Finite Group Databasebeta

← #24919613

Subgroups — #24919613

Order 10624.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128347C211yesmaximal, characteristic, complemented
2#19128063C211yesmaximal, characteristic, complemented
3#19128042C211yesmaximal, characteristic, complemented
4#19128025C211yesmaximal, characteristic, complemented
5#19127978C211yesmaximal, characteristic, complemented
6#19127945C211yesmaximal, characteristic, complemented
7#19127935C211yesmaximal, characteristic, complemented
8#19127916C211yesmaximal, characteristic, complemented
9#19127844C211yesmaximal, characteristic, complemented
10#19127834C211yesmaximal, characteristic, complemented
11#19127811C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127705C211yesmaximal, characteristic, complemented
14#19127695C211yesmaximal, characteristic, complemented
15#555×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C2×#13050578C2211yescharacteristic, complemented
19C2×#13050575C2211yescharacteristic
20C2×#13050575C2211yescharacteristic
21C2×#13050574C2211yescharacteristic
22C22×C83⋊C8C2211yescharacteristic, complemented
23#14581780C2211yescharacteristic
24#14581777—21no—
25#14581771C2211yescharacteristic
26#14581766C2211yescharacteristic
27#14581765—21no—
28#14581760—21no—
29C4×D332C2211yescharacteristic
30C22⋊C4×D83—21no—
31#14581725C2211yescharacteristic, complemented
32#14581725C2211yescharacteristic, complemented
33#14581723C2211yescharacteristic, complemented
34C4⋊(C83⋊C8)C2211yescharacteristic, complemented
35#14581715C2211yescharacteristic, complemented
36#14581715C2211yescharacteristic, complemented
37#14581715C2211yescharacteristic, complemented
38#14581713C2211yescharacteristic, complemented
39#14581712C2211yescharacteristic
40#14581712C2211yescharacteristic
41#14581712C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44#14581712C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic, complemented
47#14581703C2211yescharacteristic, complemented
48C4⋊Q8×C83C2211yescharacteristic, complemented
49#124×C83—21no—
50#124×C83—21no—
51#121×C83—21no—
52#121×C83C2211yescharacteristic
53#121×C83C2211yescharacteristic, complemented
54C4×D4×C83C2211yescharacteristic
55C4×D4×C83C2211yescharacteristic, complemented
56#117×C83—21no—
57#117×C83C2211yescharacteristic
58#117×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050590—22no—
61#13050590—41no—
62C2×Dic166C2311yescharacteristic
63C2×D332—22no—
64C2×D332—21no—
65C2×D332—41no—
66C2×D332C2311yescharacteristic
67C2×D332C2311yescharacteristic
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C4×D83—21no—
71C2×C4×D83—21no—
72#13050578—41no—
73#13050575—22no—
74#13050575—41no—
75#13050574—41no—
76C2×C83⋊C8—41no—
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C2×C83⋊C8C2311yescharacteristic
81C4⋊Dic83C2311yescharacteristic
82Dic83⋊C4—21no—
83C22⋊Dic83—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86(C2×D83)⋊C4—21no—
87(C2×D83)⋊C4—21no—
88C2×Q8×C83—21no—
89C2×D4×C83—21no—
90C2×D4×C83—21no—
91C2×D4×C83D411yescharacteristic
92C2×D4×C83C2311yescharacteristic
93C22×C4×C83—21no—
94C22×C4×C83C2311yescharacteristic
95C22×C4×C83C2311yescharacteristic
96C4⋊C4×C83—21no—
97C4⋊C4×C83D411yescharacteristic
98C4⋊C4×C83C2311yescharacteristic
99C4⋊C4×C83C2311yescharacteristic
100C4⋊C4×C83C2311yescharacteristic
101C22⋊C4×C83—21no—
102C22⋊C4×C83—21no—
103C22⋊C4×C83—21no—
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C22⋊C4×C83D411yescharacteristic
107C22⋊C4×C83D411yescharacteristic
108C42×C83—21no—
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C23×C83C2×D411yescharacteristic
113C23×C83C2×D411yescharacteristic
114C22×D83—21no—
115C22×D83—21no—
116C22×D83—42no—
117C22×D83—42no—
118C22×D83—81no—
119Q8×C83—81no—
120D4×C83—21no—
121D4×C83—21no—
122D4×C83—41no—
123D4×C83—81no—
124D4×C83—81no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—42no—
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2×D411yescharacteristic
136C2×C4×C83C2×D411yescharacteristic
137C2×C4×C83C2411yescharacteristic
138C22⋊D83—42no—
139C22⋊D83—81no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142D332—21no—
143D332—21no—
144D332—21no—
145D332—42no—
146D332#4112yes—
147C4×D83—42no—
148C4×D83—81no—
149Dic166—21no—
150Dic166—21no—
151C83⋊C8—21no—
152C83⋊C8—21no—
153C83⋊C8—21no—
154C83⋊C8—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83C22×D411yescharacteristic
165C2×D83—22no—
166C2×D83—41no—
167C2×D83—41no—
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—84no—
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83C2×#4111yescharacteristic
180C4×C83H(5,𝔽2)11yescharacteristic
181Dic83—41no—
182Dic83—41no—
183C2×C83—41no—
184C2×C83—41no—
185C2×C83—81no—
186C2×C83#54611yescharacteristic
187C2×C83#57511yescharacteristic
188C2×C83#57611yescharacteristic
189D83—42no—
190D83—81no—
191#3319—831nomaximal, Sylow
192C83#331911yescharacteristic, complemented, Sylow
193#555D8311yescharacteristic, complemented
194#546—831no—
195#489—831no—
196#485—831no—
197#482—831no—
198#474—831no—
199#468—831no—
200#466—831no—
201#463—831no—
202#449—831no—
203#447—831no—
204#442—831no—
205#435—831no—
206#418—831no—
207#416—831no—
208C2×#41—831no—
209C22×D4—831no—
210C2×SD16—831no—
211C2×D8—831no—
212C2×D8—831no—
213C2×M16—831no—
214C22×C8—831no—
215C4⋊Q8C2×D8311yescharacteristic, complemented
216#124—21no—
217#124—21no—
218#124—831no—
219#123—1661no—
220C22⋊Q8—831no—
221#121—21no—
222#121—831no—
223#121—1661no—
224#121C2×D8311yescharacteristic
225#121C2×D8311yescharacteristic, complemented
226#120—1661no—
227C4×D4—831no—
228C4×D4C2×D8311yescharacteristic
229C4×D4C2×D8311yescharacteristic, complemented
230#117—21no—
231#117C2×D8311yescharacteristic
232#117C2×D8311yescharacteristic, complemented
233C2×C22⋊C4—1661no—
234#107—831no—
235#107—831no—
236#106—831no—
237C4⋊C8—831no—
238#103—831no—
239#103—831no—
240#103—831no—
241#102—831no—
242#102—831no—
243#102—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247#102—831no—
248C22⋊C8—831no—
249C22⋊C8—831no—
250#97—831no—
251C24—1661no—
252#41—3321no—
253#41—1662no—
254C2×Q8—21no—
255C2×Q8—831no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3321no—
259C2×D4—831no—
260C2×D4—831no—
261C2×D4—1662no—
262C2×D4—1661no—
263C2×D4—1661no—
264C2×D4—1661no—
265C2×D4C22⋊D8311yescharacteristic
266C2×D4C22×D8311yescharacteristic
267C22×C4—21no—
268C22×C4—1661no—
269C22×C4—1661no—
270C22×C4C22×D8311yescharacteristic
271C22×C4C22×D8311yescharacteristic
272SD16—3321no—
273D8—3321no—
274D8—1662no—
275M16—3321no—
276C2×C8—3321no—
277C2×C8—831no—
278C2×C8—831no—
279C2×C8—831no—
280C2×C8—831no—
281C4⋊C4—21no—
282C4⋊C4—831no—
283C4⋊C4—1661no—
284C4⋊C4C22⋊D8311yescharacteristic
285C4⋊C4C22×D8311yescharacteristic
286C4⋊C4C22×D8311yescharacteristic
287C4⋊C4C22×D8311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4—1661no—
296C22⋊C4—1661no—
297C22⋊C4—1661no—
298C22⋊C4C22⋊D8311yescharacteristic
299C22⋊C4C22⋊D8311yescharacteristic
300C42—21no—
301C42—21no—
302C42C22×D8311yescharacteristic
303C23—21no—
304C23—3322no—
305C23—3322no—
306C23—6641no—
307C23—1661no—
308C23—1661no—
309C23C2×C22⋊D8311yescharacteristic
310C23C2×C22⋊D8311yescharacteristic
311Q8—81no—
312Q8—1661no—
313Q8—1661no—
314D4—21no—
315D4—21no—
316D4—41no—
317D4—81no—
318D4—81no—
319D4—3322no—
320D4—3322no—
321D4—832no—
322D4—6641no—
323D4—1661no—
324D4—1661no—
325D4—1661no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—3322no—
335C2×C4—6641no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4C2×C22⋊D8311yescharacteristic
339C2×C4C2×C22⋊D8311yescharacteristic
340C2×C4C2×C22⋊D8311yescharacteristic
341C2×C4C2×C22⋊D8311yescharacteristic
342C2×C4C23×D8311yescharacteristic
343C8—1661no—
344C8—1661no—
345C8—1661no—
346C8—1661no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—3321no—
357C22—3321no—
358C22—3321no—
359C22—3321no—
360C22—3321no—
361C22—6644no—
362C22—1662no—
363C22C22×C22⋊D8311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—3321no—
372C4—3321no—
373C4#41×D8311yescharacteristic
374C4#1458183811yescharacteristic
375C2—41no—
376C2—41no—
377C2—81no—
378C2—3322no—
379C2—6641no—
380C2#1912833911yescharacteristic
381C2#1912849911yescharacteristic
382C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#555×C83C2
socleC22×C83C22×D4
hypercenter#555D83
hypocenterC83#3319
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #19128347C2semidirect
2. #19128063C2semidirect
3. #19128042C2semidirect
4. #19128025C2semidirect
5. #19127978C2semidirect
6. #19127945C2semidirect
7. #19127935C2semidirect
8. #19127916C2semidirect
9. #19127844C2semidirect
10. #19127834C2semidirect
11. #19127811C2semidirect
12. #19127773C2semidirect
13. #19127705C2semidirect
14. #19127695C2semidirect
15. #555×C83C2semidirect
18. C2×#13050578C22semidirect
22. C22×C83⋊C8C22semidirect
31. #14581725C22semidirect
32. #14581725C22semidirect
33. #14581723C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. #14581715C22semidirect
36. #14581715C22semidirect
37. #14581715C22semidirect
38. #14581713C22semidirect
46. C22⋊(C83⋊C8)C22semidirect
47. #14581703C22semidirect
48. C4⋊Q8×C83C22semidirect
53. #121×C83C22semidirect
55. C4×D4×C83C22semidirect
58. #117×C83C22semidirect
192. C83#3319semidirect
193. #555D83semidirect
215. C4⋊Q8C2×D83semidirect
225. #121C2×D83semidirect
229. C4×D4C2×D83semidirect
232. #117C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283472
#191280632
#191280422
#191280252
#191279782
#191279452
#191279352
#191279162
#191278442
#191278342
#191278112
#191277732
#191277052
#191276952
#555×C832
#331983