Finite Group Databasebeta

← #24918318

Subgroups — #24918318

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#19128267C211yesmaximal, characteristic, complemented
3#19128041C211yesmaximal, characteristic, complemented
4#19128027C211yesmaximal, characteristic, complemented
5#19128021C211yesmaximal, characteristic, complemented
6#19128013C211yesmaximal, characteristic, complemented
7#19127952C211yesmaximal, characteristic, complemented
8#19127949C211yesmaximal, characteristic, complemented
9#19127937C211yesmaximal, characteristic, complemented
10#19127850C211yesmaximal, characteristic, complemented
11#19127849C211yesmaximal, characteristic, complemented
12#19127845C211yesmaximal, characteristic, complemented
13#19127836C211yesmaximal, characteristic, complemented
14#19127667C211yesmaximal, characteristic, complemented
15#465×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic, complemented
17C2×#13050590C2211yescharacteristic
18C2×D4×D83C2211yescharacteristic
19C2×#13050577C2211yescharacteristic
20C2×D664C2211yescharacteristic
21C2×#13050575C2211yescharacteristic
22C2×#13050574C2211yescharacteristic, complemented
23#14581788—21no—
24#14581778—21no—
25#14581770—21no—
26#14581768C2211yescharacteristic
27#14581767—21no—
28#14581767C2211yescharacteristic
29#14581767C2211yescharacteristic
30#14581766C2211yescharacteristic
31#14581766C2211yescharacteristic, complemented
32#14581765—21no—
33#14581765—21no—
34#14581763C2211yescharacteristic
35#14581762—21no—
36#14581760—21no—
37#14581750C2211yescharacteristic
38C4⋊C4×D83C2211yescharacteristic, complemented
39#14581726C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, complemented
41#14581723C2211yescharacteristic, complemented
42#14581716C2211yescharacteristic, complemented
43#14581714C2211yescharacteristic
44#14581714C2211yescharacteristic, complemented
45#14581713C2211yescharacteristic
46#14581712C2211yescharacteristic
47#14581712C2211yescharacteristic, complemented
48(C2×D83)⋊C8C2211yescharacteristic
49(C2×D83)⋊C8C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51(C2×D83)⋊C8C2211yescharacteristic, complemented
52C2×SD16×C83C2211yescharacteristic
53C22×C8×C83C2211yescharacteristic, complemented
54C22⋊Q8×C83C2211yescharacteristic, complemented
55#121×C83C2211yescharacteristic
56#106×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58#102×C83C2211yescharacteristic
59C23×D83—21no—
60#13050591—41no—
61#13050591—41no—
62#13050590—41no—
63#13050590—41no—
64C2×Dic166D411yescharacteristic
65C2×D332—21no—
66C2×D332—21no—
67C2×D332—21no—
68C2×D332D411yescharacteristic
69C2×D332C2311yescharacteristic
70C2×D332C2311yescharacteristic
71C2×C22⋊D83—21no—
72C2×C22⋊D83—21no—
73C2×C22⋊D83—21no—
74C2×C22⋊D83—21no—
75C2×C22⋊D83D411yescharacteristic
76C2×C22⋊D83D411yescharacteristic
77D4×D83—41no—
78D4×D83—41no—
79C2×C4×D83—21no—
80C2×C4×D83—21no—
81C2×C4×D83C2311yescharacteristic
82C2×C4×D83C2311yescharacteristic
83#13050577—41no—
84D664—41no—
85#13050575—41no—
86#13050574—41no—
87C2×C83⋊C8C2311yescharacteristic
88C2×C83⋊C8C2311yescharacteristic
89C4⋊Dic83C2311yescharacteristic
90C4⋊Dic83C2311yescharacteristic
91Dic83⋊C4—21no—
92Dic83⋊C4—21no—
93C22⋊Dic83—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97(C2×D83)⋊C4—21no—
98(C2×D83)⋊C4—21no—
99(C2×D83)⋊C4—21no—
100C4×Dic83—21no—
101C2×Q8×C83C2311yescharacteristic
102C2×D4×C83—21no—
103C2×D4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105SD16×C83—41no—
106C2×C8×C83—41no—
107C2×C8×C83C2311yescharacteristic
108C2×C8×C83C2311yescharacteristic
109C4⋊C4×C83—21no—
110C4⋊C4×C83C2311yescharacteristic
111C4⋊C4×C83C2311yescharacteristic
112C22⋊C4×C83—21no—
113C22⋊C4×C83—21no—
114C23×C83—21no—
115C23×C83C2×D411yescharacteristic
116C22×D83—21no—
117C22×D83—21no—
118C22×D83—42no—
119C22×D83—81no—
120C22×D83—81no—
121C22×D83C2×D411yescharacteristic
122C22×D83C2×D411yescharacteristic
123Q8×C83—21no—
124Q8×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—81no—
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×C83C2411yescharacteristic
135C22⋊D83—41no—
136C22⋊D83—41no—
137C22⋊D83—42no—
138C22⋊D83—81no—
139C22⋊D83—81no—
140C22⋊D83—81no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83C2×D411yescharacteristic
144C2×Dic83C2×D411yescharacteristic
145D332—21no—
146D332—21no—
147D332—21no—
148D332—21no—
149D332—41no—
150D332—42no—
151D332—81no—
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×C83C22×D411yescharacteristic
171C2×D83—21no—
172C2×D83—21no—
173C2×D83—21no—
174C2×D83—21no—
175C2×D83—21no—
176C2×D83—21no—
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—41no—
186C4×C83—41no—
187C4×C83—41no—
188C4×C83H(5,𝔽2)11yescharacteristic
189C4×C83H(5,𝔽2)11yescharacteristic
190Dic83—21no—
191Dic83—21no—
192Dic83—41no—
193Dic83—41no—
194C2×C83—41no—
195C2×C83—81no—
196C2×C83#53411yescharacteristic
197C2×C83#57611yescharacteristic
198C2×C83#57711yescharacteristic
199D83—41no—
200D83—41no—
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#465D8311yescharacteristic, complemented
215#450—831no—
216#449—831no—
217#449—831no—
218#447—831no—
219#408—831no—
220C2×#41—831no—
221C2×#41—831no—
222C22×D4—831no—
223C2×SD16—831no—
224C2×SD16C2×D8311yescharacteristic
225C2×D8—831no—
226C2×D8—831no—
227C2×M16—831no—
228C22×C8C2×D8311yescharacteristic, complemented
229#127—1661no—
230#124—1661no—
231C22⋊Q8—831no—
232C22⋊Q8—1661no—
233C22⋊Q8C2×D8311yescharacteristic, complemented
234#121—831no—
235#121—831no—
236#121—831no—
237#121—831no—
238#121—831no—
239#121—1661no—
240#121—1661no—
241#121—1661no—
242#121C2×D8311yescharacteristic
243#120—1661no—
244#120—1661no—
245#117—831no—
246C2×C4⋊C4—831no—
247#107—831no—
248#107—831no—
249#106—831no—
250#106C2×D8311yescharacteristic, complemented
251#103—831no—
252#103C2×D8311yescharacteristic, complemented
253#102—831no—
254#102—831no—
255#102—831no—
256#102—831no—
257#102—831no—
258#102C2×D8311yescharacteristic
259C22⋊C8—831no—
260C22⋊C8—831no—
261C22⋊C8—831no—
262C22⋊C8—831no—
263C24—1661no—
264#41—3321no—
265#41—3321no—
266#41—3321no—
267#41—3321no—
268C2×Q8—831no—
269C2×Q8C22×D8311yescharacteristic
270C2×D4—21no—
271C2×D4—3321no—
272C2×D4—3321no—
273C2×D4—831no—
274C2×D4—831no—
275C2×D4—831no—
276C2×D4—831no—
277C2×D4—831no—
278C2×D4—1661no—
279C2×D4—1661no—
280C2×D4—1661no—
281C2×D4—1661no—
282C2×D4—1661no—
283C2×D4—1661no—
284C2×D4—1661no—
285C2×D4C22×D8311yescharacteristic
286C22×C4—831no—
287C22×C4—831no—
288C22×C4—1661no—
289C22×C4—1661no—
290C22×C4C22×D8311yescharacteristic
291SD16—41no—
292SD16—3321no—
293D8—3321no—
294D8—3321no—
295M16—3321no—
296C2×C8—41no—
297C2×C8—831no—
298C2×C8—831no—
299C2×C8C22×D8311yescharacteristic
300C2×C8C22×D8311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—831no—
303C4⋊C4—831no—
304C4⋊C4—1661no—
305C4⋊C4—1661no—
306C4⋊C4C22×D8311yescharacteristic
307C4⋊C4C22×D8311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
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—3322no—
320C23—831no—
321C23—831no—
322C23—6641no—
323C23—6641no—
324C23—1661no—
325C23—1661no—
326C23D4×D8311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—3321no—
330D4—21no—
331D4—21no—
332D4—81no—
333D4—3321no—
334D4—3322no—
335D4—3321no—
336D4—3321no—
337D4—3322no—
338D4—6641no—
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—3322no—
353C2×C4—831no—
354C2×C4—831no—
355C2×C4—6641no—
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—3321no—
375C22—3321no—
376C22—3321no—
377C22—3321no—
378C22—3321no—
379C22—3321no—
380C22—6644no—
381C22—1661no—
382C22—1661no—
383C22—1661no—
384C22—1661no—
385C22—1661no—
386C22—1661no—
387C22C2×D4×D8311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—3321no—
393C4—3321no—
394C4—1661no—
395C4—1661no—
396C4#1458183711yescharacteristic
397C4#1458183811yescharacteristic
398C2—41no—
399C2—81no—
400C2—3321no—
401C2—3321no—
402C2—6641no—
403C2—6641no—
404C2#1912825911yescharacteristic
405C2#1912850311yescharacteristic
406C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#465×C83C2
socleC22×C83C22×D4
hypercenter#465D83
hypocenterC83#3198
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #19128281C2semidirect
2. #19128267C2semidirect
3. #19128041C2semidirect
4. #19128027C2semidirect
5. #19128021C2semidirect
6. #19128013C2semidirect
7. #19127952C2semidirect
8. #19127949C2semidirect
9. #19127937C2semidirect
10. #19127850C2semidirect
11. #19127849C2semidirect
12. #19127845C2semidirect
13. #19127836C2semidirect
14. #19127667C2semidirect
15. #465×C83C2semidirect
16. C2×#13050591C22semidirect
22. C2×#13050574C22semidirect
31. #14581766C22semidirect
38. C4⋊C4×D83C22semidirect
39. #14581726C22semidirect
40. #14581725C22semidirect
41. #14581723C22semidirect
42. #14581716C22semidirect
44. #14581714C22semidirect
47. #14581712C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. (C2×D83)⋊C8C22semidirect
53. C22×C8×C83C22semidirect
54. C22⋊Q8×C83C22semidirect
56. #106×C83C22semidirect
57. #103×C83C22semidirect
204. C83#3198semidirect
214. #465D83semidirect
228. C22×C8C2×D83semidirect
233. C22⋊Q8C2×D83semidirect
250. #106C2×D83semidirect
252. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282812
#191282672
#191280412
#191280272
#191280212
#191280132
#191279522
#191279492
#191279372
#191278502
#191278492
#191278452
#191278362
#191276672
#465×C832
#319883