—
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—3322no—
354C2×C4—6641no—
355C2×C4—1661no—
356C2×C4—1661no—
357C2×C4—1661no—
358C2×C4—1661no—
359C2×C4—1661no—
360C2×C4C2×D33211yescharacteristic
361C2×C4C2×D33211yescharacteristic
362C2×C4C2×D33211yescharacteristic
363C2×C4C23×D8311yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—3321no—
378C22—3321no—
379C22—3321no—
380C22—3321no—
381C22—3321no—
382C22—3321no—
383C22—3321no—
384C22—3321no—
385C22—3321no—
386C22—6644no—
387C22C22×D33211yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—3321no—
392C4—3321no—
393C4—3321no—
394C4—3321no—
395C4—3321no—
396C4#1458183711yescharacteristic
397C4#1458183711yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—6641no—
402C2—6641no—
403C2—6641no—
404C2#1912826311yescharacteristic
405C2#1912850611yescharacteristic
406C2#1912851311yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#408×C83C2
socleC22×C83C22×D4
hypercenter#408D83
hypocenterC83#3198
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #19128279C2semidirect
2. #19128265C2semidirect
3. #19128043C2semidirect
4. #19128026C2semidirect
5. #19128020C2semidirect
6. #19128015C2semidirect
7. #19127954C2semidirect
8. #19127947C2semidirect
9. #19127937C2semidirect
10. #19127930C2semidirect
11. #19127854C2semidirect
12. #19127848C2semidirect
13. #19127848C2semidirect
14. #19127837C2semidirect
15. #408×C83C2semidirect
16. C2×#13050590C22semidirect
25. #14581771C22semidirect
32. #14581766C22semidirect
39. C2×C4⋊Dic83C22semidirect
40. #14581726C22semidirect
41. #14581726C22semidirect
42. #14581724C22semidirect
43. #14581724C22semidirect
44. #14581717C22semidirect
45. #14581717C22semidirect
50. #14581714C22semidirect
51. #14581714C22semidirect
53. C2×M16×C83C22semidirect
54. C22×C8×C83C22semidirect
57. C22⋊C8×C83C22semidirect
58. C22⋊C8×C83C22semidirect
204. C83#3198semidirect
219. #408D83semidirect
227. C2×M16C2×D83semidirect
228. C22×C8C2×D83semidirect
261. C22⋊C8C2×D83semidirect
262. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282792
#191282652
#191280432
#191280262
#191280202
#191280152
#191279542
#191279472
#191279372
#191279302
#191278542
#191278482
#191278372
#408×C832
#319883