Finite Group Databasebeta

← #24919418

Subgroups — #24919418

Order 10624.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 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#19128031C211yesmaximal, characteristic, complemented
4#19128019C211yesmaximal, characteristic, complemented
5#19127988C211yesmaximal, characteristic, complemented
6#19127952C211yesmaximal, characteristic, complemented
7#19127928C211yesmaximal, characteristic, complemented
8#19127916C211yesmaximal, characteristic, complemented
9#19127844C211yesmaximal, characteristic, complemented
10#19127834C211yesmaximal, characteristic, complemented
11#19127811C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127700C211yesmaximal, characteristic, complemented
14C2×#14581712C211yesmaximal, characteristic, complemented
15#547×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C2×#13050578C2211yescharacteristic, complemented
19C2×#13050575C2211yescharacteristic
20C2×#13050575C2211yescharacteristic
21C2×#13050574C2211yescharacteristic, complemented
22C22×C83⋊C8C2211yescharacteristic
23#14581780C2211yescharacteristic
24#14581777—21no—
25#14581771C2211yescharacteristic
26#14581766C2211yescharacteristic
27#14581765—21no—
28#14581760—21no—
29C4×D332C2211yescharacteristic
30C22⋊C4×D83—21no—
31#14581723C2211yescharacteristic, complemented
32#14581723C2211yescharacteristic, 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
48C2×#41×C83—21no—
49C4⋊Q8×C83C2211yescharacteristic, complemented
50#123×C83—21no—
51#123×C83—21no—
52C22⋊Q8×C83—21no—
53#121×C83C2211yescharacteristic
54#121×C83C2211yescharacteristic, complemented
55C4×D4×C83C2211yescharacteristic
56C4×D4×C83C2211yescharacteristic, complemented
57C2×C4⋊C4×C83C2211yescharacteristic
58C2×C4⋊C4×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—
88#41×C83—44no—
89C2×Q8×C83—21no—
90C2×D4×C83—21no—
91C2×D4×C83—21no—
92C2×D4×C83D411yescharacteristic
93C2×D4×C83C2311yescharacteristic
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83—21no—
97C22×C4×C83C2311yescharacteristic
98C22×C4×C83C2311yescharacteristic
99C4⋊C4×C83—21no—
100C4⋊C4×C83—21no—
101C4⋊C4×C83—21no—
102C4⋊C4×C83D411yescharacteristic
103C4⋊C4×C83C2311yescharacteristic
104C4⋊C4×C83C2311yescharacteristic
105C4⋊C4×C83C2311yescharacteristic
106C22⋊C4×C83—21no—
107C22⋊C4×C83—21no—
108C22⋊C4×C83—21no—
109C22⋊C4×C83D411yescharacteristic
110C22⋊C4×C83D411yescharacteristic
111C42×C83C2311yescharacteristic
112C23×C83—21no—
113C23×C83C2×D411yescharacteristic
114C23×C83C2×D411yescharacteristic
115C22×D83—21no—
116C22×D83—21no—
117C22×D83—42no—
118C22×D83—42no—
119C22×D83—81no—
120Q8×C83—42no—
121D4×C83—22no—
122D4×C83—21no—
123D4×C83—21no—
124D4×C83—42no—
125D4×C83—42no—
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—21no—
133C2×C4×C83—42no—
134C2×C4×C83—42no—
135C2×C4×C83—42no—
136C2×C4×C83C2×D411yescharacteristic
137C2×C4×C83C2×D411yescharacteristic
138C2×C4×C83C2×D411yescharacteristic
139C2×C4×C83C2×D411yescharacteristic
140C2×C4×C83C2411yescharacteristic
141C22⋊D83—42no—
142C22⋊D83—81no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145D332—21no—
146D332—21no—
147D332—21no—
148D332—42no—
149D332#4112yescomplemented
150C4×D83—42no—
151C4×D83—81no—
152Dic166—21no—
153Dic166—21no—
154C83⋊C8—21no—
155C83⋊C8—21no—
156C83⋊C8—21no—
157C83⋊C8—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83AGL(1,ℤ/8)12yes—
167C22×C83C22×D411yescharacteristic
168C2×D83—22no—
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—84no—
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83C2×#4111yescharacteristic
183C4×C83H(5,𝔽2)11yescharacteristic
184Dic83—41no—
185Dic83—41no—
186C2×C83—22no—
187C2×C83—41no—
188C2×C83—81no—
189C2×C83#54611yescharacteristic
190C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C83#57711yescharacteristic
192D83—42no—
193D83—81no—
194#3306—831nomaximal, Sylow
195C83#330611yescharacteristic, complemented, Sylow
196#547D8311yescharacteristic, complemented
197#546—831no—
198#489—831no—
199#483—831no—
200#481—831no—
201#476—831no—
202#469—831no—
203#465—831no—
204#463—831no—
205#449—831no—
206#447—831no—
207#442—831no—
208#435—831no—
209#417—831no—
210C2×#102—831no—
211C2×#41—21no—
212C2×#41—831no—
213C22×D4—831no—
214C2×SD16—831no—
215C2×D8—831no—
216C2×D8—831no—
217C2×M16—831no—
218C22×C8—831no—
219C4⋊Q8C2×D8311yescharacteristic, complemented
220#124—831no—
221#123—21no—
222#123—21no—
223#123—1661no—
224C22⋊Q8—21no—
225C22⋊Q8—831no—
226#121—831no—
227#121—1661no—
228#121C2×D8311yescharacteristic
229#121C2×D8311yescharacteristic, complemented
230#120—1661no—
231C4×D4—831no—
232C4×D4C2×D8311yescharacteristic
233C4×D4C2×D8311yescharacteristic, complemented
234C2×C4⋊C4C2×D8311yescharacteristic
235C2×C4⋊C4C2×D8311yescharacteristic, complemented
236C2×C22⋊C4—1661no—
237#106—831no—
238#106—831no—
239#106—831no—
240C4⋊C8—831no—
241#103—831no—
242#103—831no—
243#103—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247#102—831no—
248#102—831no—
249#102—831no—
250#102—831no—
251C22⋊C8—831no—
252C22⋊C8—831no—
253#97—831no—
254C24—1661no—
255#41—44no—
256#41—3321no—
257#41—1662no—
258C2×Q8—21no—
259C2×Q8—831no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—3321no—
263C2×D4—831no—
264C2×D4—831no—
265C2×D4—1662no—
266C2×D4—1661no—
267C2×D4—1661no—
268C2×D4—1661no—
269C2×D4C22⋊D8311yescharacteristic
270C2×D4C22×D8311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—1661no—
275C22×C4—1661no—
276C22×C4C22×D8311yescharacteristic
277C22×C4C22×D8311yescharacteristic
278SD16—3321no—
279D8—3321no—
280D8—1662no—
281M16—3321no—
282C2×C8—3321no—
283C2×C8—831no—
284C2×C8—831no—
285C2×C8—831no—
286C2×C8—831no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—831no—
291C4⋊C4—1661no—
292C4⋊C4C22⋊D8311yescharacteristic
293C4⋊C4C22×D8311yescharacteristic
294C4⋊C4C22×D8311yescharacteristic
295C4⋊C4C22×D8311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—1661no—
300C22⋊C4—1661no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4C22⋊D8311yescharacteristic
305C22⋊C4C22⋊D8311yescharacteristic
306C42C22×D8311yescharacteristic
307C23—21no—
308C23—3322no—
309C23—3322no—
310C23—6641no—
311C23—1661no—
312C23—1661no—
313C23C2×C22⋊D8311yescharacteristic
314C23C2×C22⋊D8311yescharacteristic
315Q8—42no—
316Q8—1661no—
317Q8—1661no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—3322no—
324D4—3322no—
325D4—832no—
326D4—6641no—
327D4—1661no—
328D4—1661no—
329D4—1661no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—3322no—
341C2×C4—6641no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4C2×C22⋊D8311yescharacteristic
345C2×C4C2×C22⋊D8311yescharacteristic
346C2×C4C2×C22⋊D8311yescharacteristic
347C2×C4C2×C22⋊D8311yescharacteristic
348C2×C4C23×D8311yescharacteristic
349C8—1661no—
350C8—1661no—
351C8—1661no—
352C8—1661no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3321no—
362C22—3321no—
363C22—3321no—
364C22—3321no—
365C22—3321no—
366C22—6644no—
367C22—1662no—
368C22#1458181812yes—
369C22C22×C22⋊D8311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—3321no—
378C4—3321no—
379C4#41×D8311yescharacteristic
380C4#1458183811yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—3322no—
385C2—6641no—
386C2#1912833911yescharacteristic
387C2C2×#1458181811yescharacteristic
388C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#547×C83C2
socleC22×C83C22×D4
hypercenter#547D83
hypocenterC83#3306
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #19128347C2semidirect
2. #19128063C2semidirect
3. #19128031C2semidirect
4. #19128019C2semidirect
5. #19127988C2semidirect
6. #19127952C2semidirect
7. #19127928C2semidirect
8. #19127916C2semidirect
9. #19127844C2semidirect
10. #19127834C2semidirect
11. #19127811C2semidirect
12. #19127773C2semidirect
13. #19127700C2semidirect
14. C2×#14581712C2semidirect
15. #547×C83C2semidirect
18. C2×#13050578C22semidirect
21. C2×#13050574C22semidirect
31. #14581723C22semidirect
32. #14581723C22semidirect
33. #14581723C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. #14581715C22semidirect
36. #14581715C22semidirect
37. #14581715C22semidirect
38. #14581713C22semidirect
46. C22⋊(C83⋊C8)C22semidirect
47. #14581703C22semidirect
49. C4⋊Q8×C83C22semidirect
54. #121×C83C22semidirect
56. C4×D4×C83C22semidirect
58. C2×C4⋊C4×C83C22semidirect
149. D332#41semidirect
195. C83#3306semidirect
196. #547D83semidirect
219. C4⋊Q8C2×D83semidirect
229. #121C2×D83semidirect
233. C4×D4C2×D83semidirect
235. C2×C4⋊C4C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283472
#191280632
#191280312
#191280192
#191279882
#191279522
#191279282
#191279162
#191278442
#191278342
#191278112
#191277732
#191277002
C2×#145817122
#547×C832
#330683