Finite Group Databasebeta

← #24918319

Subgroups — #24918319

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#19128281C211yesmaximal, characteristic, complemented
2#19128268C211yesmaximal, characteristic, complemented
3#19128039C211yesmaximal, characteristic, complemented
4#19128026C211yesmaximal, characteristic, complemented
5#19128018C211yesmaximal, characteristic, complemented
6#19128013C211yesmaximal, characteristic, complemented
7#19127953C211yesmaximal, characteristic, complemented
8#19127944C211yesmaximal, characteristic, complemented
9#19127936C211yesmaximal, characteristic, complemented
10#19127927C211yesmaximal, characteristic, complemented
11#19127845C211yesmaximal, characteristic, complemented
12#19127845C211yesmaximal, characteristic, complemented
13#19127835C211yesmaximal, characteristic, complemented
14#19127666C211yesmaximal, characteristic, complemented
15#450×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic, complemented
17C2×#13050590C2211yescharacteristic
18C2×#13050575C2211yescharacteristic
19C2×#13050575C2211yescharacteristic
20C2×#13050573C2211yescharacteristic, complemented
21C2×C8×D83C2211yescharacteristic, complemented
22#14581788—21no—
23#14581779—21no—
24#14581770—21no—
25#14581768C2211yescharacteristic, complemented
26#14581767—21no—
27#14581767C2211yescharacteristic, complemented
28#14581766C2211yescharacteristic
29#14581765—21no—
30#14581765—21no—
31#14581763C2211yescharacteristic
32#14581763C2211yescharacteristic
33#14581763C2211yescharacteristic
34#14581763C2211yescharacteristic
35#14581761—21no—
36#14581761—21no—
37#14581751C2211yescharacteristic
38C4⋊C4×D83C2211yescharacteristic, complemented
39#14581726C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, complemented
41#14581724C2211yescharacteristic, complemented
42#14581723C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581716C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic, complemented
46#14581714C2211yescharacteristic, complemented
47#14581713C2211yescharacteristic
48#14581713C2211yescharacteristic
49C22⋊(C83⋊C8)C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51(C2×D83)⋊C8C2211yescharacteristic, complemented
52C22×D4×C83C2211yescharacteristic
53C2×SD16×C83C2211yescharacteristic
54C2×SD16×C83C2211yescharacteristic
55C22⋊Q8×C83C2211yescharacteristic
56#102×C83C2211yescharacteristic
57#102×C83C2211yescharacteristic
58C22⋊C8×C83C2211yescharacteristic
59#13050591—41no—
60#13050591—41no—
61#13050590—41no—
62#13050590—41no—
63C2×Dic166D411yescharacteristic
64C2×D332—21no—
65C2×D332—21no—
66C2×D332D411yescharacteristic
67C2×D332C2311yescharacteristic
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72C2×C22⋊D83—21no—
73C2×C22⋊D83D411yescharacteristic
74C2×C22⋊D83D411yescharacteristic
75C2×C4×D83—21no—
76C2×C4×D83—21no—
77C2×C4×D83C2311yescharacteristic
78C2×C4×D83C2311yescharacteristic
79#13050575—41no—
80#13050575—41no—
81#13050573—41no—
82C2×C83⋊C8C2311yescharacteristic
83C2×C83⋊C8C2311yescharacteristic
84C8×D83—41no—
85C4⋊Dic83C2311yescharacteristic
86C4⋊Dic83C2311yescharacteristic
87C4⋊Dic83C2311yescharacteristic
88Dic83⋊C4—21no—
89Dic83⋊C4—21no—
90C22⋊Dic83—21no—
91C22⋊Dic83—21no—
92C22⋊Dic83—21no—
93C22⋊Dic83—21no—
94C22⋊Dic83—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97(C2×D83)⋊C4—21no—
98C4×Dic83—21no—
99C24×C83—21no—
100C2×Q8×C83C2311yescharacteristic
101C2×D4×C83—21no—
102C2×D4×C83—41no—
103C2×D4×C83—41no—
104C2×D4×C83C2311yescharacteristic
105C2×D4×C83C2311yescharacteristic
106C22×C4×C83C2311yescharacteristic
107SD16×C83—41no—
108SD16×C83—41no—
109C2×C8×C83C2311yescharacteristic
110C2×C8×C83C2311yescharacteristic
111C4⋊C4×C83—21no—
112C4⋊C4×C83C2311yescharacteristic
113C22⋊C4×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—42no—
117C23×C83—81no—
118C23×C83—81no—
119C23×C83C2×D411yescharacteristic
120C22×D83—21no—
121C22×D83C2×D411yescharacteristic
122C22×D83C2×D411yescharacteristic
123Q8×C83—21no—
124Q8×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—21no—
128D4×C83—21no—
129D4×C83—42no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2411yescharacteristic
136C22⋊D83—41no—
137C22⋊D83—41no—
138C22⋊D83—81no—
139C22⋊D83—81no—
140C22⋊D83—81no—
141C22⋊D83—81no—
142C22⋊D83—81no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83C2×D411yescharacteristic
147C2×Dic83C2×D411yescharacteristic
148D332—21no—
149D332—21no—
150D332—41no—
151D332—42no—
152D332—81no—
153C4×D83—21no—
154C4×D83—21no—
155C4×D83—21no—
156C4×D83—21no—
157C4×D83—42no—
158C4×D83—81no—
159Dic166—41no—
160C8×C83—21no—
161C8×C83—21no—
162C83⋊C8—21no—
163C83⋊C8—21no—
164C22×C83—21no—
165C22×C83—21no—
166C22×C83—21no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83—41no—
170C22×C83—41no—
171C22×C83—41no—
172C22×C83—41no—
173C22×C83—84no—
174C22×C83C22×D411yescharacteristic
175C2×D83—21no—
176C2×D83—21no—
177C2×D83—21no—
178C2×D83—21no—
179C2×D83—21no—
180C2×D83—21no—
181C2×D83—41no—
182C2×D83—41no—
183C2×D83—41no—
184C4×C83—21no—
185C4×C83—41no—
186C4×C83—41no—
187C4×C83H(5,𝔽2)11yescharacteristic
188C4×C83H(5,𝔽2)11yescharacteristic
189Dic83—21no—
190Dic83—21no—
191Dic83—41no—
192Dic83—41no—
193Dic83—41no—
194C2×C83—41no—
195C2×C83—81no—
196C2×C83—81no—
197C2×C83#53411yescharacteristic
198C2×C83#57611yescharacteristic
199C2×C83#57711yescharacteristic
200D83—41no—
201D83—41no—
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#450D8311yescharacteristic, complemented
216#449—831no—
217#449—831no—
218#447—831no—
219#408—831no—
220C2×#41—831no—
221C2×#41—831no—
222C22×D4C2×D8311yescharacteristic
223C2×SD16C2×D8311yescharacteristic
224C2×SD16C2×D8311yescharacteristic
225C2×D8—831no—
226C2×D8—831no—
227C2×M16—831no—
228C22×C8—831no—
229#127—1661no—
230#124—1661no—
231C22⋊Q8—831no—
232C22⋊Q8—1661no—
233C22⋊Q8C2×D8311yescharacteristic
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#102C2×D8311yescharacteristic
258#102C2×D8311yescharacteristic
259C22⋊C8—831no—
260C22⋊C8—831no—
261C22⋊C8—831no—
262C22⋊C8C2×D8311yescharacteristic
263C24—21no—
264#41—3321no—
265#41—3321no—
266#41—3321no—
267#41—3321no—
268C2×Q8—831no—
269C2×Q8C22×D8311yescharacteristic
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—831no—
274C2×D4—831no—
275C2×D4—831no—
276C2×D4—831no—
277C2×D4—1661no—
278C2×D4—1661no—
279C2×D4—1661no—
280C2×D4—1661no—
281C2×D4—1661no—
282C2×D4—1661no—
283C2×D4—1661no—
284C2×D4C22×D8311yescharacteristic
285C2×D4C22×D8311yescharacteristic
286C22×C4—831no—
287C22×C4—831no—
288C22×C4—1661no—
289C22×C4—1661no—
290C22×C4C22×D8311yescharacteristic
291SD16—41no—
292SD16—41no—
293D8—3321no—
294D8—3321no—
295M16—3321no—
296C2×C8—3321no—
297C2×C8—831no—
298C2×C8—831no—
299C2×C8C22×D8311yescharacteristic
300C2×C8C22×D8311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—831no—
303C4⋊C4—831no—
304C4⋊C4—831no—
305C4⋊C4—1661no—
306C4⋊C4—1661no—
307C4⋊C4C22×D8311yescharacteristic
308C22⋊C4—21no—
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—21no—
319C23—21no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—831no—
324C23—831no—
325C23—1661no—
326C23D4×D8311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—3321no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—3321no—
336D4—3322no—
337D4—3321no—
338D4—3321no—
339D4—6641no—
340D4—6641no—
341D4—6641no—
342D4—6641no—
343D4—6641no—
344D4—6641no—
345D4—1661no—
346D4—1661no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—3322no—
352C2×C4—831no—
353C2×C4—831no—
354C2×C4—6641no—
355C2×C4—1661no—
356C2×C4—1661no—
357C2×C4—1661no—
358C2×C4—1661no—
359C2×C4—1661no—
360C2×C4—1661no—
361C2×C4—1661no—
362C2×C4D4×D8311yescharacteristic
363C2×C4C23×D8311yescharacteristic
364C8—21no—
365C8—21no—
366C8—1661no—
367C8—1661no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—84no—
378C22—3321no—
379C22—3321no—
380C22—3321no—
381C22—1661no—
382C22—1661no—
383C22—1661no—
384C22—1661no—
385C22—1661no—
386C22—1661no—
387C22C2×D4×D8311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—3321no—
392C4—3321no—
393C4—3321no—
394C4—1661no—
395C4—1661no—
396C4#1458183711yescharacteristic
397C4#1458183711yescharacteristic
398C2—41no—
399C2—81no—
400C2—81no—
401C2—3321no—
402C2—3321no—
403C2—6641no—
404C2#1912826111yescharacteristic
405C2#1912850511yescharacteristic
406C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
hypercenter#450D83
hypocenterC83#3198
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. #19128281C2semidirect
2. #19128268C2semidirect
3. #19128039C2semidirect
4. #19128026C2semidirect
5. #19128018C2semidirect
6. #19128013C2semidirect
7. #19127953C2semidirect
8. #19127944C2semidirect
9. #19127936C2semidirect
10. #19127927C2semidirect
11. #19127845C2semidirect
12. #19127845C2semidirect
13. #19127835C2semidirect
14. #19127666C2semidirect
15. #450×C83C2semidirect
16. C2×#13050591C22semidirect
20. C2×#13050573C22semidirect
21. C2×C8×D83C22semidirect
25. #14581768C22semidirect
27. #14581767C22semidirect
38. C4⋊C4×D83C22semidirect
39. #14581726C22semidirect
40. #14581725C22semidirect
41. #14581724C22semidirect
42. #14581723C22semidirect
43. #14581716C22semidirect
44. #14581716C22semidirect
45. #14581714C22semidirect
46. #14581714C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. (C2×D83)⋊C8C22semidirect
204. C83#3198semidirect
215. #450D83semidirect

Maximal subgroups

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

subgroupindex
#191282812
#191282682
#191280392
#191280262
#191280182
#191280132
#191279532
#191279442
#191279362
#191279272
#191278452
#191278352
#191276662
#450×C832
#319883