Finite Group Databasebeta

← #24919619

Subgroups — #24919619

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#19128416C211yesmaximal, characteristic, complemented
2#19128060C211yesmaximal, characteristic, complemented
3#19128039C211yesmaximal, characteristic, complemented
4#19128022C211yesmaximal, characteristic, complemented
5#19127981C211yesmaximal, characteristic, complemented
6#19127949C211yesmaximal, characteristic, complemented
7#19127935C211yesmaximal, characteristic, complemented
8#19127913C211yesmaximal, characteristic, complemented
9#19127846C211yesmaximal, characteristic, complemented
10#19127835C211yesmaximal, characteristic, complemented
11#19127811C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127706C211yesmaximal, characteristic, complemented
14#19127697C211yesmaximal, characteristic, complemented
15#546×C83C211yesmaximal, characteristic, complemented
16C2×#13050577C2211yescharacteristic, complemented
17C2×#13050575C2211yescharacteristic
18C2×#13050575C2211yescharacteristic
19C2×#13050574C2211yescharacteristic
20C22×C83⋊C8C2211yescharacteristic, complemented
21#14581792C2211yescharacteristic, complemented
22#14581778—21no—
23#14581778—21no—
24#14581766C2211yescharacteristic
25#14581766C2211yescharacteristic, complemented
26#14581763—21no—
27C4×D332C2211yescharacteristic, complemented
28#14581751C2211yescharacteristic
29#14581751C2211yescharacteristic, complemented
30#14581750—21no—
31#14581725C2211yescharacteristic, complemented
32#14581725C2211yescharacteristic, complemented
33#14581723C2211yescharacteristic, complemented
34C4⋊(C83⋊C8)C2211yescharacteristic, complemented
35#14581716C2211yescharacteristic, complemented
36#14581716C2211yescharacteristic, complemented
37#14581716C2211yescharacteristic, complemented
38#14581713C2211yescharacteristic
39#14581713C2211yescharacteristic
40#14581713C2211yescharacteristic
41#14581713C2211yescharacteristic
42#14581713C2211yescharacteristic
43#14581713C2211yescharacteristic
44#14581712C2211yescharacteristic, complemented
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic, complemented
47#14581703C2211yescharacteristic, complemented
48C2×#41×C83C2211yescharacteristic
49C22×D4×C83C2211yescharacteristic
50#124×C83C2211yescharacteristic
51#123×C83—21no—
52C22⋊Q8×C83C2211yescharacteristic
53#121×C83—21no—
54#121×C83C2211yescharacteristic
55#120×C83—21no—
56C4×D4×C83C2211yescharacteristic
57C4×D4×C83C2211yescharacteristic
58C2×C22⋊C4×C83—21no—
59C2×Dic166—21no—
60C2×D332C2311yescharacteristic
61C2×C22⋊D83—21no—
62C2×C22⋊D83—21no—
63C2×C4×D83—21no—
64#13050577—41no—
65#13050575—22no—
66#13050575—41no—
67#13050574—41no—
68C2×C83⋊C8—41no—
69C2×C83⋊C8C2311yescharacteristic
70C2×C83⋊C8C2311yescharacteristic
71C2×C83⋊C8C2311yescharacteristic
72C2×C83⋊C8C2311yescharacteristic
73C4⋊Dic83—21no—
74C4⋊Dic83C2311yescharacteristic
75C4⋊Dic83C2311yescharacteristic
76C4⋊Dic83C2311yescharacteristic
77C22⋊Dic83—21no—
78C22⋊Dic83—21no—
79(C2×D83)⋊C4—21no—
80(C2×D83)⋊C4—21no—
81(C2×D83)⋊C4—21no—
82C4×Dic83—21no—
83C4×Dic83—21no—
84C24×C83—21no—
85#41×C83—22no—
86#41×C83—41no—
87C2×Q8×C83C2311yescharacteristic
88C2×D4×C83—22no—
89C2×D4×C83—21no—
90C2×D4×C83—21no—
91C2×D4×C83—21no—
92C2×D4×C83—41no—
93C2×D4×C83D411yescharacteristic
94C2×D4×C83C2311yescharacteristic
95C2×D4×C83C2311yescharacteristic
96C22×C4×C83—21no—
97C22×C4×C83—21no—
98C22×C4×C83C2311yescharacteristic
99C22×C4×C83C2311yescharacteristic
100C4⋊C4×C83—21no—
101C4⋊C4×C83D411yescharacteristic
102C4⋊C4×C83C2311yescharacteristic
103C22⋊C4×C83—21no—
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C22⋊C4×C83—21no—
108C22⋊C4×C83D411yescharacteristic
109C22⋊C4×C83D411yescharacteristic
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—42no—
114C23×C83—42no—
115C23×C83—81no—
116C23×C83C2×D411yescharacteristic
117C23×C83C2×D411yescharacteristic
118C22×D83—21no—
119Q8×C83—21no—
120Q8×C83—21no—
121D4×C83—21no—
122D4×C83—21no—
123D4×C83—21no—
124D4×C83—41no—
125D4×C83—42no—
126D4×C83—42no—
127D4×C83—81no—
128D4×C83#4112yes—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—42no—
136C2×C4×C83—81no—
137C2×C4×C83C2×D411yescharacteristic
138C2×C4×C83C2×D411yescharacteristic
139C2×C4×C83C2×D411yescharacteristic
140C2×C4×C83C2×D411yescharacteristic
141C2×C4×C83C2411yescharacteristic
142C22⋊D83—81no—
143C22⋊D83—81no—
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147D332—21no—
148D332—21no—
149C4×D83—42no—
150Dic166—81no—
151C83⋊C8—21no—
152C83⋊C8—21no—
153C83⋊C8—21no—
154C83⋊C8—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—22no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—84no—
168C22×C83C22×D411yescharacteristic
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—41no—
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83C2×#4111yescharacteristic
179C4×C83H(5,𝔽2)11yescharacteristic
180Dic83—41no—
181Dic83—41no—
182Dic83—41no—
183C2×C83—41no—
184C2×C83—41no—
185C2×C83—42no—
186C2×C83—81no—
187C2×C83#54611yescharacteristic
188C2×C83#57511yescharacteristic
189C2×C83#57611yescharacteristic
190D83—81no—
191#3319—831nomaximal, Sylow
192C83#331911yescharacteristic, complemented, Sylow
193#555—831no—
194#546D8311yescharacteristic, complemented
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×#41C2×D8311yescharacteristic
209C22×D4C2×D8311yescharacteristic
210C2×SD16—831no—
211C2×D8—831no—
212C2×D8—831no—
213C2×M16—831no—
214C22×C8—831no—
215C4⋊Q8—831no—
216#124—1661no—
217#124—1661no—
218#124C2×D8311yescharacteristic
219#123—21no—
220C22⋊Q8C2×D8311yescharacteristic
221#121—21no—
222#121—831no—
223#121—831no—
224#121—1661no—
225#121C2×D8311yescharacteristic
226#120—21no—
227C4×D4—831no—
228C4×D4C2×D8311yescharacteristic
229C4×D4C2×D8311yescharacteristic
230#117—831no—
231#117—831no—
232#117—1661no—
233C2×C22⋊C4—21no—
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—21no—
252#41—22no—
253#41—41no—
254C2×Q8—1661no—
255C2×Q8C22×D8311yescharacteristic
256C2×D4—22no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—831no—
262C2×D4—1661no—
263C2×D4—1661no—
264C2×D4C22⋊D8311yescharacteristic
265C2×D4C22×D8311yescharacteristic
266C2×D4C22×D8311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
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—831no—
284C4⋊C4—831no—
285C4⋊C4—1661no—
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—1661no—
301C42—1661no—
302C42C22×D8311yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—1661no—
309C23C2×C22⋊D8311yescharacteristic
310C23C2×C22⋊D8311yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—6641no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—42no—
319D4—42no—
320D4—81no—
321D4—6641no—
322D4—6641no—
323D4—1661no—
324D4—1661no—
325D4#1305058912yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—3322no—
335C2×C4—1661no—
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—22no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—84no—
360C22—3321no—
361C22—3321no—
362C22—3321no—
363C22C22×C22⋊D8311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—3321no—
371C4—3321no—
372C4—3321no—
373C4C2×#1305058911yescharacteristic
374C4#1458183711yescharacteristic
375C2—41no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—6641no—
380C2#1912834511yescharacteristic
381C2#1912849911yescharacteristic
382C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#546×C83C2
socleC22×C83C22×D4
hypercenter#546D83
hypocenterC83#3319
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #19128416C2semidirect
2. #19128060C2semidirect
3. #19128039C2semidirect
4. #19128022C2semidirect
5. #19127981C2semidirect
6. #19127949C2semidirect
7. #19127935C2semidirect
8. #19127913C2semidirect
9. #19127846C2semidirect
10. #19127835C2semidirect
11. #19127811C2semidirect
12. #19127773C2semidirect
13. #19127706C2semidirect
14. #19127697C2semidirect
15. #546×C83C2semidirect
16. C2×#13050577C22semidirect
20. C22×C83⋊C8C22semidirect
21. #14581792C22semidirect
25. #14581766C22semidirect
27. C4×D332C22semidirect
29. #14581751C22semidirect
31. #14581725C22semidirect
32. #14581725C22semidirect
33. #14581723C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. #14581716C22semidirect
36. #14581716C22semidirect
37. #14581716C22semidirect
44. #14581712C22semidirect
46. C22⋊(C83⋊C8)C22semidirect
47. #14581703C22semidirect
192. C83#3319semidirect
194. #546D83semidirect

Maximal subgroups

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

subgroupindex
#191284162
#191280602
#191280392
#191280222
#191279812
#191279492
#191279352
#191279132
#191278462
#191278352
#191278112
#191277732
#191277062
#191276972
#546×C832
#331983